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+arXiv:2301.11979v1  [cond-mat.str-el]  27 Jan 2023
+DFT, L(S)DA, LDA+U, LDA+DMFT..., whether we do approach
+to a proper description of optical response for strongly correlated
+systems?
+A.S. Moskvin1
+1Ural Federal University, Ekaterinburg, 620083 Russia
+Аннотация
+I present a critical overview of so-called "ab initio"DFT (density fuctional theory) based
+calculation schemes for the description of the electronic structure, energy spectrum, and optical
+response for strongly correlated 3d oxides, in particular, crystal-field and charge transfer transitions
+as compared with an "old" cluster model that does generalize crystal-field and ligand-field theory.
+As a most instructive illustration of validity of numerous calculation techniques I address the
+prototypical 3d insulator NiO predicted to be a metal in frames of a standard LDA (local density
+approximation) band theory.
+1
+
+INTRODUCTION
+The electronic states in strongly correlated 3d oxides manifest both significant localization
+and dispersional features. One strategy to deal with this dilemma is to restrict oneself to
+small many-electron clusters embedded to a whole crystal, then creating model effective
+lattice Hamiltonians whose spectra may reasonably well represent the energy and dispersion
+of the important excitations of the full problem. Despite some shortcomings the method did
+provide a clear physical picture of the complex electronic structure and the energy spectrum,
+as well as the possibility of a quantitative modeling.
+However, last decades the condensed matter community faced an expanding flurry of
+papers with the so called ab initio calculations of electronic structure and physical properties
+for strongly correlated systems such as 3d compounds based on density functional theory
+(DFT). The modern formulation of the DFT originated in the work of Hohenberg and
+Kohn [1], on which based the other classic work in this field by Kohn and Sham [2]. The Kohn-
+Sham equation, has become a basic mathematical model of much of present-day methods for
+treating electrons in atoms, molecules, condensed matter, solid surfaces, nanomaterials, and
+man-made structures [3]. Of the top three most cited physicists in the period 1980-2010, the
+first (Perdew: 65 757 citations) and third (Becke: 62 581 citations) were density-functional
+theorists [4].
+However, DFT still remains, in some sense, ill-defined: many of DFT statements were
+ill-posed or not rigorously proved. Most widely used DFT computational schemes start
+with a "metallic-like"approaches making use of approximate energy functionals, firstly LDA
+(local density approximation) scheme, which are constructed as expansions around the
+homogeneous electron gas limit and fail quite dramatically in capturing the properties of
+strongly correlated systems. The LDA+U and LDA+DMFT (DMFT, dynamical mean-
+field theory) [5] methods are believed to correct the inaccuracies of approximate DFT
+exchange correlation functionals. The main idea of these computational approaches consists
+in a selective description of the strongly correlated electronic states, typically, localized
+d or f orbitals, using the Hubbard model, while all the other states continue to be
+treated at the level of standard approximate DFT functionals. At present the LDA+U
+and LDA+DMFT methods are addressed to be most powerful tools for the investigation
+of strongly correlated electronic systems, however, these preserve many shortcomings of
+2
+
+the DFT-LDA approach. Despite many examples of a seemingly good agreement with
+experimental data (photoemission and inverse-photoemission spectra, magnetic moments,...)
+claimed by the DFT community, both the questionable starting point and many unsolved
+and unsoluble problems give rise to serious doubts in quantitative and even qualitative
+predictions made within the DFT based techniques. In a certain sense the cluster based
+calculations seem to provide a better description of the overall electronic structure of
+insulating 3doxides and its optical response than the DFT based band structure calculations,
+mainly due to a clear physics and a better account for correlation effects (see, e.g.,
+Refs. [6, 7]).
+The paper is organized as follows. In Sec.II we do present a short critical overview of the
+DFT and the DFT based technique with a focus on the NiO oxide. Sec.III is devoted to a
+short overview of the cluster model approaches to a proper semiquantitative description of
+the optical response in strongly correlated 3d oxides with a focus on the NiO oxide. A short
+summary is made in Sec.IV.
+SHORT OVERVIEW OF THE DFT BASED TECHNIQUE
+Hohenberg-Kohn-Sham DFT
+Density functional theory finds its roots in the approach which Thomas and Fermi
+elaborated shortly after the creation of quantum mechanics [8, 9]. The Thomas-Fermi theory
+of atoms may be interpreted as a semiclassical approximation, where the energy of a system
+is written as a functional of the one-particle density.
+Justifying earlier attempts directed at generalizing the Thomas-Fermi theory, Hohenberg
+and Kohn [1] in 1964 advanced a theorem: "For any system of interacting particles in an
+external potential v(r), the external potential is uniquely determined (except for a constant)
+by the ground state density n0(r) which states that the exact ground-state energy is a
+functional of the exact ground-state one-particle density. Unfortunately, it does not tell
+how to construct this functional, i.e., it is an existence theorem for the energy-density
+functional. This explains the fact of why so much effort has been dedicated to the task of
+obtaining approximate functionals for the description of the ground-state properties of many-
+particle systems. Contrary to wavefunction theory, where the objective is to approximate
+3
+
+the wavefunction, in DFT we choose to make approximations for the functional.
+However, DFT still remains, in some sense, ill-defined: many of the DFT statements
+were ill-posed or not rigorously proved. Indeed, the HK theorem is the constellation of two
+statements: (i) the mathematically rigorous HK lemma, which demonstrates that the same
+ground state density cannot correspond to two different potentials of an external field, and
+(ii) the hypothesis of the existence of the universal density functional. However, the HK
+lemma cannot provide justification of the universal density functional for fermions [10]. In
+other words, each external field determines a unique density, and each density determines
+a unique external field on the basis of the HK lemma. However, the rule for the last
+correspondence can be nonuniversal, as the rule in general depends on the concrete form of
+the density. The existence of this nonuniversality violates the HK theorem, although the HK
+lemma is believed to be undoubtedly correct [10].
+Furthermore, there are more serious critics. Sarry and Sarry [11] claim that the proof of
+the HK theorem is not correct. The authors do emphasize that for a strict many-particle
+calculation only the direct mapping: external potential ⇒ ground state wave function ⇒
+electron density
+v(r) ⇒ Ψ0(r) ⇒ ρ0(r)
+is justified while the inverse mapping
+ρ0(r) ⇒ Ψ0(r) ⇒ v(r)
+claimed by the HK theorem can be validated only for single-particle self-consistent
+calculations.
+The DFT exploits the one-to-one correspondence between the single-particle electron
+density and an external potential acting upon the system and relies on the existence of
+a universal functional F[ρ(r)] which can be minimized in order to find the ground state
+energy. However,the correspondence theorem establishes the existence of the functional only
+in principle, and provides no unique practical recipe for its construction. The construction
+of the functional F[ρ(r)] in the HK-DFT is equivalent to the problem of finding the N-
+representability conditions of the reduced density matrix of order two [3, 12], the problem
+whose solution has not been found until now. Generally speaking the functional F[ρ(r)]
+must be N-dependent, namely, F[N, ρ(r)]. Another important aspect, closely related to N-
+representability, is the variational character that either exact or approximate functionals
+4
+
+F[N, ρ(r)] must have in order to guarantee that the energy remains an upper bound to the
+exact value.
+The Kohn-Sham (KS) theory goes further in reducing the problem of calculating ground
+state properties of a many-electron system in a local external single-particle potential to
+solving Hartree-like one-electron KS equations. Within the framework of the HKS-DFT,
+the many-body problem of interacting electrons in a static external potential is cast into
+a tractable problem of non-interacting electrons moving in an effective potential. The
+latter includes the external potential and the effects of the Coulomb interactions between
+the electrons, i.e. the Hartree term, describing the electron-electron repulsion, and the
+exchange and correlation (XC) interactions, which includes all the many-body interactions.
+Modeling the XC interactions is the main difficulty of DFT. In practical calculations, the
+XC contribution is approximated, and the results are only as good as the approximation
+used. Actually, in HKS-DFT there exist hundreds of XC-approximations for vKS
+xc (r) [3]. The
+existence of so many approximations, with so little guidance, makes it ever more difficult
+for non-specialists to separate the silver from the dross [13]. It is worth noting here that all
+the approximate functionals do not comply with the variational principle.
+The leading approximation for density functional construction is the so called local density
+approximation (LDA), which is based upon exact exchange energy for a uniform electron gas
+and only requires the density at each point in space. So the LDA taken from assuming that
+the electron density for an atom, molecule, or solid is similarly homogeneous. But molecules
+in LDA are typically overbound by about 1 eV/bond, and in the late 1980s the so-called
+generalized gradient approximations (GGAs) using both the density and its gradient at
+each point in space were elaborated whose accuracy seemed to be acceptable in chemical
+calculations [13]. All the GGAs functionals, by definition, are corrections to the LDA, they
+all revert to the uniform electron gas at zero density gradient. It should be noted that the
+local nature of the standard approximations implies an exponential decay of the inter-site
+interaction, in other words, the description of weak interactions such as long-range van der
+Waals interactions is well beyond any conventional DFT method [13].
+The DFT calculations are quite different from the usual quantum mechanical methods
+where better accuracy depends on computational resources and not on limitations stemming
+from the method itself. The Hartree-Fock (HF) results cannot be reproduced within the
+framework of Kohn-Sham (KS) theory because the single-particle densities of finite systems
+5
+
+obtained within the HF calculations are not v-representable, i.e., do not correspond to any
+ground state of a N non-interacting electron systems in a local external potential [14]. For
+this reason, the KS theory, which finds a minimum on a different subset of all densities, can
+overestimate the ground state energy, as compared to the HF result.
+In addition to the lack of compliance with N-representability conditions and difficulties
+in extending the application of the first HK theorem to finite subspaces, there are still other
+problems that beset DFT. They have to do with how to properly include symmetry (i.e.,
+properties of all operators commuting with the Hamiltonian of a given system). For instance,
+translational symmetry in crystalline solids should be applied only to a full many-electron
+function rather than to one-electron KS orbitals!
+Currently, the KS-DFT is about occupied orbitals only and is far from giving a consistent
+and quantitatively accurate description of open-shell spin systems, as the currently available
+approximate functionals show unsystematic errors in the (inaccurate) prediction of energies,
+geometries, and molecular properties.
+Strictly speaking, the DFT is designed for description of ground rather than excited states
+with no good scheme for excitations. Because an excited-state density does not uniquely
+determine the potential, there is no general analog of HK for excited states. The standard
+functionals are inaccurate both for on-site crystal field and for charge transfer excitations [13].
+The DFT based approaches cannot provide the correct atomic limit and the term and
+multiplet structure, which is crucial for description of the optical response for 3dcompounds.
+Although there are efforts to obtain correct results for spectroscopic properties depending
+on spin and orbital density this problem remains as an open one in DFT research. Clearly,
+all these difficulties stem from unsolved foundational problems in DFT and are related to
+fractional charges and to fractional spins. Thus, these basic unsolved issues in the HKS-DFT
+point toward the need for a basic understanding of foundational issues.
+In other words, given these background problems, the DFT based models should be
+addressed as semi-empirical approximate ones rather than ab initio theories. M. Levy
+introduced in 2010 the term DFA to define density functional approximation instead of
+DFT, which is believed to quite appropriately describe contemporary DFT [3]. In chemistry,
+it is traditional to refer to standard approaches as ab initio, while DFT is regarded as
+empirical. Because solid-state calculations are more demanding, for many decades DFT was
+the only possible approach. Thus, DFT calculations are referred to as ab initio in solid-state
+6
+
+physics and materials science [13]. Proceeding with a fixed approximate functional, the DFT
+is called "first principles in the sense that the user only chooses the atoms, and the computer
+predicts (correctly or not) all properties of the molecule or solid.
+LSDA
+Basic drawback of the spin-polarized approaches is that these start with a local density
+functional in the form (see, e.g. Ref.15)
+v(r) = v0[n(r)] + ∆v[n(r), m(r)](ˆσ · m(r)
+|m(r)|) ,
+where n(r), m(r) are the electron and spin magnetic density, respectively, ˆσ is the Pauli
+matrix, that is these imply presence of a large fictious local one-electron spin-magnetic field ∝
+(v↑−v↓), where v↑,↓ are the on-site LSDA spin-up and spin-down potentials. Magnitude of the
+field is considered to be governed by the intra-atomic Hund exchange, while its orientation
+does by the effective molecular, or inter-atomic exchange fields. Despite the supposedly
+spin nature of the field it produces an unphysically giant spin-dependent rearrangement of
+the charge density that cannot be reproduced within any conventional technique operating
+with spin Hamiltonians. Furthermore, a direct link with the orientation of the field makes
+the effect of the spin configuration onto the charge distribution to be unphysically large.
+However, magnetic long-range order has no significant influence on the redistribution of
+the charge density. The DFT-LSDA community needed many years to understand such a
+physically clear point.
+In general, the LSDA method to handle a spin degree of freedom is absolutely
+incompatible with a conventional approach based on the spin Hamiltonian concept. There
+are some intractable problems with a match making between the conventional formalism
+of a spin Hamiltonian and LSDA approach to the exchange and exchange-relativistic
+effects. Visibly plausible numerical results for different exchange and exchange-relativistic
+parameters reported in many LSDA investigations (see, e.g., Refs. [16]) evidence only a
+potential capacity of the LSDA based models for semiquantitative estimations, rather than
+for reliable quantitative data. It is worth noting that for all of these "advantageous"instances
+the matter concerns the handling of certain classical N´eel-like spin configurations (ferro-,
+antiferro-, spiral,...) and search for a compatibility with a mapping made with a conventional
+7
+
+quantum spin Hamiltonian. It’s quite another matter when one addresses the search of the
+charge density redistribution induced by a spin configuration as, for instance, in multiferroics.
+In such a case the straightforward application of the LSDA scheme can lead to an unphysical
+overestimation of the effects or even to qualitatively incorrect results due to an unphysically
+strong effect of a breaking of spatial symmetry induced by a spin configuration (see, e.g.
+Refs. [17] and references therein).
+Going beyond LSDA:LDA+U, LDA+DMFT, LDA+U+V
+It is commonly accepted now that the standard DFT-LDA(GGA) approach is insufficient
+to describe the electronic structure of the Mott insulators.
+Apparent weaknesses of the DFT approach were exposed especially after the discovery
+in 1986 of the copper-oxide superconductors, as it failed to yield the fact that the parent
+compound La2CuO4 is an antiferromagnetic insulator. This difficult period for the DFT-
+LDA method as many decided was partially ended in the early and mid 1990s especially
+when an orbital dependent Hubbard-type U was incorporated in the exchange correlation
+functional of the localized 3delectrons within the LDA+U method, while the other electrons
+are still described at the LDA level [5].
+Attempts to go beyond LSDA are based on the self-interaction-corrected density
+functional theory SIC-DFT, the LDA+U method, and the GW approximation [5]. These
+methods represent corrections of the single-particle Kohn-Sham potential in one way or
+another and lead to substantial improvements over the LSDA results for the values of the
+energy gap and local moment. Within the SIC-DFT and LDA+U methods the occupied and
+unoccupied states are split by the Coulomb interaction U, whereas within the LSDA this
+splitting is caused by the Stoner parameter J, which is typically one order of magnitude
+smaller than U. Therefore, compared with the LSDA, the novel methods capture more
+correctly the physics of transition-metal oxides and improve the results for the energy gap
+and local moment significantly.
+An important drawback of the LDA+U method is that it requires U as a starting
+parameter. Even though several schemes for the determination of U exist, it is almost always
+chosen such that it reproduces the experimental value of a specific property of the electronic
+structure, most often the band gap. Usually the LDA+U calculations imply account of
+8
+
+the on-site d-d correlations with Udd parameter and do neglect the ligand p-p correlations
+though Udd parameter is only twice as large as Upp in oxides [6, 7]. The predictive power
+of the novel methods crucially relies on a reliable assessment of the interactions, however,
+the value of the interaction parameters, such as Udd, Upp, depends on the choice of the
+downfolded model, namely, the orbitals treated in the model as well as the basis functions
+employed, as the screened interaction is determined by the various screening processes that
+are not considered in the model. Therefore a careful analysis is needed to make a proper
+model and choose appropriate parameters. By fitting, one usually finds higher accuracy for
+systems similar to those fitted, but usually greater inaccuracies far away.
+All efforts to account for the correlations beyond LDA encounter an insoluble problem
+of double counting (DC) of interaction terms which had just included into Kohn-Sham
+single-particle potential. A well defined analytical expression for the DC potential cannot
+be formulated in the context of LDA+U or other technique going beyond LDA [18]. How to
+choose the DC correction potential in a manner that is both physically sound and consistent
+is unknown. Thus, one has to resort to numerical criteria to fix the value of the DC correction.
+However, there is currently no universal and unambiguous expression for DC correction,
+and different formulations are used for different classes of materials. Moreover, different
+methods for fixing the double counting can drive the result from Mott insulating to almost
+metallic [18, 19].
+The LDA+DMFT approach combines band structure theory within the DFT-LDA with
+many-body theory as provided by dynamical mean-field theory (DMFT) [5]. Within DMFT,
+a lattice model is mapped onto an effective impurity problem embedded in a medium which
+has to be determined self-consistently, e.g., by quantum Monte-Carlo (QMC) simulations.
+This mapping becomes exact in the limit of infinite dimensions.
+The LDA+U and LDA+DMFT methods are believed to correct the inaccuracies of
+approximate DFT exchange correlation functionals. The main idea of the both computational
+approaches consists in a selective description of the strongly correlated electronic states,
+typically, localized d or f orbitals, using the Hubbard model, while all the other states
+continue to be treated at the level of standard approximate DFT functionals. At present
+the LDA+U and LDA+DMFT methods are addressed to be most powerful tools for
+the investigation of strongly correlated electronic systems, however, these preserve many
+shortcomings of the basic DFT-LDA approach.
+9
+
+Current theoretical studies of electronic correlations in transition metal oxides typically
+only account for the local repulsion between d-electrons even if oxygen ligand p-states are
+an explicit part of the effective Hamiltonian. Interatomic correlations such as Vpd between
+d- and (ligand) p-electrons, as well as the on-site and inter-site interaction between p-
+electrons (Upp and Vpp), are usually neglected. Strictly speaking, LDA+DMFT scheme
+should incorporate both Upp, Vpp, Vpd and Vdd interactions [20]. To this end we need a
+proper procedure for their calculation, however, this makes the double counting problem
+significantly more sophisticated.
+NiO as a main TMO system for so-called ab initio studies
+An ongoing challenge during the last 60 years has been the development of a theoretical
+model that could offer an accurate description of both the electric and magnetic phenomena
+observed in NiO. Nickel oxide is one of the prototypical compounds that has highlighted the
+importance of correlation effects in transition metal oxides (TMO). However, despite several
+decades of studies there is still no literature consensus on the detailed electronic structure
+of NiO. Although exhibiting a partially filled 3dband and predicted by simple band theory
+to be a good conductor, NiO has a relatively large band gap (about 4 eV) that cannot be
+accounted for in the LDA calculations.
+NiO has long been viewed as a prototype "Mott insulator" [21] with the gap formed
+by intersite cation-cation d-d charge transfer (CT) transitions, however, this view was
+later replaced by that of a "CT insulator"with the gap formed by anion-cation p-d CT
+transitions [22].
+Strictly speaking, the DFT is designed for description of ground rather than excited
+states. Nevertheless research activity in the condensed matter DFT community is focused
+on the single-particle excitation properties of the TMOs, in particular, the photoemission
+spectra and energy gap.
+The XPS combined with bremsstrahlung-isochromat spectroscopy (BIS) shows a gap
+between the top of the valence band and the bottom of the conducting band of 4.3 eV for
+NiO [23]. Namely this value appears to be in the focus of the so-called ab initio DFT-LDA
+based calculations for NiO. However, the later studies [24] have shown that the exact value of
+this conductivity gap is subject to the band position chosen to define the highest valence and
+10
+
+lowest conducting levels, obtaining values that range from 3.20 to 5.67 eV (!). Experimental
+data, in particular, oxygen x-ray emission (XES) and absorption (XAS) spectra [25] point
+to strong matrix element effects, that makes reliable estimates of the energy gap to be very
+ambiguous adventure.
+The standard DFT-LDA band theory predicts NiO to be a metal. LSDA [26] predicts
+NiO to be an insulator (with severe underestimated gap of 0.3 eV) only in antiferromagnetic
+state (!?). The later GW [27] and LDA+U [28] calculations yielded the larger gap of 3.7 eV.
+First LDA+DMFT calculation performed by Ren et al. [29] yielded the value of 4.3 eV. The
+authors claimed: "The overall agreement between the calculated single-particle spectrum and
+the experimental data is surprisingly good". However, they do neglect the matrix element
+effect, p-d covalency, Upp, Vpd, and Vdd, that de facto does invalidate their conclusion. Part
+of these effects, in particular, p − d covalency was taken into account later [30], but with a
+severe reinterpretation of the DOS. Again, the authors claim: "...we were able to provide
+a full description of the valence-band spectrum and, in particular, of the distribution of
+spectral weight between the lower Hubbard band and the resonant peak at the top of the
+valence band. However, to this day the LDA+DMFT results for NiO strongly depend on
+the choice of the DC correction potential driving the result from Mott insulating to metallic
+state [18, 19].
+It is rather surprising how little attention has been paid to the DFT based calculations
+of the TMO optical properties. Lets turn to a very recent paper by Roedl and Bechstedt [31]
+on NiO and other TMOs, whose approach is typical for DFT community. The authors
+calculated the dielectric function ǫ(ω) for NiO within the DFT-GGA+U+∆ technique and
+claim:"The experimental data agree very well with the calculated curves" (!?). However,
+this seeming agreement is a result of a simple fitting when the two model parameters U and
+∆ are determined such (U = 3.0, ∆ = 2.0 eV) that the best possible agreement concerning the
+positions and intensities of the characteristic peaks in the experimental spectra is obtained.
+In addition, the authors arrive at absolutely unphysical conclusion: "The optical absorption
+of NiO is dominated by intra-atomic t2g → eg transitions" (!?).
+Nekrasov et al. [19] realized the DMFT calculation of the optical conductivity for NiO.
+Just another correlation parameter was chosen: U = 8 eV. The authors claim a general
+agreement both with optical and the X-ray experiments. In the calculations, they found
+that the main contribution to optical conductivity is due to intra-orbital optical transitions.
+11
+
+Inter-orbital optical transitions give less than 5% of the optical conductivity intensity in
+the frequency range used in the calculations. However, as usual they did neglect a number
+of important on-site and inter-site correlation parameters and all the effects due to optical
+matrix elements that does invalidate their conclusion. Furthermore, the DFT-LDA based
+schemes do not provide the correct atomic limit and the term and multiplet structure. Hence
+these cannot correctly describe both the d-d crystal field and p-d and d-d charge transfer
+transitions. However, some authors [32] suppose that in future this problem probably can be
+solved within the LDA+DMFT.
+Surveying these and other literature data we can argue that the conventional DFT based
+technique cannot provide a proper description of the optical response for strongly correlated
+3dcompounds. As up till now, in future the optical properties of the Mott or charge transfer
+insulators will be considered within the framework of cluster approaches initially based on
+quantum-chemical calculations.
+CLUSTER MODEL IN NIO
+Cluster model approach does generalize and advance crystal-field and ligand-field theory.
+The method provides a clear physical picture of the complex electronic structure and
+the energy spectrum, as well as the possibility of a quantitative modelling. In a certain
+sense the cluster calculations might provide a better description of the overall electronic
+structure of insulating 3doxides than the band structure calculations, mainly due to a better
+account for correlation effects, electron-lattice coupling, and relatively weak interactions
+such as spin-orbital and exchange coupling. Cluster models have proven themselves to be
+reliable working models for strongly correlated systems such as transition-metal and rare-
+earth compounds. These have a long and distinguished history of application in optical and
+electron spectroscopy, magnetism, and magnetic resonance. The author with colleagues has
+successfully demonstrated great potential of the cluster model for description of the p-d
+and d-d charge transfer transitions and their contribution to optical and magneto-optical
+response in 3doxides such as ferrites, manganites, cuprates, and nickelates [33].
+Cluster models do widely use the symmetry for atomic orbitals, point group symmetry,
+and advanced technique such as Racah algebra and its modifications for point group
+symmetry [34]. From the other hand the cluster model is an actual proving-ground for various
+12
+
+calculation technique from simple quantum chemical MO-LCAO (molecular orbital-linear-
+combination-of-atomic-orbitals) method to a more elaborate LDA+MLFT (MLFT, multiplet
+ligand-field theory) [35] approach.
+Cluster models traditionally combined quantum chemical MO-LCAO calculations [34]
+based on atomic Hartree-Fock orbitals with making use parameters fitted to experiments.
+Several authors obtained model parameters by performing an LDA calculation for the cluster
+and using its Kohn-Sham MOs. First comprehensive description of the electronic structure
+of the NiO6 cluster was performed by Fujimori and Minami [36]. Effective transfer and
+overlap integrals were evaluated from LCAO parameters of NiO found by Mattheiss [37]
+by fitting their APW energy-band results. The localized approach has been shown to
+successfully explain the photoemission, optical-absorption, and isochromat spectra of NiO.
+Recently, Haverkort et al. [35] suggested a sort of generalization of conventional ligand-
+field model with the DFT-based calculations within a so-called "ab initio"LDA+MLFT
+technique. They start by performing a DFT calculation for the proper, infinite crystal
+using a modern DFT code which employs an accurate density functional and basis set
+[e.g., linear augmented plane waves (LAPWs)]. From the (self-consistent) DFT crystal
+potential they then calculate a set of Wannier functions suitable as the single-particle basis
+for the cluster calculation. The authors compared the theory with experimental spectra
+(XAS, nonresonant IXS, photoemission spectroscopy) for SrTiO3, MnO, and NiO and found
+overall satisfactory agreement, indicating that their ligand-field parameters are correct to
+better than 10%. However, as in Ref. [36] the authors have been forced to treat on-site
+correlation parameter Udd and orbitally averaged (spherical) ∆pd parameter as adjustable
+ones. Comparing the novel LDA+MFLT technique with that of Fujimori and Minami [36]
+one should note very similar level of their quantitative conclusions. Despite the involvement
+of powerful calculation techniques the numerical results of the both approaches seem to
+be more like semiquantitative ones. In such a situation we should transfer the center of
+gravity of the cluster approaches more and more to elaboration of physically sound and clear
+semiquantitative models that are maximally take into account all the symmetry requirements
+on one hand and refer to experiment on the other.
+Hereafter, we do present a most recent and most comprehensive such a cluster model
+approach to the description of the p-d and d-d CT transitions in NiO [38] that nicely
+illustrates great potential of the model that does combine simple physically clear ligand-
+13
+
+field analysis, its semiquantitative predictions with a regular appeal to experimental data.
+We believe that such an approach should precede and accompany any detailed numerical
+calculation providing its physical validation.
+Starting with an octahedral NiO6 complex with the point symmetry group Oh we deal
+with five Ni 3dand eighteen oxygen O 2p atomic orbitals forming both the hybrid Ni 3d-O
+2p bonding and antibonding eg and t2g molecular orbitals (MO), and the purely oxygen
+nonbonding a1g(σ), t1g(π), t1u(σ), t1u(π), t2u(π) orbitals. The nonbonding t1u(σ) and t1u(π)
+orbitals with the same symmetry are hybridized due to the oxygen-oxygen O 2pπ - O
+2pπ transfer. The relative energy position of different nonbonding oxygen orbitals is of
+primary importance for the spectroscopy of the oxygen-3d-metal charge transfer. This is
+firstly determined by the bare energy separation ∆ǫ2pπσ = ǫ2pπ − ǫ2pσ between O 2pπ and O
+2pσ electrons. Since the O 2pσ orbital points towards the two neighboring positive 3d ions,
+an electron in this orbital has its energy lowered by the Madelung potential as compared
+with the O 2pπ orbitals, which are oriented perpendicular to the respective 3d-O-3d axes.
+Thus, the Coulomb arguments favor the positive sign of the π − σ separation ǫpπ − ǫpσ
+whose numerical value can be easily estimated in the frames of the well-known point charge
+model, and appears to be of the order of 1.0 eV. In a first approximation, all the γ(π) states
+t1g(π), t1u(π), t2u(π) have the same energy. However, the O 2pπ-O 2pπ transfer and overlap
+yield the energy correction to the bare energies with the largest value and a positive sign for
+the t1g(π) state. The energy of the t1u(π) state drops due to a hybridization with the cation
+4pt1u(π) state.
+The ground state of NiO610− cluster, or nominally Ni2+ ion corresponds to t6
+2ge2
+g
+configuration with the Hund 3A2g(F) ground term. Typically for the octahedral MeO6
+clusters [33] the nonbonding t1g(π) oxygen orbital has the highest energy and forms the first
+electron removal oxygen state while the other nonbonding oxygen π-orbitals, t2u(π), t1u(π),
+and the σ-orbital t1u(σ) have a lower energy with the energy separation ∼ 1 eV inbetween
+(see Fig. 1).
+The p-d CT transition in NiO10−
+6
+center is related to the transfer of O 2p electron to the
+partially filled 3deg-subshell with the formation on the Ni-site of the (t6
+2ge3
+g) configuration of
+nominal Ni+ ion isoelectronic to the well-known Jahn-Teller Cu2+ ion. Yet actually instead
+of a single p-d CT transition we arrive at a series of O 2pγ→ Ni 3deg CT transitions
+forming a complex p-d CT band. It should be noted that each single electron γ→eg p-d
+14
+
+Рис. 1: (Color online) Spectra of the intersite d-d, p-d CT transitions and on-site crystal field d-d
+transitions in NiO. Strong dipole-allowed σ−σ d-d and p-d CT transitions are shown by thick solid
+uparrows; weak dipole-allowed π − σ p-d transitions by thin solid uparrows; weak dipole-forbidden
+low-energy transitions by thin dashed uparrows, respectively. Dashed downarrows point to different
+electron-hole relaxation channels, dotted downarrows point to photoluminescence (PL) transitions,
+I1,2 are doublet of very narrow lines associated with the recombination of the d-d CT exciton.
+The spectrum of the crystal field d-d transitions is reproduced from Ref. [45]. The right hand side
+reproduces a fragment of the RIXS spectra for NiO [41].
+CT transition starting with the oxygen γ-orbital gives rise to several many-electron CT
+states. For γ=t1,2 these are the singlet and triplet terms 1,3T1, 1,3T2 for the configurations
+t6
+2ge3
+gt1,2, where t1,2 denotes the oxygen hole. The complex p-d CT band starts with the
+dipole-forbidden t1g(π)→eg, or 3A2g→1,3T1g, 1,3T2g transitions, then includes two formally
+dipole-allowed the so-called π→σ p-d CT transitions, the weak t2u(π)→eg, and relatively
+strong t1u(π)→eg CT transitions, respectively, each giving rise to 3A2g→3T2u transitions.
+15
+
+Finally the main p-d CT band is ended by the strongest dipole-allowed σ→σ t1u(σ)→
+eg (3A2g→3T2u) CT transition. The above estimates predict the separation between the
+partial p-d bands to be ∼ 1 eV. Thus, if the most intensive CT band with a maximum
+around 7 eV observed in the RIXS spectra [39–41] to attribute to the strongest dipole-
+allowed O 2pt1u(σ)→Ni 3deg CT transition then one should expect the low-energy p-d
+CT counterparts with the maxima around 4, 5, and 6 eV respectively, which are related
+to the dipole-forbidden t1g(π)→eg, the weak dipole-allowed t2u(π)→eg, and relatively strong
+dipole-allowed t1u(π)→eg CT transitions, respectively (see Fig. 1). It is worth noting that
+the π→σ p-d CT t1u(π)−eg transition borrows a portion of the intensity from the strongest
+dipole-allowed σ→σ t1u(σ)→eg CT transition because the t1u(π) and t1u(σ) states of the
+same symmetry are partly hybridized due to the p-p covalency and overlap.
+Thus, the overall width of the p-d CT bands with the final t6
+2ge3
+g configuration occupies
+a spectral range from ∼ 4 up to ∼ 7 eV. The left hand side of Fig. 1 summarizes the
+main semiquantitative results of the cluster model predictions for the energy and relative
+intensities of the p-d CT transitions. Interestingly this assignment finds a strong support in
+the reflectance (4.9, 6.1, and 7.2 eV for the allowed p-d CT transitions) spectra of NiO [42]. A
+rather strong p(π)-d CT band peaked at 6.3 eV is clearly visible in the absorption spectra of
+MgO:Ni [43]. The electroreflectance spectra [44] which detect the dipole-forbidden transitions
+clearly point to a low-energy forbidden transition peaked near 3.7 eV missed in the reflectance
+and absorption spectra [42, 43, 45], which thus defines a p-d character of the optical CT gap
+and can be related to the onset transition for the whole complex p-d CT band. It should
+be noted that a peak near 3.8 eV has been also observed in the nonlinear absorption spectra
+of NiO [46]. At variance with the bulk NiO a clearly visible intensive CT peak near 3.6-
+3.7 eV has been observed in the absorption spectra of NiO nanoparticles [47]. This strongly
+supports the conclusion that the 3.7 eV band is related to the bulk-forbidden CT transition
+which becomes the partially allowed one in the nanocrystalline state [38]. It is worth noting
+that the hole-type photoconductivity threshold in bulk NiO has been observed also at this
+"magic" energy 3.7 eV [48], that is the t1g(π)→eg p-d CT transition is believed to produce
+itinerant holes. Indeed, the p-d CT transitions in NiO6 cluster are of so-called "anti-Jahn-
+Teller" type, that is these are transitions from orbitally nondegenerate state to the final p-d
+CT state state formed by two orbitally degenerate states that points to strong electron-lattice
+effects in excited state. The final Ni1+ 3d9(t6
+2ge3
+g) configuration is isoelectronic to Cu2+ ion in
+16
+
+cubic crystal field and presents a well-known textbook example of a Jahn-Teller center that
+implies a strong trend to the localization, while a photo-generated hole can move more or
+less itinerantly in the O 2p valence band determining the hole-like photoconductivity [48]. It
+should be noted that any oxygen π-holes have a larger effective mass than the σ-holes, that
+results in a different role of the p(π)-d and p(σ)-d CT transitions both in photoconductivity
+and, probably, the luminescence stimulation.
+A spectral feature near 6 eV, clearly visible in the NiO photoluminescence excitation
+(PLE) spectra [38] can be certainly attributed to a rather strong p(π)-d (t1u(π) → eg) CT
+transition while the spectral feature near 5 eV to a weaker p(π)-d (t2u(π) → eg) CT transition.
+Interestingly the strongest p(σ)-d (t1u(σ) → eg) CT transition at ∼ 7 eV is actually inactive
+in the PLE spectra, most likely, due to a dominating nonradiative relaxation channel for the
+oxygen t1u(σ) holes.
+However, the p-d CT model cannot explain the main low-energy spectral feature, clearly
+visible in the PLE spectra near 4 eV [38], thus pointing to manifestation of another CT-type
+mechanism. Indeed, along with the p-d CT transitions an important contribution to the
+optical response of the strongly correlated 3doxides can be related to the strong dipole-
+allowed d-d CT, or Mott transitions [33]. In NiO one expects a strong d-d CT transition
+related to the σ − σ-type eg − eg charge transfer t6
+2ge2
+g + t6
+2ge2
+g→ t6
+2ge3
+g + t6
+2ge1
+g between nnn
+Ni sites with the creation of electron NiO611− and hole NiO69− centers (nominally Ni+ and
+Ni3+ ions, respectively) thus forming a bound electron-hole dimer, or d-d CT exciton.
+The strong dipole-allowed Franck-Condon d(eg)-d(eg) CT transition in NiO manifests
+itself as a strong spectral feature near 4.5 eV clearly visible in the absorption of thin
+NiO films [49], RIXS spectra [39, 41], the reflectance spectra (4.3 eV) [42]. Such a strong
+absorption near 4.5 eV is beyond the predictions of the p-d CT model and indeed is lacking
+in the absorption spectra of MgO:Ni [43]. It should be noticed that, unlike all the above
+mentioned structureless spectra, the nonlinear absorption spectra [46] of NiO films do reveal
+an anticipated "fine" structure of the d-d CT exciton with the two narrow peaks at 4.075
+and 4.33 eV preceding a strong absorption above 4.575 eV. Interestingly the separation 0.2-
+0.3 eV between the peaks is typical for the exchange induced splittings in NiO (see, e.g.,
+the "0.24 eV" optical feature [45]). Accordingly, the 4.1 eV peak in the PLE spectra can be
+unambiguously assigned to the d-d CT transition [38].
+The charge, spin, and orbital degeneracy of the final state of this unique double anti-
+17
+
+Jahn-Teller transition 3A2g + 3A2g→2Eg + 2Eg results in a complex band observed at 4.2-4.5
+eV [38]. The exchange tunnel reaction Ni++Ni3+↔Ni3++Ni+ due to a two-electron transfer
+gives rise to the two symmetric (S- and P-) excitons having s- and p-symmetry, respectively,
+with the energy separation δ0 = 2|t| and δ1 =
+2
+3|t| for the spin singlet and spin triplet
+excitons, where t is the two-electron transfer integral whose magnitude is of the order of the
+Ni2+-Ni2+ exchange integral: t ≈ Innn. Interestingly the P-exciton is dipole-allowed while
+the S-exciton is dipole-forbidden. The anti-Jahn-Teller d-d CT exciton is prone to be self-
+trapped in the lattice due to the electron-hole attraction and a particularly strong double
+Jahn-Teller effect for both the electron and hole centers. Recombination transitions for such
+excitons produce a bulk luminescence with puzzling well isolated doublet of very narrow
+lines with close energies near 3.3 eV [38] that corresponds to a reasonable Stokes shift of 1
+eV. To the best of our knowledge it is the first observation of the self-trapping for the d-d
+CT excitons.
+Thus, we see that a simple cluster model is able to provide a semiquantitative description
+of a large body of experimental spectroscopic data, including subtle effects beyond the reach
+of any "ab initio"DFT technique. We have shown that the prototype 3doxide NiO, similar
+to perovskite manganites RMnO3 or parent cuprates such as La2CuO4 [33], should rather
+be sorted neither into the CT insulator nor the Mott-Hubbard insulator in the Zaanen-
+Sawatzky-Allen scheme [22].
+SUMMARY
+There are still a lot of people who think the Hohenberg-Kohn-Sham DFT within the
+LDA has provided a very successful ab initio framework to successfully tackle the problem
+of the electronic structure of materials. However, both the starting point and realizations
+of the DFT approach have raised serious questions. The HK "theorem"of the existence of
+a mythical universal density functional that can resolve everything looks like a way into
+Neverland, the DFT heaven is probably unattainable. Various DFAs, density functional
+approximations, local or nonlocal, will never be exact. Users are willing to pay this price
+for simplicity, efficacy, and speed, combined with useful (but not yet chemical or physical)
+accuracy [4, 13].
+The most popular DFA fail for the most interesting systems, such as strongly correlated
+18
+
+oxides. The standard approximations over-delocalize the d-electrons, leading to highly
+incorrect descriptions. Some practical schemes, in particular, DMFT can correct some of
+these difficulties, but none has yet become a universal tool of known performance for such
+systems [13].
+Any comprehensive physically valid description of the electron and optical spectra for
+strongly correlated systems, as we suggest, should combine simple physically clear cluster
+ligand-field analysis with a numerical calculation technique such as LDA+MLFT [35], and
+a regular appeal to experimental data.
+The research was supported by the Ministry of Education and Science of the Russian
+Federation, project FEUZ-2020-0054.
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+arXiv:2301.12160v1  [physics.optics]  28 Jan 2023
+Polarization-independent second-order photonic topological corner states
+Linlin Lei,1 Shuyuan Xiao,2, 3 Wenxing Liu,1 Qinghua Liao,1, ∗ Lingjuan He,1 and Tianbao Yu1, †
+1School of Physics and Materials Science,
+Nanchang University, Nanchang 330031, China
+2Institute for Advanced Study, Nanchang University, Nanchang 330031, China
+3Jiangxi Key Laboratory for Microscale Interdisciplinary Study,
+Nanchang University, Nanchang 330031, China
+1
+
+Abstract
+Recently, much attention has been paid to second-order photonic topological insulators (SPTIs), because
+of their support for highly localized corner states with excellent robustness. SPTIs have been implemented
+in either transverse magnetic (TM) or transverse electric (TE) polarizations in two-dimensional (2D) pho-
+tonic crystals (PCs), and the resultant topological corner states are polarization-dependent, which limits
+their application in polarization-independent optics. However, to achieve polarization-independent corner
+states is not easy, since they are usually in-gap and the exact location in the topological bandgap is not
+known in advance. Here, we report on a SPTI based on a 2D square-lattice PC made of an elliptic metama-
+terial, and whether the bandgap is topological or trivial depends on the choice of the unit cell. It is found that
+locations of topological bandgaps of TM and TE polarizations in the frequency spectrum can be indepen-
+dently controlled by the out-of-plane permittivity ε⊥ and in-plane permittivity ε∥, respectively, and more
+importantly, the location of in-gap corner states can also be separately manipulated by them. From this, we
+achieve topological corner states for both TM and TE polarizations with the same frequency in the PC by
+adjusting ε⊥ and ε∥, and their robustness against disorders and defects are numerically demonstrated. The
+proposed SPTI provides a potential application scenario for polarization-independent topological photonic
+devices.
+I.
+INTRODUCTION
+Recently, the concept of higher-order topological insulators (HOTIs) has been extended from
+electronic waves into classic waves[1–12]. It has been shown that HOTIs do not obey the usual
+bulk-edge correspondence but comply with the bulk-edge-corner correspondence[13–15]. For
+instance, a two-dimensional (2D) second-order topological insulator possesses one-dimensional
+(1D) gapped edge states and zero-dimensional (0D) in-gap corner states. In addition to the charac-
+teristics of strong field localization and small mode volume, 0D corner states also show excellent
+robustness against fabrication flaws[16–18]. On this basis, they have enormous application value
+in the topological cavity[18, 19], lasing[20, 21], non-linear optics[22, 23], and sensing[24]. How-
+ever, for photonic crystals (PCs), the two kinds of polarization, transverse magnetic (TM) and
+transverse electric (TE) modes, are usually studied in a separate way. One reason is either of the
+∗ [email protected]
+† [email protected]
+2
+
+two modes can be excited independently, each with its own band structure, and the other is that
+forming a common band gap (CBG) is not easy, especially the topological one. Past researches
+have shown the polarization-independent optics is potentially useful in polarization-independent
+waveguides relying full bandgaps[25], enhanced nonlinear optical effects[26], and polarization
+division multiplexing[27]. Topologically protected polarization-independent optics would give
+them additional resistance to perturbation. It is worth noting that dual-polarization second-order
+photonic topological states have been proposed by Chen et al. recently, based on a topologically
+optimized geometric structure within a square-lattice[28]. However, eigenfrequencies of topolog-
+ical states for the two polarizations are not the same, despite they have a common topological
+bandgap.
+In this paper, a 2D second-order photonic topological insulator (SPTI) is proposed, of which the
+topological states are polarization-independent. The square-lattice PC having a fishnet structure is
+made by an elliptic metamaterial. The permittivity is anisotropic and nevertheless, the geometry
+structure is rather simple compared with the previously proposed topologically optimized struc-
+ture. That the CBG is either trivial or topological depends on the choice of the unit cell (UC) for
+both TM and TE modes. The proposed SPTI can host topological edge states and corner states
+for the two modes at the same time. Our results show polarization-independent topological cor-
+ner states based a SPTI is not guaranteed by a common topological bandgap. However, we find
+that locations of bandgaps and corner states in the frequency spectrum can be manipulated inde-
+pendently by the out-of-plane permittivity ε⊥ and in-plane permittivity ε∥ for TM and TE modes,
+respectively, which gives an effective way to achieve overlapped corner states for the two modes.
+On this basis, corner states independent of polarization can be realized by choosing appropriate
+ε⊥ and ε∥. Numerical simulations further show the corner states are topologically protected, with
+strong robustness to disorders and defects. Our work shows potential applications in polarization-
+independent topological photonic devices.
+II.
+STRUCTURE DESIGN AND BAND TOPOLOGY
+For PCs, it is well known that TM bandgaps are favored in dielectric rods, while TE bandgaps
+prefer dielectric veins[29]. From this, the proposed square-lattice PC is constructed by thin dielec-
+tric veins with dielectric rods located at lattice sites, as shown in Fig. 1(a). a is the lattice constant,
+and the circle radius r and vein width d are 0.3a and 0.18a, respectively. The dielectric material is
+3
+
+anisotropic, an elliptic metamaterial with the permittivity ε = (ε∥, ε∥, ε⊥) = (16.9, 16.9, 10). Gen-
+erally, topological corner states lie in a topological bandgap[13], and hence a topological CBG
+of TM and TE polarizations is the prerequisite for polarization-independent topological corner
+states. The choice of the elliptic metamaterial is based on the consideration that bandgap locations
+of TM and TE polarizations in the frequency spectrum can be manipulated independently by ε⊥
+and ε∥, respectively. In practical, we can use the multilayer model to construct the anisotropic
+permittivity[30]. The multilayer consists of two alternative dielectrics with high and low permit-
+tivity, and it is placed horizontally in the x-y plane. According to the formulisms (16) and (17)
+proposed in ref[31], the PC slab with permittivity (16.9,16.9,10) can be approximately built by
+the high dielectric with the permittivity of 17.67 and the air layer when the filling ratio of high
+dielectric is 0.954. Herein, the calculation of band structures and numerical simulations are based
+on the finite element method using the commercial software COMSOL Multiphysics.
+FIG. 1. (a) Fishnet PC and two kinds of unit cells (UC), UC1 and UC2. (b) Band structures of TM and
+TE modes, denoted by red and blue dot-lines, respectively. Even and odd parities of UC1 (UC2) at high
+symmetric points are indicated by plus and minus symbols, colored in red and blue for TM and TE modes,
+respectively. (c) Ez field patterns of the two TM bands at the X point for UC1 and UC2. (d) Hz field patterns
+of the two TE bands at the X point for UC1 and UC2.
+4
+
+(a)
+(b)
+0.4
+TM
+TE
+Frequency(c/a)
+0.3
++
++(-)
+UC1
+(+)
+0.2
++(-)
+-(+)
+0.1
+0.0
+X(Y)
+M
+(c)
+(d)
+TM
+TE
+UC1
+UC1
+UC2
+UC2
+1st
+2nd
+1st
+2nd
++1
+-1
++1
+-1
+Ez
+HzBased on the common square lattice, two kinds of unit cells (UCs), UC1 and UC2, are selected
+in Fig. 1(a). Note that the two UCs are consistent with each other after shifting the center of one
+of the UCs by half of the period along x and y directions. Therefore, they share the same band
+structure as plotted in Fig. 1(b), with red and blue dot-lines denoting the TM and TE modes, re-
+spectively. One can find that there is a CBG indicated by the gray region lying between the first
+and second bands of TM modes. However, for the two UCs, the CBG possesses different topo-
+logical behaviors characterized by the 2D Zak phase [see Appendix A], which has the following
+form[32–34]:
+θZak
+j
+=
+�
+dkxdkyTr[ ˆA j(kx, ky)],
+(1)
+where j = x or y, and the Berry connection ˆA j = i⟨u(k)|∇kj|u(k)⟩ with u(k) being the periodic part
+of the Bloch function. The 2D Zak phase can also be understood by the 2D bulk polarization via
+θZak
+j
+= 2πP j with
+P j = 1
+2(
+�
+n
+qn
+j mod 2),
+(−1)qn
+j = η(X j)
+η(Γ)
+(2)
+where P j is determined by the parity η associated with π rotation at Γ and X(Y) points and the
+summation is over all the occupied bands below the bandgap. Here, Px is equal to Py, namely,
+Px = Py, due to the C4 symmetry[35, 36]. Eigenfield patterns at the X point of the two bands
+for TM and TE modes are shown in Figs. 1(c) and 1(d), respectively, with the monopole an even
+parity and dipole an odd parity. As can be seen, the parities of the two bands at the X point have an
+inversion between UC1 and UC2 for both the two modes, whereas the parities at the Γ point stay
+the same. Moreover, parities of the same UC at the X point are opposite for TM and TE modes,
+which gives the same UC distinct topological properties for the two modes. Concretely, for TM
+modes, the distinct parties of UC1 at the X and Γ points give the 2D bulk polarization (Px, Py) a
+value of (0, 0) and the 2D Zak phase (θZak
+x , θZak
+y ) a value of (0, 0), while the same parity of UC2 at
+the X and T points makes (Px, Py) = (1
+2, 1
+2) and (θZak
+x , θZak
+y ) = (π, π). The opposite is true for the TE
+modes. As a result, the bandgap of UC1 is trivial and of UC2 is topological for TM modes, and it
+is reversed for TE modes.
+5
+
+FIG. 2. Projected band structures of (a) TM and (b) TE modes, with edge modes colored in red and blue,
+respectively. Eigenfields at kx = 0 show the edge modes can be well confined at the interface between
+UC1s and UC2s for both TM and TE modes. (c) Dependence of bandgaps and eigenfrequencies of one of
+the corner states on ε⊥ and ε∥ for the two modes. The area shaded in light red indicates TM bandgaps,
+while the area shaded in light blue indicates TE bandgaps. The red and blue lines denote one of the corner
+states of TM and TE modes, respectively. (d) TM corner states (colored in red) and TE corner states
+(colored in blue) under any combination of ε⊥ and ε∥ in the same parameter range of (c). The yellow
+intersecting line denotes the combinations that have overlapped corner states. The yellow points on the
+intersecting line is two of the combinations, and their anisotropic permittivity (ε∥, ε∥, ε⊥) are (16.9,16.9,10)
+and (16.375,16.375,9.7), respectively. The green points are the two points that share the same anisotropic
+permittivity (16.7,16.7,10.4) but have different eigenfrequencies.
+III.
+POLARIZATION-INDEPENDENT TOPOLOGICAL CORNER STATES
+The topological distinction between UC1 and UC2 ensures the existence of topological edge
+states[37–40]. To show this, we construct a supercell composed of five UC1s and five UC2s along
+the y direction, and projected band structures are shown in Figs. 2(a) and 2(b) for TM and TE
+modes, respectively. In the calculation, periodic boundary conditions are applied to the x direction.
+6
+
+a
+0.3
+(b)
+0.3
+Frequency(c/a)
+0.2
+0.2
+0.1
+0.1
++1
+TM
+TE
+0.0
+0.0
+0
+1
+-1
+0
+k(π/a)
+k,=0
+kx(元/a)
+k,=0
+(c)
+16.4 16.6 16.8 17.0 17.2 174
+(d)
+TMcornerstates
+0.28
+0.266
+TEcornerstates
+0.264
+Frequency(c/a)
+0.27
+0.262
+(e)
+0.260
+Frequency(c/
+0.26
+0.258
+0.256
+0.25
+0.254
+TM bandgap
+（16.375,9.7)
+TM corner state
+0.252
+0.24
+TE bandgap
+(16.9,10)
+TE corner state
+(16.7,10.4)
+3
+9.4
+9.6
+9.8
+10.010.210.410.6
+81FIG. 3. (a) Schematic of the finite-size box-shaped PC, with 15×15 UC1s surrounded by 6-layer UC2s. (b)
+Eigenfrequencies of the box-shaped PC. TM and TE modes are denoted by pentagons and circles, with their
+corner states colored in red and blue, respectively. Edge modes are shown as cyan. (c) Eigenfields of the
+overlapped edge and corner modes.
+FIG. 4. (a) Box-shaped PC with four disorders (red dots) around four corners of the internal PC composed
+UC1s. The enlarged view shows one of the four disorders, with 10% decrease in radius and 0.1a deviation
+from the lattice site along x and y directions. Eigenfields of four corner modes of (b) TM and (c) TE modes,
+under the influence of the disorders.
+As can be seen, there is one in-gap edge state for both the two modes, which does not occupy the
+7
+
+(a)
+(b
+EC1
+EC2
+0.25895(c/a)
+0.25897(c/a)
+TEC3
+TEC4
+0.25876(c/a
+0.25897(c/a)
+0.25897(c/a)
+O
+Ez
+Hz(a)
+(b)
+0.28
+TM bulk
+TE bulk
+6-layer Uc2s
+TM edge
+TEedge
+0.27
+★
+TMcorner
+TE corner
+requency(c/a)
+0.26
+0.25883(c/a)
+L5x15UC1s
+0.25
+★
+0.24
+0.23
+0
+10
+20
+30
+40
+50
+60
+Solution number
+(c)
+TMC2
++1
+5209(c/a
+0.25883
+5883
+-1
+TEC1
+TEC2
+TEC3
+TEC4
++1
+OHz
+0.25883(c/a)
+-1
+0.24836(c/a)
+0.25881(c/a)
+0.25883(c/a)
+0.25883(c/a)FIG. 5. (a) Box-shaped PC with defects produced by removing five UC1s in the center and four UC2s near
+the edge of the PC. Eigenfields of four corner modes of (b) TM and (c) TE modes, under the influence of
+the defects.
+whole bulk bandgap and canbe confined at the interface between UC1s and UC2s. Since therer is a
+C4 symmetry for the PC, we can define a corner topological index: Qc = 1
+4([X1] + 2[M1] + 3[M2]),
+where [Πp] = #Πp − #Γp and #Πp is defined as the number of bands below the bandgap with
+rotation eigenvalues Πn
+p = e[2πi(p−1)/n] for p=1, 2, 3, 4. For the nontrivial TM and TE cases, they
+both have [X1] = −1, [M1] = −1, [M2] = 0. Therefore, the corner topological index is Qc = 1
+4
+for both the two modes, indicating 1
+4 fractionalized corner states at each of the four corners[40]. It
+is noteworthy that the existence of polarization-independent corner states is not guaranteed by the
+CBG. In Fig. 2(c), we change ε⊥ and ε∥ in the certain range near (16.9, 16.9, 10) to solely adjust
+the positions of supercell bandgaps in the frequency spectrum for TM and TE modes, respectively.
+Specifically, for the TM band gap, we increase ε⊥ from 9.4 to 10.6 and keep ε∥ at any value, while
+for the TE band gap, we increase ε∥ from 16.3 to 17.5 and keep ε⊥ at an arbitrary value. As can
+be seen, the positions of the two bandgaps descend as the corresponding permittivity increases,
+and the TM bandgap (light red area) is completely embedded in the TE bandgap (light blue area),
+forming the CBG. We also calculate the eigenfrequencies of TM (red line) and TE (blue line)
+corner states from the finite-size box-shaped PC shown in Fig. 3(a), and find that they are in
+the CBG and the variation trend of the corner states with the permittivity is the same as that of
+the bandgaps. Since the two kinds of polarized corner states are independent of each other, in
+order to search for the overlapped ones, we plot their eigenfrequencies under any combination of
+ε⊥ and ε∥ in Fig. 2(d). It can be observed that corner states of the two modes do not coincide
+with each other except on the yellow intersecting line. The yellow points on the intersecting
+8
+
+(a)
+(b)
+EC1
+EC2
+0.25881(c/a)
+0.25883(c/a)
+TEC3
+TEC4
+0.25883(c/a)
+0.25883(c/a)
++1
+E7
+Hzline are two of the combinations that have the overlapped corner states, and the corresponding
+anisotropic permittivities (ε∥, ε∥, ε⊥) are (16.9,16.9,10) and (16.375,16.375,9.7). As a contrast,
+green points are the two points that share the same anisotropic permittivity (16.7,16.7,10.4) but
+have different eigenfrequencies. Therefore, the anisotropic permittivity provides an additional
+freedom to manipulate the location of corner states of the two modes, making the corner states
+either polarization-independent or polarization-separable [see Appendix B].
+To verify the existence of the polarization-independent corner states, a box-shaped PC of finite
+size is constructed, which is composed of 15 × 15 UC1s surrounded by six-layer UC2s, as shown
+in Fig. 3(a). The calculated eigenfrequencies of TM and TE modes based on the anisotropic per-
+mittivity (16.9,16.9,10) are shown in Fig. 3(b). As can be seen, both of them show gapped edge
+modes and four in-gap corner modes. Red and blue dotted lines go through the overlapped cor-
+ner and edge states, respectively. In Fig. 3(c), eigenfields of these topological states indicate that
+the edge modes can be well confined along the whole interface between UC1s and UC2s, while
+the corner states are highly localized at the corners of the internal PC formed by the UC1s. Re-
+markably, topological corner states for the two modes do share the same eigenfrequencies, and the
+common eigenfrequency of the corner states is 0.25883(c/a). This is different from the previously
+reported dual-polarization topological corner states, which possess the topological CBG, but their
+eigenfrequencies are not overlapped at all[28].
+FIG. 6. (a) Eigenfrequencies of the box-shaped PC with an anisotropic permittivity of (16.375,16.375,9.7),
+showing overlapped corner states of TM and TE modes. Pentagons and circles denote TM and TE modes,
+and their corner states are colored in red and blue, respectively. (b). Eigenfields of the corner states of TM
+modes. (c) Eigenfields of the corner states of TE modes.
+The polarization-independent photonic corner states are topologically protected due to their
+9
+
+(b)
+(c)
+2
+0.28
+M
+TEC1
+TEC2
+★
+TM bulk/edge
+★
+TM corner
+TE bulk/edge
+a
+0.27
+TEcorner
+0.26247(c/a)
+0.26244(c/a)
+0.26247(c/a)
+0.26247(c/a)
+★食鱼食★★★★
+★★★★★
+FMC3
+FMC4
+TEC3
+TEC4
+0.25
+0
+2
+4
+6
+8
+10
+12
+14
+0.26252(c/a)
+0.26256(c/a)
+0.26247(c/a)
+0.26247(c/a)
+Solution number
++1
++1
+0
+-1
+Ez
+HzFIG. 7. (a) Box-shaped PC with a disorder (red dot) located at the left bottom corner of the internal PC
+composed UC1s. The enlarged view shows the single disorder, with 10% decrease in radius and 0.1a
+deviation from the lattice site along x and y directions. Eigenfields of four corner modes of (b) TM and (c)
+TE modes, under the influence of the disorder.
+FIG. 8. (a) Eigenfrequencies of the box-shaped PC with an anisotropic permittivity of (16.7,16.7,10.4),
+which shows corner states of TM and TE modes are not overlapped. Pentagons and circles denote TM and
+TE modes, and their corner states are colored in red and blue, respectively. (b). Eigenfields of the corner
+states of TM modes. (c) Eigenfields of the corner states of TE modes.
+topology origin[41, 42]. To verify this, we introduce four disorders marked by red dots around
+the four corners into the instead perfect PC, as shown in Fig. 4(a). The enlarged view in Fig. 4(a)
+exhibits the single disorder, with 10% decrease in radius and 0.1a deviation from the lattice site
+along x and y directions. Eigenfields of the corner states of TM and TE modes are shown in
+Figs. 4(b) and 4(c), respectively, from which we can see that the corner states still exist with
+negligible offsets of the eigenfrequencies. Beyond that, defects, produced by removing five UC1s
+10
+
+(b)
+(c)
+0.28
+TEC1
+TEC2
+TM bulk/edge
+★
+TM corner
+TEbulk/edge
+0.27
+TE corner
+0.25416(c/a)
+025411c3
+0.26017(c/a)
+0.26020(c/a)
+TMC3
+TEC3
+M4
+TEC4
+★
+0.25
+0
+2
+4
+6
+8
+10
+12
+14
+0.25416(c/a
+0.25423(c/
+0.26020(c/a)
+0.26020(c/a
+Solution number
++1
++1
+O
+-1
+Ez
+Hz(a)
+(b)
+(c)
+TMC1
+TEC1
+TEC2
+0.26225(c/a)
+0.26245(c/a)
+0.26244(c/a)
+0.26247(c/a)
+TMC3
+TMC4
+TEC3
+TEC4
+0.26249(c/a)
+0.26255(c/a)
+0.26247(c/a)
+0.26427(c/a)
++1
+O1
++1
+0
+-1
+Ez
+Hzand four UC2s in the center and near the edge of the PC respectively, are also introduced, as shown
+in Fig. 5(a). As can be seen in Figs. 5(b) and 5(c), since the defects are far away from the corners,
+the eigenfrequencies of the corner states for the TM and TE modes remain unchanged, although
+the defects have a more destructive effect on the PC structure[43].
+IV.
+CONCLUSION
+In summary, a polarization-independent SPTI is achieved, based on a 2D square-lattice PC. The
+dielectric is an elliptic metamaterial, and the geometric structure is rather simple nevertheless. By
+selecting appropriate geometric parameters and anisotropic permittivity, a CBG is can be obtained
+for TM and TE modes. That the CBG of a certain UC is either trivial or topological depends on the
+polarization modes. Topological corner states of TM and TE modes can coexist in the CBG, but
+only the combinations of in-plane permittivity ε∥ and out-of-plane permittivity ε⊥ that lie on the
+intersecting line in the eigenfrequency-permittivity space can make them overlapped. Numerical
+simulations further show they have strong robustness to disorders and defects. The proposed
+scheme can also be extended to corner states induced by the quadrupole topological phase in
+square-lattices, pseudo-spin and valley-spin degrees of freedom. Our work would pave the way
+toward designing high-performance polarization-independent topological photonic devices, such
+as the polarization-independent topological laser and coupled cavity-waveguide.
+ACKNOWLEDGMENTS
+The work was jointly supported by the National Natural Science Foundation of China (12064025,
+12264028) and Natural Science Foundation of Jiangxi Province (20212ACB202006) and Major
+Discipline Academic and Technical Leaders Training Program of Jiangxi Province (20204BCJ22012).
+Appendix A: Tight-binding model
+The tight-binding model gives the topological phase transition between the UC1 and UC2 a
+well description, in which one can take the dielectric rods as lattice sites for TM modes while the
+air holes act the part for TE modes. The Hamiltonain has the following form,
+11
+
+H = −
+�
+ij
+tijc†
+i cj,
+(A1)
+where tij is the hopping amplitude between the nearest lattice sites and c†
+i (cj) is the creation (an-
+nihilation) operator. As there is only one lattice cite in UCs, the one band below the photonic
+bandgap can be expressed as
+E = −t0(eikx + e−ikx + eiky + e−iky) = −2t0(cos kx + cos ky).
+(A2)
+Look at TM modes first, for UC1, the lattice site choosed as the inversion center is at the center
+of the UC1, and the inversion operator is I = 1. Hence, parities at Γ and X points are the same.
+For UC2, lattice sites are at the four corners and the inversion operator I = e±i(kx+ky) hinges on
+which lattice site is referenced. Thus, the parity is +1 at the Γ = (0, 0) point, while it is -1 at the
+X = (π, 0) point[34].
+For TE modes, lattice sites of UC1 choosed as the inversion center are at the four corners, since
+the air holes instead of the dielectric rods act the role of lattice sites. For UC2, the lattice site
+choosed as the inversion center is at the center of UC2. As a consequence, parities at Γ and X
+points are oppostie for UC1, while they are the same for the UC2. The results are consistence with
+parities showed in Fig. 1(b).
+Appendix B: Switch between polarization-independent and polarization-separable corner states
+Here, we would like to show another anisotropic permittivity lying on the intersecting line that
+can achieve polarization-independent topological corner states. The anisotropic permittivity is
+(16.375, 16.375, 9.7), as indicated in the Fig. 2(e). Fig. 6(a) shows the calculated eigenfrequencies,
+from which we can see that the corner states of the two modes can be overlapped under this
+permittivity. Eigenfrequencies and eigenfields of the corner states are shown in Figs. 6(b) and
+6(c), and one can see the overlapped eigenfrequency is 0.26247(c/a). In Fig. 7, if we introduce a
+single disorder into the box-shaped PC, the corner states still survive with litte frequency shit, but
+monopole and quadrupole of TM modes no longer exist due to the broken of the C4 symmetry of
+the box-shaped PC.
+Noting that if the anisotropic permittivity is off the intersecting line, polarization-independent
+corner states will be changed into polarization-separable corner states. As shown in the Fig. 2(e),
+12
+
+if the anisotropic permittivity is (16.7,16.7,10.4), eigenfrequencies of the corner states of the two
+modes are apart from each other. In detail, we plot the eigenfrequencies in Fig. 8(a), and one can
+find that none of the four corner states of the two modes are the same, and the maximum frequency
+difference between the two modes is 0.00606(c/a) as shown in Figs. 8(b) and 8(c).
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+15
+
+This figure "fig_1.png" is available in "png"� format from:
+http://arxiv.org/ps/2301.12160v1
+
+This figure "fig_2.png" is available in "png"� format from:
+http://arxiv.org/ps/2301.12160v1
+
+This figure "fig_3.png" is available in "png"� format from:
+http://arxiv.org/ps/2301.12160v1
+
+This figure "fig_4.png" is available in "png"� format from:
+http://arxiv.org/ps/2301.12160v1
+
+This figure "fig_5.png" is available in "png"� format from:
+http://arxiv.org/ps/2301.12160v1
+
+This figure "fig_6.png" is available in "png"� format from:
+http://arxiv.org/ps/2301.12160v1
+
+This figure "fig_7.png" is available in "png"� format from:
+http://arxiv.org/ps/2301.12160v1
+
+This figure "fig_8.png" is available in "png"� format from:
+http://arxiv.org/ps/2301.12160v1
+

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+arXiv:2301.03503v1  [hep-th]  9 Jan 2023
+IPARCOS-23-001
+Embedding Unimodular Gravity in String Theory
+Luis J. Garay∗
+Departamento de F´ısica Te´orica and IPARCOS,
+Universidad Complutense de Madrid, 28040 Madrid, Spain
+Gerardo Garc´ıa-Moreno†
+Instituto de Astrof´ısica de Andaluc´ıa (IAA-CSIC),
+Glorieta de la Astronom´ıa, 18008 Granada, Spain
+Abstract
+Unimodular Gravity is a theory displaying Weyl rescalings of the metric and transverse (volume-
+preserving) diffeomorphisms as gauge symmetries, as opposed to the full set of diffeomorphisms
+displayed by General Relativity. Recently, we presented a systematic comparison of both theories,
+concluding that both of them are equivalent in everything but the behaviour of the cosmological
+constant under radiative corrections. A careful study of how Unimodular Gravity can be embedded
+in the string theory framework has not been provided yet and was not analyzed there in detail.
+In this article, we provide such an explicit analysis, filling the gap in the literature. We restrict
+ourselves to the unoriented bosonic string theory in critical dimension for the sake of simplicity,
+although we argue that no differences are expected for other string theories. Our conclusions are
+that both a Diff and a WTDiff invariance principle are equally valid for describing the massless
+excitations of the string spectrum.
+∗ [email protected]
+† [email protected]
+1
+
+CONTENTS
+I. Introduction
+2
+II. Unimodular Gravity and General Relativity: Matching global degrees of freedom
+5
+III. String perturbation theory in trivial backgrounds
+10
+IV. Strings in general backgrounds
+16
+A. Determination of the Weyl anomaly
+17
+B. Including string-loop corrections
+21
+C. EFTs for the theory
+23
+V. Conclusions
+25
+Acknowledgments
+26
+References
+27
+I.
+INTRODUCTION
+Unimodular gravity (UG) is a theory which is so similar to General Relativity (GR) that
+one may wonder to what extent both of them are equivalent. Recently we presented a sys-
+tematic comparison of both theories in all the regimes and situations in which a potential
+difference might appear, which was still lacking [1]. We concluded that for all of the possible
+regimes analyzed there, both theories are equivalent except for the behaviour of the cosmo-
+logical constant. Whereas the cosmological constant is radiatively stable in UG [2] (it is
+simply an integration constant of the equations of motion), in GR it is radiatively unstable.
+In this way, if one uses technical naturalness in the sense introduced by ’t Hooft [1, 3, 4] as
+a guiding principle toward building theories, UG theories are much more desirable than GR
+theories since the cosmological constant is technically natural.
+There are mainly three arguments used to argue that the low-energy limit of string theory
+is given by the effective field theory (EFT) consisting of GR coupled to some other fields.
+First of all, when one analyzes the massless spectrum (leaving aside the tachyon field) of
+bosonic string theory propagating on top of flat spacetime one finds that for oriented strings
+2
+
+it contains a graviton, a Kalb-Ramond field, and a dilaton; and for unoriented strings
+it contains only a graviton and a dilaton.
+In principle, for computing observables only
+involving massless states, one expects that one can write down an effective action which
+simply involves fields that account for these massless excitations, i.e., a graviton-field hµν,
+(possibly) a Kalb-Ramond field Bµν, and a dilaton field Φ.
+As usual, the fundamental
+observable considered is the S-matrix.
+Now, we come to the arguments used to argue that GR “emerges naturally” as the low-
+energy description of such degrees of freedom. First of all, it has been argued that the only
+self-consistent way of coupling the graviton (massless spin-2 representation of the Poincar´e
+group) to itself is through GR. In that way, having a massless spin-2 field in the spectrum,
+one necessarily guesses that the non-linear structure of the theory needs to be GR up to
+potential higher-derivative corrections arising in the EFT. However, we argued [1, 5] that
+the self-coupling of UG gravitons (those displaying linerarized WTDiff gauge-invariance) to
+themselves also gives rise to the full non-linear UG in a consistent way, although the coupling
+of the graviton to itself is through the traceless part of the energy-momentum tensor, instead
+of the full one. Hence, this first argument does not allow one to discern whether UG or GR
+is preferred from the string point of view since one is as legitimate as the other.
+The second argument comes from the analysis of string scattering amplitudes, which was
+already revisited in [1]. One can compute within string perturbation theory the scattering
+amplitudes for graviton asymptotic states. The result is that, to the lowest order in α′ and
+at string tree level, one obtains the same scattering amplitudes obtained in GR. The point is
+that UG scattering amplitudes are exactly the same as the GR scattering amplitudes [6, 7].
+In that sense, GR is not preferred over UG from the point of view of scattering amplitudes
+either, as it was concluded in [1].
+The final argument comes from analyzing perturbatively in α′ the non-linear sigma model
+that arises from coupling the string degrees of freedom to an arbitrary background metric (or
+conformal structure), Kalb-Ramond field, and dilaton field generated by the string degrees
+of freedom themselves. For such a model, the Weyl symmetry of the worldsheet, which is
+potentially anomalous, needs to be handled carefully. However, although in flat spacetime
+and zero background fields it simply constrains the dimension of spacetime to be 26 (critical
+dimension), in this case constraints also appear for the spacetime fields entering the sigma
+model construction. Such constraints arise from imposing a cancellation of the Weyl anomaly
+3
+
+to make it a sensible theory.
+The equations that arise are basically Einstein equations,
+although interpreted as β-functionals. Both GR and UG give rise to Einstein equations,
+hence from this point of view we show that it is possible to write both a GR and UG-like
+EFT for the massless degrees of freedom of the string. Moreover, both actions are also
+consistent with the previous argument since they reproduce all the scattering amplitudes
+involving massless states of the string. The only difference that seems to appear, is that,
+whereas in the GR EFT the cosmological constant is a coupling constant that needs to be
+set to zero, in the UG EFT it is an integration constant that needs to be set to zero. In
+other words, UG contains the space of theories which is GR with all possible values of the
+cosmological constant within a single theory.
+The α′-expansion on its own points toward a zero cosmological constant. However, once
+we include string loop corrections, the situation changes.
+We will revisit the Fischler-
+Susskind approach [8–10]towards including the lowest order string loop correction in the
+picture. In this way, an arbitrary cosmological constant is generated through the string-
+loop corrections in the EFT. In this way, the EFT that we need to write down within the
+GR EFT to include the string-loop corrections contains an arbitrary cosmological constant,
+which is exactly what happens with the UG EFT, although in the former case it is a coupling
+constant whereas in the latter it is an integration constant. In this way, we conclude that
+both the UG and the GR EFTs can account for the low-energy description of massless string
+states with the only difference arising in the nature of the cosmological constant.
+It is worth remarking that this analysis gives further evidence for UG as a sensible classical
+theory of gravitation according to the criteria invoked by Weinberg in [11]. According to
+Weinberg, one of the key aspects that needs to be addressed to regard UG as a reasonable
+classic theory of gravitation is to understand whether it can be obtained as a low energy
+limit of a quantum theory of gravitation. By embedding UG within the framework of string
+theory, here we answer here in the affirmative.
+The remain of this article is structured as follows. In Section II we introduce the frame-
+work of UG, making special emphasis on the existence of a priviliged background volume
+form and the existence of an additional global degree of freedom with respect to GR. Then,
+we introduce a modification of GR in which a new global degree of freedom, precisely the
+cosmological constant is assigned to a (D+1)-form field, to make clear the difference between
+UG and the standard formulation of GR. In Sec. III we review the basics of the quantiza-
+4
+
+tion of strings in flat spacetimes and explain why UG and GR are both valid as the low
+energy description of string theory from the point of view of scattering amplitudes involving
+massless particles. In Sec. IV we move on to analyze strings in general backgrounds. In
+Subsec. IV A we rederive the consistency conditions (Weyl anomaly cancellation) from the
+perturbative α′ expansion of the sigma model. Some of the details of the computation that
+are well explained in the literature and not relevant for our purposes are skipped and we
+refer the reader to the literature at those points. In Subsec. IV B we introduce the Susskind-
+Fischler approach for cancelling some of the divergences arising from string loops, with the
+divergences of the sigma model on the trivial genus worldsheet. The main novelty that this
+mechanism introduces is a cosmological-constant-like term in the β functions. We close this
+section by analyzing in Subsec. IV C how these consistency conditions can be derived from
+an effective action once they are interpreted as equations of motion for the background fields.
+We emphasize the consistency of this approach when computing scattering amplitudes in-
+volving the massless excitations. we close this section. In Sec. V we summarize the results
+and draw the conclusions that can be taken from our analysis. We also point interesting
+future lines of work that seem promising in virtue of our analysis presented here.
+Notation and conventions: Our convention for the signature of the metric is (−, +, ..., +)
+for the (D+1)-dimensional target space metric and (−, +) for the worldsheet metric. Tensor
+objects will be represented by bold symbols, whereas their components in a given basis will
+be written with the same (not bold) symbol and indices, e.g., the Minkowski metric η
+will be represented in components as ηµν. We will use Greek letters for spacetime indices
+(µ, ν, ...) whereas we will reserve lower case latin indices (a, b, ...) for the worldsheet indices.
+Curvature quantities like the Riemann tensor are defined following Misner-Thorne-Wheeler’s
+conventions [12] and we will specify explicitly the metric it depends on, e.g. Rα
+βγδ(g). We
+also represent the (D + 1)-dimensional Newton’s constant as κ2 = 16πG.
+II.
+UNIMODULAR GRAVITY AND GENERAL RELATIVITY: MATCHING
+GLOBAL DEGREES OF FREEDOM
+It is well accepted that metric theories of gravity, those in which the fundamental object
+describing the gravitational field at a given point is a metric, are suitable for describing
+gravitational experiments to great accuracy [13]. The metric at a given point of the spacetime
+5
+
+is completely specified by the lightcone at that point up to a conformal factor. Although
+the conformal structure of the spacetime is allowed to fluctuate both in UG and GR, the
+difference arises in the conformal factor. Whereas in UG the conformal factor is fixed to be
+a fiducial (non-dynamical) volume form that we represent as ω =
+1
+(D+1)!ω(x)dx0 ∧ ... ∧ dxD
+and hence it does not have any dynamics, in GR it is also dynamical like the lightcone itself.
+Naively, one could conclude that this reduction in the number of independent components
+of the metric may lead to a reduction of the independent degrees of freedom of the theory.
+However, it reduces the gauge symmetries of the theory to only transverse diffeomorphisms
+(those preserving the background volume form) and hence it is not surprising that the theory
+displays the same number of local degrees of freedom as GR does. Actually, it displays
+an additional global degree of freedom associated with the cosmological constant. In this
+section we will introduce the basic formulation of UG, emphasizing the presence of this new
+additional global degree of freedom. Furthermore, we will present a formulation of GR closer
+in spirit to UG, since the cosmological constant appears as a combination of an arbitrary
+integration constant and the renormalized cosmological constant entering the action and we
+still have the invariance under the full set of diffeomorphisms.
+Let us begin with the standard formulation of UG. UG is a theory in which the group of
+gauge transformations is WTDiff (Weyl rescalings of the metric and Transverse Diffeomor-
+phisms) instead of the whole group of Diffs (Diffeomorphisms), see [1] for further details. In
+order to define such a theory, we need to use the non-dynamical volume form that we have
+already introduced ω. It is useful to introduce the Weyl-invariant auxiliary metric
+˜gµν = gµν
+�ω2
+|g|
+�
+1
+D+1
+.
+(1)
+In this way, every curvature scalar built from the auxiliary metric ˜gµν inherits the invariance
+under Weyl rescalings and is also invariant under transverse-diffeomorphism transformations
+by construction. The simplest action principle that one can think for an UG-like theory is
+the UG version of the Einstein-Hilbert action:
+SUG =
+1
+2κ2
+�
+dD+1xωR (g) .
+(2)
+We can also add a coupling to some matter fields which need to couple to the auxiliary metric,
+i.e., the matter action will be of the form Sm (˜g, Φ), so that it remains Weyl-invariant (note
+that the matter fields are not affected by Weyl transformations). The equations of motion
+6
+
+of this theory are the traceless Einstein equations:
+Rµν(˜g) −
+1
+D + 1R(˜g)˜gµν = κ2
+�
+Tµν(˜g) −
+1
+D + 1T(˜g)˜gµν
+�
+.
+(3)
+Upon using the Bianchi identities, they become Einstein equations with the cosmological
+constant entering as an integration constant [1]
+Rµν(˜g) − 1
+2R(˜g)˜gµν + Λ˜gµν = κ2Tµν(˜g),
+(4)
+provided that ˜∇µT µν (˜g) = 0.
+It is clear that the Weyl invariance is trivial in the sense that its gauge fixing is trivial,
+we simply need to fix the volume form given by the determinant of the metric
+�
+|g| to be
+the background volume form. Actually, this can be done also at the level of the action. The
+result is still a local action for the metric which does not contain any mention to the Weyl
+symmetry. In that sense, the resulting action is the most minimalistic action that one can
+conceive for a metric field. If one tried to make a gauge fixing of the remaining degrees of
+freedom, one would end up with a non-local action for the actual physical degrees of freedom
+encoded in the field gµν.
+In this way, it seems clear that both theories display the same number of local degrees
+of freedom of GR, except for the cosmological constant that we will analyze now. To put
+it in other words, leaving aside the cosmological constant, from the point of view of initial
+conditions, the same amount of initial data are needed to specify a solution to the equations.
+The cosmological constant in this case appears with a difference, it is an additional global
+degree of freedom. The simplest way to see this is from the point of view of such constant
+being an integration constant. This means that it is a constant that parametrizes the space
+of solutions, which is separate from the initial data required in GR. In that sense, it is a
+constant to be fixed by initial conditions which makes the space of solutions of UG bigger
+than the GR space of solutions, precisely by this cosmological constant as an integration
+constant. This analysis can be made much more precise by making a Hamiltonian analysis
+of the theory, as it has been done in [14], reaching the same conclusions.
+We have concluded that UG is equivalent to GR, up to a global degree of freedom which
+is precisely playing the role of the cosmological constant. To make it more explicit, we will
+introduce now an additional field in GR that accounts for this global degree of freedom, to
+sharpen the difference. We need to introduce a (D +1)-form field which is the differential of
+7
+
+a D-form [15, 16]. Explicitly, we want to introduce a D + 1 form F which is the differential
+of a D-form A. In components, this reads:
+Fµ0...µD = ∇[µ0Aµµ1...µD ].
+(5)
+We can write down the action principle which is the Einstein-Hilbert action with an arbitrary
+cosmological constant and a Maxwell-like term for F , namely:
+S =
+1
+2κ2
+�
+dD+1x√−g
+�
+−2Λ + R(g) −
+K
+(D + 1)!Fµ0...µDF µ0...µD
+�
+,
+(6)
+where K is simply a coupling constant which can be both positive or negative. The equations
+of motion for the F -field are
+∇µ0F µ0...µD = 0.
+(7)
+In a (D + 1)-dimensional manifold, a completely antisymmetric volume form like F needs
+to be proportional to the ǫ pseudotensor. Hence, the equations of motion simply fixed the
+proportionality function to be a constant, i.e.
+Fµ0...µD = c√−gǫµ0...µD.
+(8)
+From the point of view of the initial value problem, this constant c is precisely a global degree
+of freedom that needs to be fixed in terms of initial conditions. From that point of view, it
+is akin to the cosmological constant in UG, since it is completely fixed in terms of the initial
+conditions. We can sharpen the analogy by examining how does this constant c enter the
+equations of motion for the metric. The energy-momentum tensor once we evaluate the F
+form on shell, behaves exactly as a cosmological constant [15, 16]. Assuming the existence of
+additional matter fields, the equations of motion for the gravitational field take the following
+form:
+Rµν(g) − 1
+2R(g)gµν + Λeffgµν = κ2Tµν(g),
+(9)
+where the constant Λeff is expressed in terms of the action as
+Λeff = Λ + NDKc2,
+(10)
+with ND an irrelevant numerical factor depending on the spacetime dimension. In this way,
+the cosmological constant entering the equations of motion for the metric are a combination
+of an initial condition c and the cosmological constant Λ entering the action.
+8
+
+From a purely classical point of view, we have presented a theory akin to GR, exhibiting
+the whole set of diffeomorphisms as gauge symmetries and containing an additional global
+degree of freedom encoded in a (D + 1)-form. The equations of motion for this volume form
+enforce that it is proportional to the Levi-Civita pseudotensor, with the proportionality
+constant been called here c. The constant of proportionality enters the equations of motion
+for the metric as an effective cosmological constant. In this way, it plays a similar role to
+the one played by the global degree of freedom of UG. Independently of the value that we
+assign to the cosmological constant entering the action Λ, the resulting effective cosmological
+constant entering Einstein equations Λeff is given by a combination of Λ and c. In terms of
+the initial conditions, it is possible to adjust c in order to make Λeff take any desired value.
+This formulation of GR with the additional (D + 1) form field is equivalent to UG, in the
+sense that it displays the same amount of degrees of freedom, both local and global, and the
+global degree of freedom plays the role of a cosmological constant.
+At the quantum level, both formulations seem to be different from the point of view of
+radiative corrections. The reason behind this mismatch is that, whereas in UG the cos-
+mological constant does not receive any radiative corrections and this makes it technically
+natural [1, 2]1, in this formulation of GR, the cosmological constant in the action Λ does re-
+ceive radiative corrections, and hence it is not technically natural. However, the cosmological
+constant relevant for the dynamics is the effective one Λeff that combines the renormalized
+Λ with the initial value constant c. It is possible to obtain any value for the cosmological
+constant Λeff independently of the potentially huge radiative corrections that Λ may receive.
+The equivalence once quantum corrections are included into the picture is unclear. Whether
+this formulation is then completely equivalent to UG at the semiclassical level is something
+that deserves a separate and detailed study on its own.
+Our point here was mainly to introduce a formulation within the GR setup that is close
+to the UG version, so that both theories can be compared easily. We have made explicit
+the difference existing in the global degrees of freedom of UG and GR (UG contains the
+whole space of GR with arbitrary values of the cosmological constant coupling). This only
+difference in the two theories, will be also the only difference appearing from the point of
+1 We note that technical naturalness is a definition that only applies to coupling constants appearing in
+the action. In that sense it is not completely legitimate to say that in UG the cosmological constant is
+technically natural since it is not a coupling constant. However, making an abuse of language we find it
+convenient to say that it is technically natural.
+9
+
+view of regarding UG as the low energy EFT for massless string states.
+III.
+STRING PERTURBATION THEORY IN TRIVIAL BACKGROUNDS
+This section contains a review of the quantization of strings in a flat background as well
+as the computation of string scattering amplitudes for gravitons from string theory. This is
+well-known material that can be found in any textbook [17, 18]. Also we think that a reader
+unfamiliarized with string theory might find here a quick introduction to the arguments
+presented in the literature leading to the conclusion that GR is the EFT describing the
+excitation in massless degrees of freedom. We find convenient to make such introduction
+here to expand the discussion presented in [1] about how the scattering amplitudes can be
+equivalently obtained from a GR and a UG-like EFT.
+The starting point of our discussion of perturbative string theory is the action describing
+relativistic strings propagating in flat spacetime. For relativistic free particles it is natural to
+consider the action to be the proper time of the particle trajectory i.e., the embedding of the
+worldline in the target space. In the same way, for strings it is natural to consider the area
+swept out by the worldsheet to replace the proper time of the particle trajectory. For that
+purpose, let us introduce a coordinate system in the worldsheet, a pair σa (a = 0, 1) which
+correspond to the time coordinate σ0 ∈ (−∞, ∞) and a spatial coordinate σ1. Furthermore,
+we will restrict our attention to closed strings (those giving rise to graviton excitations) in
+which the points at σ1 and σ1 + 2π are identified. If we endow the (D + 1) dimensional flat
+spacetime with coordinates Xµ, we look for an action such that the area density swept by
+the string is expressed in terms of derivatives of the embedding Xµ(τ, σ). We notice that
+the induced metric on the worldsheet is given by
+hab = ηµν∂aXµ∂bXν.
+(11)
+If we take the action to be the area swept out by the string, we write down the Nambu-Goto
+action as
+SNG[X] = −
+1
+2πα′
+�
+d2σ
+√
+−h.
+(12)
+The constant α′ represents the string tension, i.e., the energy density per unit length. Al-
+though this action is perfectly reasonable classically, from the point of view of quantization
+10
+
+is problematic. This is because it is not quadratic in its variables: we have a square root
+appearing explicitly in the action.
+To circumvent this problem, one can work with the
+Polyakov action, which is given by
+SP[X, γ] = −
+1
+4πα′
+�
+d2σ√−γγab∂aXµ∂bXνηµν.
+(13)
+In this action, there is an additional configuration variable γab which is a metric in the
+worldsheet. Now, this action is clearly quadratic in the Xµ variables over which we will
+path-integrate to quantize the theory. To see the equivalence among these two actions, we
+can compute the equations of motion for the γab variable. Actually, following the standard
+conventions, we can define a two-dimensional energy-momentum tensor as the variation of
+the Polyakov action with respect to the worldsheet metric, i.e. γab:
+Tab = −
+1
+√−γ
+δSP
+δγab =
+1
+4πα′
+�
+∂aXµ∂bXν − 1
+2γabγad∂cXµ∂dXν
+�
+ηµν.
+(14)
+The Polyakov action does not contain any derivatives of the metric γab, and hence the
+equations of motion for the metric can be regarded as a constraint Tab = 0 (as a consequence,
+strictly speaking it is not a dynamical variable). Actually, this constraint can be used to
+solve γab in terms of the Xµ variables. When we plug the result back into the Polyakov
+action, we find the Nambu-Goto action we began with.
+It is worth pausing at this point and discussing the continuous symmetries of the theory:
+• Poincar´e invariance. This is a global symmetry on the worldsheet
+Xµ → Λµ
+νXν + c��.
+(15)
+• Reparametrization invariance or diffeomorphism invariance in the worldsheet σa →
+˜σa(σ). Whereas the Xµ fields transform as worldsheet scalars, γab transforms as a
+two-index covariant tensor:
+Xµ(σ) → Xµ(˜σ) = Xµ(σ),
+(16)
+γab(σ) → ˜γab(˜σ) = ∂σc
+∂˜σa
+∂σd
+∂˜σb γcd(σ).
+(17)
+• Weyl invariance of the worldsheet metric γab. This transformation leaves invariant the
+Xµ coordinates and the metric gets a local rescaling
+Xµ(σ) → Xµ(σ),
+(18)
+γab(σ) → e2φ(σ)γab(σ).
+(19)
+11
+
+We can distinguish now between oriented and unoriented strings. The former have a well
+defined transformation law under the parity transformation σ1 → 2π − σ1. We will focus on
+the unoriented strings for the sake of simplicity.
+Not all the symmetries that we have introduced are directly preserved through the process
+of quantization. Actually, the Weyl symmetry is anomalous, as it is well known. However,
+in this case the Weyl symmetry is a gauge symmetry that we must insist on preserving
+at the quantum level to remove unphysical states. We will further discuss this point later
+when we deal with strings in general backgrounds. For the time being, let us focus on the
+quantization of the theory through a path-integral procedure.
+Let us illustrate the quantization of the theory through a path-integral procedure as
+well as the spectrum that the theory displays.
+Let us define the generating functional
+following the usual Faddeev-Popov procedure. First of all, we would write down the action
+in Euclidean space, in order to make the quantization procedure sensible. We write down
+the generating functional as
+Z =
+1
+V (gauge)
+�
+DγDXe−SP [X,γ],
+(20)
+where V (gauge) represents the volume of the gauge group. We recall that we have the Weyl
+rescalings of the metric and diffeomorphisms as gauge symmetries of our theory. Hence,
+we need to avoid counting more than once physical configurations and that is the reason
+for taking the quotient by the volume of the gauge group. As usual, we will introduce a
+Faddeev-Popov determinant ∆F P[γ] to take this volume into account.
+The integral over the gauge orbits cancels with the volume of the gauge group and we
+reach the expression for the generating functional which is
+Z[γ] =
+�
+DX∆F P[γ]e−SP [X,γ].
+(21)
+Choosing a convenient normalization for the action, we can rewrite the Faddeev-Popov
+determinant as
+∆F P[γ] =
+�
+DbDce−Sg[b,c],
+(22)
+where b and c are ghosts Grassman-values that anticommute and
+Sg = 1
+2π
+�
+d2σ√γbab∇acb.
+(23)
+12
+
+At this point, we have reduced the evaluation of the path integral for the bosonic string
+theory to the evaluation of the path integral:
+Z =
+�
+DbDcDXe−SP [γ,X]−Sg[γ,b,c],
+(24)
+which is the CFT of D + 1 scalar fields (the Xµ) and the bc-ghost system [17, 19]. If the
+theory is going to preserve the Weyl invariance, we need the theory to have a total zero
+central charge. This is precisely the consistency condition that we mentioned would appear.
+Weyl invariance means that the trace of the two-dimensional energy momentum tensor needs
+to vanish. In two-dimensions, the trace of the energy-momentum tensor is determined by
+the central charge and the trace anomaly
+⟨T a
+a⟩ = − c
+12R [γ] .
+(25)
+The system of the Xµ-scalars and the bc-ghost system is linear, and hence the total central
+charge is the sum of the central charges of the two systems independently:
+c = cg + cX.
+(26)
+The bc-ghost system [17] has a central charge cg = −26 while each scalar field gives a
+contribution of 1 to the central charge cX = D + 1. Ensuring Weyl-invariance means that
+we need the spacetime dimension to be 26. This is the well-known way in which the critical
+dimension of bosonic string theory emerges.
+Now that we have ensured how to preserve the gauge invariance at the quantum level in
+order to make the theory consistent, it is time to talk about the spectrum of the strings.
+Our point is simply to illustrate that the spectrum of the closed unoriented bosonic string
+contains a tachyon, a dilaton, and a graviton. Hence, for this purpose, we can skip the
+detailed BRST analysis and focus only on the states generated by the X-fields which are
+the “physical fields”.
+In order to characterize the spectrum, the simplest way to do it is to use the so called
+state-operator map for CFTs [20, 21], in which states are replaced by operator insertions
+that generate them by acting in a neighbourhood of the vacuum. For this purpose, it is
+first easier to use complex coordinates σ → (z, ¯z) on the worldsheet. Furthermore, we now
+need the operators to be gauge invariant. The diffeomorphism invariance can be ensured
+by integrating local operators O(z, ¯z) over the worldsheet, i.e. constructing operators of the
+13
+
+form
+V =
+�
+d2zO(z, ¯z),
+(27)
+with V standing for vertex operators. Weyl invariance is ensured by choosing the operators
+O to transform adequately under Weyl rescalings, i.e., having a suitable weight. The measure
+of integration, d2z has a conformal weight (−1, −1) under such rescalings. Hence, O needs
+to be a primary operator of the CFT with weight (+1, +1) to compensate it.
+The kind of operators that give rise to the lowest energy states of the string are eip·X
+and Pµν∂Xµ∂Xνeip·X, with p a given momentum that we endow the string with and Pµν
+the polarization tensor [17, 18].
+The operator eip·X gives rise to the tachyon, since we
+need to impose that −p2 = −4/α′ < 0 for the operator to be Weyl invariant. The operator
+Pµν∂Xµ∂Xνeip·X corresponds to the dilaton (pure trace part of Pµν) and the symmetric part
+of Pµν gives rise to the graviton, since p2 = 0 (massless condition) and pµPµν = 0 (transverse
+condition) needs to be imposed to ensure the Weyl invariance. The antisymmetric part does
+not appear for unoriented strings since it corresponds to the Kalb-Ramond excitation [17].
+Up to this point, we have analyzed the spectrum of the closed unoriented bosonic string
+theory and found that the massless states correspond to the dilaton and the graviton. The
+Polyakov action per se does not give rise to interactions. We will now make a small digression
+on how interactions among the massless states arise in string theory. There is a term that we
+can add to the Polyakov action which is an Einstein-Hilbert term that is purely topological
+in two-dimensions
+Sint = λ
+4π
+�
+d2σ√γR(γ) = 2λ(1 − g),
+(28)
+being g the genus of the worldsheet and λ a coupling constant which we assume to be small
+in order to do perturbation theory. Hence, if we add this term to the string action, we will
+get
+Z =
+�
+topologies
+�
+DXDγe−SP −Sint =
+∞
+�
+g=0
+e−2λ(1−g)
+�
+DXDγe−SP .
+(29)
+If we call eλ = gs, as it is common, this gives a good expansion as long as gs ≪ 1. The whole
+series is known to be a divergent series as the standard perturbative series in QFT [22]. In
+addition to this problem, there is a harder problem which is the finiteness of each of the
+terms in the series, i.e., the path integral over the different geometries. For a fixed topology,
+14
+
+the path integral with the Polyakov action requires to compute a sum over the moduli
+of conformally inequivalent surfaces.
+In general, for higher loop orders (i.e.
+non-trivial
+topologies) this requires to perform an integral over a moduli space that is not obviously
+convergent, although some results in the literature point toward its finiteness [23].
+Now it comes to the point of computing some observables. The observable to compute
+in string theory is the string S-matrix. This means, we plug some “in” state of the free
+string spectrum and compute the probability amplitude of generating another “out” state of
+free string spectrum. These states are generated by introducing their corresponding vertex
+operators.
+For our purposes of analyzing how GR or UG might emerge from string theory, we are
+interested in computing the scattering amplitude involving m gravitons with momenta pi and
+polarization tensors ei which we represent as A(m)(p1, e1; p2, e2; ...pm, em). This is computed
+as a suitable path integral for the Polyakov action SP that schematically reads [17, 18]
+A(m)(1h1, 2h2, ..., mhm) = 1
+g2s
+1
+Vgauge
+�
+DXDg e−SP[X,g]
+m
+�
+i=1
+Vi(pi, hi),
+(30)
+where Vi represents the vertex operator associated with a graviton insertion with a given
+spin and momentum. To begin with, we particularize the amplitude for three gravitons and
+we find
+A(p1, e1; p2, e2; p3, e3) = igs(α′)6
+2
+(2π)26δ26 (p1 + p2 + p3) e1µνe2αβe3γδT µαγT νβδ,
+(31)
+where
+T µαγ = pµ
+23ηαγ + pα
+31ηγµ + pγ
+12ηµα + α′
+8 pµ
+23pα
+31pγ
+12,
+(32)
+pµ
+ij = pµ
+i − pµ
+j . The terms of order O(α′) in T µαγ contribute as O(p4) to the amplitude.
+If we focus just on the lowest order terms O(p2), this amplitude is equivalent to the ones
+computed at tree level from the Einstein-Hilbert action upon the identification κ = gs(α′)6.
+The same agreement is found with amplitudes involving an arbitrary number of gravitons:
+if we neglect the higher-order contribution from the string amplitude, they agree with those
+computed from the Einstein-Hilbert action [17, 18], with the same identification of κ and
+the string constants.
+As it has been already discussed in the literature [6, 7], the tree-level scattering amplitudes
+of gravitons computed in GR and UG are identical.
+Hence, from the point of view of
+15
+
+scattering amplitudes, string theory does not point toward GR in a univocal way: both UG
+and GR are equivalent from a low-energy effective field theory point of view. This result
+was already advanced in [1] and we have reproduced here the analysis in more detail for the
+sake of completeness. We will come back to this analysis later, when we introduce the low
+energy EFTs for the massless degrees of freedom of the string: both the UG and the GR-like
+actions.
+IV.
+STRINGS IN GENERAL BACKGROUNDS
+Up to now, we have only considered strings propagating in flat spacetime. However, the
+spectrum of the strings contains some excitations which typically interact among themselves
+and could lead to the generation of a non-trivial background. In particular, it contains a
+graviton and, necessarily, gravitons need to interact gravitationally. At low energies, all the
+excitations that matter are the massless ones. In the same way a laser is a coherent state of
+photons, we expect that a coherent state of gravitons might look like a curved background
+and a string propagating on top of it needs to be described appropiately. The same comment
+applies to the dilaton field. As such, we can write down the most general renormalizable
+action including those fields, which is the following non-linear σ-model
+S[X, γ] = SP[X, γ] + SD[X, γ] = −
+1
+4πα′
+�
+d2σ√−γ
+�
+γabGµν(X)∂aXµ∂bXν + α′R (γ) Φ(X)
+�
+,
+(33)
+where Gµν(X) represents a metric (graviton excitations), Φ(X) represents the dilaton back-
+ground field, and R[γ] represents the Ricci-scalar of the two-dimensional metric. This term
+breaks explicitly the Weyl invariance in the worldsheet. This term is of a higher dimension
+than the Weyl-invariant terms, and it does not require to be normalized with a dimensionful
+constant. In virtue of the expansion in α′ that we will perform, we will cancel the tree-level
+contribution to the anomaly of this last term with the one-loop contribution of the classically
+Weyl-invariant terms. The result of this procedure is a reasonable effective field theory for
+the massless degrees of freedom of the string.
+There are two missing terms that still give rise to a renormalizable theory. The first of
+these terms is the coupling to the Kalb-Ramond field. However, if we focus on unoriented
+strings, we can skip it since the divergences of the rest of the terms do not require this term
+16
+
+to be renormalized. In case we deal with oriented strings, this term gives a contribution to
+the conformal anomaly [17].
+The additional term that we can add to the action corresponds to a coupling to the
+background tachyon field T(X)
+ST = 1
+4π
+�
+d2σ√−γT (X) .
+(34)
+In principle this term is needed to cancel some of the quadratic divergences arising from
+vacuum to vacuum diagrams. However, if we use a renormalization scheme such that those
+divergences are absent (for example, dimensional regularization), we can safely skip those
+terms. Hence, we will work with a renormalization scheme fullfilling this property. Fur-
+thermore, it is worth mentioning that supersymmetry in the worldsheet ensures that those
+quadratic divergences are absent in superstrings due to the characteristic cancellation among
+fermionic and bosonic degrees of freedom, with independence of the renormalization scheme.
+A.
+Determination of the Weyl anomaly
+Anomalies always appear when there are two symmetries that the theory displays at the
+classical level, but it is not possible to quantize such theory preserving both of them. This
+means, there is a trade-off between the two symmetries and it is only possible to preserve
+one of them in the process. For example, the chiral anomaly is a trade-off between the vector
+and axial currents for massless fermion fields. If we use a regularization procedure which
+automatically preserves one of those currents, then straightforwardly the other current will
+be anomalous.
+In the case of the chiral anomaly, it is standard to use a regularization
+scheme that preserves gauge invariance and hence yields to the conservation of the vector
+current, leading to an anomalous axial current. In the case of Weyl invariance for strings, we
+are using a regularization scheme that preserves diffeomorphism invariance, while the Weyl
+symmetry becomes potentially anomalous. We need to ensure that the non-linear sigma
+model is chosen in such a way that it gives rise to a Weyl-invariant theory. In a language
+closer to particle physics, this means that we need to choose our theory in such a way that
+we cancel the potential gauge anomalies, which in this case corresponds to choosing the
+background fields in such a way that the theory is not Weyl-anomalous.
+In the case of
+the Standard Model, since it corresponds to a chiral gauge theory, arbitrary matter fields
+17
+
+would lead to an anomalous theory. However, the matter content is such that the potential
+anomaly is absent. This is precisely what we have done in the previous section to fix the
+target space dimension to be 26; otherwise, the Weyl-symmetry becomes anomalous. In
+this case, we expect constraints also on the background fields entering the non-linear sigma
+models, i.e., constraints that the Gµν(X) and the Φ(X) fields need to obey.
+We want now to write down the most general form that the Weyl anomaly can display.
+Following D’Hoker [24], it is possible to show that the structure of the anomaly for unoriented
+strings in a curved background needs to be of the form
+⟨T a
+a ⟩ = βG
+µν(X)∂aXµ∂bXνγab + βΦ (X) R (γ) + βV
+µ (X)gabD∗
+a∂bXµ,
+(35)
+where D∗
+a represents the covariant derivative on the product space of the cotangent space
+of the worldsheet and the tangent space of the target space, and it can be explicitly written
+down as
+D∗
+a∂bXµ = ∂a∂bXµ − Γc
+ab∂cXµ + Γµ
+νρ∂aXν∂bXρ,
+(36)
+where Γc
+ab are the Christoffel symbols of the metric γab and Γµ
+νρ represent the Christoffel
+symbols of the metric Gµν. The last term in the Weyl anomaly, βV can be removed through
+a transformation on the Xµ fields, since we are always able to perform a local transformation
+on the Xµ fields at the same time that we perform a Weyl-rescaling of the metric. This
+leaves only two independent β functionals: βG and βΦ2.
+Hence we need to determine the β functionals obtained from the action (33). We want
+to study perturbatively this action order by order in the α′ expansion, which is done by
+assuming that the background fields Gµν(X), Φ(X) vary smoothly with respect to the scale
+α′. It is conventional to do the computations in the background field formalism. In this for-
+malism, we decompose the fields Xµ in a background part Xµ
+0 and its quantum fluctuations
+Y µ
+Xµ (σ) = Xµ
+0 (σ) + Y µ (σ) ,
+(37)
+where the integration is now performed with respect to the quantum fluctuations instead of
+Xµ. We define the effective action Γ[X0, g] following [25] as
+e−Γ[X0,g] =
+�
+DY e
+−
+�
+S(X0,Y )−S(X0)−� d2σY µ(σ) δS
+δXµ
+0
+�
+,
+(38)
+2 For oriented strings there will be another β-functional associated with the Kalb-Ramond field.
+18
+
+which is the generating functional of the Feynman diagrams relevant for the computation of
+the β-functionals.
+At this point, it is better to pause and mention a crucial step in the computations. The
+coordinate difference does not transform in a covariant way under changes of coordinates.
+Hence, in order to obtain results that are manifestly covariant, it is better to do the com-
+putation in variables that are manifestly covariant at intermediate steps. This can be done
+as follows. Imagine that the coordinates Xµ
+0 correspond to a given point p0 and the coor-
+dinates Xµ = Xµ
+0 + Y µ to a point p. If both points are close enough, there exists only one
+geodesic with respect to Gµν connecting both of them. Hence, we can replace the coordinate
+difference Y µ which characterizes the point p by the tangent vector tµ of the geodesic at the
+point p0, which transforms covariantly under changes of coordinates. Hence, it is better to
+use this vector as the integration variable in the path integral.
+In fact, we can use this tangent vector tµ to perform a covariant Taylor expansion based
+on Xµ
+0 of an arbitrary tensor living in the target manifold. To put it explicitly, any tensor
+Tµ1...µn(X) can be expanded as
+Tµ1...µn(X0 + t) =
+∞
+�
+k=0
+T (k)
+µ1...µnν1...νk(X0)tν1 . . . tνk,
+(39)
+where each of the terms T (k)
+µ1...µnν1...νk is a combination of covariant derivatives of the tensor
+Tµ1...µn and contractions with curvature tensors evaluated at X0. This expansion can be
+achieved with the help of the normal coordinate expansion although we emphasize that it
+remains valid in an arbitrary coordinate system since it is a tensor expression. We are inter-
+ested in the expansion of the tensors Gµν, Φ(X) (the latter is a trivial tensor, i.e. a scalar),
+and the object ∂a (Xµ
+0 + Y µ). These expansions can be obtained after a straightforward
+computation, see [25] for details:
+Gµν(X) = Gµν(X0) + 1
+3Rµρσνtρtσ + ...,
+(40)
+Φ(X) = Φ(X0) + ∇µΦ(X0)tµ + 1
+2∇µ∇νΦ(X0)tµtν + ...,
+(41)
+∂a (Xµ
+0 + Y µ) = ∂aXµ
+0 + ∇atµ + 1
+3Rµ
+νρσ∂aXσ
+0 tνtρ + ...,
+(42)
+where Rµ
+νρσ represents the Riemann tensor associated with Gµν.
+We are not ready to perform the diagrammatic computation yet. There is a problem
+arising from the fact that the term that gives us the propagator for the quantum fields over
+19
+
+which we integrate, tµ, contains an arbitrary metric in front of it, i.e. we need to invert
+a term that looks like Gµν(X0)∇atµ∇btν. The way to deal with this problem and obtain a
+simple propagator is to introduce a vielbein eA
+µ(X0) which fulfills the property
+eA
+µ(X0)eB
+ν(X0)ηAB = Gµν(X0),
+(43)
+with ηAB a Lorentzian metric. In this way, we can rewrite all the vector expressions in the
+non-holonomic basis eA
+µ and get a trivial propagator for the tA = eA
+µtµ fields. This comes
+with a subtlety, because now the derivatives ∇a involve the spin-connection of the spacetime
+ω AB
+µ
+; for example,
+∇atA = ∂atA + ω AB
+µ
+∂aXµ
+0 tCηBC.
+(44)
+Obtaining a trivial propagator means breaking the SO(D, 1) invariance that the theory
+displays, but since we are working in a formalism that is explicitly gauge covariant, we
+automatically know that there will always be contributions in the diagrammatic expansion
+that make the theory explicitly gauge covariant in intermediate steps. Up to this point,
+collecting all the information, we have performed the following expansion for the Polyakov
+piece of the action:
+SP = SP[X0] +
+1
+2πα′
+�
+d2σ√γγabGµν(X0)∂aXµ
+0 ∇btν
+(45)
++
+1
+4πα′
+�
+d2σ√γγab �
+ηAB∇atA∇btB�
+(46)
++
+1
+3πα′
+�
+d2σ√γγabRµABC∂aXµ
+0 tAtB∇btC
+(47)
++
+1
+12πα′
+�
+d2σ√γγabRABCDtBtC∇a∇atA∇btD,
+(48)
+and for the dilaton part we have the trivial structure:
+SD[X0 + t] =SD[X0] − 1
+8π
+�
+d2σ√γ∇AΦ(X0)tA
+(49)
+−
+1
+16π
+�
+d2σ√γ∇A∇BΦ(X0)tAtB + ... .
+(50)
+We recall that we can safely impose the equations of motion for the classical fields and safely
+drop the linear terms. This is tantamount to a legitimate field redefinition.
+Now we can determine the trace anomaly, see Eq. (35) from the effective action introduced
+above. The computation requires to go to the next higher order in loops in the dilaton field,
+20
+
+since the piece of the action for the dilaton field α′ comes with an additional α′ with respect
+to the other field. The computation is rather lengthy and hence we do not reproduce it
+here [25]. We simply write down the result as
+βG
+µν = Rµν (G) − ∇µ∇νΦ + O(α′),
+(51)
+βΦ = D − 26
+6
++ α′ �
+−R (G) + 2∇2Φ + (∇Φ)2�
++ O
+�
+α′2�
+.
+(52)
+A comment is in order now. If we are dealing with a flat worldsheet, the vanishing of βG
+is enough to ensure the Weyl invariance at the quantum level, as long as we are working in
+D = 26 dimensions, the critical dimension (see Eq. (35). Hence, in principle, we expect that
+the same applies to non-flat worldsheets, i.e. that the condition βΦ = 0 is not independent
+of βG = 0. Actually, we have a non-trivial constraint coming from the Bianchi identity
+∇µ
+�
+Rµν (G) − 1
+2R (G) Gµν
+�
+= 0.
+(53)
+This ensures that we have to the computed order the following identity whenever βG
+µν = 0
+∇µβG
+µν = ∇νβΦ = 0,
+(54)
+as can be seen by direct calculation. This implies that βΦ is a constant as long as βG = 0.
+By continuity, this automatically implies at this level that βΦ = 0 for D = 26 [25]. From
+now on we will restrict ourselves to work in D = 26 and make a comment on strings on
+non-critical dimension later.
+B.
+Including string-loop corrections
+At this point, we have only focused on the zeroth-order in the gs-expansion. Although
+it is clear that string loops should modify the results, it is not completely clear how those
+corrections must be included. One of the most accepted proposals is the Fischler-Susskind
+approach [8–10]. The idea behind such mechanism is that string loop divergences can be
+absorbed through a renormalization of the background fields in the non-linear sigma models.
+Let us illustrate this explicitly for unoriented closed bosonic strings. For the purpose of this
+section, it is simpler to work with a a sharp cut-off as regularization scheme.
+The divergences in string loops appear when we have to sum over conformally inequivalent
+surfaces of a fixed topology (i.e. genus). For a fixed but arbitrary topology (i.e. we focus
+21
+
+here on non-trivial topologies), this sum is an integral over a finite-dimensional parameter
+space, the so-called Teichm¨uller space [17, 18].
+These integrals are divergent, but these
+divergences arise from handles that shrink to zero size. These divergences are equivalent to
+the divergences coming from inserting a local operator on the trivial-genus worldsheet. In a
+flat spacetime, the divergence appearing for the torus topology can be eliminated through
+the insertion of an operator log Λ
+2π γabηµν∂aXµ∂bXν, with a suitable coefficient. Here Λ is a
+suitable cut-off in the Teichm¨uller space.
+If we move to a curved geometry Gµν with a non-trivial zero mode of the dilaton field Φ,
+we need to substitute the metric Gµν and include a relative factor e−Φ to account for the
+dependence of the path integral on the topology of the surface. We recall that the asymptotic
+value of the dilaton field λ = ⟨Φ⟩ is identified with the string coupling constant gs = eλ
+through an exponential relation, as it can be seen by comparison of the actions in Eq. (33)
+and Eq. (28) [17, 18]. Explicitly for the first non-trivial order (torus topology) we have the
+following divergences:
+δSloop = log Λ
+2π
+�
+d2σ√−γγabe−ΦGµν(X)∂aXµ∂bXν.
+(55)
+The e−Φ factor ensures that, when evaluated on the trivial topology on the worldsheet, it
+captures the divergences in the torus. If the dilaton field displays a non-trivial background
+profile Φ(X), not only a zero mode λ, we expect that replacing Φ with Φ(X) would lead to
+a first term in an α′ expansion of the term. This term modifies the β-functional (we will
+refer from now on to those β-functionals modified due to the the presence of string loop
+corrections as ˜β) associated with the metric through the addition of a term δβG
+µν to the
+functional βG
+µν above
+˜βG
+µν = βG
+µν + δβG
+��ν,
+(56)
+which looks like a cosmological constant term, i.e.
+δβG
+µν = Ce−ΦGµν,
+(57)
+where C is an arbitrary constant that arises in the renormalization procedure. On equal
+footing, an additional contribution to the dilaton, which we call δβΦ will also appear, al-
+though it is hard to evaluate explicitly.
+Instead, it is easier to obtain it by applying a
+consistency argument [8–10]. As we have argued above, in principle the vanishing of the
+22
+
+modified ˜βΦ-functional through string loop corrections is not independent of the vanishing
+of the ˜βG
+µν functional. As we have seen, in the CFT computation, it being constant is pre-
+cisely a consequence of the vanishing of the remaining β-functionals. By this consistency
+condition, it is possible to derive an equation for the ˜βΦ-function.
+Taking the divergence of the ˜βG
+µν and simplifying it through Bianchi identities and using
+also the vanishing of ˜βG
+µν itself, we find:
+∇µ ˜βG
+µν = ∇ν
+�1
+2R (G) − ∇2Φ − 1
+2 (∇Φ)2
+�
+(58)
+This leads us to the following ˜βΦ functional for the dilaton field:
+˜
+βΦ = α′
+�
+−R (G) + 2∇2Φ + 1
+2 (∇Φ)2
+�
+,
+(59)
+which knowing that is a constant, can be safely chosen to be equal to zero. In case that we
+were dealing with strings in non-critical dimension, an additional D − 26/6 factor should be
+included arising from the bc-ghost system contribution to the Weyl-anomaly at the string
+tree level. Notice that we have introduced α′ as a dimensionful parameter. Once we have
+reached this point, it is better to pause and recapitulate what we have done until now. We
+began analyzing the α′-expansion of the sigma model describing the propagation of strings in
+arbitrary backgrounds. We determined the β-functionals of the Weyl anomaly to the lowest
+order. Then we jumped into the problem of including string-loop corrections that should
+clearly modify the constraints that the background fields should obey. For the purpose of
+including such corrections, we noticed that the divergences arising from the string loops can
+be absorbed into a renormalization of the background fields Gµν and Φ. Hence, up to this
+point we have found a set of equations that these background fields need to obey for the
+consistent propagation of the strings.
+C.
+EFTs for the theory
+The consistency equations that we found arising from the Weyl anomaly cancellation and
+the cancellation of the divergences from string loop corrections resemble a lot the equations
+of motion of a given field theory for Gµ(X) and Φ(X):
+˜βG
+µν = Rµν (G) − ∇µ∇νΦ + Ce−ΦGµν + O(α′),
+(60)
+˜βΦ = D − 26
+6
++ α′ �
+−R (G) + 2∇2Φ + (∇Φ)2�
++ O
+�
+α′2�
+.
+(61)
+23
+
+Setting C = 0 corresponds to omitting the string loop corrections. The natural question is
+then whether it is possible to obtain an effective action whose dynamics correctly reproduce
+these equations. In addition, such effective action needs to correctly account for the scatter-
+ing amplitudes involving only massless excitations of the string (to the lowest order in the
+α′ expansion) in order to be a sensible action. There are (at least) two effective actions that
+fullfill these criteria: match the scattering amplitudes involving gravitons and dilatons and
+their equations of motion give rise to the β-functionals. These two actions correspond to a
+GR-like EFT and a UG-like EFT. The GR-like EFT can be given as:
+SGR
+eff =
+1
+2κ2
+�
+dD+1X
+√
+−GeΦ
+�
+−(D − 26)
+6α′
+− 2Ce−Φ + R (G) + (∇Φ)2
+�
++ O(α′).
+(62)
+From this action principle it is straightforward to obtain the β-functionals as
+˜βΦ = −2κ2 e−Φ
+√
+−G
+δSGR
+eff
+δΦ ,
+(63)
+˜βG
+µν = 2κ2 e−Φ
+√
+−G
+�δSGR
+eff
+δGµν + 1
+2Gµν
+δSGR
+eff
+δΦ
+�
+.
+(64)
+Furthermore, it is possible to perform a field redefinition to map this action to the Einstein
+Frame [18].
+Following [1] we know that it is also possible to write down an action principle which
+reproduces the same equations of motion that Eq. (62) displays, with the cosmological con-
+stant C entering as an integration constant instead of a coupling constant. To be concrete,
+we can write down the following action principle:
+SUG
+eff =
+1
+2κ2
+�
+dD+1XωeΦ
+�
+−(D − 26)
+6α′
++ R( ˜
+G) + ( ˜∇Φ)2
+�
++ O(α′).
+(65)
+If we compute the variation with respect to Gµν we obtain the traceless version of the
+equations obtained from Eq. (62). Explicitly, if we define
+δSUG
+eff
+δGµν
+= Kµν (G) − 1
+2K (G) ,
+(66)
+for the variation of SUG
+eff we obtain the following:
+δSUG
+eff
+δGµν = Kµν( ˜
+G) −
+1
+D + 1K( ˜
+G) ˜Gµν = 0.
+(67)
+with K( ˜
+G) = ˜GµνKµν( ˜
+G). Upon taking the divergence and using the generalized Bianchi
+identities for the corresponding tensor K entering the equations (see [1] for further details)
+24
+
+we find:
+Eµν = Kµν( ˜
+G) − 1
+2K( ˜
+G) ˜Gµν + C ˜Gµν = 0.
+(68)
+Again, a suitable combination of these equations with the equation obtained from the equa-
+tion of motion for Φ we find:
+˜βΦ = −2κ2e−Φ
+ω
+δSUG
+eff
+δΦ ,
+(69)
+˜βG
+µν = 2κ2e−Φ
+ω
+�
+Eµν + 1
+2
+˜Gµν
+δSeff
+δΦ
+�
+,
+(70)
+confirming our claim that the Unimodular Gravity action (65) reproduces the β-functionals.
+Notice that this effective action does not only reproduce the β-functionals but it also repro-
+duces all of the scattering amplitudes involving massless excitations of the string (graviton
+and dilaton asymptotic states), as derived following the procedure sketched in the previous
+section. In that sense, both actions reproduce the desired properties and hence none of them
+is preferred over the other one from the perspective of using them as EFTs for the massless
+modes of the string.
+V.
+CONCLUSIONS
+We have analyzed the embedding of UG in string theory from the point of view of the
+consistent quantization of the strings in an arbitrary background. Furthermore, we have
+followed the proposal by Susskind and Fischler towards cancelling divergences arising from
+string loops with suitable counterterms in the non-linear sigma model. Our analysis here
+does not unveil any preference for UG or GR as a low energy description of string theory.
+This ties up the loose ends that were not analyzed in [1], regarding the embedding of UG
+in string theory. To put it explicitly: both UG and GR are equally valid as low energy
+descriptions of the massless modes of string theory and none of them seems to be preferred
+over the other one.
+Regarding future directions of work, we recall that our analysis here has focused on
+bosonic string theory. At first sight, the extension to superstring theory seems straightfor-
+ward although subtleties may arise in a careful study. Previous considerations of supergrav-
+ity in a UG-like context suggest that some of the vacua may spontenously break SUSY and
+hence both theories may develop a potential inequivalence at the quantum level [26–28].
+25
+
+Although there is no analysis of the global degrees of freedom in such contexts, it should be
+mentioned that it seems possible that a careful implementation of SUSY in that contexts
+requires also from a fermionic global degree of freedom, which is the responsible for the
+apparent SUSY-breaking presented there.
+A second direction of work that is worthwhile exploring is that of non-perturbative defini-
+tions of string theory and its interplay with UG. For instance, the gauge/gravity correspon-
+dence (also called usually AdS/CFT) [29–31] and matrix models [32], among them probably
+we could highlight the BFSS matrix model [33]. In such contexts, we have not explored
+whether it is easy or not to accomodate a UG principle instead of a GR principle.
+ACKNOWLEDGMENTS
+The authors would like to thank Carlos Barcel´o and Ra´ul Carballo-Rubio for collab-
+oration in early stages of this project and invaluable discussions during the preparation
+of the manuscript.
+We would also like to thank Tom´as Ort´ın for helpful conversations.
+Financial support was provided by the Spanish Government through the projects PID2020-
+118159GB-C43, PID2020-118159GB-C44, and by the Junta de Andaluc´ıa through the
+project FQM219. GGM acknowledges financial support from the grant CEX2021-001131-S
+funded by MCIN/AEI/10.13039/501100011033. GGM is funded by the Spanish Government
+fellowship FPU20/01684.
+26
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+

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+arXiv:2301.13603v1  [math.LO]  31 Jan 2023
+Limits of structures and Total NP Search Problems∗
+Ondřej Ježil
+[email protected]
+Faculty of Mathematics and Physics, Charles University†
+Abstract
+For a class of finite graphs, we define a limit object relative to some computation-
+ally restricted class of functions. The properties of the limit object then reflect how
+a computationally restricted viewer “sees” a generic instance from the class. The
+construction uses Krajíček’s forcing with random variables [7]. We prove sufficient
+conditions for universal and existential sentences to be valid in the limit, provide sev-
+eral examples, and prove that such a limit object can then be expanded to a model
+of weak arithmetic. We then take the limit of all finite pointed paths to obtain a
+model of arithmetic where the problem OntoWeakPigeon is total but Leaf (the
+complete problem for PPA) is not. This can be viewed as a logical separation of
+the oracle classes of total NP search problems, which in our setting implies standard
+nonreducibility of Leaf to OntoWeakPigeon.
+1
+Introduction
+There exist several logical constructions of limits of classes of finite structures such as
+the ultraproduct and the compactness theorem. The latter was used in [2] to prove the
+0–1 law for structures over relational vocabularies.
+In combinatorics there are also several notions of limits of finite graphs. For example
+the dense graph limit defined for a sequence of graphs {Gk}k>0 satisfying the condition
+that
+t(F, Gn) = |hom(F, G)|
+|Gn||F |
+,
+converges for every fixed connected graph F, where hom(F, G) denotes the set of all graph
+homomorphisms from F to G. This provided a framework (see [8]) to restate and find new
+proofs for results in extremal graph theory — for instance Goodman’s theorem relating
+the number of edges to the number of triangles in a graph. There are other notions of
+limits of sequences of graphs, and we refer the interested reader to [10]. Another recent
+use of limit objects for the results of extremal combinatorics was by Razborov in [11].
+∗ This work has been supported by Charles University Research Center program No.UNCE/SCI/022.
+† Sokolovská 83, Prague, 186 75, The Czech Republic
+1
+
+In this work, we define a new construction of a limit object. Given a class of finite
+graphs G, whose vertex sets are initial segments of N, we can stratify it into the sequence
+of sets {Gk}∞
+k=1 as follows
+Gk = {G ∈ G; G has {0, . . . , k − 1} as its vertex set}.
+Our construction would yield a pseudofinite structure if limk→∞|Gk| = 1, but an
+ordinary application of the compactness theorem suffices for that, we therefore generally
+care about the case, where limk→∞|Gk| = ∞.1 We call such a sequence of sets of graphs
+a wide sequence and the limit object its wide limit.
+Let F be a class of functions with some computational restrictions, for example take
+F to be the set of functions computed by decision trees of some small depth. We define
+the wide limit denoted limF Gn, where n is a technical parameter to be defined later.
+The wide limit limF Gn is a Boolean-valued graph2 — its edge relation does not only
+permit the truth values 0 and 1 but also many other values from some infinite complete
+Boolean algebra B. This algebra is in fact also a σ-algebra with a measure µ on it,
+so to any statement formulated as a first order sentence ϕ we can assign a real number
+µ([[ϕ]]) ∈ [0, 1] which measures how far is the truth value of ϕ (denoted [[ϕ]]) from the value
+0. The key method we use is arithmetical forcing with random variables, developed in [7],
+which allows us to construct models of (weak) arithmetical theories and by restricting to
+a language of graphs gives us Boolean-valued graphs. In these Boolean-valued graphs,
+validity of existential quantifiers corresponds to the ability of F to solve search problems
+over the class of graphs we are considering.
+Our limit object can be expanded to the original model Krajíček’s method would
+otherwise construct. We prove (Theorem 5.8) that the truth values of first order sentences
+concerning the object are preserved even when evaluated in the model of arithmetic
+relativized to the wide limit (under a mild condition on the family F).
+As an application of this construction, we take the limit of all finite paths starting
+at the vertex 0 relative to the class of functions computed by oracle trees of subex-
+ponential depth and obtain the Boolean-valued graph limFnb ∗PATHn which is an infi-
+nite path with only one endpoint. This object is then expanded to a Boolean-valued
+model of weak second order arithmetic K(∗PATHn, Fnb, Gnb) in which every instance of
+OntoWeakPigeon has a solution. However, the object limFnb ∗PATHn in the model
+K(∗PATHn, Fnb, Gnb) is an instance of the PPA-complete problem Leaf which does not
+have a solution. This can be seen as a logical analogue of an oracle separation of these
+two classes, which is known to hold3. We then show the result implies a separation of
+those classes under stronger notion of reducibility.
+1The case where the limit tends to some other positive number results in a structure which after
+collapsing to a two-valued boolean algebra becomes pseudofinite too.
+2Generally, we can do this with any L-structures for some first order language L. The limit object is
+then a Boolean-valued L-structure limF Gn. In this work we restrict ourselves to the language of graphs
+L = {E} to simplify the presentation.
+3OntoWeakPigeon can be reduced to WeakPigeon which is known to be in PPP [6] and it is
+known [1] that Leaf cannot be reduced to any problem in PPP.
+2
+
+There is already an established connection between complexity of search problems and
+logic (namely bounded arithmetic, see [4]). The model we construct is not known nor ex-
+pected to be a model of any theory which has been considered under these investigations.
+However, we show that open induction and open comprehension is valid in this model,
+and thus we show these principles along with the principle that OntoWeakPigeon is
+total cannot prove that the problem Leaf is total. The way the model is constructed
+also implies nonreducibility from Leaf to OntoWeakPigeon for subexponential time
+oracle machines. Moreover, one can at least in theory tweak our construction (e.g. by
+extending the family Fnb) to obtain a model of a stronger theory. This has been success-
+fully done for several models already in [7, Chapter 10, Chapter 14, Chapter 21] using
+the switching lemma.
+2
+Preliminaries
+By graphs we mean structures in a language with a single binary relation denoted E
+which is antireflexive and possibly symmetric if the graph in question is undirected. We
+will denote any particular graph by ω as it will be used in some sense as a sample of a
+discrete probability space. The edge relation of a particular graph ω will be denoted Eω.
+In the rest of this section we recall notions needed for Krajíček’s forcing construc-
+tion. Fundamental notion we use throughout the work is of nonstandard models of (true)
+arithmetic. Let Lall be the language containing the names of all relations and functions
+on the natural numbers and let ThLall(N) denote the set of true sentences in this lan-
+guage. By classical results of logic there exist Lall-structures in which all sentences from
+ThLall(N) are valid but which are not isomorphic to N. These are called nonstandard
+models (of ThLall(N)).
+All nonstandard models of ThLall(N) (and even much weaker theories) contain an
+isomorphic copy of N as an initial segment. Therefore, we can assume that in fact all
+models we encounter satisfy N ⊆ M.
+After considering a concrete nonstandard model M (of ThLall(N)) we shall call the
+elements of M\N nonstandard numbers. These can be intuitively understood as “infinite
+natural numbers”. The key feature of those elements is that all functions and relations
+from Lall are defined even on nonstandard numbers. This includes functions for coding
+sequences and sets by numbers, and therefore we can use notation like a0, . . . , an−1 even
+for a nonstandard number n. The notation then means that for each i ∈ M such that
+i < n we have an object ai coded by a number in M and that this whole sequence is
+coded by some number {ai}n−1
+i=0 ∈ M. For a nonstandard number S ∈ M coding a set
+we denote its nonstandard size (cardinality) to be |S|. In the case where we talk about
+a binary string x the notation |x| denotes the bit length of x (which is nonstandard if x
+is).
+In the next section we will fix a nonstandard model M which has the model theoretic
+property that it is ℵ1-saturated. There is a self-contained construction of such model
+in [7, Appendix]. The only consequence of the ℵ1-saturation we shall use is the following.
+Property. Let {ai}∞
+i=0 be a sequence of standard numbers. Then there exists t ∈ M \N
+3
+
+and a sequence {bi}t
+i=0 ∈ M such that for all i ∈ N it holds that ai = bi. We shall call
+the sequence of {bi}t
+i=0 the nonstandard prolongation of {ai}∞
+i=0.
+The language Lall contains symbols for all relations on N.
+Since every sequence
+of numbers can be defined by some relation it turns out that in our case there is a
+unique nonstandard prolongation which matches the definition of the wide sequence (up
+to length which can be chosen arbitrarily high). We can therefore allow ourselves to use
+nonstandard numbers as indices of any sequences of objects unambiguously.
+Any nonstandard model M can be extended to an ordered ring ZM by adding neg-
+ative elements. This ring then can be extended to a fraction field QM. We shall call
+elements of QM M-rationals. The field QM contains an isomorphic copy of Q as a sub-
+structure. We call an element in QM with absolute valued greater than all k
+1, k ∈ N,
+infinite otherwise we call it finite. We call elements in QM with absolute value smaller
+than all 1
+k, k ∈ N infinitesimal.
+We will denote the set of finite M-rationals as QM
+fin and one can check it forms an
+ordered ring.
+Lemma (The existence of a standard part). There is a function st : QM
+fin → R assigning
+to each finite M-rational a real number. st is a ring homomorphism and the kernel of st
+is exactly the ideal of infinitesimal numbers. When q is a finite M-rational we call st(q)
+its standard part.
+We shall use the structure QM analogously to how hyperreal numbers are used in
+nonstandard analysis. For more details about nonstandard analysis we recommend [3]
+to the interested reader. The following result characterizes convergence of sequences of
+rational numbers using QM.
+Theorem. Let {ai}∞
+i=0 be a sequence of rational numbers and let r ∈ R.
+Then the
+following are equivalent.
+• limi→∞ ai = r
+• For every {bi}t
+i=0, t ∈ M\N, which is a nonstandard prolongation of {ai}∞
+i=0, there
+is an nonstandard s0 ≤ t, such that for every nonstandard s ≤ s0: st(as) = r.
+It is important for forcing with random variables to consider discrete probability
+spaces of nonstandard size. We shall always use uniform distribution on the samples
+(although this is not necessary for the general construction). Thus, the probability of an
+event coded by an element A ∈ M is then just the M-rational number |A|/|S| where S
+is the set of samples of such a space.
+We conclude this section by restating classical inequalities used in this work using
+the nonstandard approach.
+Theorem (Bernoulli’s inequlity). Let y ∈ M, x ∈ QM and x ≥ −1, then
+(1 + x)y ≥ 1 + yx.
+Theorem (Exponential inequality). Let x ∈ M \ N, then
+st
+��
+1 − 1
+x
+�x�
+≤ e−1.
+4
+
+3
+Wide limits
+3.1
+The definition
+We shall define a wide limit of every sequence of the following form.
+Definition 3.1. A sequence of sets of graphs {Gk}∞
+k=1 is called a wide sequence if the
+following holds:
+• Every graph ω ∈ Gk has the vertex set {0, . . . , k − 1}.
+• limk→∞|Gk| = ∞.
+By abuse of notation we will simply talk about a wide sequence Gk instead of {Gk}∞
+k=1.
+Since a wide limit is a Boolean-valued graph, we need to construct a Boolean algebra in
+which the truth evaluation of statements shall take place.
+For the construction of the Boolean algebra we will closely follow [7, Chapter 1] albeit
+with slight changes. Let us now fix for the rest of this work an ℵ1-saturated model of
+ThLall(N) which we will denote M.
+Definition 3.2. Let n ∈ M. We define
+An = {A ⊆ {0, . . . , n − 1}; A ∈ M},
+in words An is the set of subsets of {0, . . . , n − 1} coded by an element in M. This is
+a boolean algebra and to each A ∈ An we assign an M-rational |A|/n which we call its
+counting measure.
+Even though An is a boolean algebra with a “measure” it is not a σ-algebra. Indeed,
+An contains all singletons, but the countable set of those elements in {0, . . . , n − 1} with
+only finitely many predecessors is not definable by compactness. However, having infinite
+joins and meets at our disposal allows us to interpret quantifiers in the boolean valued
+case, so we now want to ‘tweak’ this Boolean algebra.
+Definition 3.3. Let I be the ideal of An consisting of elements with infinitesimal count-
+ing measure. We define Bn = An/I. Each element in Bn is of the form A/I, where
+A ∈ An, and we define µ(A/I) = st(|A|/n). We will denote the maximal element of Bn
+by 1 and the minimal element by 0.
+One can easily check that µ is well-defined since for all A ∈ I it holds that st(|A|/n) =
+0. The measure µ is called the Loeb measure. The following then holds.
+Lemma 3.4 ( [7, Lemma 1.2.1]). Bn is a σ-algebra with a real valued measure µ. More-
+over, Bn is a complete boolean algebra.
+It is important to note that 1 ∈ Bn is the only element of Bn with measure µ(1) = 1
+and similarly 0 ∈ Bn is the only element with measure µ(0) = 0. Also, for B, B′ ∈ Bn
+the inequality B ≤ B′ implies µ(B) ≤ µ(B′).
+5
+
+We now define precisely what we mean by the family of functions F relative to which
+we will be taking the wide limit. This is still a part of Krajíček’s construction, we just
+modify it to make it compatible with our setup — where we start with a wide sequence.
+For every k ∈ N the set Gk is finite and thus can be coded by a number. Therefore,
+there is a nonstandard prolongation of this sequence, and we can consider the set coded
+by the nonstandard number Gn, which matches the definition of the wide sequence in M.
+Definition 3.5. Let {Gk}∞
+k=1 be a wide sequence and let n ∈ M \ N. We say that F is a
+family of random variables on Gn if every α ∈ F is a function coded by a number in M
+with domain Gn and taking values in M. We say α ∈ F is an F-vertex if for all ω ∈ Gn
+it holds that α(ω) ∈ {0, . . . , n − 1}. The set of all F-vertices is denoted U(F).
+If the wide sequence {Gk}∞
+k=1 and the number n ∈ M \ N is clear from context we
+just say F is a family of random variables. This is for now everything we need to recall
+from [7], and we can proceed to define the central object of our work.
+Definition 3.6 (The wide limit). Let {Gk}∞
+k=1 be a wide sequence, let n ∈ M \ N and
+let F be a family of random variables on Gn. We define the wide limit limF,n{Gk}∞
+k=1 as
+a Bn-valued structure in the language consisting of a single binary relation symbol {E}
+as follows. The universe of the wide limit is taken as the set of all F-vertices. We now
+inductively define the truth values for all {E}-sentences.
+• [[α = β]] = {ω ∈ Gn; α(ω) = β(ω)}/I
+• [[E(α, β)]] = {ω ∈ Gn; Eω(α(ω), β(ω))}/I
+• [[ − ]] commutes with ¬, ∧ and ∨
+• [[(∃x)A(x)]] = �
+α∈U(F ) [[A(α)]]
+• [[(∀x)A(x)]] = �
+α∈U(F ) [[A(α)]]
+By abuse of notation we will denote the wide limit limF,n{Gk}∞
+k=1 by limF Gn. To
+stress in which boolean valued structure is the truth evaluation [[ − ]] taking place we will
+sometimes denote the evaluation C1[[ − ]], C2[[ − ]] for boolean valued structures C1 and C2
+respectively. Furthermore, if C1[[ϕ]] = 1 for some sentence ϕ we say ϕ is valid in C1.
+Note that since Gn can be recovered from F as the domain of its elements the wide
+limit strictly speaking only depends on F. We keep Gn in the notation to cover the
+situation where we have a very general family of functions (e.g. the family of polynomial
+functions FPV) which can be applied to every wide sequence. Thus, the notation limF Gn
+means that F is restricted to those functions which take elements of Gn as an input even
+when F possibly contains other functions too.
+The variability of the parameter n may also seem unnecessary and indeed in our
+applications it is the case, but generally there are examples of wide sequences where n
+directly affects the wide limit.
+6
+
+Example 3.7. Let Fconst be the family of all constant functions with domain Gn and
+range anywhere in M. Let
+Gk =
+�
+{({0, . . . , k − 1}, E); |E| = 2, (0, 1) ∈ E}
+k even
+{({0, . . . , k − 1}, E); |E| = 1, (0, 1) ̸∈ E}
+k odd
+then
+lim
+Fconst Gn[[E(0, 1)]] =
+�
+1
+n even
+0
+n odd.
+3.2
+An example of a wide limit relative to shallow decision trees
+Now we shall define the first nontrivial family of random variables relative to which we
+shall take wide limits of several sequences. The functions in the family will be computed
+by shallow decision trees. So the shape of the wide limit reflects what can ‘superloga-
+rithmic’ trees witness in the wide sequence with probability arbitrarily close to 1.
+Definition 3.8. Let Trud be a family of labeled rooted binary trees in M of the following
+form. At each vertex the tree is labeled by an element of {0, . . . , n − 1} × {0, . . . , n − 1}
+and the two outgoing edges incident to it are labeled as 0 and 1 respectively. The leaves
+are labeled by an element of M. The depth of the tree is bounded by a number of a
+form n1/t (rounded to the nearest element of M) for some t ∈ M \ N.
+A computation of a T ∈ Trud on some ω ∈ Gn is defined as follows. Start at the root
+and interpret each label (i, j) of the vertex as a question whether the pair (i, j) is in
+the edge set Eω and follow a path through T reading 1 as a positive answer and 0 as a
+negative answer. The label of the leaf visited at the end of the path is the output of T
+on ω, denoted T(ω).
+We define Frud to be the set of all functions computed by a tree T ∈ Trud.
+The simplest wide sequence we shall consider is the following sequence of sets of
+undirected graphs with exactly one edge.
+Definition 3.9. EDGEk = {({0, . . . , k − 1}, E); |E| = 1}
+Since any ω ∈ EDGEk has only 1 edge in all potential k · (k − 1)/2 edges, it is not
+likely a shallow tree will find the edge. This is the idea behind the proof of the following
+theorem.
+Theorem 3.10.
+lim
+Frud
+EDGEn[[(∃x)(∃y)E(x, y)]] = 0
+Proof. Let α, β ∈ U(Frud), we proceed by proving that
+[[E(α, β)]] = 0
+which is enough to prove the theorem since even an infinite disjunction of the values 0
+is 0.
+7
+
+Let α and β be computed by T ∈ Trud and S ∈ Trud respectively. Let the depth of
+both T and S be at most n1/t, where t ∈ M \N. Walk down T from the root and always
+prolong the path along the edge labeled 0. On this path we have a set of at most n1/t
+different pairs of vertices and a label of the leaf lT .
+We do the same for S, and we find another set of at most n1/t pairs of vertices and
+a label of the leaf lS. lS and lT are then combined to one last pair (lS, lT ). Now we just
+need to compute the probability that none of these 2n1/t + 1 pairs of vertices are not in
+the edge set Eω.
+There are
+�n
+2
+�
+different graphs in EDGEn and
+�n−4n1/t−2
+2
+�
+graphs which fulfill our
+requirements. The probability is thus
+�n−4n1/t−2
+2
+�
+�n
+2
+�
+= (n − 4n1/t − 2)(n − 4n1/t − 3)
+n(n − 1)
+≥ (n − 4n1/t − 3)2
+n2
+≥
+�
+1 − 4n1/t + 3
+n
+�2
+≥
+�
+1 − 8n1/t + 6
+n
+�
+after taking the standard part of the last line we get st(1 − 8n1/t+6
+n
+) = 1. Therefore,
+µ([[E(α, β)]]) = 0 and [[E(α, β)]] = 0.
+3.3
+Sufficient conditions for validity of existential and universal sen-
+tences
+To analyze wide limits we need ideally to know the values of sentences which describe
+properties whose complexity we are interested in. Generally this can be hard, so in this
+section we prove sufficient conditions at least for the validity of universal and existential
+sentences.
+We will start with the simpler condition for the validity of universal sentences. This
+is important also because we would like to know that a wide limit of a wide sequence
+of graphs (i.e. antireflexive {E}-structures) is also a graph and that a wide limit of a
+wide sequence of undirected graphs (directed graphs with E symmetric) is an undirected
+graph. All of these properties are expressible as universal sentences.
+Theorem 3.11. Let Gk be a wide sequence and let F be any family of random variables.
+Let ϕ(x0, . . . , xl−1) be an open {E}-formula and assume that
+lim
+k→∞ Pr
+ω∈Gk
+[ω |= (∀x)ϕ(x)] = 1.
+Then limF Gn[[(∀x)ϕ(x)]] = 1.
+8
+
+Proof. By induction in M we have that st(Prω∈Gn[ω |= (∀x)ϕ(x)]) = 1. Therefore, we
+have for every tuple of F-vertices α that [[ϕ(α)]] = 1. Now
+[[(∀x)ϕ(x)]] =
+�
+α∈U(F )l
+[[ϕ(α)]]
+=
+�
+α∈U(F )l
+1
+= 1.
+Corollary 3.12. Let Gk be a wide sequence and F any family of random variables.
+• If all ω ∈ Gk, k ∈ N, are directed graphs ({E}-structures satisfying that E is antire-
+flexive) then limF Gn is a Boolean-valued {E}-structure in which the antireflexivity
+of E is valid (i.e. limF Gn is a Boolean-valued graph).
+• If all ω ∈ Gk, k ∈ N, are undirected graphs (directed graphs where E is symmetric)
+then limF Gn is an {E}-structure in which both antireflexivity and symmetry of E
+is valid. (i.e. limF is a Boolean-valued undirected graph)
+Now to give a sufficient condition for the validity of an existential sentence (∃x)ϕ(x)
+we use the auxiliary value of density of ϕ(x0, . . . , xl−1) defined as the probability that a
+random graph ω ∈ Gk and a random tuple a ∈ {0, . . . , k − 1}l satisfy ω |= ϕ(a) and show
+that the limiting density gives a lower bound for the measure of [[(∃x)ϕ(x)]].
+Theorem 3.13. Let Gk be a wide sequence and let F be a family of random variables
+which contains all constant functions. Let ϕ(x0, . . . , xl−1) be an open {E}-formula and
+let p ∈ [0, 1]. Assume that
+lim
+k→∞ Pr
+ω∈Gk
+a
+[ω |= ϕ(a)] ≥ p,
+where a is sampled uniformly over all elements of {0, . . . , k − 1}l. Then
+µ(lim
+F Gn[[(∃x)ϕ(x)]]) ≥ p.
+In particular if p = 1 then limF Gn[[(∃x)ϕ(x)]] = 1.
+Proof. Consider an array C indexed by ω ∈ Gn and a ∈ {0, . . . , n − 1}l such that
+Cω,a =
+�
+1
+ω |= ϕ(a)
+0
+otherwise.
+By the assumption and induction in M we have that
+st
+�
+1
+nl|Gn|
+�
+ω∈Gn
+�
+a
+Cω,a
+�
+≥ p.
+9
+
+We now claim that there exists a specific b ∈ {0, . . . , n−1}l such that st(Prω∈Gn[ω |=
+ϕ(b)]) ≥ p. Assume for contradiction that the claim is false. Then
+1
+|Gn|nl
+�
+ω∈Gn
+�
+a
+Cω,α = 1
+nl
+�
+a
+Pr
+ω∈Gn[ω |= ϕ(a)]
+≤ Pr
+ω∈Gn[ω |= ϕ(a0)],
+where we pick a0 such that it maximizes Prω∈Gn[ω |= ϕ(a0)].
+But after taking the
+standard part of the inequality we obtain that
+st
+�
+1
+nl|Gn|
+�
+ω∈Gn
+�
+a
+Cω,a
+�
+≤ st( Pr
+ω∈Gn[ω |= ϕ(a0)]) < p.
+Which is a contradiction and so the claim is true. Let γb be a tuple of constant
+functions which is at every sample equal to b. We have
+[[(∃x)ϕ(x)]] =
+�
+α∈U(F )l
+[[ϕ(α)]]
+≥ [[ϕ(γb)]]
+and by taking µ of this inequality we finally obtain that µ([[(∃x)ϕ(x)]]) ≥ p.
+In the following example we use Theorem 3.13 to show that in the wide limit of graphs
+which have exactly one large clique and no other edges the nonexistence of a standard
+sized clique cannot be valid relative to any F with all constants.
+Example 3.14. Consider the wide sequence
+SK1/2
+k
+= {({0, . . . , k − 1}, E); E has a clique of size ⌊k/2⌋ and no other edges}.
+We will check that for an {E}-formula ϕl(x) which states that x forms a clique of size l
+we have
+lim
+k→∞
+Pr
+ω∈SK1/2
+k
+a
+[ω |= ϕl(a)] ≥ (1/2)l.
+Notice that we can compute the probability for a fixed a such that ai ̸= aj whenever
+i ̸= j, since the ratio of tuples containing some vertex twice is infinitesimal. So we have
+Pr
+ω∈SK1/2
+k
+[ω |= ϕl(a)] =
+l−1
+�
+i=0
+�
+1 − k − ⌊k/2⌋
+k − i
+�
+≥
+�
+1 − k − ⌊k/2⌋
+k − l
+�l
+≥
+�
+1 −
+1
+2(1 − l/k) −
+1
+k − l
+�l
+10
+
+and since l ∈ N we just take the limit of the inner expression. But one can see that
+limk→∞(1 − l/k) = 1 and that limk→∞(1/(k − l)) = 1.
+Now by Theorem 3.13 we obtain that for any F that contains all constants we have
+lim
+F SK1/2
+n [[(∃x)ϕl(x)]] > 0.
+The following example demonstrates that Theorem 3.11 cannot be generalized to a
+similar hypothesis as Theorem 3.13.
+Example 3.15. Let Gk consist of all undirected graphs on the vertex set {0, . . . , k − 1}
+with exactly ⌈ k(k−1)
+2 log(k)⌉edges. One can see that
+lim
+k→∞ Pr
+ω���Gk
+x,y
+[ω |= ¬E(x, y)] = 1,
+but in fact limFrud Gn[[(∀x)(∀y)¬E(x, y)]] = 0.
+Let t ∈ M \ N such that n1/t is not bounded above by a standard number. Let T be
+a tree which queries on all paths a fixed set of n1/t different potential edges. If we prove
+that any such set in Gn has to contain at least one edge with probability infinitesimally
+close to 1 then we can construct Frud-vertices α and β using T such that [[E(α, β)]] = 1
+by simply taking T and labeling each leaf on a path which finds an edge with one of the
+vertices incident to this edge.
+Let S be the set of potential edges queried by T and let m =
+�n
+2
+�
+. Now we have
+Pr
+ω∈Gn[S contains no edge in ω] =
+(m − n1/t)!(m − ⌈ m
+log n⌉!)
+m!(m − ⌈
+m
+log m⌉ − n1/t)!
+=
+n1/t−1
+�
+i=0
+m − ⌈ m
+log n⌉ − i
+m − i
+≤
+�
+1 −
+⌈ m
+log n⌉
+m
+�n1/t
+≤
+�
+1 −
+1
+2 log n
+�n1/t
+standard part of which is for all k ∈ N bounded above by
+st
+��
+1 −
+1
+2 log n
+�k·2 log n�
+≤ e−k
+which tends to 0 as k → ∞.
+11
+
+4
+A wide limit of Leaf instances relative to oracle trees
+The class of total NP search problems TFNP, first defined in [9], consists of all relations
+on binary strings P(x, y) such that:
+• (verifiability in polynomial time) There is a polynomial time machine M which,
+given x, y, can decide whether P(x, y) holds.
+• (totality) There exists a polynomial p and for every x there exists at least one y
+satisfying |y| ≤ p(|x|) such that P(x, y) holds.
+Two particular problems are relevant for us.
+The problem Leaf is formulated as follows. An instance is given by a number k and
+a undirected graph ω on the vertex set {0, . . . , 2|k| − 1}, presented by a Boolean circuit
+of polynomial size in |k| computing its neighborhood function, such that degω(0) = 1
+and ∀v : degω(v) ≤ 2. The task is then to find some nonzero v with degω(v) = 1. The
+corresponding combinatorial principle being the handshaking lemma, which assures the
+problem is total.
+The problem OntoWeakPigeon is formulated as follows.
+An instance is given
+by a number k and two functions A : {0, . . . , 2|k| − 1} → {0, . . . , 2|k|−1 − 1} and B :
+{0, . . . , 2|k|−1 − 1} → {0, . . . , 2|k| − 1}, each presented by a Boolean circuit of polynomial
+size in |k|. The task is then to find some x such that B(A(x)) ̸= x or some y such
+that A(B(y)) ̸= y. The corresponding combinatorial principle being the bijective weak
+pigeonhole principle, which assures the problem is total. The domain of A is twice as
+large as its range, so B and A cannot form a pair of inverse functions between their
+respective domains.
+So far, we presented what is called ‘type 1’ problem in [1]. We are interested about the
+‘type 2’ problems which replace the input function(s) with oracle(s). So in the ‘type 2’
+Leaf problem, the input is a pair (α, x) where α is an oracle which describes the neighbor
+function on G with vertex set {0, . . . , 2|x| − 1}. For the ‘type 2’ OntoWeakPigeon
+problem, the input is a triple (α, β, x), where α and β are oracles describing the functions
+α : {0, . . . , 2|x| − 1} → {0, . . . , 2|x|−1 − 1} and β : {0, . . . , 2|x|−1 − 1} → {0, . . . , 2|x| − 1}.
+The associated computational models for the ‘type 1’ problems are Turing machines
+and for the ‘type 2’ problems oracle Turing machines.
+4.1
+The wide limit and oracle trees
+The wide sequence ∗PATHk (pointed paths on k vertices) consists of all undirected graphs
+ω on the vertex set {0, 1, . . . , k − 1} which are isomorphic to a path with k − 1 edges
+connecting all vertices and degω(0) = 1. Graphs in ∗PATHk are ‘the hardest instances
+of Leaf’ so we can expect the wide limit to reflect the complexity of these instances
+relative to the family F we choose.
+Since each ω ∈ ∗PATHk has only k − 1 edges we can proceed similarly to the proof
+of Theorem 3.10 to get the following.
+Lemma 4.1. limFrud ∗PATHn[[(∃x)(∃y)E(x, y)]] = 0
+12
+
+To get a result which reflects the properties of the wide sequence more faithfully we
+will define a new family of random variables on ∗PATHn.
+Definition 4.2. We define Tnb as the set of all labeled rooted trees of the following shape:
+• Each non-leaf node is labeled by some v ∈ {0, . . . , n − 1}.
+• For each {u, w} ⊆ {0, . . . , n − 1} and a node v there is an outgoing edge from v
+labeled {u, w} (it can be that u = w).
+• Each leaf is labeled by some m ∈ M.
+• The depth of the tree is defined as the maximal number of edges in a path from
+the root, and we require it is at most n1/t (rounded to the nearest element of M)
+for some t ∈ M \ N.
+The computation of such a tree in Tnb on ω ∈ ∗PATHn is defined as follows. We
+build a path by starting at the root and interpreting every vertex labeled by some v as
+a question ‘what are the neighbors of the vertex v?’ and we follow the output edge with
+the answer and continue analogously until we find a leaf. The label of the leaf is defined
+to be the output of the computation.
+We define Fnb to be the set of all functions on ∗PATHn which are computed by some
+T ∈ Tnb.
+The trees computing the functions in Fnb can be thought of as a protocol describing
+the behavior of a machine M communicating with an oracle describing a particular
+ω ∈ ∗PATHn. In the study of total NP search problems presented by oracles, we usually
+denote the size of the object by some 2|x| where x is an additional input to the problems.
+If 2|x| = n then n1/t = 2|x|/t which for t ∈ M \ N corresponds to protocols describing
+non-uniform subexponential-time computations. If we prove that no tuple of Fnb-vertices
+satisfies some open {E}-formula in limFnb ∗PATH we also prove that subexponential-time
+oracle machines cannot solve the corresponding type 2 problem on a non-diminishing
+fraction of the inputs. In the rest of this section we proceed to prove that limFnb ∗PATHn
+has no vertex with degree 1 other than the vertex 0.
+To do so, we will consider computations of trees on samples with different nonstandard
+lengths. For the rest of this section we put Gm = ∗PATHm for all m ∈ M, but we can
+assume m to be smaller than n. We define T (m)
+nb
+to be the subset of Tnb consisting of
+all the trees that have the vertex labels from {0, . . . , m − 1}. For trees in T (m)
+nb
+we can
+extend the definition of a computation to input graphs from Gm in a straight forward
+way.
+Definition 4.3. We say a tree T ∈ T (m)
+nb
+fails on ω ∈ Gm if the output of T on ω has
+degree 2.
+Definition 4.4. Let m ∈ M, v ∈ {0, . . . , m − 1} and {u, w} ⊆ {0, . . . , m − 1} we define
+Gv:{u,w}
+m
+= {ω ∈ Gm; ω |= E(v, u) ∧ E(v, w)}.
+13
+
+Lemma 4.5. Let m ∈ M and let u, v and w be distinct elements of {1, . . . , m−1}. Then
+there are bijections:
+Gv:{u,w}
+m
+∼= Gm−2 × {L, R}
+Gv:{u,0}
+m
+∼= Gm−2
+Gv:{u}
+m
+∼= Gm−1
+G0:{u}
+m
+∼= Gm−1
+Proof. For the first case a bijection can be given as follows.
+Contract u, v and w to
+just one vertex min{u, v, w} and if u is closer to 0 than w pick L otherwise pick R
+and relabel the remaining vertices using a function ‘new’ which has a property that if
+u′, v′ remain and u′ < v′ as numbers then new(u′) < new(w′) and the range of new is
+{0, . . . , m−2}. This can be inverted by first renaming the vertices using new−1 and then
+replacing min{u, v, w} by a path (u, v, w) with the orientation given either by L or R.
+The second bijection is almost the same, but the orientation is clear since u is always
+the neighbor further from 0 since the other neighbor is 0.
+The third and fourth bijections are given by just removing one end of the graph and
+relabeling.
+Definition 4.6. Let m ∈ M and v ∈ {0, . . . , m − 1}.
+Let u and w be elements of
+{0, . . . , m − 1} \ {v} and let T ∈ T (m)
+nb
+be a tree with the root labeled v. By Tv:{u,w} we
+denote the induced subtree whose root is the vertex neighboring the root of T via the
+edge labeled {u, w}.
+Lemma 4.7. Let m ∈ M. Let T ∈ T (m)
+nb
+be a tree with the root labeled v ̸= 0. For
+each u and w which are distinct elements of {0, . . . , m − 1} \ {v} there exists a tree
+˜Tv:{u,w} ∈ T (m−2)
+nb
+of the same depth as Tv:{u,w} such that
+Pr
+ω∈Gm[Tv:{u,w} fails | ω |= E(v, u) ∧ E(v, w)] =
+Pr
+ω∈Gm−2[ ˜Tv:{u,w} fails].
+If T has the root labeled 0 then there exists a tree ˜T0:{u} ∈ T (m−1)
+nb
+of the same depth
+as T0:{u} such that
+Pr
+ω∈Gm[T0:{u} fails | ω |= E(0, u)] =
+Pr
+ω∈Gm−1[ ˜T0:{u} fails].
+Proof. In the case where the root is labeled by v ∈ {1, . . . , m − 1} we can construct
+the tree ˜Tv:{u,w} by simply relabeling vertices of Tv:{u,w}. We use the relabeling function
+‘new’ from the proof of Lemma 4.5. Now for every ω ∈ Gm there is by the first bijection in
+Lemma 4.5 a uniquely determined ω′ ∈ Gm−2. The computation of ˜Tv:{u,w} on ω′ is then
+of the same shape as the computation of Tv:{u,w} on ω assuming ω |= E(v, u) ∧ E(v, w).
+And ˜Tv:{u,w}(ω′) has the same degree in ω′ as Tv:{u,w}(ω) does in ω.
+The case where the root is labeled by 0 is analogous, but we instead use the relabeling
+from the fourth bijection in Lemma 4.5.
+14
+
+Lemma 4.8. Let T ∈ T (m)
+nb
+of depth d ∈ M and let d ≤ m. Then we have
+Pr
+ω∈Gm[T fails] ≥
+d
+�
+i=0
+�
+1 −
+2
+m − 2i − 2
+�
+.
+Proof. We proceed by induction on d. The case d = 0 follows from
+Pr
+ω∈Gm[T fails] ≥
+�
+1 −
+1
+m − 1
+�
+≥
+�
+1 −
+2
+m − 2
+�
+.
+Now for the inductive case we assume the lemma holds for d − 1, and prove it for d. If
+the root of T is labeled 0 we proceed as follows. For a given T let u0 be the vertex which
+minimizes the value Prω∈Gm[T fails | E(0, u0)] which exists by the least number principle
+in M. Then by Lemma 4.7 and the induction hypothesis
+Pr
+ω∈Gm[T fails] ≥
+Pr
+ω∈Gm[T fails | E(0, u0)]
+=
+Pr
+ω∈Gm−1[ ˜T0:{u0} fails]
+≥
+d−1
+�
+i=0
+�
+1 −
+2
+m − 2i − 3
+�
+≥
+d
+�
+i=0
+�
+1 −
+2
+m − 2i − 2
+�
+.
+Now for the case where the root of T is labeled by nonzero v. First we note that
+Pr
+ω∈Gm[v has degree 2 ∧ ¬E(v, 0)] = 1 −
+2
+m − 1.
+Now we choose distinct u0, w0 such that they minimize
+Pr
+ω∈Gm[Tv:{u0,w0} fails | E(v, u0) ∧ E(v, w0)].
+Then by the Lemma 4.7 and the induction hypothesis we have
+Pr
+ω∈Gm[T fails] ≥
+�
+1 −
+2
+m − 1
+�
+Pr
+ω∈Gm[Tv:{u0,w0} fails | E(v, u0) ∧ E(v, w0)]
+=
+�
+1 −
+2
+m − 1
+�
+Pr
+ω∈Gm−2[ ˜Tv:{u0,w0} fails]
+≥
+�
+1 −
+2
+m − 1
+� d−1
+�
+i=0
+�
+1 −
+2
+m − 2i − 4
+�
+≥
+�
+1 −
+2
+m − 1
+�
+d
+�
+i=1
+�
+1 −
+2
+m − 2i − 2
+�
+≥
+d
+�
+i=0
+�
+1 −
+2
+m − 2i − 2
+�
+.
+15
+
+Lemma 4.9. Let T ∈ Tnb, then st (Prω∈Gn[T fails]) = 1.
+Proof. The depth of T is bounded by n1/t for some t ∈ M \ N. We have by Lemma 4.8
+that
+Pr
+ω∈Gn[T fails] ≥
+n1/t
+�
+i=0
+�
+1 −
+2
+n − 2i − 2
+�
+(1)
+≥
+�
+1 −
+2(n1/t + 1)
+n − 2n1/t − 2
+�
+(2)
+and the standard part of this lower bound is 1.
+Finally, in the next theorem we prove that a formalization of ‘there is a nonzero
+vertex of degree 1’ is not valid in limFnb ∗PATHn and in fact its boolean value is 0.
+Theorem 4.10.
+lim
+Fnb ∗PATHn[[(∃v)(∃u)(∀w)(v ̸= 0 ∧ E(v, u) ∧ (E(v, w) → w = u))]] = 0
+Proof. By expanding the left-hand side of the statement we get
+�
+α∈U(Fnb)
+�
+β∈U(Fnb)
+�
+γ∈U(Fnb)
+[[α ̸= 0 ∧ E(α, β) ∧ (E(α, γ) → γ = β)]].
+Therefore, it is enough if we prove that for each Fnb-vertices α and β there exists an
+Fnb-vertex γ such that
+[[α ̸= 0 ∧ E(α, β) ∧ (E(α, γ) → γ = β)]] = 0.
+For any α, β ∈ U(Fnb) we can append the tree computing β to every leaf of a tree
+computing α. This is still a tree in Tnb as its depth is at most twice the maximum of
+depths of the original trees. By relabeling the leaves of the resulting tree we can obtain
+a tree computing a function
+γ(ω) =
+�
+v
+if degω(α(ω)) = 1 and v is the only neighbor of α(ω)
+w
+if degω(α(ω)) = 2, w is a neighbor of α(ω) and w ̸= β(ω).
+This is obviously an Fnb-vertex. Let us assume for contradiction that
+[[α ̸= 0 ∧ E(α, β) ∧ (E(α, γ) → γ = β)]] > 0.
+By definition this gives us
+0 < st
+�
+Pr
+ω∈Gn[α(ω) ̸= 0 ∧ Eω(α(ω), β(ω)) ∧ (E(α(ω), γ(ω)) → γ(ω) = β(ω))]
+�
+≤ st
+�
+Pr
+ω∈Gn[α(ω) ̸= 0 ∧ degω(α(ω)) = 1]
+�
+,
+but this is in contradiction with Lemma 4.9.
+16
+
+5
+The expanded model with a Leaf instance without a so-
+lution and with total OntoWeakPigeon
+As a part of the proof of Theorem 4.10 we proved what can be reformulated as the
+statement that oracle instances of Leaf are not in oracle time O(2f(|x|)) with f ∈ o(|x|1/c)
+for every c ∈ N even when we just require it to be correct on any nondiminishing ratio
+of inputs as |x| grows. In this section we proceed to compare strength of (type 2) NP
+search problems not only with oracle FP but also with other NP search problems via
+relative consistency of their totality and nontotality. We will show that there is a model
+of weak second order arithmetic in which the problem Leaf is not total even though
+OntoWeakPigeon is.
+5.1
+The structures K(F, G)
+We will now recall the construction of second order models of weak arithmetic K(F, G)
+defined in [7, Chapter 5]. We will take the liberty to define them as an extension of
+the definition of a wide limit to obtain structures K(Gn, F, G) 4 which under the right
+conditions result in a structure in some sublanguage of Lall with two sorts: numbers and
+bounded sets of numbers which contains the wide limit as an object of the second sort.
+Definition 5.1. Let L ⊆ Lall. This determines a language L2 which we get by adding
+to L second order variables X, Y, . . . whose intended interpretation are bounded sets and
+the equality symbol for second order variables (denoted the same as the first order one).
+All second order variables are treated as function symbols and can form terms with the
+first order terms as arguments.
+We will also use the second order variables as relation symbols, and we define the
+atomic formula X(x0, . . . , xk−1) simply to be evaluated as the formula X(x0, . . . , xk−1) ̸=
+0.
+Now we assume we fix a number n, a wide sequence Gk and a family of random
+variables on Gn which all together determine a wide limit limF Gn.
+Definition 5.2. We define Mn ⊆ M to be the subset of M consisting of all numbers
+bounded above by 2n1/t for some t ∈ M \ N.
+Definition 5.3. We define Ln ⊆ Lall to contain all relation symbols from Lall and all
+functions from Lall for which their values on any element of Mn is still in Mn. We say
+F is Ln-closed if for every function symbol f ∈ Ln we have that f(α0, . . . , αk−1) ∈ F.
+Note that Mn is then a substructure of the Ln-reduct of M.
+Definition 5.4. We say that G is a family of random functions (on Gn) if every Θ ∈ G
+assigns to each ω ∈ Gn a function Θω ∈ Mn.
+4This notation is just making some parameters of the construction explicit, the models constructed
+can be obtained by the original method without first constructing the wide limit. Our contribution is in
+observing that the truth values of first order sentences concerning the wide limit is preserved between
+the wide limit and the structure K(Gn, F, G).
+17
+
+We say G is F-compatible if for every α ∈ F, Θ ∈ G we have that the function Θ(α)
+defined as
+Θ(α)(ω) =
+�
+Θω(α(ω))
+if α(ω) ∈ dom(Θω)
+0
+otherwise
+is in fact in F.
+Definition 5.5. Let F be an Ln-closed family of random variables with values in Mn.
+Let G be an F-compatible family of random functions. We define K(Gn, F, G) to be
+a Bn-valued L2
+n structure with first order sort of the universe F and second order sort
+of the universe G. The valuation of formulas is then given by the following inductive
+definition:
+• [[α = β]] = {ω ∈ Gn; α(ω) = β(ω)}/I, where α, β ∈ F
+• [[R(α0, . . . , αk−1)]] = {ω ∈ Gn; ω |= R(α0(ω), . . . , αk−1(ω))}/I, where α0, . . . , αk−1
+are from F and is R a relation symbol in Ln
+• [[Θ = Ξ]] = {ω ∈ Gn; Θω = Ξω}/I, where Θ, Ξ ∈ G
+• [[(∀x)A(x)]] = �
+α∈F [[A(α)]]
+• [[(∃x)A(x)]] = �
+α∈F [[A(α)]]
+• [[(∀X)A(X)]] = �
+Θ∈G [[A(Θ)]]
+• [[(∃X)A(X)]] = �
+Θ∈G [[A(Θ)]].
+5.2
+Preservation of sentences concerning the wide limit
+We will now prove (under a mild condition on F) that in a structure K(Gn, F, G) which
+represents the wide limit limF Gn by a second order object are the values of all sentences
+regarding the object the same as in the wide limit. This lets us construct models in
+which an object elementary equivalent to the wide limit might be desired.
+Definition 5.6. We say that the edge relation of the wide limit limF Gn is represented
+in G by Γ if Γ ∈ G and for all α, β ∈ U(F) we have that
+K(Gn, F, G)[[Γ(α, β)]] = lim
+F Gn[[E(α, β)]].
+Definition 5.7. We say a family of random variables F has restrictable ranges if for
+every α ∈ F and m ∈ Mn there is ˜αm ∈ F such that
+˜αm(ω) =
+�
+α(ω)
+α(ω) < m
+0
+otherwise.
+18
+
+Theorem 5.8. Let ϕ be a {E}-sentence.
+Let F be Ln-closed and have restrictable
+ranges and let G be F-compatible. Let the edge relation of the wide limit limF Gn be
+represented in G by Γ. We define ˜ϕ(Γ) to be the L2
+n-sentence obtained by replacing
+all the occurrences of the relation symbol E by Γ, keeping the structure of the logical
+connectives and replacing all quantifiers (∀x)(. . . ) by (∀x)(x < n → (. . . )) and (∃x)(. . . )
+by (∃x)(x < n ∧ . . . ).
+Then we have that for all {E}-sentences that
+lim
+F Gn[[ϕ]] = K(Gn, F, G)[[ ˜ϕ(Γ)]].
+Proof. We will prove that for all {E}-formulas ϕ(x) and all α ∈ F we have that
+lim
+F Gn[[ϕ(α)]] = K(Gn, F, G)[[ ˜ϕ(Γ, α)]].
+We proceed by induction on the complexity of the formula.
+The case for atomic
+formulas is clear and the step for logical connectives also since [[ − ]] commutes with
+them.
+With the induction step for negation in hand it is now enough to prove the
+inductive step for the universal quantifier.
+We assume that the statement works for a formula of lower complexity ϕ(y, x). By
+the restrictability of ranges in F we get that for all β ∈ F there is ˜βn ∈ U(F) such that
+K(Gn, F, G)[[ ˜ϕ(Γ, ˜βn, α)]] ≤ K(Gn, F, G)[[β < n → ˜ϕ(Γ, β, α)]].
+Now we have that for all α ∈ U(F)
+K(Gn, F, G)[[(∀y) ˜ϕ(Γ, y, α)]] =
+�
+α∈F
+K(Gn, F, G)[[β < n → ˜ϕ(Γ, β, α)]]
+=
+�
+˜βn∈U(F )
+K(Gn, F, G)[[ ˜ϕ(Γ, ˜βn, α)]]
+=
+�
+˜βn∈U(F )
+lim
+F Gn[[ϕ(˜βn, α)]]
+= lim
+F Gn[[(∀y)ϕ(y, α)]].
+5.3
+Failure of totality of Leaf
+Now we are in a situation that lets us construct a model of weak second order arithmetic
+that contains an instance of the problem Leaf without a solution. Consider a suitable
+family Gnb in which we can define not only the wide limit itself but instances of some
+other search problem. We can then ask: ‘Do all these instances have a solution?’ This
+is a way to compare the strength of the different total NP search problems by relative
+unprovability results. We will pick the family Gnb such that validity of totality of some
+search problem P implies the nonexistence of a suitable reduction from Leaf to P.
+19
+
+Definition 5.9. Let Gnb be the family of all random functions on ∗PATHn such that for
+each Θ ∈ Gnb there exists a tuple (γ0, . . . , γm−1) ∈ M so that γi ∈ Fnb and
+Θ(α)(ω) =
+�
+γα(ω)(ω)
+α(ω) < m
+0
+otherwise.
+In the models Mn we are working with there is a pairing function ⟨i, j⟩ which can
+code pairs of numbers by a single number thus we can represent functions of any finite
+arity by functions from Gnb.
+One can understand the tuples which compute the random functions from Gnb as
+tuples of protocols describing the computations of subexponential oracle machines. Such
+a tuple defines a function which is at every index of the tuple computed using queries to
+some ω ∈ ∗PATHn. If we prove that every instance of a search problem P represented
+by such a tuple has a solution in K(∗PATHn, Fnb, Gnb), and we know that Leaf in
+K(∗PATHn, Fnb, Gnb) is not total, which implies nonexistence of a subexponential oracle
+machine which converts solutions of P to solutions of Leaf even on any standard fraction
+of instances from ∗PATHn and thus a nonexistence of a many-one reduction from Leaf
+to P as defined in [1].
+Lemma 5.10.
+1. Fnb has restrictable ranges
+2. Fnb is Ln-closed
+3. Gnb is Fnb-compatible
+4. Gnb represents the edge relation of limFnb ∗PATHn.
+Proof. 1, 2: Here we can proceed simply by relabeling the leaves of the trees computing
+the functions from Fnb.
+3: Assume that Θ ∈ Gnb is computed by a tuple (γ0, . . . , γm−1). By induction in
+M there exists t ∈ M \ N such that ∀i ∈ {0, . . . , m − 1} the depth of γi is at most
+n1/t. Therefore, for all α ∈ Fnb we have that Θ(α) has also depth at most n1/t′ for some
+t′ ∈ M \ N. Thus, Gnb is Fnb-compatible.
+4: Let γ⟨i,j⟩ ∈ Fnb be computed by a tree in Tnb which queries i and outputs 1
+if the neighbor set contains j otherwise it outputs 0. Let Γ be computed by a tuple
+(γ⟨i,j⟩)n−1
+i,j=0. Then we have
+K(∗PATHn, Fnb, Gnb)[[Γ(α, β)]] = lim
+Fnb ∗PATHn[[E(α, β)]].
+Definition 5.11. The L2
+n-formula ϕLeaf(X, Y, m) is defined as the disjunction of the
+following formulas
+(X(0) ̸= Y (0) ∨ X(0) = 0)
+(∃x)((x < m) ∧ (X(x) > m − 1 ∨ Y (x) > m − 1))
+(∃x)((x < m) ∧ ((X(x) = x ∧ Y (x) ̸= x) ∨ (X(x) ̸= x ∧ Y (x) = x)))
+(∃x)((x < m) ∧ (Y (X(x)) ̸= x ∧ X(X(x)) ̸= x) ∨ (X(Y (x)) ̸= x ∧ Y (Y (x)) ̸= x)))
+(∃x)((0 < x < m) ∧ (X(x) = Y (x) ∧ X(x) ̸= x)),
+20
+
+this formula formalizes that if X and Y are functions representing the neighbor set of
+each x < m as {X(x), Y (x)} \ {x} and 0 has only one neighbor then there has to exist
+another y < x which also has only one neighbor.
+Theorem 5.12.
+K(∗PATHn, F, G)[[(∃X)(∃Y )(∃m)¬ϕLeaf(X, Y, m)]] = 1
+Proof. We can find Θ1, Θ2 ∈ Gnb such that for each v ∈ {0, . . . , n − 1} we have that
+{Θ1(v)(ω), Θ2(v)(ω)} is the neighbor set of v on ω ∈ ∗PATHn. (We can just query v and
+split the answer between Θ1 and Θ2.)
+By Theorem 4.10 we know that limF Gn has one degree 1 vertex and all other vertices
+of degree 2 and by Lemma 5.10 we know that it can be represented by some Γ ∈ Gnb.
+Furthermore, we can verify that
+[[(Γ(α, β)) ≡ (Θ1(α) = β ∨ Θ2(α) = β)]] = 1,
+thus Θ1 and Θ2 do not satisfy the last disjunct of ϕLeaf otherwise it would be in contra-
+diction with Theorem 4.10. By their construction and the definition of ∗PATHk we have
+that (Θ1, Θ2, n) does not satisfy the other disjuncts either.
+5.4
+Totality of OntoWeakPigeon
+Definition 5.13. The L2
+n formula ϕOntoWeakPigeon(X, Y, m) is defined as the disjunc-
+tion of the following formulas
+(∃x)((x < 2m) ∧ (X(x) > m − 1))
+(∃y)((y < m) ∧ Y (y) > m − 1))
+(∃x)((x < 2m) ∧ Y (X(x)) ̸= x)
+(∃y)((y < m) ∧ X(Y (y)) ̸= y)
+it formalizes the bijective weak pigeonhole principle which claims that any pair of func-
+tions
+X :{0, . . . , 2m − 1} → {0, . . . , m − 1}
+Y :{0, . . . , m − 1} → {0, . . . 2m − 1}
+is not a pair of inverse bijections.
+To prove that ϕOntoWeakPigeon(X, Y, m) is valid in K(∗PATHn, Fnb, Gnb) we will
+construct a tree which finds some x such that Yω(Xω(x)) ̸= x or Xω(x) > m − 1 with
+probability infinitesimally close to one.
+Definition 5.14. Let Θ, Ξ ∈ Gnb, and ζ ∈ Fnb. We say that a tree T ∈ Tnb fails for
+(Θ, Ξ, ζ) on ω if
+Θω(T(ω)) < ζ(ω)
+and
+Ξω(Θω(T(ω))) = T(ω).
+In words if T does not witness the failure of Ξ being the inverse function to Θ.
+21
+
+Lemma 5.15. Let Θ, Ξ ∈ Gnb and ζ ∈ Fnb. Then there is a tree T such that
+st
+�
+Pr
+ω∈Gn[T fails for (Θ, Ξ, ζ)]
+�
+= 0.
+Proof. Without loss of generality we may assume that ζ is actually constant, and its
+value is r ∈ Mn which we pick to be the least possible output of ζ on any sample.
+Furthermore, let Θ be computed by (θ0, . . . , θ2r−1) and Ξ by (ξ0, . . . , ξr−1).
+We construct T by stages and at each stage it will have some potential output. First
+we notice that at the beginning stage there is at least one i ∈ {0, . . . , 2r − 1} such that
+the probability that θi < r or ξθi = i is at most 1
+2. The tree T0 is thus the constant tree
+always outputting i.
+Assume Td−1 have been constructed and pick any path p ∈ Td−1. If p did not fail we
+leave it as it is otherwise we extend Td−1 along this path and after extending all such
+paths this will become the new stage Td. The path p has a leaf with some label i. We
+can check whether i fails by first appending the tree θi to this path and then to each
+new leaf (labeled with a number < r) appending ξθi, let the leaves which confirm the
+nonfailure of i be labeled by i. Now consider a path p′ extending p without determined
+output. We claim that there is j ∈ {0, . . . , 2r − 1} such that
+Pr
+Gn[θj < r ∧ ξθj = j | p′ is compatible with ω] ≤ 1
+2,
+where p′ being compatible with ω means that the computation along p′ agrees with the
+edge labels which would be chosen according to ω.
+To prove the claim we notice that along p′ it was confirmed that already d-many
+distinct elements of {0, . . . , 2r − 1} are in bijection with some d-many elements of the
+set {0, . . . , r − 1}. Therefore, to fail further there are only at most (r − d)-many other
+values j′ in {0, . . . , 2r−1} for which it holds that ξθj′ = j′. By an analogous argument to
+the proof of Theorem 3.13 this is enough to show that at least for one of them the claim
+holds since r−d
+2r ≤ 1
+2. Thus, we let j to be the label of the leaf of p′ which concludes the
+construction.
+Therefore, by construction for each d ∈ Mn, d < 2r we have
+Pr
+ω∈Gn[Td fails for (Θ, Ξ, ζ)] ≤ 2−d.
+If r is in Mn \ N then we put T = Tt′ for any nonstandard t′ such that the depth of
+T is still bounded by some n1/t, where t ∈ Mn \ N. Otherwise, we put T = T2r−1 and
+since this tree can go through the whole range of Θ it can never fail.
+Theorem 5.16.
+K(∗PATHn, Fnb, Gnb)[[(∀X)(∀Y )(∀m)ϕOntoWeakPigeon(X, Y, m)]] = 1
+Proof. By Lemma 5.15 we can construct for each (Θ, Ξ, ζ) a tree T which computes some
+function α which validates the third disjunct of ϕOntoWeakPigeon.
+22
+
+Theorem 5.17. Let ϕ(x) be an L2
+n-formula with parameters from Fnb and Gnb. Then
+for every m ∈ Mn the open comprehension principle
+(∃X)(∀y < m)(X(y) ≡ ϕ(y))
+and the open induction principle
+¬ϕ(0) ∨ ϕ(m) ∨ (∃x < m)(ϕ(x) ∧ ¬ϕ(x + 1))
+are both valid in K(∗PATH, Fnb, Gnb).
+Proof. This can be proven completely analogously to [7, Lemma 20.2.5].
+Compiling the results we have about K(∗PATH, Fnb, Gnb) we get the following.
+Corollary 5.18. In the structure K(∗PATH, Fnb, Gnb) the following are valid:
+• open induction with parameters from Fnb and Gnb
+• open comprehension with parameters from Fnb and Gnb
+• every instance of OntoWeakPigeon has a solution
+• there is an instance of Leaf which does not have a solution.
+Concluding remarks
+We have to note that the problem OntoWeakPigeon has not been considered in
+the context of oracle NP search problems and the proof of Theorem 5.16 cannot be
+adapted to prove that every instance of the stronger WeakPigeon 5 has a solution
+in K(∗PATH, Fnb, Gnb) because the presence of the inverse function is essential to the
+construction of the witness.
+A stronger problem called SourceOrSink is well established in the study of NP
+search problems (it is the complete problem for PPAD, see [1]) and can be formulated
+as follows: Given a directed graph ω on the vertex set {0, . . . , 2|x| − 1} with the property
+that any vertex v has outdegree bounded by 1 and indegree also bounded by 1 and the
+indegree of the zero vertex is 0 find a nonzero vertex which is either a source or a sink.
+In the type 2 setting the problem is given by a tuple (α, β, x), where x is a binary string
+and α and β functions presented by an oracle with domain {0, . . . , 2|x| − 1} computing
+the potential successor or predecessor of a given vertex.
+It was established in [1] that Leaf is not many-one reducible to SourceOrSink and
+therefore this nonreducibility may be reflected in our model K(∗PATH, Fnb, Gnb). The
+way SourceOrSink is presented is similar to how OntoWeakPigeon is presented,
+and thus a similar strategy could be potentially used to solve the following problem.
+Problem. Let ϕSourceOrSink(X, Y, m) be the formula formalizing that (X, Y, m) as an
+instance of SourceOrSink has a solution. Is it true that
+K(∗PATH, Fnb, Gnb)[[(∀X)(∀Y )(∀m)ϕSourceOrSink(X, Y, m)]] = 1?
+5The problem to witness that α : {0, . . . , 2|x| − 1} → {0, . . . , 2|x|−1 − 1} is not injective.
+23
+
+Acknowledgement
+This work is based on the author’s master’s thesis [5] which was completed under the
+supervision of Jan Krajíček. The author also thanks Eitetsu Ken for comments on a draft
+of this paper.
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+24
+

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+Searching for Lindbladians obeying local conservation laws and showing thermalization
+Devashish Tupkary,1, ∗ Abhishek Dhar,2, † Manas Kulkarni,2, ‡ and Archak Purkayastha3, 4, 5, §
+1Institute for Quantum Computing and Department of Physics and Astronomy,
+University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
+2International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
+3Department of Physics, Indian Institute of Technology, Hyderabad 502285, India
+4Centre for complex quantum systems, Aarhus University, Nordre Ringgade 1, 8000 Aarhus C, Denmark
+5School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
+We investigate the possibility of a Markovian quantum master equation (QME) that consistently
+describes a finite-dimensional system, a part of which is weakly coupled to a thermal bath. In order
+to preserve complete positivity and trace, such a QME must be of Lindblad form. For physical con-
+sistency, it should additionally preserve local conservation laws and be able to show thermalization.
+First, we show that the microscopically derived Redfield equation (RE) violates complete positivity
+unless in extremely special cases. We then prove that imposing complete positivity and demanding
+preservation of local conservation laws enforces the Lindblad operators and the lamb-shift Hamil-
+tonian to be ‘local’, i.e, to be supported only on the part of the system directly coupled to the
+bath. We then cast the problem of finding ‘local’ Lindblad QME which can show thermalization
+into a semidefinite program (SDP). We call this the thermalization optimization problem (TOP).
+For given system parameters and temperature, the solution of the TOP conclusively shows whether
+the desired type of QME is possible up to a given precision. Whenever possible, it also outputs a
+form for such a QME. For a XXZ chain of few qubits, fixing a reasonably high precision, we find
+that such a QME is impossible over a considerably wide parameter regime when only the first qubit
+is coupled to the bath. Remarkably, we find that when the first two qubits are attached to the bath,
+such a QME becomes possible over much of the same paramater regime, including a wide range of
+temperatures.
+I.
+INTRODUCTION
+A small finite-dimensional quantum system, a part of
+which is weakly coupled to a macroscopic thermal bath,
+is expected to thermalize to the temperature of the bath.
+Describing this dynamics is relevant across various fields
+in quantum science and technology, including quantum
+information and thermodynamics [1], quantum optics [2],
+quantum chemistry [3], engineering [4] and biology [5].
+In absence of coupling to the macroscopic thermal bath,
+the dynamics of the density matrix of the system is gov-
+erned by the Heisenberg equation of motion. This unitary
+evolution is Markovian.
+When coupled to the macro-
+scopic thermal bath, the dynamics becomes non-unitary,
+described by a quantum master equation (QME) [6–8].
+We investigate whether it is possible to have a physically
+consistent Markovian QME describing such dynamics. In
+order to do so, we are led to introduce the “thermaliza-
+tion optimization problem” (TOP). This is a semidefinite
+program (SDP), the output of which conclusively shows
+whether, for given system parameters and temperature,
+such a QME is possible, up to given precision. When-
+ever possible, the output also yields one possible form
+for such a QME. Whenever impossible, it means that,
+for such parameters, the dynamics cannot be described
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+§ [email protected]
+FIG. 1. Schematic of the setup we consider: an arbitrary fi-
+nite dimensional system described by Hamiltonian HS, a part
+of which is weakly coupled to a thermal bath at inverse tem-
+perature β. The Hilbert space of the system, HS is divided
+into a part HL which directly couples to the bath, and the
+remaining part HM.
+by any Markovian QME even at weak system-bath cou-
+pling, and therefore must have some non-Markovian char-
+acter. The SDP can be solved using standard packages
+in high-level computing.
+We note that, while SDP is
+widely used in many branches of quantum information
+and communication [9, 10], and also in quantum chem-
+istry [11, 12], it has been combined with open quantum
+system techniques in only few previous works [13–15], in
+very different contexts.
+It was shown by Gorini, Kossakowski, Sudarshan, and
+arXiv:2301.02146v1  [quant-ph]  5 Jan 2023
+
+Hilbert space Hl
+Hilbert space HM
+Hilbert space Hs = HL  HM, Hamiltonian Hs2
+Lindblad (GKSL) [16–18] that any QME that preserves
+complete positivity and trace of the density matrix, and
+describes Markovian dynamics has to be of the form
+∂ρ
+∂t = i[ρ, HS + HLS] + D(ρ),
+D(ρ) =
+d2−1
+�
+λ=1
+γλ
+�
+LλρL†
+λ − 1
+2{L†
+λLλ, ρ}
+�
+,
+γλ ≥ 0,
+(1)
+which is commonly called the Lindblad equation.
+In
+Eq. (1), ρ is the density matrix of the system, d is the
+Hilbert space dimension of the system, HS is the sys-
+tem Hamiltonian, HLS is the Lamb shift Hamiltonian,
+Lλ are the Lindblad operators, γλ are the rates, and
+D(ρ) is called the “dissipator” term. The preservation
+of complete positivity condition is enforced by demand-
+ing γλ ≥ 0. Lindblad equations have been extensively
+used in studying both theoretical and experimental se-
+tups [6–8, 19–22].
+Given this enormous scope of application, it is of
+paramount importance to assess the conditions under
+which such a Markovian description emerges from a more
+microscopic theory.
+The standard way to microscopi-
+cally obtain a Markovian QME is to consider the global
+Hamiltonian of the system weakly coupled to baths, and
+to trace out the baths perturbatively to the leading or-
+der. Starting with this microscopic viewpoint, it becomes
+clear that only having an equation in the Lindblad form
+is not sufficient, there are some additional fundamental
+requirements for physical consistency [23].
+In particu-
+lar, one must preserve local conservation laws, and if the
+steady state is unique, the system is not driven and all
+baths have same temperature, the system must thermal-
+ize to the temperature of the baths. It would be useful to
+have a QME which, by construction, is of Lindblad form
+and satisfy these additional requirements. In this paper,
+we systematically go about searching for such a QME for
+a setup where a part of the system is coupled to a single
+bath (see Fig. 1). This is done in three steps, each step
+having important consequences:
+1. The microscopically derived quantum master equa-
+tion to the leading order in system-bath coupling
+is the so-called Redfield equation (RE) [6].
+The
+RE has been shown to preserve local conservation
+laws and be able to show thermalization [23]. Here,
+we provide an explicit, model independent proof
+that the RE necessarily violates complete positiv-
+ity unless the Redfield dissipator happens to act
+“locally”, meaning it is identity on the part of the
+system that is not directly coupled to the bath. Al-
+though the violation of complete positivity by RE
+has been previously demonstrated in specific exam-
+ples [24–30], we are unaware of any previous work
+with such a model independent explicit proof.
+2. We then prove that enforcing complete positivity
+condition γλ ≥ 0 and preservation of local conser-
+vation laws necessarily requires the Lindblad oper-
+ators and the Lamb shift Hamiltonian to be “lo-
+cal”. That is, they must act only on the part of
+the system coupled to the bath, and be identity
+on the part of the system that is not connected to
+the bath. This directly rules out the possibility of
+any so-called ‘global’ Lindblad equation, such as
+the eigenbasis Lindblad equation [6, 8], the Uni-
+versal Lindblad Equation [31] to be consistent with
+local conservation laws.
+3. Given the restriction of the dissipator and the
+Lamb shift Hamiltonian to be “local’, we devise
+a numerical technique using SDP to check conclu-
+sively in a case-by-case basis whether such a QME
+can show thermalization in a particular situation.
+We call this the TOP. We use this method to study
+the case of a XXZ chain of few qubits with a part
+of it coupled to a bath. If the bath is coupled only
+to the first qubit, our method conclusively shows
+that over a large regime of system parameters and
+temperature, no such QME exists.
+However, re-
+markably if the bath is coupled to two qubits of
+the chain, up to a chosen precision, our method
+shows that a Marovian QME respecting all condi-
+tions becomes possible over a considerable regime
+of parameters, including a wide range of tempera-
+tures. Note that the RE for the XXZ chain does
+show thermalization and preserve local conserva-
+tion laws [23], even when only one qubit is attached
+to a bath. But it is not completely positive.
+This work is organized as follows. In Sec. II we ex-
+plain the setup studied in this work, derive the Redfield
+equation for our setup, and show that it will necessarily
+violate complete positivity, unless the Redfield dissipa-
+tor happens to act “locally”.
+In Sec. III we consider
+quantum master equations preserving complete positiv-
+ity and obeying local conservation laws, and show that
+such equations must have a dissipator and Lamb shift
+operator that acts only on the part of the system cou-
+pled to the bath. In Sec. IV we discuss the possibility of
+QMEs respecting complete positivity, local conservation
+laws and being able to show thermalization. We intro-
+duce the TOP, and use it in the specific case of the few
+site XXZ chain with one or two sites attached to bath.
+In Sec. V we summarize our results, and discuss future
+directions. Certain details are delegated to the appen-
+dices.
+II.
+THE MODEL AND REDFIELD
+DESCRIPTION
+Our setup is described schematically in Fig. 1. The
+full Hamiltonian be given by
+H = HS + ϵHSB + HB,
+(2)
+where HS and HB are the Hamiltonians of the system
+and bath respectively, ϵ ≪ 1 is a small dimensionless pa-
+
+3
+rameter controlling system-bath coupling strength, and
+HSB is the system-bath coupling Hamiltonian. At initial
+time, the system is considered to be in an arbitrary ini-
+tial state ρ(0), while the bath is in a thermal state with
+inverse temperature β
+ρtot(0) = ρ(0) ⊗ ρB,
+ρB = e−βHB
+ZB
+.
+(3)
+Starting with this initial state, the whole set-up of the
+system and the bath is evolved with the full Hamiltonian
+H, and the bath degrees of freedom are traced out to
+obtain the state of the system,
+ρ(t) = TrB
+�
+e−iHtρtot(0)eiHt�
+,
+(4)
+where TrB(. . .) denotes trace over bath degrees of free-
+dom.
+The Eq. (4), by construction, is a completely
+positive trace preserving (CPTP) map from ρ(0) to
+ρ(t) [6, 7].
+Without any loss of generality, we assume
+TrB(HSBρB) = 0, where TrB(. . .) denotes trace over
+bath degrees of freedom [6, 7].
+The effective equation
+of motion for the system density matrix written to the
+leading order in system-bath coupling strength ϵ is the
+RE, given by, [6],
+∂ρ
+∂t =i[ρ(t), HS]
++ϵ2
+� ∞
+0
+dt′ TrB[HSB, [HSB(−t′), ρ(t) ⊗ ρB]],
+(5)
+where
+HSB(t) = ei(HS+HB)tHSBe−i(HS+HB)t
+(6)
+and ρB is the state of the bath. In complete generality,
+we can write the system-bath coupling Hamiltonian as
+HSB =
+�
+l
+(SlB†
+l + S†
+l Bl),
+(7)
+where Sl and Bl are operators on the system and bath
+respectively, and l can be summed over as many indices as
+required for HSB. Using Eq.(7) in Eq.(5) and simplifying,
+we have
+∂ρ
+∂t = i[ρ(t), HS] + ϵ2� �
+l
+�
+S†
+l , S(1)
+l
+ρ(t)
+�
+−
+�
+S†
+l , ρ(t)S(2)
+l
+�
++ H.c
+�
+,
+(8)
+where
+S(1)
+l
+=
+�
+m
+� ∞
+0
+dt′ Tr
+�
+BlB†
+m(−t′)ρB
+�
+Sm(−t′)
++
+�
+m
+� ∞
+0
+dt′ Tr
+�
+BlBm(−t′)ρB
+�
+S†
+m(−t′)
+S(2)
+l
+=
+�
+m
+� ∞
+0
+dt′ Tr
+�
+B†
+m(−t′)BlρB
+�
+Sm(−t′)
++
+�
+m
+� ∞
+0
+dt′ Tr
+�
+Bm(−t′)BlρB
+�
+S†
+m(−t′).
+(9)
+Since the actual microscopic evolution is given by a
+CPTP map [see Eqs. (3), (4)], one might naively expect
+that the evolution obtained from the microscopically de-
+rived RE respects complete positivity. However, as we
+prove in the next subsection in generality, unless in ex-
+tremely special cases, the RE violates complete positivity.
+A.
+Violation of complete positivity in Redfield
+equation
+1.
+Choosing an operator Basis
+As shown in Fig. 1, we consider only a part of the sys-
+tem is coupled to the bath.
+Let us denote HL as the
+Hilbert space of that part of the system that couples to
+the bath, and let HM be the Hilbert space of the re-
+maining part of the system. In mathematical terms, this
+means that any operator OM in the Hilbert space HM
+commutes with the system-bath coupling Hamiltonian
+HSB,
+[OM, HSB] = 0.
+(10)
+The system Hamiltonian can then be written as
+HS = HL + HLM + HM,
+(11)
+where the Hamiltonian HL is in Hilbert space HL, the
+Hamiltonian HM is in Hilbert space HM, and HLM gives
+the coupling between the two Hilbert spaces. Note that
+we do not consider this coupling to be small.
+Let the dimension of HL and HM be dL and dM re-
+spectively. Then, one can pick an orthonormal basis of
+operators {fi} and {gj} on HL and HM respectively,
+where 1 ≤ i ≤ d2
+L and 1 ≤ j ≤ d2
+M, and where or-
+thonormality is defined according to the Hilbert Schmidt
+inner product given by ⟨A, B⟩ = Tr[A†B]. One can al-
+ways choose this basis such that fd2
+L = IL/√dL and
+gd2
+M = IM/√dM, where IM and IL are the identity op-
+erators on those spaces. Such a basis is required by the
+GKSL theorem [6, 7, 16, 18]. Taking the tensor prod-
+uct of these two basis, one can obtain an orthonormal
+basis {Fk} = {fi} ⊗ {gj} for operators on HS, with
+Fd2
+Ld2
+M = IS/
+√
+d, where d = dLdM is the dimension of
+the system Hilbert space. Without loss of generality, the
+Lindblad equation [Eq. (1)] written in this basis is given
+by
+∂ρ
+∂t = i[ρ, HS + HLS] +
+d2−1
+�
+α,˜α=1
+Γα˜α
+�
+F˜αρF †
+α − {F †
+αF˜α, ρ}
+2
+�
+,
+(12)
+where complete positivity of ρ is preserved iff Γ is positive
+semidefinite [6, 7]. Eq. (12) can be turned into Eq. (1)
+by diagonalizing the matrix Γ.
+The complete positivity of RE can be checked by tak-
+ing the RE to the same form as Eq.(12) and checking if
+the corresponding Γ is positive semidefinite. To do so,
+
+4
+let us relabel the indices so that Fi = fi ⊗ IM/√dM for
+1 ≤ i ≤ d2
+L − 1. This allows us to expand the system
+operators in Eq. (8) as,
+Sl =
+d2
+�
+α=1
+alαFα,
+S†
+l =
+d2
+�
+α=1
+a′
+lαFα,
+S(1)
+l
+=
+d2
+�
+α=1
+blαFα,
+S(1)†
+l
+=
+d2
+�
+α=1
+b′
+lαFα,
+S(2)
+l
+=
+d2
+�
+α=1
+clαFα,
+S(2)†
+l
+=
+d2
+�
+α=1
+c′
+lαFα,
+(13)
+where alα = a′
+lα = 0, ∀
+d2
+L ≤ α ≤ d2 −1 since Sl and S†
+l
+are identity on HM. Substituting Eq. (13) into Eq. (8),
+we obtain
+∂ρ
+∂t = i[ρ, HS] − ϵ2 �
+l
+d2
+�
+α,˜α=1
+�
+a��
+lαbl˜α[F †
+α, F˜αρ]
++ c′∗
+lαa′
+l˜α[ρF †
+α, F˜α] + b∗
+lαal˜α[ρF †
+α, F˜α] + a′∗
+lαc′
+l˜α[F †
+α, F˜αρ]
+�
+.
+(14)
+Using some straightforward algebra (see Appendix A),
+Eq. (14) can be simplified to
+∂ρ
+∂t = i[ρ, HS + HLS] +
+d2−1
+�
+α,˜α=1
+Γα˜α
+�
+F˜αρF †
+α − {F †
+αF˜α, ρ}
+2
+�
+,
+(15)
+where Γα˜α is a (d2 −1)×(d2 −1) hermitian matrix given
+by
+Γα˜α = ϵ2 �
+l
+(a∗
+lαbl˜α + c′∗
+lαa′
+l˜α + b∗
+lαal˜α + a′∗
+lαc′
+l˜α),
+(16)
+where x∗ in Eq. (16) denotes the complex conjugate of x,
+and the expression of HLS that appears in Eq. (15) can
+be found in Appendix A. Since Eq. (16) is of the form in
+Eq. (12), the condition for preserving complete positivity
+of ρ is equivalent to Γ being positive semidefinite.
+2.
+The dissipator
+Let us now look at Γα˜α from Eq. (16). Recall from
+Sec. II A 1, that if α ≥ d2
+L, then alα = a′
+lα = 0. Therefore,
+Γα˜α = 0 when α, ˜α ≥ d2
+L, and has the following structure:
+Γ =
+�
+Γα,˜α<d2
+L
+Γα<d2
+L,˜α≥d2
+L
+Γα≥d2
+L,˜α<d2
+L
+0
+�
+.
+(17)
+In general, the off-diagonal blocks Γα<d2
+L,˜α≥d2
+L, and
+Γα≥d2
+L,˜α<d2
+L will not be identically zero.
+The RE will
+preserve complete positivity if and only if the matrix
+Γ is positive semidefinite. To move forward we will re-
+quire the following Lemma concerning positive semidefi-
+nite matrices.
+Lemma 1 Let M be any positive semidefinite matrix,
+such that Mjj = 0. Then, ∀i it must be the case that
+Mij = Mji = 0. Thus, if a positive semidefinite matrix
+has a zero as its jth diagonal element, then the entire jth
+row and jth column must consist of zeros.
+Proof : For a matrix M to be positive semidefinite, it
+is necessary that every 2 × 2 matrix (M ′) of the form
+M ′ =
+� Mii Mij
+Mji Mjj
+�
+(18)
+is positive semidefinite. Otherwise there would exist a
+vector |v′⟩ such that ⟨v′| M ′ |v′⟩ < 0. Then, there would
+exist a vector |v⟩ which is non-zero only on the ith and
+jth entries, such that ⟨v| M |v⟩ = ⟨v′| M ′ |v′⟩ < 0.
+If
+Mjj = 0, the eigenvalues of M ′ are given by solving the
+characteristic equation: λ2 − Miiλ − MijMji = 0. Since
+M is hermitian, for both eigenvalues to be non-negative,
+it must be the case that Mij = Mji = 0.
+Applying Lemma 1 to Eq. (17), we find that Γ can
+be positive semidefinite only if the off-diagonal blocks
+Γα<d2
+L,˜α≥d2
+L and Γα≥d2
+L,˜α<d2
+L are zero. Since this is not
+generically true for RE, we have shown that the RE equa-
+tion for any generic setup where the bath only couples
+to a part of the system violates complete positivity. An
+important exception here are situations where the RE
+happens to be such that the off-diagonal blocks in Γ are
+identically zero. In such cases, the RE dissipator consists
+of operators of the form fi ⊗ IM/√dM, and acts only
+the part of the system connected to the bath. However,
+such situations can be expected to arise in only some ex-
+tremely special cases. Even in such situations, RE may
+or may not preserve complete positivity, depending on
+Γα<d2
+L,˜α<d2
+L. Thus we have rigorously shown that for se-
+tups where the bath couples to only a part of the system,
+the RE will violate complete positivity unless its dissipa-
+tor is “local” and acts only on the part of the system
+that is connected to the bath. We also see from Eq. (16),
+that both the positive and negative eigenvalues of the Γ
+matrix can be the same order in system-bath coupling,
+i.e O(ϵ2).
+We stress that although it is known that RE does not
+conserve complete positivity, most previous works show
+this via specific examples (for instance, [24, 25, 30, 32]).
+To the best of our knowledge, we are not aware of a
+proof of violation of complete positivity via the explicit,
+model independent construction as shown above. In Ap-
+pendix B, we give a concrete example of a two-qubit XXZ
+system with the first qubit connected to the bath. We
+indeed find that the matrix Γ for this specific example
+has the expected structure from Eq. (17), and negative
+eigenvalues.
+A natural question that arises, is how one could recover
+complete positivity of RE via a suitable approximation to
+the RE. In order to recover complete positivity, one must
+change Γα˜α to some �Γα˜α that is positive semidefinite. By
+the above discussion, for any ˜α ≥ d2
+L, this will require
+either (i) making �Γαα > 0, ∀α > d2
+L or (ii) making �Γα˜α =
+
+5
+0 ∀α < d2
+L. We will see in the next section (Sec. III) that
+any equation with (i), will violate local conservation laws.
+III.
+LINDBLAD DESCRIPTIONS OBEYING
+LOCAL CONSERVATION LAWS
+A.
+Local conservation laws
+Let us make precise what we mean by preservation of
+local conservation laws. For our setup, since the bath
+only acts on HL part of the system, any operator on
+HM commutes with the system-bath coupling Hamilto-
+nian HSB (see Eq. (10), Fig. 1). So, writing down the
+Heisenberg equation of motion with respect to the full
+Hamiltonian H [Eq. (2)], and using Eq. (10), we see that
+the dynamical equation for the expectation value of any
+observable OM on HM is given by
+d
+dt ⟨IL ⊗ OM⟩ = −i ⟨[IL ⊗ OM, HS]⟩
+(19)
+where ⟨X⟩ = Tr[Xρ]. Any effective QME obtained by
+integrating out the bath should satisfy this property. We
+call QMEs satisfying this property as ones preserving lo-
+cal conservation laws. The justification for this name be-
+comes clear if we look at an operator in HM that would
+remain conserved if there is coupling with HL, i.e, if HLM
+in Eq.(11) is zero. One such operator is the Hamiltonian
+HM. The dynamical equation for expectation value of
+HM gives the energy continuity equation
+d
+dt ⟨HM⟩ = JL→M,
+JL→M = −i ⟨[HM, HLM]⟩ .
+(20)
+Here, JL→M can be interpreted as the energy current
+from the region L to the region M (see Fig. 1). In steady
+state, the rate of change of any system operator is zero.
+From above equation, this gives, JL→M = 0 in steady
+state. Thus, steady state energy current inside the sys-
+tem is zero. This is a statement of local conservation of
+energy and is one of the fundamental physical require-
+ments for a system coupled to a single bath that follows
+from the more general requirement Eq. (19).
+Importantly, the RE, i.e, Eq. (5), can be shown to sat-
+isfy Eq. (19) [23] and thereby preserves local conservation
+laws. We can write any QME to leading order in system
+bath coupling as
+∂ρ
+∂t = L0(ρ) + ϵ2L2(ρ),
+(21)
+where L0(ρ) = i[ρ, HS] and L2(ρ) contains both the dis-
+sipator and the Lamb-shift Hamiltonian. Computing the
+left hand size of Eq. (19) using Eq. (21), and comparing
+with the right hand size of Eq. (19), we obtain [23],
+Tr[(IM ⊗ OM)L2(ρ)] = 0.
+(22)
+This is a necessary condition for satisfying local conser-
+vation laws. If we now further restrict the QME to be of
+Lindblad form, i.e, of the form Eq. (1), thereby respecting
+complete positivity, we obtain the following theorem.
+Theorem 2 Any QME of Lindblad form Eq. (1) (thereby
+satisfying complete positivity) that also satisfies local con-
+servation laws must have the Lindblad operators and the
+Lamb-shift Hamiltonian acting only on the part of the
+system connected to the bath. That is, Lλ = LL
+λ ⊗ IM,
+HLS = HL
+LS ⊗IM, where LL
+λ, HL
+LS act only on the Hilbert
+space HL which is coupled to the bath by system-bath cou-
+pling Hamiltonian HSB, and IM is the identity on the
+remaining of the system Hilbert space HM.
+In the next subsection, we give the proof of this theorem.
+B.
+Proof of theorem 2
+We start by writing the most general Lindblad equa-
+tion in the basis of Sec. II A 1,
+∂ρ
+∂t = i[ρ, HS + �HLS] +
+d2−1
+�
+α,˜α=1
+�Γα˜α
+�
+F˜αρF †
+α − {F †
+αF˜α, ρ}
+2
+�
+(23)
+where �HLS is some Lamb shift Hamiltonian, and �Γ is a
+positive semidefinite matrix. As mentioned before, this
+form can be reduced to the form of Eq.(1) by transform-
+ing to a basis where matrix �Γ is diagonal. So, it suffices to
+work with this form. Since Fi = fi ⊗ IM for 1 ≤ i < d2
+L,
+the condition for the Lindblad operators to act only on
+HL then translates to the matrix �Γ being of the form
+�Γ =
+� �Γα,˜α<d2
+L 0
+0
+0
+�
+.
+(24)
+In the following we prove that in order to preserve local
+conservation laws the matrix �Γ must be of this form.
+1.
+The restriction on �Γ
+Writing down the evolution of any observable IL ⊗OM
+for Eq. (23), we have
+d ⟨IL ⊗ OM⟩
+dt
+= −i ⟨[IL ⊗ OM, HS]⟩ − i ⟨[IL ⊗ OM, �HLS]⟩
++
+d2
+L−1
+�
+α=1
+d2−1
+�
+˜α=d2
+L
+�Γα˜α
+2
+⟨[IL ⊗ OM, F †
+αF˜α]⟩
++
+d2−1
+�
+α=d2
+L
+d2
+L−1
+�
+˜α=1
+�Γα˜α
+2
+⟨[F †
+αF˜α, IL ⊗ OM]⟩
++
+d2−1
+�
+α,˜α=d2
+L
+�
+�Γα˜α
+�
+⟨F †
+α(IL ⊗ OM)F˜α⟩ − 1
+2 ⟨(IL ⊗ OM)F †
+αF˜α⟩
+− 1
+2 ⟨F †
+αF˜α(IL ⊗ OM)⟩
+�
+(25)
+
+6
+where ⟨X⟩ denotes the expectation value of X given by
+Tr[Xρ], and we use the fact that [Fi, IL ⊗ OM] = 0 when
+i < d2
+L. We can combine all commutators in above equa-
+tion into a single commutator with an effective Lamb-
+shift like Hamiltonian, which we denote �H(2)
+LS. This gives
+d ⟨IL ⊗ OM⟩
+dt
+= −i ⟨[IL ⊗ OM, HS + �H(2)
+LS]⟩
++
+d2−1
+�
+α,˜α=d2
+L
+�Γα˜α
+�
+⟨F †
+α(IL ⊗ OM)F˜α⟩ − 1
+2 ⟨(IL ⊗ OM)F †
+αF˜α⟩
+− 1
+2 ⟨F †
+αF˜α(IL ⊗ OM)⟩
+�
+(26)
+where �H(2)
+LS is some hermitian operator. Comparing with
+Eq. (19), and noting that Tr[Mρ] = 0
+∀ρ implies M =
+0, we obtain the condition for satisfying Eq. (19) as
+− i[IL ⊗ OM, �H(2)
+LS] +
+d2−1
+�
+α,˜α=d2
+L
+�Γα˜α
+�
+F †
+α(IL ⊗ OM)F˜α
+− 1
+2(IL ⊗ OM)F †
+αF˜α − 1
+2F †
+αF˜α(IL ⊗ OM)
+�
+= 0
+∀OM.
+(27)
+We will now relabel the indices α, ˜α for the sake of con-
+venience.
+Recall from Sec. II A 1, that α, ˜α which ap-
+pear in the above expression can be equivalently writ-
+ten as α → (αL, αM), ˜α → (˜αL, ˜αM), and vice-versa,
+where Fα = fαL ⊗ gαM . Therefore, �d2−1
+α=1 is equivalent
+to �d2
+M−1
+αM=1
+�d2
+L
+αL=1. We can also expand �H(2)
+LS as
+�H(2)
+LS =
+d2
+�
+α=1
+ναFα =
+d2
+L
+�
+αL=1
+d2
+M
+�
+αM=1
+ναL,αM (fαL ⊗gαM ). (28)
+This basis aids in taking a partial trace over the L part of
+the system. Performing this partial trace and using the
+orthonormality of fi operators, we can rewrite Eq. (27)
+as
+− i
+d2
+M
+�
+αM=1
+νd2
+L,αM [OM, gαM ] +
+d2
+M−1
+�
+αM,˜αM=1
+�ΛαM,˜αM
+�
+g†
+αM OMg˜αM − 1
+2OMg†
+αM g˜αM − 1
+2g†
+αM g˜αM OM
+�
+= 0,
+∀OM.
+(29)
+where
+�ΛαM,˜αM =
+d2
+L
+�
+αL,˜αL=1
+δαL,˜αL�Γ(αL,αM),(˜αL,˜αM)
+(30)
+We show in Appendix C that Eq. (29) implies
+d2
+M−1
+�
+αM=1
+�ΛαM,αM =
+d2
+M−1
+�
+αM=1
+d2
+L
+�
+αL=1
+�Γ(αL,αM),(αL,αM) = 0 (31)
+Since we require �Γ to be positive semidefinite, it cannot
+have negative values on the diagonal. Therefore, Eq. (31)
+implies �Γ(αL,αM),(αL,αM) = 0 for 1 ≤ αL ≤ d2
+L and 1 ≤
+αM ≤ d2
+M − 1 .
+Equivalently, �Γαα = 0 for α ≥ d2
+L.
+Applying Lemma 1 to this case, we see that �Γα,˜α can be
+non-zero only when both α, ˜α < d2
+L, and therefore �Γ is a
+matrix of the form in Eq. (24). This concludes the first
+part of the proof.
+2.
+The restriction on �
+HLS
+Given this structure for �Γ from Eq. (24), we now in-
+vestigate the restrictions on the Lamb shift Hamilto-
+nian �HLS. Since �Γ has to obey Eq. (24), we find that
+�HLS = �H(2)
+LS = �
+αL,αM ναL,αM (fαL ⊗ gαM ) in Eq. (27).
+Then, our condition for satisfying local conservation laws
+from Eq. (27) is given by,
+−i
+d2
+L
+�
+αL=1
+d2
+M
+�
+αM=1
+ναL,αM [IL ⊗OM, (fαL ⊗gαM )] = 0
+∀OM.
+(32)
+which further implies
+− i
+d2
+L
+�
+αL=1
+d2
+M
+�
+αM=1
+ναL,αM fαL ⊗ [OM, gαM ] = 0
+∀OM. (33)
+Multiplying both sides by f †
+αL ⊗ IM, and tracing out the
+L part of the system, we obtain
+− i
+d2
+M
+�
+αM=1
+ναL,αM [OM, gαM ] = 0
+∀αL, OM.
+(34)
+This can happen only if
+d2
+M
+�
+αM=1
+ναL,αM gαM ∝ IM
+∀αL
+(35)
+Using Eq. (35) in Eq. (28), and recalling that �HLS =
+�H(2)
+LS, we obtain �HLS = �H(L)
+LS ⊗ IM. This concludes the
+second part of the proof.
+C.
+Remarks on Theorem 2
+Theorem 2 says that for a QME to preserve complete
+positivity and obey local conservation laws, both the
+Lamb shift Hamitonian and the dissipator must only act
+on the part of the system connected to the bath. Such a
+Lindblad equation is often termed a ‘local Lindblad equa-
+tion’. Theorem 2 thus says that only local Lindblad equa-
+tions are consistent with local conservation laws. The RE
+preserves local conservation laws without having a local
+dissipator, but it does so at the cost of losing complete
+
+7
+positivity.
+Any global form of Lindblad equation, like
+the eigenbasis Lindblad equation [6, 8] and the universal
+Lindblad Equation [31], violates local conservation laws,
+while preserving complete positivity. One main reason
+such global forms of Lindblad equations are often derived
+under various approximations is that they can be proven
+to show thermalization. However, general statements re-
+lated to thermalization in local Lindblad equations are
+usually difficult to make. In the next section, we discuss
+a numerical technique which allows to study whether, in
+a given set-up, a QME consistent with Theorem 2 is pos-
+sible such that it also shows thermalization.
+IV.
+ON THE POSSIBILITY OF
+THERMALIZATION WITH LOCAL
+DISSIPATORS
+A.
+Condition for satisfying thermalization
+We start by making precise what we mean by thermal-
+ization. Going back to the form of the QME in Eq. (21),
+the setup is said to show thermalization if
+lim
+ϵ→0
+�
+lim
+t→∞ et(L0+ϵ2L2)ρ(0)
+�
+= ρth,
+ρth =
+e−βHS
+Tr[e−βHS],
+(36)
+irrespective of the initial state of the system ρ(0). Phys-
+ically, it means that if the system is weakly coupled to
+a thermal bath for a long time, and then the system-
+bath coupling is slowly switched-off, the state of the sys-
+tem will be the Gibbs state at the temperature of the
+bath, irrespective of the system’s initial state. If there
+is no explicit time-dependence in the Hamiltonian, as in
+Eq. (2), this statement can be proven starting with the
+initial state of the full set-up in Eq. (3), and assuming
+that the steady state is unique [23]. Given a QME of the
+form in Eq.(21), in Ref. [23], we showed that the follow-
+ing condition needs to be satisfied in order to guarantee
+thermalization,
+⟨Ei| L2(ρth) |Ei⟩ = 0
+∀ i.
+(37)
+where |Ei⟩ is the eigenvector of the system Hamiltonian
+HS with eigenvalue Ei.
+The derivation of this condi-
+tion can also be found in Appendix D. Given a system
+Hamiltonian HS, the inverse temperature β of the bath
+and a partition of the system Hilbert space into the part
+HL which is attached to a bath and the remainder of
+Hilbert space HM (see Fig.1), we would like to find a
+QME respecting the restrictions in theorem 2 and satis-
+fying Eq.(37). As we will see later by example, such a
+QME is not guaranteed to be possible. In the next sub-
+section, we provide a numerical way to conclusively check
+if such a QME is possible in a given setup.
+B.
+The thermalization optimization problem
+For the setup shown in Fig. 1, the most general form
+of QME respecting the restrictions in theorem 2 for sat-
+isfying complete positivity and local conservation laws
+is
+∂ρ
+∂t = i[ρ, HS + ϵ2H(L)
+LS ⊗ IM] + ϵ2
+d2
+L−1
+�
+αL,˜αL=1
+Γ(L)
+αL,˜αL
+�
+(f˜αL ⊗ IM)ρ(fαL ⊗ IM)†
+− {(fαL ⊗ IM)†(f˜αL ⊗ IM), ρ}
+2
+�
+.
+(38)
+Here H(L)
+LS is a Lamb shift Hamiltonian that acts on the
+HL part of the system, Γ(L) is (d2
+L − 1) ⊗ (d2
+L − 1) matrix
+that must be positive semidefinite, dL is the dimension
+of the Hilbert space HL that is directly coupled with the
+bath. We include the factor of ϵ2 in front of Γ(L) and
+H(L)
+LS explicitly.
+The system Hamiltonian HS and the
+Hilbert space HL are assumed to be given. The task is
+then to find H(L)
+LS and Γ(L), such that Eq.(37) is satisfied
+to a given precision, for a given inverse temperature β.
+To this end, we introduce the quantity
+τ =
+�
+i
+|⟨Ei| L2(ρth) |Ei⟩| ,
+(39)
+where L2 consist of all terms in Eq. (38) which are mul-
+tiplied by ϵ2, i.e all terms except the commutator with
+the system Hamiltonian. Then, we can cast the task in
+terms of an optimization problem given by :
+minimize :
+τ by varying H(L)
+LS , Γ(L),
+subject to :
+H(L)
+LS is hermitian,
+Tr(Γ(L)) = 1,
+Γ(L) ≥ 0,
+(40)
+where we use Γ(L) ≥ 0 to denote Γ(L) being positive
+semidefinite. The condition Tr(Γ(L)) = 1 is imposed to
+avoid the trivial solution H(L)
+LS , Γ(L) = 0, which trivially
+gives the global minimum τ = 0. Since we want Γ(L)
+to be a non-zero positive semi-definite matrix, it must
+have a positive trace. We have fixed that trace to one
+in some arbitrarily chosen energy unit.
+This does not
+cause loss of generality since the strength of system-bath
+coupling is explicitly governed by ϵ2 in Eq.(38). We call
+the optimization problem in Eq. (40) the “thermalization
+optimization problem”(TOP). Let τopt be the optimal
+value obtained from solving TOP. Given a tolerance δ,
+if τopt < δ,
+the desired QME is possible,
+else,
+the desired QME is impossible,
+(41)
+up to the precision δ.
+Most interestingly, we find that the TOP in Eq. (40)
+can be written as a SDP. The background and theoreti-
+cal framework of SDP is discussed in Appendix E 1. The
+
+8
+TOP is proven to be a SDP in Appendix E 2. In particu-
+lar, any choice of H(L)
+LS and Γ(L) in Eq. (40) can be used
+to obtain an upper bound on τopt. Then, the theoreti-
+cal framework of SDP can be used to construct a “dual”
+problem to the optimization problem in Eq. (40). This
+dual problem can then be used to obtain a lower bound on
+τopt. If the lower bound and upper bound match, then
+this guarantees that one has found the global optimal
+value of τ . Our above described approach is transpar-
+ent, and simple to use, since the optimization problem in
+Eq. (40) can be directly put into standard packages for
+disciplined convex optimization like the CVX MATLAB
+package [33].
+In particular, CVX itself automatically
+constructs the dual problem, outputs τopt, and gives one
+choice of H(L)
+LS , and Γ(L) which yields the output value of
+τopt. Thus, if τopt < δ, it not only says that the desired
+type of QME is possible but also it outputs one possible
+candidate for such a QME. If τopt ≥ δ, the desired type
+of QME is impossible.
+We would like to point out here that, in a microscopic
+derivation, given the temperature of the baths, H(L)
+LS and
+Γ(L) would depend only on the bath spectral functions
+and system-bath coupling Hamiltonian.
+So, the TOP
+can be thought of as varying over all possible bath spec-
+tral functions and system-bath coupling Hamiltonians to
+find τopt. Thus, if τopt ≥ δ, we can conclusively say that,
+for the chosen system parameters and temperature, un-
+der no choice of bath spectral function and system-bath
+coupling Hamiltonian, can a Markovian QME be derived
+which simultaneously satisfies complete positivity, local
+conservation laws and shows thermalization up to the
+chosen precision. In the next subsection, we look at the
+TOP in an open XXZ qubit chain.
+C.
+Open XXZ qubit chain as an example
+We study the possibility of having a Lindblad descrip-
+tion satisfying local conservation laws and showing ther-
+malization in an open XXZ qubit chain system with some
+of the qubits attached to baths. The system Hamiltonian
+for this setup is given by
+HS =
+N
+�
+ℓ=1
+ω(ℓ)
+0
+2 σ(ℓ)
+z
+−
+N−1
+�
+ℓ=1
+gℓ
+�
+σ(ℓ)
+x σ(ℓ+1)
+x
++ σ(ℓ)
+y σ(ℓ+1)
+y
++ ∆ℓσ(ℓ)
+z σ(ℓ+1)
+z
+�
+,
+(42)
+where σ(ℓ)
+x,y,z denotes the Pauli matrices acting on the
+ℓth qubit, and ω(ℓ)
+0 , gℓ, and gℓ∆ℓ represent the mag-
+netic field, the overall qubit-qubit coupling strength and
+the anisotropy respectively. The first NL qubits are at-
+tached to a bath, while the remaining NM = N − NL
+qubits are not attached to any bath.
+We use the for-
+malism of Sec. IV B to investigate thermalization in this
+set-up for various values of NL and NM.
+In order to
+do so, we first need to construct the basis for opera-
+FIG. 2. τopt vs g, for NL = 1, with ω(ℓ)
+0
+= 1, ∆ℓ = 1, β = 1
+and gℓ = g for all ℓ. The tolerance chosen is δ = 10−6. We find
+that τopt ≫ δ, conclusively showing that, for such setups, no
+QME can simultaneously preserve complete positivity, obey
+local conservation laws, and show thermalization up to the
+precision set by the tolerance.
+FIG. 3.
+τopt vs β, for NL = 1, with ω(ℓ)
+0
+= 1, ∆ℓ = 1,
+gℓ = 0.1 for all ℓ. The tolerance chosen is δ = 10−6, and is
+plotted as the dashed horizontal line. We find that τopt ≫ δ,
+indicating that, for such setups, it is not possible to have a
+QME simultaneously preserving complete positivity, obeying
+local conservation laws, and showing thermalization up to the
+precision set by the tolerance.
+tors in Hilbert space of the first NL qubits, so that the
+most general form of the desired QME can be written
+as in Eq. (38).
+For the ℓth qubit, we choose the ba-
+sis {−σ(ℓ)
+z /
+√
+2, σ(ℓ)
++ , σ(ℓ)
+− , I(ℓ)
+2 /
+√
+2}, where σ(ℓ)
++
+= (σ(ℓ)
+x
++
+iσ(ℓ)
+y )/2, σ(ℓ)
+− = (σ(ℓ)
+x
+− iσ(ℓ)
+y )/2, and I(ℓ)
+2
+is the identity
+operator for the qubit Hilbert space. The basis for the
+first NL qubits is obtained by direct product of the basis
+of each of the qubits. We construct the TOP [Eq.(40)]
+in this basis, which we then directly input in the CVX
+MATLAB package to obtain τopt. We set an ad-hoc value
+of the tolerance δ = 10−6 [see Eq.(41)]. In a typical cal-
+culation from a weak system-bath coupling QME, usually
+the error due to neglecting higher order terms would be
+larger than such a low value of tolerance.
+
+-NM=1
+θNM = 2
+×NM=3
++NM=4
+10
+10~3
+100
+10-1
+10~2
+910
+*NM=1
+θNM=2
+10-5
+×NM =3
++NM= 4
+100
+10-1
+101
+39
+1.
+Single qubit attached to bath
+First we consider the case where the first qubit of the
+chain is coupled to a bath, so, NL = 1. In Fig. (2), we
+plot τopt vs g for NM = 1, 2, 3, fixing ω(ℓ)
+0
+= 1, ∆ℓ = 1 for
+all ℓ and β = 1. We find that τopt ≫ δ when NL = 1, and
+NM = 1, 2, 3. Thus, no Markovian QME can simultane-
+ously preserve complete positivity, obey local conserva-
+tion laws, and satisfy thermalization for such setups in
+the chosen range of parameters. This explicit example
+also directly rules out the possibility of having a general
+form of Markovian QME that is guaranteed to meet all
+the fundamental requirements. We see from Fig. (2) that
+τopt increases with g. This is consistent with previous re-
+sults showing that such local Lindblad equations become
+a good description when the coupling between the sys-
+tem qubits are weak.
+Additionally, we see that for a
+given value of g, τopt decreases as NM increases. This,
+interestingly, seems to indicate that for long chains, local
+Lindblad with dissipator acting on only one qubit might
+be able to describe the thermal state up to a reason-
+ably good precision. However, more detailed studies are
+required to make any conclusive statement in this direc-
+tion.
+Intuitively, memory effects, and hence Markovianity of
+open system dynamics, depend on temperature. So, we
+might expect that at a different temperature, τopt might
+be smaller than δ.
+In Fig. (3), we plot τopt vs β for
+NL = 1, fixing ω(ℓ)
+0
+= 1, ∆ℓ = 1, gℓ = 0.1 for all ℓ. We
+find that τopt ≫ δ, for almost the entire range of β chosen,
+showing that for these parameters, no Markovian QME
+can simultaneously preserve complete positivity, obey lo-
+cal conservation laws, and satisfy thermalization. But,
+we see some interesting features. Firstly, as before, we
+see that τopt decreases as NM increases. Secondly, we see
+that τopt varies non-monotonically with β. At very low
+temperatures, τopt decreases tends to decay below δ. In
+Fig. (3), for NM = 4 and β = 10, τopt < δ. At high
+temperatures also, τopt decreases. This suggests that at
+very low and very high temperatures, it is possible to
+obtain a local Lindblad equation that shows thermaliza-
+tion. That this is indeed true can be checked indepen-
+dently. At such extremes of temperatures, for at least
+some choices of baths and system-bath couplings, local
+Linblad equations can be microscopically derived [34, 35].
+This explains the non-monotonic dependence of τopt on
+β. Since very high and very low temperatures can allow
+for a local Lindblad description, it is then intuitive that
+departure from such behavior is maximum when β is of
+the order of system time scales. Indeed it is close to such
+values of β, i.e, β ∼ ω(ℓ)
+0
+= 1, that the highest value of
+τopt is seen in Fig. (3).
+We would like to point out here that, τopt ≥ δ does not
+mean the system coupled to a thermal bath is unable
+to thermalize for such parameters. Unless in extremely
+special cases, it is almost always possible to find a bath
+spectral function and system-bath coupling which ensure
+FIG. 4. τopt vs g, for NL = 2, with ω(ℓ)
+0
+= 1, ∆ℓ = 1, gℓ = g
+for all ℓ and β = 1. The tolerance chosen is δ = 10−6, and is
+plotted as the dashed horizontal line. We find that τopt ≪ δ
+for smaller values of g, indicating that, for such setups, it is
+possible to have a QME simultaneously preserving complete
+positivity, obeying local conservation laws, and showing ther-
+malization up to the precision set by the tolerance.
+FIG. 5. τopt vs β, for NL = 2, with ω(ℓ)
+0
+= 1, ∆ℓ = 1, gℓ = 0.1
+for all ℓ. The tolerance chosen is δ = 10−6, and is plotted as
+the dashed horizontal line. We find that τopt ≪ δ for smaller
+values of g, indicating that, for such setups, it is possible to
+have a QME simultaneously preserving complete positivity,
+obeying local conservation laws, and showing thermalization
+up to the precision set by the tolerance. The values in the
+figure less than 10−12 are below the numerical precision of
+CVX Matlab package.
+thermalization. So τopt ≥ δ instead means that, for those
+parameters, the dynamics of approach to thermal state
+cannot be governed by a completely positive Markovian
+QME preserving local conservation laws. The dynamics
+then must have some non-Markovian character. In fact,
+as we see from above example, τopt gives an estimate of
+how close to Markovianity the open system dynamics can
+be for the chosen parameters.
+Next, we discuss the case where first two qubits are
+attached to the bath and highlight the drastic difference
+observed for similar choice of parameters.
+
+*NM
+二
+θNM = 2
+×NM=3
++NM=4
+NM=5
+10-10
+Topt
+10-20
+10-1
+10~2
+100
+gNM=
+θNM=2
+&= WN*
+ΦNM=5
+Topt
+15
+10
+10-20
+101
+10-1
+100
+310
+0.00033
+0
+0
+-0.00123 0
+0 0.00033
+0
+0
+0.00019
+0 0
+0.00001
+0
+0
+0
+0.00196
+0
+0
+-0.00179 0 0
+-0.03638 0
+0
+0 0
+0
+-0.00002 0
+0
+0
+0.00081
+0
+0
+0 0
+0
+-0.00081 0
+0 0.01307
+0
+0
+-0.00021
+-0.00123 0
+0
+0.02611
+0
+0 -0.00282 0
+0
+0.00157
+0 0
+-0.00143 0
+0
+0
+-0.00179 0
+0
+0.00172
+0 0
+0.03491
+0
+0
+0 0
+0
+0.00011
+0
+0
+0
+0
+0
+0
+0 0
+0
+0
+0
+0 0
+0
+0
+0
+0.00033
+0
+0
+-0.00282 0
+0 0.00045
+0
+0
+0.00002
+0 0
+0.00011
+0
+0
+0
+-0.03638 0
+0
+0.03491
+0 0
+0.70685
+0
+0
+0 0
+0
+0.00209
+0
+0
+0
+-0.00081 0
+0
+0 0
+0
+0.00082
+0
+0 -0.01311 0
+0
+0.00021
+0.00019
+0
+0
+0.00157
+0
+0 0.00002
+0
+0
+0.00035
+0 0
+-0.00014 0
+0
+0
+0
+0
+0
+0
+0 0
+0
+0
+0
+0 0
+0
+0
+0
+0
+0
+0.01307
+0
+0
+0 0
+0
+-0.01311 0
+0 0.26032
+0
+0
+-0.00208
+0.00001
+0
+0
+-0.00143 0
+0 0.00011
+0
+0
+-0.00014 0 0
+0.00009
+0
+0
+0
+-0.00002 0
+0
+0.00011
+0 0
+0.00209
+0
+0
+0 0
+0
+0.0001
+0
+0
+0
+-0.00021 0
+0
+0 0
+0
+0.00021
+0
+0 -0.00208 0
+0
+0.00009
+TABLE I.
+The Γ(L) obtained from CVX for NL = 2, NM = 4, ω(ℓ)
+0
+= 1, gℓ = 0.1, ∆ℓ = 1, β = 1, i.e, the values used to
+compute Fig. 6. Every entry is rounded to 5 digits after the decimal point for convenience of representation. The corresponding
+H(L)
+LS is given to be zero. This Γ(L) satisfies complete positivity, local conservation laws and thermalization up to a precision of
+δ = 10−6 for the given choice of parameters.
+2.
+Two qubits attached to bath
+In Fig. (4), we plot τopt vs gℓ = g for NL = 2, i.e, first
+two qubits attached to the bath, and NM = 1, 2, 3, 4,
+taking fixed values of ω(ℓ)
+0
+= 1, ∆ℓ = 1 and β = 1. These
+parameters are the same as in Fig. 2 and g is varied over
+the same range. Quite remarkably, in stark contrast to
+Fig. 2, we find that in this case τopt ≪ δ over a consid-
+erable range for g < 1. We then look at the behavior
+of τopt versus β, fixing ω(ℓ)
+0
+= 1, ∆ℓ = 1, gℓ = g = 0.1.
+This is shown in Fig. 5. For NM > NL, we again see
+the non-monotonic behavior. However, in stark contrast
+to Fig. 3, we find that over the entire chosen range of β
+τopt ≪ δ. Thus, for NL = 2, over a considerable range of
+parameters, a QME that simultaneously preserves com-
+plete positivity, obeys local conservation laws, and satis-
+fies thermalization up to the given precision is possible.
+This is a highly non-trivial result.
+Previously, local two-qubit Lindblad dissipators have
+been used to study energy transport in XXZ-type qubit
+chains (for example, Refs. [36, 37]). However, those local
+two-qubit Lindblad operators were constructed so as to
+thermalize the two qubits only, in absence of coupling to
+the rest of the chain. Such Lindblad description is not
+guaranteed to thermalize the whole chain to the given
+inverse temperature β of the bath [38]. Our result here
+shows that it is possible to have a two-qubit local Lind-
+blad description that can thermalize the full chain to the
+given temperature of the bath to a good approximation.
+As mentioned before, CVX also outputs a possible
+choice of Γ(L) and H(L)
+LS matrices corresponding to τopt.
+So, when τopt < δ, we get one possible candidate for the
+desired type of QME. For our choice of parameters, we
+find that CVX always outputs H(L)
+LS = 0, and a non-
+trivial value of Γ(L) that would be hard to guess other-
+wise. In Table. I, we demonstrate the Γ(L) obtained for
+NL = 2, NM = 4, ω(ℓ)
+0
+= 1, gℓ = 0.1, ∆ℓ = 1, β = 1. The
+Γ(L) matrix corresponds to the basis of operators {Fk}
+chosen as
+Fk = f⌈k/4⌉ ⊗ fk(mod 4) ⊗ IM
+(43)
+where {fi} = {−σz/
+√
+2, σ−, σ+, I2/
+√
+2}, ⌈k/4⌉ denotes
+the nearest integer greater than or equal to k/4, and
+k(mod 4) denotes the value of k modulo 4, and k goes
+from 1 to 15. We also note that the exact values of Γ(L)
+and H(L)
+LS computed by CVX may depend on the exact
+configuration of the programming environment (such as
+internal solvers used by CVX).
+For every parameter of the system, there is a differ-
+ent τopt, with a corresponding value of Γ(L) and H(L)
+LS
+given by CVX. If we want to explore a large parameter
+space of the system, it seems that we need a different
+Γ(L) and H(L)
+LS for each parameter point. Surprisingly,
+we find that this is not always required. If τopt ≪ δ for
+one set of parameters, we can substantially change pa-
+rameters of the system far from the qubits attached to
+baths, and still obtain a value of τ ≪ δ with the same
+value of Γ(L) and H(L)
+LS . This is shown in Fig. (6), where
+τ is calculated changing various parameters away from
+the two qubits coupled with the bath, fixing H(L)
+LS = 0
+and Γ(L) to be the same as in Table. I. Over the entire
+regime of chosen parameters τ ≪ δ. Note that, in con-
+trast to previous plots, this is not be the optimal value of
+τ. Nevertheless, if τ ≪ δ, we still get a completely posi-
+tive Markovian QME preserving local conservation laws
+and showing thermalization up to the chosen precision.
+Given Γ(L) and H(L)
+LS , it is much easier to just check this
+rather than finding the optimal value τopt.
+If parameters of the two qubits that are coupled to the
+bath are changed, we can no longer use the same Γ(L)
+and H(L)
+LS . For example, if we choose the same Γ(L) as
+
+11
+FIG. 6. τ vs g4, for NL = 2, NM = 4, with ω(ℓ)
+0
+= 1, ∆ℓ = 1,
+β = 1 and gℓ = 0.1 for all ℓ unless otherwise mentioned.
+τ is computed from Γ(L) and H(L)
+LS obtained from CVX for
+NL = 2, NM = 4, with ω(ℓ)
+0
+= 1, ∆ℓ = 1, β = 1 and gℓ = 0.1
+for all ℓ. The modified parameters for the plots are given by (i)
+(no parameters changed), (ii) ∆3 = 0.4, ∆4 = 1.2, (iii) ω(3)
+0
+=
+1.5, ω(4)
+0
+= 1.5, g5 = 0.3, (iv) ω(3)
+0
+= 1.5, ω(4)
+0
+= 1.5, g5 =
+0.3, ∆4 = 0.5, (v) g3 = 0.3. We find that τ ≪ δ = 10−6 even
+if parameters are changed for qubits of the system that are
+not coupled to the baths.
+in Fig. 6, and change g1 to 0.2 from 0.1, we get τ =
+0.0014 ≫ δ.
+The above observation suggests that the values of Γ(L)
+and H(L)
+LS obtained by CVX can be used to define a QME,
+independent of the parameters in the bulk of the sys-
+tem. This is consistent with underlying picture that each
+value of Γ(L) and H(L)
+LS corresponds to a different choice of
+the bath spectral function and the system-bath coupling
+Hamiltonian. If we change any parameter of the qubits
+attached to the baths, the change reflects substantially
+on the system-bath coupling Hamiltonian, so the value of
+τ changes drastically from τopt obtained with original pa-
+rameters. If we change any parameter away from the two
+qubits, the change reflects much less on the system-bath
+coupling Hamiltonian, causing τ to be of the same order
+as the original value of τopt. This presents an exciting
+prospect for studying the dynamics of the system-bath
+setup over a wide range of parameters, including a wide
+range of temperatures, with physically consistent Marko-
+vian QMEs. Such studies may also be possible for long
+chains, since local Markovian dissipation is favourable for
+tensor network based numerical techniques. Such dissi-
+pation may, also, in principle, be engineered in quantum
+computing and quantum simulation platforms, like ion
+traps [39, 40], Rydberg atoms [41, 42], superconducting
+qubits [43] and quantum dots [44].
+V.
+SUMMARY AND OUTLOOK
+Searching for a physically consistent Markvian QME
+— A physically consistent Markovian QME must satisfy
+complete positivity, obey local conservation laws and be
+able to show thermalization. In this work, we have sys-
+tematically gone about searching for such QMEs. This
+is done in three steps, and the result in each step has im-
+portant consequences. Especially, we are led to introduce
+the TOP problem, which is an optimization problem for
+finding a QME with all the above properties up to a given
+precision. The TOP opens a completely new avenue in
+the study of dissipative quantum systems.
+We consider a finite-dimensional undriven system a
+part of which is weakly coupled to a thermal bath. The
+microscopically derived QME written to leading order
+in system-bath coupling is the RE, which is known to
+obey local conservation laws and be able to show ther-
+malization [23].
+First, we show in generality that the
+RE violates complete positivity, unless in extremely spe-
+cial cases. Although there are previous works showing
+this via specific examples (for instance, [24, 25, 30, 32]),
+we are unaware of a model independent proof similar to
+ours. Next, we prove that imposing complete positivity
+and preservation of local conservation laws enforces the
+QME to be of ‘local’ form. That is, the Lindblad op-
+erators and the Lamb-shift Hamiltonian must have sup-
+port only on the part of the system directly coupled to
+the bath, and be identity elsewhere. This rules out the
+possibility of any ‘global’ forms of Lindblad equations,
+which are usually constructed to show thermalization,
+to be consistent with local conservation laws. Then, we
+ask if a ‘local’ Lindblad equation can be found which is
+able to show thermalization. We find that, the task of
+finding such a Lindblad equation can be cast as an op-
+timization problem, which we call TOP. Most interest-
+ingly, this optimization problem turns out to be a SDP.
+For given system and parameters, the SDP can be ef-
+ficiently solved using high-level programming packages
+like the CVX Matlab package. The output of the TOP
+conclusively shows whether the desired type of QME is
+possible for the chosen system parameters and tempera-
+ture, up to a chosen precision. For numerical example,
+we look at the TOP in a XXZ qubit chain of few sites,
+fixing a reasonably high precision. When only the first
+site is coupled to a bath, we find that, unless in extremes
+of temperatures, it is impossible to find a local Lindblad
+equation that is capable of showing thermalization up to
+the chosen precision.
+Discussion in light of various existing forms of QMEs
+— Various forms of QMEs have been derived it literature
+under various approximations (for example, [31, 35, 45–
+54]), along with the standard RE, local and eigenbasis
+Lindblad equations [6].
+Although the above example
+shows that there is no general form of physically consis-
+tent Markovian QME, this does not immediately make
+them unusable.
+Instead, it turns out that in each of
+these forms of QME, some elements of the system den-
+sity matrix are given correctly, while the others are not
+[23]. So, one needs to be careful in interpreting the re-
+sults from them, always keeping in mind their micro-
+scopic derivation and approximations. The RE, despite
+not being completely positive, is provably more accurate
+than all such Lindblad QMEs. To elucidate how this can
+
+X10-8
+2.5
+2.
+1.5
+(i)
+(ii)
+(ii)
+1
+(iv)
+中(v)
+0.5
+100
+9412
+happen, imagine that, in a given setup, physically, the
+population of one energy level, say, ⟨Ej| ρ |Ej⟩, is zero in
+steady state. The RE might then give a small negative
+value (say, ⟨Ej| ρ |Ej⟩ = −10−3), while any of the Lind-
+blad equations will give a larger positive value, which
+might be (say, ⟨Ej| ρ |Ej⟩ = 0.1). Either case is a prob-
+lem if we want to calculate various kinds of entropies,
+as often required in quantum information and thermo-
+dynamics. In case of RE, unphysical results can often
+be ruled out by checking the scaling with system-bath
+coupling [23, 55]. This is often more difficult in Lindblad
+QMEs, where approximations are often less controlled.
+The state obtained from the recently derived ULE [31],
+which been shown to violate local conservation laws [23],
+can be corrected to obtain results as accurate as the RE
+[48]. This re-instates the local conservation laws, at the
+cost of also re-instating the same positivity problem of
+the density matrix as in RE. In another recent work, a
+general form of QME has been derived [45] which is more
+accurate than RE, even though complete positivity of dy-
+namics is still not guaranteed.
+TOP and (non) Markovianity — In the microscopic
+picture, given the temperature of the bath, the QME is
+completely defined by the bath spectral functions and
+the type of system-bath coupling. The TOP can then be
+thought of as varying over all possible bath spectral func-
+tions and types of system-bath couplings to find the clos-
+est to satisfying thermalization the local Lindblad equa-
+tion can be. So, when TOP shows that the desired type
+of QME is impossible, it means no matter what type of
+bath is attached and how it is coupled to the system, for
+the chosen parameters, it is impossible to describe the
+dynamics via a completely positive Markovian QME sat-
+isfying local conservation laws and showing thermaliza-
+tion. The approach to thermal state must then have some
+non-Markovian character for such system parameters and
+temperature. The output of TOP, τopt, shows non-trivial
+dependence on the system parameters and the temper-
+ature. This dependence seems to capture how close to
+Markovian the dynamics can be for the chosen parame-
+ters.
+Surprises when two qubits are attached to bath — Sur-
+prisingly, we have found that, when first two qubits of
+the few-site XXZ chain are attached to a bath, solving
+the TOP shows that it is possible to find Lindbladians
+obeying local conservation laws and showing thermaliza-
+tion up to quite high precision. This holds over a con-
+siderable range of parameters, including a wide range of
+temperatures. Notably, in this entire parameter regime,
+when one qubit was coupled to a bath, such a QME was
+impossible.
+Whenever the TOP shows a QME respecting all condi-
+tions is possible, standard high-level programming pack-
+ages used to solve the SDP also outputs one possible form
+for such a QME. When two qubits are attached to the
+bath, the form of QME so obtained, which respects all the
+requirements, is quite non-trivial and would be hard to
+guess otherwise. Even more interestingly, we have found
+that if we take one such QME obtained for one choice of
+system parameters, and change some system parameters
+away from two qubits that couple to the bath, the QME
+still satisfies all the requirements. This opens several ex-
+citing possibilities that we describe below.
+Future directions — Our results open the exciting pos-
+sibility of studying the dynamics of approach to thermal
+state in open quantum many-body systems using phys-
+ically consistent Markovian QMEs, over a wide range
+of parameters, including a wide range of temperatures.
+This is particularly aided by the fact that local Lindblad
+equations are favourable for tensor network techniques.
+Studying such dynamics at finite temperatures is often
+quite challenging otherwise, requiring simulation of non-
+Markovian dynamics [56–58].
+The TOP lets us find parameters of the system where
+local Lindblad equations can show thermalization. For
+two qubits attached to bath, this range of parameters
+can be considerably large, as we have seen. It may be
+possible to design such local dissipation in quantum com-
+puting and quantum simulation platforms like ion traps
+[39, 40], Rydberg atoms [41, 42], superconducting qubits
+[43] and quantum dots [44]. Especially in ion traps and
+Rydberg atom platforms, this offers an interesting way to
+controllably prepare finite temperature states of complex
+quantum many-body systems in these platforms, which
+is presently a technological challenge. Usually, one would
+require global Lindblad dissipators to ensure that a ther-
+mal state is prepared. This would be hard to design in
+quantum simulation platforms if one wants to simulate
+complex many-body systems. The possibility of having
+local dissipation confined to two qubits offers a much eas-
+ier alternative.
+Moreover, as we have seen in the example of XXZ qubit
+chain, the dependence of the output of the TOP, τopt, on
+various parameters of the system already encode rich and
+interesting physics.
+For complex quantum many-body
+systems, one may need more scalable techniques for SDP,
+which is itself a direction of research in computer science
+[59].
+Using these techniques, the rich behavior of τopt
+with various parameters can then be studied.
+It is therefore clear that our results, especially the in-
+troduction of the TOP, leads to new paradigm within the
+fields of quantum information, computation and technol-
+ogy. Nevertheless, various questions still remain. One
+main question concerns steady-state coherences [55, 60–
+62]. When coupled to a thermal bath at any finite cou-
+pling, the system density matrix will have coherences in
+energy eigenbasis of the system [61, 62]. These coher-
+ences can be important in quantum information and ther-
+modynamics [63–66] and are given correctly to the lead-
+ing order by the RE [23, 55, 62]. However, it is not clear
+that the steady-state coherences calculated from physi-
+cally consistent Markovian QME obtained via TOP will
+be the same as those obtained from RE. Further inves-
+tigation is required in this respect, which will be carried
+out in future works.
+All code used in this work can be found at [67].
+
+13
+ACKNOWLEDGEMENTS
+MK would like to acknowledge support from the
+project 6004-1 of the Indo-French Centre for the Promo-
+tion of Advanced Research (IFCPAR), Ramanujan Fel-
+lowship (SB/S2/RJN-114/2016), SERB Early Career Re-
+search Award (ECR/2018/002085) and SERB Matrics
+Grant (MTR/2019/001101) from the Science and En-
+gineering Research Board (SERB), Department of Sci-
+ence and Technology, Government of India. AD and MK
+acknowledge support of the Department of Atomic En-
+ergy, Government of India, under Project No. RTI4001.
+AP acknowledges funding from the European Research
+Council (ERC) under the European Unions Horizon 2020
+research and innovation program (Grant Agreement No.
+758403). A.P also acknowledges funding from the Dan-
+ish National Research Foundation through the Center of
+Excellence “CCQ” (Grant agreement no.: DNRF156).
+Appendix A: Casting Eq.(14) to Eq.(15)
+In this appendix we show the steps for taking Eq. (14)
+to the form of Eq. (15) which is more amenable to study-
+ing issues related to conservation of complete positivity.
+We start with Eq. (14), which we recall to be
+∂ρ
+∂t = i[ρ, HS] − ϵ2 �
+l
+d2
+�
+α,˜α=1
+�
+a∗
+lαbl˜α[F †
+α, F˜αρ]
++ c′∗
+lαa′
+l˜α[ρF †
+α, F˜α] + b∗
+lαal˜α[ρF †
+α, F˜α] + a′∗
+lαc′
+l˜α[F †
+α, F˜αρ]
+�
+.
+(A1)
+This can be rewritten as
+∂ρ
+∂t = i[ρ, HS] + ϵ2 �
+l
+d2
+�
+α,˜α=1
+�
+a∗
+lαbl˜α
+�
+F˜αρF †
+α − {F †
+αF˜α, ρ}
+2
+− [F †
+αF˜α, ρ]
+2
+�
++ c′∗
+lαa′
+l˜α
+�
+F˜αρF †
+α − {F †
+αF˜α, ρ}
+2
++ [F †
+αF˜α, ρ]
+2
+�
++ b∗
+lαal˜α
+�
+F˜αρF †
+α − {F †
+αF˜α, ρ}
+2
++ [F †
+αF˜α, ρ]
+2
+�
++ a′∗
+lαc′
+l˜α
+�
+F˜αρF †
+α − {F †
+αF˜α, ρ}
+2
+− [F †
+αF˜α, ρ]
+2
+��
+,
+(A2)
+where, A, B
+=
+AB + BA is the anti-commutator.
+Next,
+we
+note
+that
+the
+summation
+�d2
+α,˜α=1
+in
+above
+equation,
+can
+be
+written
+as
+�d2
+α,˜α=1
+=
+�
+α=˜α=d2 + �d2−1
+α=1,˜α=d2 + �d2−1
+˜α=1,α=d2 + �d2−1
+α,˜α=1 .
+Us-
+ing this, and the fact that Fd2 = IS/
+√
+d commutes with
+all operators, we combine all commutator terms and
+write them as as i[ρ, HS + HLS] to obtain
+∂ρ
+∂t = i[ρ, HS + HLS] +
+d2−1
+�
+α,˜α=1
+Γα˜α
+�
+F˜αρF †
+α − {F †
+αF˜α, ρ}
+2
+�
+.
+(A3)
+Here
+HLS = ϵ2 �
+l
+�
+d2
+�
+α,˜α=1
+�a∗
+lαbl˜α
+2i
+− c′∗
+lαa′
+l˜α
+2i
+− b∗
+lαal˜α
+2i
++ a′∗
+lαc′
+l˜α
+2i
+�
+F †
+αF˜α
++
+d2−1
+�
+α=1
+(a∗
+lαbl,d2 + c′∗
+lαa′
+ld2 + b∗
+lαal,d2 + a′∗
+lαc′
+l,d2)
+2i
+√
+d
+F †
+α
+−
+d2−1
+�
+˜α=1
+(a∗
+ld2bl˜α + c′∗
+l,d2a′
+l˜α + b∗
+l,d2al˜α + a′∗
+l,d2c′
+l˜α)
+2i
+√
+d
+F˜α
+�
+(A4)
+and
+Γα˜α = ϵ2 �
+l
+(a∗
+lαbl˜α + c′∗
+lαa′
+l˜α + b∗
+lαal˜α + a′∗
+lαc′
+l˜α),
+(A5)
+α, ˜α going from 1 to d2 − 1. This is Eq. (15) given in the
+main text.
+Appendix B: An example of RE violating complete
+positivity
+In this section, we will present a simple example of the
+discussion in Sec. II. Our setup consists of a two-qubit
+XXZ qubit chain, where only the first qubit is connected
+to the bath modelled by an infinite number of bosonic
+modes.
+Let H be the Hamiltonian of the full set-up,
+given by
+H = HS + ϵ HSB + HB,
+(B1)
+where
+HS = ω0
+2 (σ(1)
+z
++ σ(2)
+z )
+− g(σ(1)
+x σ(2)
+x
++ σ(1)
+y σ(2)
+y
++ ∆σ(1)
+z σ(2)
+z )
+HSB =
+∞
+�
+r=1
+(κr ˆB†
+rσ(1)
+− + κ∗
+r ˆBrσ(1)
++ )
+HB =
+∞
+�
+r=1
+Ωr ˆB†
+r ˆBr
+(B2)
+
+14
+where σ(ℓ)
+x,y,z denotes the Pauli matrices acting on the ℓth
+qubit, σ(ℓ)
++
+= (σ(ℓ)
+x
++ iσ(ℓ)
+y )/2, σ(ℓ)
+−
+= (σ(ℓ)
+x
+− iσ(ℓ)
+y )/2,
+ˆBr is bosonic annihilation operator for the rth mode of
+the bath. Here, ω0, g, and g∆ represent the magnetic
+field, the overall qubit-qubit coupling strength and the
+anisotropy respectively. The RE for this setup can be
+computed to be [23]
+∂ρ
+∂t = i[ρ(t), HS] + ϵ2�
+[S†, S(1)ρ(t)] − [S†, ρ(t)S(2)]
++ H.c
+�
+(B3)
+with
+S† = σ(1)
++ ,
+S = σ(1)
+−
+S(1) =
+4
+�
+j,k=1
+|Ej⟩ ⟨Ej| σ(1)
+− |Ek⟩ ⟨Ek| D(j, k),
+S(2) =
+4
+�
+j,k=1
+|Ej⟩ ⟨Ej| σ(1)
+− |Ek⟩ ⟨Ek| C(j, k)
+,
+(B4)
+and
+C(j, k) = J(Ekj)n(Ekj)
+2
+− iP
+� ∞
+0
+dω J(ω)n(ω)
+ω − Ekj
+,
+D(j, k) = eβ(Ekj−µℓ)J(Ekj)n(Ekj)
+2
+− iP
+� ∞
+0
+dω eβ(ω−µ)J(ω)n(ω)
+ω − Ekj
+,
+J(ω) =
+∞
+�
+k=1
+2π |κk|2 δ(ω − Ωk),
+n(ω) = [eβω − 1]−1.
+(B5)
+In above, J(ω) is called the bath spectral function. Let us
+consider bosonic baths described by Ohmic spectral func-
+tions with Gaussian cut-offs, J(ω) = ωe−(ω/ωc)2Θ(ω),
+where Θ(ω) is the Heaviside step function, and ωc is the
+cut-off frequency. The above operators can then be com-
+puted numerically.
+The next step is to choose the basis fi and gj for oper-
+ators on HL and HM. For the general case, one can start
+with any set of linearly independent operators that forms
+a basis and includes the identity operator, and then ap-
+ply the Gram Schmidt orthonormalization procedure to
+produce an orthonormal basis that includes the normal-
+ized identity operator. For our case, one can easily verify
+that the set {−σ(i)
+z /
+√
+2, σ(i)
+− , σ(i)
++ , I(i)
+2 /
+√
+2} suffices, where
+i = 1 for {fi} and i = 2 for {gj}, and I2 is the identity
+operator.
+The basis for the full system {Fi} can be constructed
+from the above basis as described in subsection
+II A 1,
+and is given by
+Fi = fi ⊗ I(2)
+2
+2
+(for i = 1, 2, 3)
+F3i+j = fi ⊗ gj
+(for i = 1, 2, 3, 4 and j = 1, 2, 3)
+F16 = I4
+2
+(B6)
+Any operator X can be expanded in terms of the above
+basis as X = �
+α xαFα, where xα = ⟨Fα, X⟩ = Tr(F †
+αX).
+Thus, expanding S, S†, S(1), S(2) [Eq. (B4)], one can
+evaluate all the coefficients in Eq. (13).
+Finally, one
+can compute the matrix Γ according to Eq. (16). The
+matrix Γ for this example, with parameters chosen as
+g = 0.1, ω0 = 1, ωc = 10, β = 1, µ = −0.5, ∆ = 1 is given
+by
+Γ = ϵ2
+�
+�����������������������
+0
+0
+0
+0
+0
+0
+0
+0 0 0 0 0 0 0 0
+0
+0
+1.542 + 3.428i 0
+0
+0.014 + 0.047i
+0
+0 0 0 0 0 0 0 0
+0 1.542 − 3.428i
+0
+0 −0.18 − 0.007i
+0
+0.18 + 0.007i 0 0 0 0 0 0 0 0
+0
+0
+0
+0
+0
+0
+0
+0 0 0 0 0 0 0 0
+0
+0
+−0.18 + 0.007i 0
+0
+0
+0
+0 0 0 0 0 0 0 0
+0 0.014 − 0.047i
+0
+0
+0
+0
+0
+0 0 0 0 0 0 0 0
+0
+0
+0.18 − 0.007i
+0
+0
+0
+0
+0 0 0 0 0 0 0 0
+0
+0
+0
+0
+0
+0
+0
+0 0 0 0 0 0 0 0
+0
+0
+0
+0
+0
+0
+0
+0 0 0 0 0 0 0 0
+0
+0
+0
+0
+0
+0
+0
+0 0 0 0 0 0 0 0
+0
+0
+0
+0
+0
+0
+0
+0 0 0 0 0 0 0 0
+0
+0
+0
+0
+0
+0
+0
+0 0 0 0 0 0 0 0
+0
+0
+0
+0
+0
+0
+0
+0 0 0 0 0 0 0 0
+0
+0
+0
+0
+0
+0
+0
+0 0 0 0 0 0 0 0
+0
+0
+0
+0
+0
+0
+0
+0 0 0 0 0 0 0 0
+�
+�����������������������
+(B7)
+
+15
+We see that the above matrix has the expected struc-
+ture of Eq. (17),
+Γ =
+�
+Γα,˜α<4
+Γα<4,˜α≥4
+Γα≥4,˜α<4
+0
+�
+(B8)
+and crucially, Γα<4,˜α≥4 ̸= 0. Therefore, as per the GKSL
+theorem, this RE will not preserve complete positivity.
+This example was computed using QuTiP [68, 69].
+Appendix C: Effective Lindblad equation satisfying
+local conservation laws
+In this appendix, we will show that Eq. (29) implies
+Eq. (31). The condition for a QME preserving complete
+positivity and obeying local conservation laws is given by
+Eq. (29), which we recall to be
+− i[OM, H′] +
+d2
+M−1
+�
+αM,˜αM=1
+�ΛαM,˜αM
+�
+g†
+αM OMg˜αM
+− 1
+2OMg†
+αM g˜αM − 1
+2g†
+αM g˜αM OM
+�
+= 0,
+∀OM,
+(C1)
+where we write
+H′ =
+d2
+M
+�
+αM=1
+νd2
+L,αM gαM ,
+(C2)
+for convenience. To move forward we will use the opera-
+tor vector correspondence from Ref. 9, where the vector-
+ized version of the operator X is given by vec(X), and
+can be constructed using linearity and
+vec(|i⟩ ⟨j|) = |i⟩ ⊗ |j⟩∗ ,
+(C3)
+where |j⟩∗ denotes the complex conjugate of |j⟩. We will
+apply Eq. (C3) to Eq. (C1), using the identity (Eq. 1.132
+of Ref. 9)
+vec(A0BAT
+1 ) = (A0 ⊗ A1)vec(B).
+(C4)
+Applying the vec operation on both sides of Eq. (C1),
+we obtain
+− i vec(IMOMH′) + i vec(H′OMIM) +
+d2
+M−1
+�
+αM,˜αM=1
+�ΛαM,˜αM
+�
+vec(g†
+αM OMg˜αM ) − 1
+2vec(IMOMg†
+αM g˜αM )
+− 1
+2vec(g†
+αM g˜αM OMIM)
+�
+= 0,
+∀OM.
+(C5)
+Eq. (C5) can be simplified using Eq. (C4) to obtain,
+�
+− iIM ⊗ (H′)T + iH′ ⊗ IM
+d2
+M−1
+�
+αM,˜αM=1
+�ΛαM,˜αM
+�
+g†
+αM ⊗ gT
+˜αM − 1
+2IM ⊗ (g†
+αM g˜αM )T
+− 1
+2(g†
+αM g˜αM ) ⊗ IM
+��
+vec(OM) = 0,
+∀OM.
+(C6)
+Eq. (C6) is of the form M vec(OM) = 0 for all hermi-
+tian OM. Since hermitian matrices (such as OM) form
+a basis for the entire space of operators, this implies
+M vec(X) = 0 for all operators X. This is because one
+can expand X as a linear combination of hermitian op-
+erators (such as OM). Now, M vec(X) = 0 ∀ X implies
+M = 0. Therefore, Eq. (C6) implies
+M = − iIM ⊗ (H′)T + iH′ ⊗ IM +
+d2
+M−1
+�
+αM,˜αM=1
+�ΛαM,˜αM
+�
+g†
+αM ⊗ gT
+˜αM − 1
+2IM ⊗ (g†
+αM g˜αM )T
+− 1
+2(g†
+αM g˜αM ) ⊗ IM
+�
+= 0.
+(C7)
+If M = 0, then Tr(M) = 0. Taking the trace of Eq. (C7),
+and using the orthonormality of {gi} along with the fact
+that Tr(gi) = δi,d2
+M , we obtain
+Tr(M) = −
+d2
+M−1
+�
+αM=1
+�ΛαM,αM dM = 0
+(C8)
+which implies
+d2
+M−1
+�
+αM=1
+�ΛαM,αM =
+d2
+M−1
+�
+αM=1
+d2
+L
+�
+αL=1
+�Γ(αL,αM),(αL,αM) = 0. (C9)
+which is Eq. (31) in the main text.
+Appendix D: The condition for thermalization
+From fundamental principles of quantum statistical
+mechanics, we expect the system to thermalize when cou-
+pled to baths at equal temperatures. The exact condition
+that QME’s must obey to satisfy thermalization has been
+derived in Ref. 23. For the sake of completeness, we recall
+that discussion here.
+Let the total system Hamiltonian be given by H =
+HS + ϵHSB + HB, where ϵ is a dimensionless parame-
+ter controlling the strength of the system bath coupling,
+and HSB is the system bath coupling Hamiltonian. We
+proceed by obtaining an order-by-order solution to the
+steady state of our QME. Any QME describing our setup
+
+16
+can be expanded in the so-called time-convolution-less
+form [6],
+∂ρ(t)
+∂t
+=
+∞
+�
+m=0
+ϵ2mL2m(t)[ρ(t)],
+(D1)
+where L could in general be time-dependent operators
+and L0(t)[ρ(t)] = i[ρ(t), HS]. For quantum master equa-
+tions written to second-order in system-bath coupling,
+the above summation can be truncated at second order.
+Denoting L2m ≡ limt→∞ L2m(t), the steady state ρSS
+can be given by
+ρSS = lim
+t→∞ et(L0+ϵ2L2)ρ(0),
+(D2)
+which is assumed to be unique. The steady state satisfies
+0 =
+∞
+�
+m=0
+ϵ2mL2m[ρSS].
+(D3)
+We can then perform an expansion of ρSS in the even
+powers of ϵ as
+ρSS =
+∞
+�
+m=0
+ϵ2mρ(2m)
+SS
+(D4)
+Using Eq. (D4) in Eq. (D3), we can obtain an order by
+order solution of ρSS. At the zeroth order in ϵ, we obtain
+[ρ(0)
+SS, HS] = 0.
+(D5)
+Assuming that the Hamiltonian has no degeneracies,
+Eq. (D5) implies that ρ(0)
+SS is diagonal in the energy eigen-
+basis,
+ρ(0)
+SS =
+�
+i
+pi |Ei⟩ ⟨Ei| .
+(D6)
+where |Ei⟩ is an eigenstate of the system. At second order
+in ϵ (m = 1), we obtain the following two equations,
+⟨Ei| L2[ρ(0)
+SS] |Ei⟩ = 0,
+∀i
+(D7)
+i(Ei − Ej) ⟨Ei| ρ(2)
+SS |Ej⟩
++ϵ2 ⟨Ei| L2[ρ(0)
+SS] |Ej⟩ = 0,
+∀i ̸= j
+(D8)
+Since ρ(0)
+SS is diagonal in the energy eigenbasis, Eq. (D7)
+determines the diagonal elements of ρ(0)
+SS.
+Having ob-
+tained ρ(0)
+SS, Eq. (D8) then determines the off-diagonal
+elements of ρ(2)
+SS.
+Note from above equations that the
+leading order diagonal elements of ρSS are independent
+of ϵ.
+It can also be shown that the leading order off-
+diagonal elements of ρSS in the energy eigenbasis of the
+system scale as ϵ2. As discussed in the main text, the
+QME thermalizes if
+lim
+ϵ→0 ρSS = ρth
+(D9)
+where ρth is the Gibbs state of the system given by
+ρth =
+e−βHS
+Tr[e−βHS].
+(D10)
+We then conclude that the thermalization in this sense
+is a statement about leading order diagonal elements of
+ρSS. Substituting Eq. (D9) in Eq. (D7), we obtain the
+following condition on L2 for the system to thermalize,
+⟨Ei| L2[ρth] |Ei⟩ = 0
+∀i.
+(D11)
+Appendix E: Semidefinite Programming (SDP)
+1.
+Basic Theory
+In this section, we present the theoretical framework of
+semidefinite programming (SDP). We follow the defini-
+tion of SDPs given in page 57 of Ref. 9. In what follows,
+we will use Φ and Ψ to denote hermitian preserving lin-
+ear maps. We will also use Φ† to denote the “adjoint
+map” [9], which is defined as the unique linear map that
+satisfies
+⟨A, Φ(B)⟩ = ⟨Φ†(A), B⟩
+(E1)
+where
+⟨A, B⟩ = Tr(A†B)
+(E2)
+denotes the Hilbert Schmidt inner product.
+An SDP is defined by the tuple (Φ, Ψ, A, B, C), where
+Φ, Ψ are hermitian-preserving linear maps, and A, B, C
+are hermitian operators. The “primal” problem of the
+SDP is given by
+maximize :
+⟨A, X⟩
+w.r.t. X
+subject to :
+Φ(X) = B, Ψ(X) ≤ C, X ≥ 0,
+(E3)
+where the inequalities represent matrix inequalities. I.e,
+A ≥ B is equivalent to A − B ≥ 0 and implies A − B
+is positive semidefinite. We will use the notation Xf to
+denote any “feasible” value of X that satisfies the three
+constraints in Eq. (E3), and P to denote the maximum
+value of ⟨A, X⟩ attained in Eq. (E3) (assuming there is
+atleast one X which satisfies constraints).
+For every “primal” problem, there exists a “dual”
+problem given by
+minimize :
+⟨B, Y ⟩ + ⟨C, Z⟩
+w.r.t. Y, Z
+subject to :
+Φ†(Y ) + Ψ†(Z) ≥ A,
+Y is hermitian, Z ≥ 0.
+(E4)
+We will use the notation (Yf, Zf) to denote any “fea-
+sible” value of Y, Z that satisfies the three constraints
+in Eq. (E4), and D to denote the minimum value of
+⟨B, Y ⟩ + ⟨C, Z⟩ attained in Eqs. (E4) (assuming atleast
+some (Y, Z) satisfies constraints).
+
+17
+FIG. 7.
+Schematic representing weak duality for the SDP
+given in Eqs. (E3) and (E4), according to Eq. (E6).
+X(j)
+f
+and (Y (j)
+f
+, Z(j)
+f ) represents any feasible input to the primal
+and dual problems respectively. Any such inputs yield upper
+and lower bounds on the solutions of the primal and dual
+problems.
+Semidefinite programs have a notion of duality asso-
+ciated with them, which relates properties of the primal
+and the dual problems. In particular, it can be shown
+that
+P ≤ D
+(E5)
+a property known as “weak duality”.
+In most situa-
+tions, it can be shown that P = D, i.e, equality holds
+in Eq. (E5). This condition is known as “strong dual-
+ity”.
+By weak duality and the definition of our primal and
+dual problems, using Eq. (E3),(E4), and (E5), we obtain
+⟨A, Xf⟩ ≤ P ≤ D ≤ ⟨B, Yf⟩ + ⟨C, Zf⟩ .
+(E6)
+From Eq. (E6), any feasible choice of inputs to the pri-
+mal and dual problem (Xf, Yf, Zf) leads to lower and
+upper bounds on the optimal values of the primal and
+dual problems [see Fig. (7)]. In particular, if we obtain
+⟨A, Xf⟩ = ⟨B, Yf⟩ + ⟨C, Zf⟩, equality holds throughout
+in Eq. (E6). This property can therefore be exploited to
+obtain exact solutions to the primal problem of an SDP.
+We will show in Sec. E 2 that the thermal optimiza-
+tion problem (TOP) in Eq. (40) can be reduced to the
+standard form of SDP [Eq. (E3)].
+2.
+Reducing the thermal optimization problem
+(TOP) to standard form
+Recall that the TOP was given by [Eq.(40)]
+minimize : τ
+subject to : H(L)
+LS is hermitian, Tr(Γ(L)) = 1, Γ(L) ≥ 0.
+(E7)
+See Eq. (39) for definition of τ and Eq. (38) for defini-
+tion of H(L)
+LS , Γ(L). In this subsection, we will show how
+the TOP from Eq. (E7) can be reduced to the standard
+form of an SDP. We note that the standard form of SDP
+in Eq. (E3) is not yet suitable for this purpose. There-
+fore, we replace A → −A, C → −C, Ψ → −Ψ, Y → −Y,
+in Eq. (E3) and Eq. (E4), leaving Φ, B, X and Z un-
+changed. Since maximizing any function is the same as
+minimizing its negative, we obtain the new “primal” form
+as
+minimize :
+⟨A, X⟩
+subject to :
+Φ(X) = B, Ψ(X) ≥ C, X ≥ 0,
+(E8)
+FIG. 8.
+Schematic representing weak duality for the SDP
+given in Eqs. (E8) and (E9), according to Eq. (E10). X(j)
+f
+and (Y (j)
+f
+, Z(j)
+f ) represents any feasible input to the primal
+and dual problems respectively. Any such inputs yield upper
+and lower bounds on the solutions of the primal and dual
+problems.
+where we use �P to denote the minimum value of ⟨A, X⟩
+obtained in Eq. (E8). The new “dual” form is written as
+maximize :
+⟨B, Y ⟩ + ⟨C, Z⟩
+subject to :
+Φ†(Y ) + Ψ†(Z) ≤ A,
+Y is hermitian, Z ≥ 0,
+(E9)
+where we use �D to denote the minimum value of ⟨B, Y ⟩+
+⟨C, Z⟩ obtained in Eq. (E9). Eq. (E6) is then transformed
+into [see Fig (8)],
+⟨A, Xf⟩ ≥ �P ≥ �D ≥ ⟨B, Yf⟩ + ⟨C, Zf⟩ .
+(E10)
+We will now show how to reduce the TOP from
+Eq. (E7) to Eq. (E8), via a series of changes to the opti-
+mization problem in Eq. (E8). We do so in three steps.
+Step 1: Our first step is to write down a primal op-
+timization problem whose solution (minimum value at-
+tained) is equal to
+||K||1 = Tr(
+√
+K†K).
+(E11)
+for any hermitian matrix K. Let Πp and Πn be projectors
+onto the positive and negative eigenspaces of K. In this
+case,
+||K||1 = Tr(ΠpKΠp) − Tr(ΠnKΠn).
+(E12)
+Let us now consider the optimization problem given by,
+minimize :
+��
+I 0
+0 I
+�
+,
+�
+P
+.
+.
+Q
+��
+subject to :
+Ψ1
+�
+P
+.
+.
+Q
+�
+=
+�
+P
+0
+0 Q
+�
+≥
+�
+K
+0
+0
+−K
+�
+,
+�
+P
+.
+.
+Q
+�
+≥ 0,
+(E13)
+where we use dots to represent arbitrary blocks of the
+matrices which can always be set to zero without af-
+fecting the objective function or constraints. Note that
+Ψ1 in Eq. (E13) is a map that replaces the off-diagonal
+blocks with null matrices, leaving the diagonal blocks un-
+changed. It is easy to see that Eq. (E13) is of the form
+Eq. (E8) (with Φ and B omitted i.e., no equality con-
+straint). Thus Eq. (E13) is an SDP.
+
+D
+(Yf),zf1)
+8
+P
+α(f1),zf1)
+D
+8
+(y(2), z(2)
++8
+p18
+The dual problem to the primal problem in Eq. (E13)
+is given by
+maximize :
+��
+K
+0
+0
+−K
+�
+,
+� ¯P
+.
+.
+¯Q
+��
+subject to :
+Ψ†
+1
+� ¯P
+.
+.
+¯Q
+�
+=
+� ¯P
+0
+0
+¯Q
+�
+≤
+�
+I 0
+0 I
+�
+� ¯P
+.
+.
+¯Q
+�
+≥ 0.
+(E14)
+where Ψ†
+1 turns out to be the same map as Ψ1 using
+Eq. (E1). It can be seen that Eq. (E14) is of the form
+Eq. (E9) (again with Φ and B omitted).
+We will now show that the optimal values attained in
+both the primal and dual problems in Eqs. (E13) and
+(E14) is equal to ||K||1. To show this, note that setting
+Pf = ΠpKΠp,
+Qf = −ΠnKΠn
+(E15)
+(where Pf and Qf denote ‘feasible’ choices of P and Q
+respectively) in Eq. (E13) allows us to obtain ||K||1 in
+the primal objective function. Furthermore, setting
+¯Pf = Πp,
+¯Qf = Πn
+(E16)
+in Eq. (E14) allows us to obtain ||K||1 in the dual objec-
+tive function. Thus, we have explicitly constructed fea-
+sible choices of inputs to the primal and dual problems
+of Eqs. (E13) and (E14) respectively, that yield ||K||1
+in the primal and dual objective functions. Therefore,
+according to Eq. (E10), the optimal values attained in
+the primal and dual problems are both exactly equal to
+||K||1.
+Step 2: In the first step we constructed an SDP whose
+solution is equal to ||K||1, for a fixed K. We will now
+construct an SDP which computes the minimum value of
+||K||1, subject to some constraints on K. We will first
+recast the problem in Eq. (E13) as
+minimize :
+Tr(P) + Tr(Q)
+subject to :
+P ≥ K, Q ≥ −K, P, Q ≥ 0.
+(E17)
+Eq. (E13) computes
+||K||1 =
+�
+i
+|Kii|.
+(E18)
+when K is diagonal. Let G be a linear, hermitian preserv-
+ing map that always outputs a diagonal matrix. Then,
+the optimization problem given by
+minimize :
+Tr(P) + Tr(Q)
+subject to :
+P ≥ G(R), Q ≥ −G(R),
+Φ(R) = B,
+P, Q, R ≥ 0,
+(E19)
+computes minR≥0,Φ(R)=B
+�
+i |G(R)ii|. We will now begin
+to connect the above formalism to TOP from Eq. (E7).
+We will show how R can be chosen to reflect the opti-
+mization over H(L)
+LS and Γ(L), and Φ can be chosen to
+reflect the trace constraint on Γ(L). Then, we will spec-
+ify a map G that takes R as input (i.e, H(L)
+LS and Γ(L)
+LS ),
+and outputs a diagonal matrix such that the objective
+function computes
+τ =
+�
+i
+|⟨Ei| L2(ρth) |Ei⟩| .
+(E20)
+Step
+3:
+In
+the
+thermal
+optimization
+problem
+[Eq. (E7)], we have an optimization over Γ(L) ≥ 0, and
+hermitian H(L)
+LS . We use the fact that any hermitian ma-
+trix H(L)
+LS can be written as a H(L)
+LS
+= S − T, where
+S, T ≥ 0. Moreover S − T for any S, T ≥ 0 is always
+hermitian. We will now replace the hermitian H(L)
+LS with
+the difference of positive matrices S − T. This is needed,
+since semidefinite programs can only handle optimization
+over positive semidefinite variables. Furthermore, let us
+identify Γ(L) with some matrix U. Now consider the map
+G that acts as follows :
+G
+�
+�
+S
+.
+.
+. T
+.
+.
+.
+U
+�
+� ≡ G(S, T, U) ≡
+�
+�
+�
+�
+�
+⟨E1| L2(ρth) |E1⟩
+0
+. . .
+0
+0
+⟨E2| L2(ρth) |E2⟩ . . .
+0
+...
+. . .
+...
+...
+0
+. . .
+0
+⟨Ed| L2(ρth) |Ed⟩ ,
+�
+�
+�
+�
+�
+(E21)
+where the map constructs L2 according to Eq. (38) after
+setting H(L)
+LS = S − T, and Γ(L) = U. Now, we consider
+a specific case of the optimization problem in Eq. (E19),
+for the choice of G in Eq. (E21). We obtain,
+minimize :
+Tr(P) + Tr(Q)
+subject to :
+P ≥ G
+�
+�
+S
+.
+.
+. T
+.
+.
+.
+U
+�
+� , Q ≥ −G
+�
+�
+S
+.
+.
+. T
+.
+.
+.
+U
+�
+� ,
+Φ1
+�
+�
+S
+.
+.
+. T
+.
+.
+.
+U
+�
+� = Tr(U) = 1, P, Q, S, T, U ≥ 0.
+(E22)
+
+19
+Since G always outputs a diagonal matrix [see Eq. (E21)],
+Eq. (E22) computes
+min
+�
+i
+������
+G
+�
+�
+S
+T
+U
+�
+�
+ii
+������
+subject to :
+S, T, U ≥ 0, Tr(U) = 1
+(E23)
+Since H(L)
+LS can always be written as S−T, and Γ(L) as U,
+Eq. (E23) [and therefore Eq. (E22) ] is identical to the
+thermal optimization problem in Eq. (E7).
+Therefore,
+Eq. (E22) computes τopt.
+All that remains is converting Eq. (E22) to the stan-
+dard form Eq. (E8).
+Eq. (E22) can be obtained from
+Eq. (E22) after choosing
+A =
+�
+�
+�
+�
+�
+I
+I
+0
+0
+0
+�
+�
+�
+�
+� ,
+B = 1,
+X =
+�
+�
+�
+�
+�
+P
+.
+.
+.
+.
+.
+Q .
+.
+.
+.
+.
+S
+.
+.
+.
+.
+. T
+.
+.
+.
+.
+.
+U
+�
+�
+�
+�
+� ,
+C = 0,
+Ψ
+�
+�
+�
+�
+�
+P
+.
+.
+.
+.
+.
+Q .
+.
+.
+.
+.
+S
+.
+.
+.
+.
+. T
+.
+.
+.
+.
+.
+U
+�
+�
+�
+�
+� =
+�
+P − G(S, T, U)
+0
+0
+Q + G(S, T, U)
+�
+,
+Φ
+�
+�
+�
+�
+�
+P
+.
+.
+.
+.
+.
+Q .
+.
+.
+.
+.
+S
+.
+.
+.
+.
+. T
+.
+.
+.
+.
+.
+U
+�
+�
+�
+�
+� = Tr(U)
+(E24)
+Recall that it is helpful to think of P, Q as variables
+needed to compute the objective function τ [Eq. (E7)],
+S, T are variables that give rise to H(L)
+LS = S − T, and U
+is a variable that encodes Γ(L). We have therefore shown
+that the TOP [Eq. (E7)] is an SDP.
+It is to be noted that it is not necessary to reduce
+the TOP to the standard form of an SDP in order to
+use CVX [33]. Infact, the TOP from Eq. (E7) can be
+directly implemented into CVX, which itself handles the
+construction of the dual problem automatically in the
+background.
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+Smoothing the H0 tension with a dynamical dark energy model
+Safae Dahmani,∗ Amine Bouali,† Imad El Bojaddaini,‡ Ahmed Errahmani,§ and Taoufik Ouali¶
+Laboratory of Physics of Matter and Radiation, Mohammed I University, BP 717, Oujda, Morocco
+(Dated: January 12, 2023)
+The discrepancy between Planck data and direct measurements of the current expansion rate H0 and the
+matter fluctuation amplitude S8 has become one of the most intriguing puzzles in cosmology nowadays. The
+H0 tension has reached 4.2σ in the context of standard cosmology i.e ΛCDM. Therefore, explanations to this
+issue are mandatory to unveil its secrets. Despite its success, ΛCDM is unable to give a satisfying explanation
+to the tension problem.
+Unless some systematic errors might be hidden in the observable measurements,
+physics beyond the standard model of cosmology must be advocated. In this perspective, we study a phantom
+dynamical dark energy model as an alternative to ΛCDM in order to explain the aforementioned issues. This
+phantom model is characterised by one extra parameter, Ωpdde, compared to ΛCDM. We obtain a strong
+positive correlation between H0 and Ωpdde, for all data combinations. Using Planck measurements together
+with BAO and Pantheon, we find that the H0 and the S8 tensions are 3σ and 2.6σ, respectively. By introducing
+a prior on the absolute magnitude, MB, of the SN Ia, the H0 tension decreases to 2.27σ with H0 = 69.76+0.75
+−0.82
+km s−1 Mpc−1 and the S8 tension reaches the value 2.37σ with S8 = 0.8269+0.011
+−0.012.
+Keywords: dark energy, H0 tension, S8 tension.
+I.
+INTRODUCTION
+Supernova Type Ia (SN Ia) observation [1, 2] reports
+an unexpected cosmic acceleration of the expansion of
+the current Universe. This observation was corroboratted
+latter by other observations such as the cosmic microwave
+background (CMB) [3, 4], the large scale structure [5, 6]
+and the baryonic acoustic oscillations (BAO) [7, 8]. The
+standard model of cosmology successfully describes this
+late time cosmic acceleration by introducing a new exotic
+component in the budget of the Universe dubbed dark
+energy (DE). In the context of the standard model of
+cosmology, called ΛCDM, the major part of the content
+of the Universe is dominated by DE which is in the
+form of a cosmological constant, Λ, and the cold dark
+matter (CDM). In addition, various observational data
+give preference to ΛCDM for a vast range of redshifts z
+[9–13]. However, this model faces many problems, among
+them the “Hubble tension”, related to the current Hubble
+rate H0 and the σ8 tension due to the matter fluctuation
+amplitude.
+The Hubble tension appears when comparing the value
+measured indirectly by calibrating theoretical models in
+the early Universe i.e. at high-redshift and that measured
+directly using cosmological distances and redshifts by
+observing space objects. Generally, the value obtained at
+high-redshifts is lower than that obtained at low-redshifts.
+The value predicted at high-redshifts i.e.
+by cosmic
+microwave
+background
+measurements
+assuming
+the
+ΛCDM model, is H0 = 67.37 ± 0.54 km s−1 Mpc−1
+[11] while the one determined by the Cepheid calibrated
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+§ [email protected]
+¶ [email protected]
+supernovae Ia, is H0 = 73.2 ± 1.3 km s−1 Mpc−1 [14].
+It is clear that there is significant discrepancy between
+these values qualified as a tension.
+This tension is at
+about 4.2σ level.
+Recent studies have shown that this
+tension depends directly on the SN Ia absolute magnitude,
+MB [15–19].
+In fact the SH0ES project measures the
+absolute peak magnitude (MB = −19.244 ± 0.037 mag
+[16]) of SN Ia, while the value of H0 can be estimated
+by the magnitude-redshift relation of SN Ia in the range
+z ∈ [0.023, 0.15] [20]. The same studies indicated that to
+test any model that modifies the late-time of the Universe,
+it is necessary to use a prior on the absolute magnitude of
+supernovae type Ia, MB, instead of using the prior on H0
+from SH0ES for a correct statistical analysis and to avoid
+misleading results.
+On the other hand, the tension between the value of
+the matter fluctuation amplitude σ8 found by CMB
+measurements and that from large-scale observations
+in the late Universe rises another problem in ΛCDM.
+The parameter that quantifies the matter fluctuations
+is defined by S8
+=
+σ8
+�
+(Ωm,0/0.3), representing a
+combination of σ8 and the matter density Ωm,0 at the
+present time.
+Constraints from Planck and those from
+local measurements are in tension at more than 2σ.
+Indeed, while the constrained S8 from Planck data is
+S8 = 0.832 ± 0.013 [11], smaller values are found from
+local measurements, e.g. S8 = 0.762+0.025
+−0.024 obtained by
+KV450 (KiDS+VIKING-450) and DES-Y1 (Dark Energy
+Survey Year 1) combined [21]. This discrepancy could be
+an evidence of new physics beyond the standard model of
+cosmology [22–24].
+Several theoretical approaches have been proposed to
+solve these tension problems, such as extensions of the
+ΛCDM model, DE–DM interactions and decaying DM
+[26–36]. These approaches have also shown that changing
+the properties of DE e.g. by introducing the early Dark
+Energy [37] and the phantom Dark Energy where the
+equation of state (EoS) parameter is slightly less than
+arXiv:2301.04200v1  [astro-ph.CO]  10 Jan 2023
+
+2
+−1, can increase the value of H0 and consequently can
+alleviate the Hubble tension compared to ΛCDM [38–47].
+These conclusions motivated us to address these issues
+in the context of a particular dynamical dark energy (DDE)
+model where the EoS and the energy density are given re-
+spectively by [48]
+pde = −(ρde + α
+3 ),
+(1.1)
+and
+ρde(z) = ρde,0 − α ln (1 + z),
+(1.2)
+where ρde,0 is the present DE density, α is a positive con-
+stant that distinguishes this model from ΛCDM. Hence
+ρde tends to the standard cosmological constant Λ at the
+present (z = 0). This phantom dynamical dark energy
+model induces an abrupt event in the future where the dark
+energy density dominates all other forms of energy density.
+However, in the past, this dark energy density decreases
+and the energy density of dark matter dominates the bud-
+get of the Universe.
+According to Eqs. (1.1) and (1.2) the EoS parameter is
+given by
+ωde = −(1 +
+α
+3(ρde,0 − α ln (1 + z))).
+(1.3)
+For positive values of α, we get ωde < −1 and the model
+describes a phantom dark energy. For negative values of α,
+ωde > −1, the model describes a quintessence dark energy
+and mimics ΛCDM in the limit α → 0. In the following,
+we will focus on the phantom case i.e. α > 0.
+The Friedmann equation of a Universe filled by CDM and
+DE can be written as [48–50]
+E2(z) = Ωm,0(1+z)3 +Ωde,0 −Ωpdde ln (1 + z), (1.4)
+where E = H(z)
+H0 , H0 is the current Hubble rate, Ωm,0 is
+the actual matter density, Ωde,0 = 8πG
+3H2
+0 ρde,0 and Ωpdde =
+8πG
+3H2
+0 α is a dimensionless parameter characterizing our
+phantom DDE model.
+From Eq (1.4), the model predicts more dark matter in the
+past as z → ∞. However, in the future, this model is
+dominant by the DE and it is characterized by a particu-
+lar behaviour. Indeed, its Hubble rate H diverges while its
+derivative ˙H remains constant. This abrupt event has been
+well studied in [48–50] and dubbed as Little Sibling of the
+Big Rip since it smooths the big rip singularity in the fu-
+ture.
+In this paper, we study the effect of this phantom dy-
+namical dark energy model (PDDE) on both tensions,
+namely the H0 and S8 tensions, and we compare the results
+with those of ΛCDM. To this aim, we perform a Markov
+Chain Monte Carlo (MCMC) [51] analysis, using the last
+datasets.
+This paper is organized as follows: in Sec. II, we describe
+the methodology followed and the data used in our anal-
+ysis. In Sec. III, we present the results and discussions,
+while in Sec. IV we analyze the effect on the power spec-
+trum. Finally, Sec. V is dedicated to conclusions.
+II.
+METHODOLOGY AND DATASETS
+To constrain our theoretical model we employ the χ2
+statistics
+χ2 = [Pobs − Pth]2
+σ2
+P
+,
+(2.1)
+where Pobs, Pth and σ2
+P indicate the observed values, the
+predicted values and the standard deviation, respectively.
+The model with a small value of χ2 fits better the observa-
+tional data and is considered as the best. We also use the
+Akaike Information Criterion (AIC) [52], which is widely
+used in cosmology [53, 54] to compare cosmological mod-
+els with different free parameters numbers
+AIC = −2 ln (Lmax) + 2N,
+(2.2)
+where L is the likelihood and N is the number of free
+parameters. The model with a small value of AIC is the
+most supported by observational data. In this work, we
+calculate △AIC = AICP DDE − AICΛCDM and we
+consider ΛCDM as the reference model i.e. △AIC = 0.
+Furthermore, a positive (negative) value of △AIC indi-
+cates that ΛCDM (PDDE) is the most preferred model by
+observational data.
+To run the MCMC [51] we use the MontePython code
+[57], which interfaces with CLASS [58] in which we have
+implemented our PDDE fluid. We consider 7-dimensional
+parameters space, consisting of six standard cosmological
+parameters ωb, ωcdm, H0, ns, τreio and ln (1010As) which
+correspond to the physical densities of baryons and CDM,
+the Hubble constant, the scalar spectral index, the optical
+depth and the power spectrum amplitude, respectively plus
+the additional parameter Ωpdde characterizing our PDDE
+model. The priors of these free parameters are mentioned
+in Table I. To avoid non-adiabatic instabilities at the
+perturbation evolution, we employ the Parameterized
+Post-Friedmann (PPF) [59] approach.
+In this work, we use the following observational data:
+Planck18: The CMB temperature measurements (low-ℓ
+TT) and polarization (low-ℓ EE) at low multipoles 2 ⩽ ℓ ⩽
+29. We also use temperature and polarization combined
+(high-ℓ TT TE EE) at higher multipoles 30 ⩽ ℓ ⩽ 2500. In
+addition we use the lensing constraint [11].
+BAO: The Baryon Acoustic Oscillation measurements at
+different redshifts z, BOSS DR12 from the CMASS (at
+z = 0.57) and LOWZ galaxies (at z = 0.32) [60], 6dFGS
+(at z = 0.106) [61] and SDSS DR7 (at z = 0.15) [62].
+Pantheon: The luminosity distance from 1048 Supernovae
+
+3
+Type Ia (SN Ia) in the range z ∈ [0.01, 2.3] [20]. The SN
+Ia data directly give measures of mb(z) for each z, where
+mb(z) is the apparent magnitude. For a given cosmolog-
+ical model this parameter can be calculated theoretically
+by
+mb(z) = 5log10[dL(z)
+Mpc ] + MB + 25,
+(2.3)
+where, dL(z) = (1 + z)
+´ z
+0
+cdz′
+H(z′) is the luminosity
+distance and MB is the absolute magnitude which will be
+considered as a free parameter in our analysis.
+Prior on MB: The SN measurements from the SH0ES
+project give a Gaussian prior on the absolute magnitude as
+MB = −19.244 ± 0.037 mag [16].
+The total χ2 of the combined data is
+χ2
+tot = χ2
+P lanck18 + χ2
+BAO + χ2
+P antheon + χ2
+MB. (2.4)
+Table I. A prior imposed on different parameters for the ΛCDM
+and PDDE models
+Parameters
+Prior
+Ωbh2
+[0.005, 0.1]
+Ωch2
+[0.01, 0.99]
+Ωpdde
+[0, 1]
+H0
+[40, 100]
+τreio
+[0.001, 0.8]
+ns
+[0.8, 1.2]
+ln (1010As)
+[2, 4]
+MB
+default priora
+a For the absolute magnitude parameter we used MontePython v3.5
+default prior.
+III.
+RESULTS AND DISCUSSIONS
+We perform an MCMC analysis to obtain con-
+straints on cosmological parameters of the PDDE model
+and compare them with those of ΛCDM. First of all,
+we employ the PDDE model with three different data
+combinations,
+namely Planck18,
+Planck18+BAO and
+Planck18+BAO+Pantheon, in order to make a comparison
+with ΛCDM and get a general insight of the analysis. In the
+second analysis, we include a prior on MB from SHOES
+to Planck18+BAO+Pantheon datasets.
+A.
+Planck18, BAO and Pantheon datasets.
+Table II shows the mean values and their correspond-
+ing errors at 68% C.L. for all considered parameters using
+Planck18, Planck18+BAO and Planck18+BAO+Pantheon.
+Fig. 1 shows the 2D and 1D posterior distributions for the
+PDDE model for the aforementioned datasets.
+Using Planck data alone, we get a large value of H0 =
+77.5±1.1 km s−1 Mpc−1 for PDDE and H0 = 67.93+0.58
+−0.63
+km s−1 Mpc−1 for ΛCDM. This last value is in ten-
+sion of about 3.6σ with the local measurement of R20
+i.e.
+HR20
+0
+= 73.2 ± 1.3 km s−1 Mpc−1.
+While for
+the PDDE model the tension is reduced to ∼ 2.5σ. De-
+spite this we notice that our model deviates strongly from
+ΛCDM i.e. Ωpdde = 0, where the value of Ωpdde obtained
+(Ωpdde = 0.4901+0.031
+−0.022) is around 15σ away from 0, which
+shows that our late model strictly shifted towards the more
+phantom regime (i.e. ωde << −1). Therefore, Planck18
+alone does not provide any sientific conclusion to our late-
+time dark energy transition. When we add the BAO data
+the Ωpdde value decreases to 0.08478+0.024
+−0.085. We also see
+a small decrease by 7.1% for H0 i.e. H0 = 69.13+0.79
+−1.1
+km s−1 Mpc−1 for PDDE and a small increase by 0.2%
+for ΛCDM i.e.
+H0 = 68.07 ± 0.45 km s−1 Mpc−1.
+The tension of H0 remains around ∼ 2.5σ for the PDDE
+model and increases to ∼ 3.7σ for ΛCDM compared to
+HR20
+0
+. The significant difference between HΛCDM
+0
+and
+HP DDE
+0
+tensions is actually not enough to come out with
+conclusive results about the H0 tension because analyz-
+ing this tension in light of any late-time model like PDDE
+should necessarily involve analyzing the Pantheon SN Ia
+sample [15].
+Adding Pantheon data to Planck18+BAO
+dataset, we observe a decreasing values of Ωpdde and H0
+to 0.0586+0.017
+−0.059 km s−1 Mpc−1 and 68.76+0.59
+−0.76 km s−1
+Mpc−1, respectively for PDDE. The tension in this case
+is still significant with 3σ for PDDE. However, this value
+is less than that of ΛCDM that gives 3.7σ. These con-
+clusions can be justified by the positive correlation ob-
+served in the (Ωpdde, H0) plan as can be seen in Fig. 1.
+We also compare the absolute magnitude, MB, value with
+SH0ES calibration i.e. MB = −19.244 ± 0.037 mag, we
+notice that our model can considerably smooth the MB
+tension, where the tension with SH0ES calibration is at
+about 3.6σ, while for ΛCDM is at about 4.2σ (see Fig.
+2). On the other hand, in the context of PDDE model,
+we obtain a relatively high value of σ8 i.e. 0.9038+0.0094
+−0.0094
+and a noticeable decrease in the current density of matter
+i.e. Ωm = 0.2358+0.0079
+−0.0082. This gives a small value of
+S8 i.e. 0.8011+0.015
+−0.014. According to KV450+DES-Y1, S8
+tension is at 1.4σ for PDDE and at 2.6σ for ΛCDM. Us-
+ing Planck18+BAO and Planck18+BAO+Pantheon, we get
+S8 = 0.831 ± 0.011 (S8 = 0.832+0.011
+−0.012) and 0.8312+0.011
+−0.012
+(0.832 ± 0.011) for ΛCDM (PDDE), respectively. S8 ten-
+sion is at 2.6σ (2.6σ) and 2.5σ (2.6σ) for ΛCDM (PDDE).
+We deduce that PDDE attenuates the S8 tension compared
+to the ΛCDM when constrained by Planck data only.
+Table III shows the χ2 for each data combination. Fur-
+thermore, △χ2
+tot = χ2(P DDE)
+tot
+− χ2(ΛCDM)
+tot
+, △AIC =
+AICP DDE − AICΛCDM and AIC are also shown in Ta-
+ble III. Using Planck18 together with BAO and Pantheon
+datasets we get a positive value of △χ2
+tot and △AIC. The
+inclusion of these data gives preference to ΛCDM.
+
+4
+Table II. Summary of the mean±1σ of the cosmological parameters for the ΛCDM and PDDE models, using Planck18, Planck18+BAO and
+Planck18+BAO+Pantheon datasets.
+Data
+Planck18a
+Planck18+BAO
+Planck18+BAO+Pantheon
+Model
+ΛCDM
+PDDE
+ΛCDM
+PDDE
+ΛCDM
+PDDE
+100Ωbh2
+2.238+0.016
+−0.017
+2.244 ± 0.016
+2.24+0.014
+−0.013
+2.236 ± 0.015
+2.241 ± 0.014
+2.236 ± 0.015
+Ωch2
+0.1199 ± 0.0013 0.1191 ± 0.0013
+0.1196+0.00099
+−0.00098
+0.12 ± 0.0011 0.1196+0.00096
+−0.001
+0.12 ± 0.001
+Ωpdde
+-
+0.4901+0.031
+−0.022
+-
+0.0847+0.024
+−0.085
+-
+0.0586+0.017
+−0.059
+H0 [km s−1 Mpc−1]
+67.93+0.58
+−0.63
+77.5 ± 1.1
+68.07 ± 0.45
+69.13+0.79
+−1.1
+68.09+0.46
+−0.45
+68.76+0.59
+−0.76
+τreio
+0.054 ± 0.0081
+0.05211+0.0088
+−0.0081
+0.0545+0.0076
+−0.0077
+0.0528+0.0075
+−0.008
+0.055+0.0074
+−0.0079
+0.053 ± 0.0076
+ns
+0.965 ± 0.0046
+0.9674+0.0044
+−0.0046
+0.9661+0.0039
+−0.0041
+0.964 ± 0.0041 0.9663+0.0039
+−0.0038
+0.965+0.0042
+−0.0039
+ln (1010As)
+3.044+0.015
+−0.016
+3.037+0.017
+−0.016
+3.044+0.015
+−0.016
+3.041 ± 0.015
+3.045 ± 0.015
+3.042 ± 0.015
+Ωm
+0.308+0.0081
+−0.008
+0.2358+0.0079
+−0.0082
+0.3065+0.0058
+−0.0061
+0.2982+0.0094
+−0.0082
+0.3063+0.0057
+−0.0062
+0.3013+0.007
+−0.0066
+σ8
+0.823 ± 0.0066
+0.9038+0.0094
+−0.0094
+0.8226 ± 0.0066
+0.835+0.0094
+−0.013
+0.8227+0.0062
+−0.0066
+0.831+0.0082
+−0.01
+S8
+0.834 ± 0.014
+0.8011+0.015
+−0.014
+0.831 ± 0.011
+0.832+0.011
+−0.012
+0.8312+0.011
+−0.012
+0.832 ± 0.011
+MB [mag]
+-
+-
+-
+-
+−19.41+0.013
+−0.012
+−19.39+0.015
+−0.017
+H0 tensionb
+3.6σ
+2.525σ
+3.72σ
+2.531σ
+3.7σ
+3σ
+MB tension
+-
+-
+-
+-
+4.2σ
+3.6σ
+S8 tension
+2.6σ
+1.4σ
+2.6σ
+2.6σ
+2.5σ
+2.6σ
+a We used the “lite” version of high-ℓ likelihood.
+b To calculate the tension between two values of H0 obtained from different data (d1, d2), we use the following expression [63, 64]:
+T(H0) = |H0(d1) − H0(d2)|/(
+�
+σ(H0(d1))2 + σ(H0(d2))2), where H0 and σ are the mean and the variance of the posterior of Hubble rate (the same
+for MB and S8).
+Table III. The best-fit χ2 per experiment for the ΛCDM and PDDE models.
+Datasets
+Planck18
+Planck18+BAO
+Planck18+BAO+Pantheon
+Model
+ΛCDM
+PDDE
+ΛCDM
+PDDE
+ΛCDM
+PDDE
+Planck high-ℓ TTTEEE lite 583.41
+582.37
+583.96
+584.26
+583.5
+583.96
+Planck low-ℓ EE
+396.23
+395.68
+395.98
+395.86
+396.26
+395.84
+Planck low-ℓ TT
+23.44
+22.31
+23.27
+23.22
+23.36
+23.43
+Planck lensing
+8.78
+8.73
+8.81
+8.8
+8.801
+8.81
+bao boss dr12
+-
+-
+3.73
+3.69
+3.88
+3.92
+bao smallz 2014
+-
+-
+1.48
+1.53
+1.41
+1.43
+Pantheon
+-
+-
+-
+-
+1025.84
+1025.80
+χ2
+tot
+1011.88 1009.12 1017.25 1017.39 2043.09
+2043.23
+△χ2
+tot
+0
+−2.76
+0
++0.14
+0
++0.14
+AIC
+1029.88 1029.12 1035.25 1037.39 2063.09
+2065.23
+△AIC
+0
+−0.76
+0
++2.14
+0
++2.14
+In the following, we will focus on the data combination
+Planck18+BAO+Pantheon as it is the only suitable combi-
+nation to study the tension in the framework of the late-
+time model PDDE.
+B.
+Adding MB prior.
+To combine the SH0ES results with the other cosmo-
+logical data, we take into account the SN Ia peak absolute
+magnitude MB rather than the H0 parameter. For this,
+we introduce a prior on MB from the SN measurements,
+MB = −19.244 ± 0.037. In Fig. 3, we show the 2D
+and 1D posterior distributions at 68.3% and 95.4% C.
+L. for all cosmological parameters of the ΛCDM and
+PDDE models.
+The mean values, the error at 68%
+C.L. and χ2 per experiment are given in Table IV and
+Table V, respectively.
+When adding the MB prior to
+Planck18+BAO+Pantheon, the Ωpdde parameter reaches
+the value 0.1087+0.052
+−0.061 and the Hubble rate increases to
+H0 = 69.76+0.75
+−0.82.
+This increase can be observed also
+for the absolute magnitude where MB = −19.37+0.017
+−0.018,
+compared to the same datasets without MB prior, as can
+be noticed from the strong positive correlation in the
+
+5
+0.115
+0.119
+0.124
+Ωcdmh2
+0
+0.284
+0.569
+Ωpdde
+67.3
+74.3
+81.3
+H0
+2.96
+3.02
+3.09
+ln1010As
+0.951
+0.967
+0.983
+ns
+0.001
+0.0402
+0.0794
+τ reio
+0.212
+0.266
+0.321
+Ωm
+0.812
+0.873
+0.934
+σ8
+0.755
+0.811
+0.868
+S8
+-19.4
+-19.4
+-19.3
+MB
+2.19
+2.24
+2.3
+Ωbh2
+-19.4
+-19.4
+-19.3
+MB
+0.115
+0.119
+0.124
+Ωcdmh2
+0
+0.284
+0.569
+Ωpdde
+67.3
+74.3
+81.3
+H0
+2.96
+3.02
+3.09
+ln1010As
+0.951
+0.967
+0.983
+ns
+0.001
+0.0402
+0.0794
+τ reio
+0.212
+0.266
+0.321
+Ωm
+0.812
+0.873
+0.934
+σ8
+0.755
+0.811
+0.868
+S8
+Planck18
+Planck18+BAO
+Planck18+BAO+Pantheon
+Figure 1. The 2D and 1D posterior distributions at 68.3% and 95.4% C.L. for the PDDE model using different combinations of data (Planck,
+Planck+BAO and Planck+BAO+Pantheon). The local measurement of H0 = 73.2 ± 1.3km s−1 Mpc−1 and S8 = 0.762+0.025
+−0.024 obtained by
+R20 and KV450+DES-Y1 respectively, are represented by the orange band.
+{Ωpdde, H0}, {Ωpdde, MB} and {H0, MB} plans (see
+Fig. 3). We also notice that the H0 tension is reduced
+to a lower value of about ∼ 2.27σ and the MB tension
+reduced to ∼ 3.06σ for the PDDE model. For ΛCDM,
+we obtain H0 = 68.58+0.43
+−0.44 and MB = −19.39 ± 0.012
+with a tension of about 3.36σ and 3.7σ, respectively.
+Therefore, we conclude that the PDDE model is able to
+make a slight attenuation of the H0 and MB tensions
+using Planck18+BAO+Pantheon+MB datasets compared
+to ΛCDM. On the other hand, the prior on MB reduces
+the value of S8 to 0.8212 ± 0.011 (0.8269+0.011
+−0.012) for
+ΛCDM (PDDE), respectively, compared to the same
+datasets without MB prior.
+A negative correlation can
+also be seen in Fig.3 between S8 and MB. According to
+KV450+DES-Y1, the S8 tension is at 2.16σ and 2.37σ
+for ΛCDM and PDDE, respectively. We notice that the
+ΛCDM model, reduces the S8 tension compared to PDDE
+when
+constrained
+by
+Planck18+BAO+Pantheon+MB
+datasets.
+Table V shows the χ2 per experiment using
+Planck18+BAO+Pantheon+MB datasets. We get a nega-
+tive value for △χ2
+tot and △AIC, i. e. △χ2
+tot = −3.01 and
+△AIC = −1.01, while in the previous section, positive
+values were found for the same datasets without MB prior.
+The negative value of △χ2
+tot is mainly related to the MB
+prior from SH0ES data with △χ2
+MB = −3.59. Conse-
+quently the PDDE model provides a slightly better fit for
+Planck18+BAO+Pantheon+MB datasets than ΛCDM.
+
+6
+ΛCDM
+PDDE
+SH0ES
+-19.6
+-19.5
+-19.4
+-19.3
+-19.2
+-19.1
+-19.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+MB
+P/Pmax
+-19.5
+-19.4
+-19.3
+-19.2
+MB
+66.8
+68.9
+71.1
+73.2
+H0
+-19.5
+-19.4
+-19.3
+-19.2
+MB
+ΛCDM
+PDDE
+Figure 2. The left panel shows 1D posterior distributions for the absolute magnitude, MB. The right panel shows 68% and 95% constraints
+on (H0, MB) plan using Planck18+BAO+Pantheon datasets. The local measurement of H0 = 73.2 ± 1.3km s−1 Mpc−1 and MB =
+−19.244 ± 0.037 mag obtained by SH0ES, are represented by the grey band and the orange band respectively.
+Table IV. Summary of the mean±1σ of cosmological parameters for the ΛCDM and PDDE models, using Planck18+BAO+Pantheon+MB
+datasets.
+Data
+Planck18+BAO+Pantheon+MB
+Model
+ΛCDM
+PDDE
+100Ωbh2
+2.25 ± 0.014
+2.242 ± 0.015
+Ωch2
+0.1185+0.00093
+−0.00098 0.1196 ± 0.0011
+Ωpdde
+-
+0.1087+0.052
+−0.061
+H0 [km s−1 Mpc−1]
+68.58+0.43
+−0.44
+69.76+0.75
+−0.82
+τreio
+0.05758+0.0073
+−0.0085
+0.05414+0.0077
+−0.0081
+ns
+0.9688+0.0039
+−0.004
+0.9661+0.0041
+−0.0042
+ln (1010As)
+3.049+0.015
+−0.016
+3.043+0.015
+−0.016
+Ωm
+0.3+0.0054
+−0.0056
+0.292 ± 0.0071
+σ8
+0.8213+0.0064
+−0.0068
+0.8382+0.011
+−0.012
+S8
+0.8212 ± 0.011
+0.8269+0.011
+−0.012
+MB [mag]
+−19.39 ± 0.012
+−19.37+0.017
+−0.018
+H0 tension
+3.36σ
+2.27σ
+MB tension
+3.7σ
+3.06σ
+S8 tension
+2.16σ
+2.37σ
+IV.
+EFFECT ON THE CMB POWER SPECTRUM.
+In the top panel of Fig. 4, we show the effect of the
+phantom dynamical dark energy model and the ΛCDM
+model on the CMB temperature power spectrum using
+the results obtained by Planck18+BAO+Pantheon+MB
+dataset. We notice that in the CMB temperature power
+spectrum, the effect of the PDDE model is visible at large
+scales 2 < ℓ < 30 but at higher multipoles ℓ this model
+is indistinguishable from ΛCDM. This conclusion agrees
+with that of several model of this type of dark energy (see
+for example [43, 45]).
+We also show the current mat-
+ter power spectrum P(z), for the ΛCDM and the PDDE
+models for different values of Ωpdde using the results ob-
+tained in Tab. II and Tab. III. The bottom left and right
+panels of Fig. 4 correspond to the amplitude of the mat-
+ter power spectrum for different k-modes running approxi-
+mately from the current Hubble horizon, k = 3.33×10−4h
+Mpc−1 to k ∼ 1h Mpc−1. The bottom panels of Fig. 4 are
+obtained using datasets under consideration without MB
+prior (left panel) and with MB prior (right panel). In the
+bottom-left panel of Fig. 4, we notice a clear difference
+
+7
+0.116
+0.12
+0.123
+Ωcdmh2
+67.4
+69.9
+72.3
+H0
+3
+3.05
+3.1
+ln1010As
+0.952
+0.965
+0.978
+ns
+0.0285
+0.0547
+0.0808
+τ reio
+-19.4
+-19.4
+-19.3
+MB
+0.272
+0.294
+0.316
+Ωm
+0.801
+0.838
+0.876
+σ8
+0.791
+0.828
+0.865
+S8
+0
+0.152
+0.304
+Ωpdde
+2.19
+2.24
+2.3
+Ωbh2
+0
+0.152
+0.304
+Ωpdde
+0.116
+0.12
+0.123
+Ωcdmh2
+67.4
+69.9
+72.3
+H0
+3
+3.05
+3.1
+ln1010As
+0.952
+0.965
+0.978
+ns
+0.0285
+0.0547
+0.0808
+τ reio
+-19.4
+-19.4
+-19.3
+MB
+0.272
+0.294
+0.316
+Ωm
+0.801
+0.838
+0.876
+σ8
+0.791
+0.828
+0.865
+S8
+ΛCDM
+PDDE
+Figure 3.
+The 2D and 1D posterior distributions at 68.3% and 95.4% C.L. for the ΛCDM and PDDE models using
+Planck18+BAO+Pantheon+MB datasets. The local measurement of H0 = 73.2 ± 1.3 km s−1 Mpc−1 and S8 = 0.762+0.025
+−0.024 obtained
+by SH0ES and KV450+DES-Y1 respectively, are shown by the orange band.
+between ΛCDM and PDDE using Planck18 datasets alone.
+This difference is due to the increase of the value of Ωpdde,
+particularly when we use Planck18 alone. This observa-
+tion is justified by the high value of σ8 for PDDE model
+and the positive correlation observed in the plan (Ωpdde,
+σ8) (see Fig. 1). However, this difference becomes less
+observable by using Planck + BAO and Planck + BAO +
+Pantheon datasets. The phantom dark energy model has
+a low effect on the matter power spectrum. This result
+is also shown in the reference [49]. The addition of MB
+prior to Planck18+BAO+Pantheon datasets slightly distin-
+guishes the PDDE model from the ΛCDM model. The
+different effect between ΛCDM and PDDE on the ampli-
+tude of the matter power spectrum is observed clearly in
+the range of smallest modes.
+V.
+CONCLUSIONS
+In this work, we have studied the effect of a phantom
+dynamical dark energy (PDDE) model on the cosmo-
+logical parameters, particularly its capability of relieving
+the H0 and S8 tensions.
+The equation of state of this
+model depends on the redshift z and deviates from the
+ΛCDM model by a positive constant. This PDDE model
+is specified by introducing an abrupt event in the future
+labeled in the literature the Little Sibling of the Big Rip
+as it smooths the big rip singularity. The Boltzman code
+CLASS has been modified to implement the parameter
+
+8
+Table V. The χ2 per experiment for the ΛCDM and PDDE models
+Dataset
+Planck18+BAO+Pantheon+MB
+Model
+ΛCDM
+PDDE
+Planck high-ℓ TTTEEE lite 585.005
+582.857
+Planck low-ℓ EE
+396.53
+396.29
+Planck low-ℓ TT
+22.75
+23.45
+Planck lensing
+8.84
+8.77
+bao boss dr12
+3.38
+4.025
+bao smallz 2014
+1.99
+2.64
+Pantheon
+1025.65
+1026.7
+MB prior
+16.29
+12.7
+χ2
+tot
+2060.47
+2057.46
+△χ2
+tot
+0
+−3.01
+AIC
+2080.47
+2079.46
+△AIC
+0
+−1.01
+characterizing the PDDE model and a first Markov
+Chain Monte Carlo analysis has been performed using
+the dataset combinations Planck18, Planck18+BAO and
+Planck18+BAO+Pantheon.
+We have found that when
+using Planck18 data alone and Planck18+BAO, a mislead-
+ing reduction of the tension is noticed. In fact, finding a
+late-time solution of the H0 tension implies an analysis
+of the SN measurements, i.e.
+Pantheon data.
+Adding
+Pantheon data shows a persistent 3σ tension for H0 and
+2.6σ for S8. Although the H0 tension for PDDE is reduced
+in comparison with ΛCDM, it is clear that a late-time
+model can not lead to a solution to this H0 discrepancy.
+In a second analysis,
+we have added a prior on
+MB that was obtained by the SH0ES project,
+i.e.
+MB
+=
+−19.244 ± 0.037 mag.
+As shown in Ta-
+ble IV, the PDDE model reduces the H0 tension to
+2.27σ and the S8 tension to 2.37σ when combin-
+ing Planck18+BAO+Pantheon datasets with the MB
+prior,
+i.e
+Planck18+BAO+Pantheon+MB.
+Further-
+more,
+the PDDE model provides a slightly better
+fit
+to
+Planck18+BAO+Pantheon+MB
+datasets
+with
+∆χ2 = −3.01 and ∆AIC = −1.01 (see Table V).
+The distinction of the PDDE model over the standard
+model of cosmology is clearly observed in our work,
+for a wide range of data combinations.
+These findings
+agree with the fact that phantom dark energy models are
+supported by observations and can be an alternative of
+ΛCDM to solve problems related to the fine-tuning, the
+coincidence and the tensions under consideration if more
+investigations with regards to these models are done.
+Particularly, other phantom dark energy models such as
+the little rip [65] can be employed and many scenarios
+for the structure of the Universe such as the inclusion of
+massive neutrinos and the modification of the space-time
+curvature can be tested. We will focus on these subjects in
+our future works.
+
+9
+101
+102
+103
+0
+1000
+2000
+3000
+4000
+5000
+6000
+( + 1)CTT/2  [ K2]
+CDM
+PDDE
+Planck Data
+2
+20
+35
+10 4
+10 3
+10 2
+10 1
+100
+k [hMpc
+1]
+102
+103
+104
+P(k) [Mpc/h]3
+CDM
+PDDE
+CDM
+PDDE
+CDM
+PDDE
+10 4
+10 3
+10 2
+10 1
+100
+k [hMpc
+1]
+102
+103
+104
+P(k) [Mpc/h]3
+CDM
+PDDE
+Figure 4. The CMB Temperature power spectrum (top panel) for the ΛCDM (dashed line) and PDDE (continuous line) models using the
+best-fit obtained by Planck18+BAO+Pantheon+MB datasets. The bottom panels correspond to the matter power spectrum using different
+combinations of data. The left panel is for Planck18 (blue lines), Planck18+BAO (red lines) and Planck18+BAO+Pantheon (green lines)
+datasets. The right panel is for Planck18+BAO+Pantheon+MB datasets.
+
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+

6dE4T4oBgHgl3EQf1w1z/content/tmp_files/2301.05293v1.pdf.txt

@@ -0,0 +1,968 @@
+HTTE: A Hybrid Technique For Travel Time
+Estimation In Sparse Data Environments
+immediate
+Nikolaos Zygouras1, [email protected] Nikolaos Panagiotou1, [email protected]
+Yang Li3, [email protected] Dimitrios Gunopulos1, [email protected] Leonidas
+Guibas2, [email protected]
+1
+Abstract
+Travel time estimation is a critical task, useful to many urban appli-
+cations at the individual citizen and the stakeholder level. This paper
+presents a novel hybrid algorithm for travel time estimation that lever-
+ages historical and sparse real-time trajectory data. Given a path and a
+departure time we estimate the travel time taking into account the histor-
+ical information, the real-time trajectory data and the correlations among
+different road segments. We detect similar road segments using historical
+trajectories, and use a latent representation to model the similarities. Our
+experimental evaluation demonstrates the effectiveness of our approach.
+1
+Introduction
+The increasing population density in modern cities is leading to massively in-
+creasing commuting demands for citizens. This strongly motivates the need for
+faster and more efficient navigation tools in the city. To be truly useful such
+systems need to be able to monitor and accurately predict the traffic condi-
+tions across the entire city road network in real-time, to respond to abrupt or
+unexpected condition changes. Accurate travel time estimation for a path in
+the road network is important for tools that help individual citizens plan their
+travel; equally stakeholders and city/traffic authorities can exploit such tools
+for efficient route planning and automatic detection of traffic anomalies. Sev-
+eral works have used data from static sensors, including loop detectors Kwon
+et al. (2003) and CCTV cameras Zhan et al. (2015) to address the travel time
+estimation for a path. Such sensors are typically located at several junctions
+across the city monitoring the traffic condition. The prevalence of such solu-
+tions has diminished since their first appearance. The increased capital cost of
+1(1)National and Kapodistrian University of Athens (2)Stanford University (3) Tsinghua-
+Berkeley Shenzhen Institute
+1
+arXiv:2301.05293v1  [cs.LG]  12 Jan 2023
+
+installing and maintaining such devices and their limited and static coverage
+of the road network in combination with the inherent inaccuracy in calculating
+the travel time from the output of these sensors (i.e. number of vehicles, speed
+and video frames) limit their application in practice.
+Recently researchers have used trajectory data in order to perform travel
+time estimation, thus taking advantage of the widespread use of mobile devices
+that are equipped with Global Position System (GPS) technology. Thus, such
+mobile devices are transformed into important moving and ubiquitous sensors
+reporting the traffic condition at different parts of the road network.
+However, not all such data are available in real-time for a variety of reasons.
+Sensors may be offline or smartphones may report their locations infrequently
+or in batches.
+Additionally, in several cases the application has access to a
+small number of sensors. Taxi or bus companies, for instance, have data for the
+vehicles in their fleet only. Over time, such applications can compile massive
+historical data with impressive coverage, although at any given time the coverage
+of the map is sparse.
+The goal of this study is to estimate the time that is required to travel a
+given query path considering a particular departure time in real time even when
+very patchy real time coverage of the network is available. To accomplish this
+we propose a hybrid model that considers efficiently the recent and historical
+trajectories generated by a sample of vehicles. In our settings, the travel time
+estimation is a challenging problem for the following reasons:
+1. Data sparsity: for the majority of the road segments we do not have any
+information regarding their recent traffic condition, since only the trajectories
+of a small subset of vehicles moving in the road network is available. Therefore,
+our setting is different from industry situations where an extensive real-time
+coverage of the traffic conditions may be available.
+2. Noisy data: the travel time reports are extremely noisy. A driver may stop at
+a traffic light and spend a while waiting, while another driver crosses the junction
+without stopping at the traffic light. This would generate two divergent travel
+time reports for the same road segment. Also, some drivers may drive faster or
+slower than others adding further complexity in the measurements.
+3. Unpredictable dynamics of traffic: there are many traffic irregularities and
+anomalies that may occur in the road network (i.e. an accident, a social event
+etc.) that could affect the expected and the orderly traffic behaviour of the road
+network.
+4. Response time: it is crucial to create a model able to answer users’ queries
+instantly and at the same time update its state in real time considering the
+recent traffic condition.
+To address the aforementioned challenges, we propose a Hybrid Travel Time
+Estimation framework, referred as HTTE. The framework achieves the estima-
+tion of the travel time for a given query path (laying on top of the road network),
+using data from moving vehicles. The proposed framework is capable of pro-
+viding predictions in real-time by exploiting the similarity of the road segments
+and by considering travel time reports provided by recent as well as historical
+trajectories. The contributions of this work can be summarized as follows:
+2
+
+• A latent representation for road segments: In order to treat the data sparsity
+problem in individual segments, we take advantage of the available traffic
+information from other segments with similar traffic behavior. We provide a
+mechanism for learning a latent representation for the road segments. This
+representation describes their traffic behaviour. Thus, road segments with
+similar traffic behaviour will be placed close in this latent space.
+• A Hybrid Estimation Model: We develop a streaming and hybrid estimation
+model that captures the recent traffic reports, the periodicity of the time
+series and the correlations among different road segments. Our framework
+models large areas of the city jointly and not the road segments individually,
+addressing this way the data sparsity problem.
+• We evaluate our method under realistic settings using data from buses and
+taxis and compare it with state of the art techniques. We show that hybrid
+techniques such as the one we propose outperform techniques that use only
+historical or only real-time and near real-time data. Experiments show that
+incorporating pathlets can improve the query efficiency up to 14 times with
+slight degradation in accuracy. This modification allows HTTE to work with
+interactive applications on a much larger scale.
+2
+Related Work
+Travel Time Estimation Using Static Sensor Data: A variety of tech-
+niques have been proposed in the literature for estimating the road segments’
+traffic flow or speed, exploiting static sensor data. Among these techniques, Qu
+et al. (2008) describes a matrix decomposition method, for estimating the traffic
+flow in Beijing, Li et al. (2013) proposes an extension of Probabilistic Principal
+Component Analysis and Kernel Principal Component Analysis that captures
+spatial and temporal dependencies and Wang et al. (2016b) uses deep convo-
+lutional neural networks for predicting the road segments’ speed. A DeepNN
+architecture able to capture spatial/temporal relations between road segments
+was proposed in Li et al. (2018b) using speeds from static sensors.
+Travel Time Estimation Using Dynamic Mobile Sensor Data: Many
+studies have explored the travel time estimation problem using moving sensors.
+In Zhan et al. (2013) the authors estimated the travel time in links, employing
+least-square optimization on taxi trip data that contained endpoint locations
+and trip metadata. A Bayesian mixture model was introduced in Zhan et al.
+(2016a) that estimated the short-term average urban link travel times with par-
+tial information available. The correlations between the travel times of nearby
+links and different time slots are crucial for inferring the traffic state of a partic-
+ular link Niu et al. (2014); Zhang et al. (2016). Online methods that determine
+the time required by a bus to reach a specified bus stop were proposed in Gal
+et al. (2017), Gal et al. (2018) and Yu et al. (2011). In Wang et al. (2016a) the
+authors propose a method that estimates the travel time by identifying near-
+neighbor trajectories, with similar origin and destination. The final estimation
+3
+
+of the travel time is the weighted average of the neighbors travel times. The au-
+thors in Jenelius and Koutsopoulos (2013) state that the travel time estimation
+can be approximated by the sum of the segments’ traversal time and a delay
+penalty that occurs at the links between the segments. In Zhan et al. (2016b)
+the authors propose a hybrid framework that incorporates (i) road network data,
+(ii) POI (Points of Interest), (iii) GPS trajectories and (iv) weather informa-
+tion to estimate the travel speed and the traffic volume. In Li et al. (2017) the
+authors proposed a technique that estimates the travel time using a small num-
+ber of GPS-equipped cars available, discovering local traffic patterns over a set
+of frequent paths, a.k. a. pathlets. At query time, a trajectory is decomposed
+into pathlets, whose recent travel time is estimated using pattern matching with
+recent travel time observations. A spatio-temporal hidden Markov model that
+models correlations among different traffic time series was proposed in Yang
+et al. (2013) taking into account the sparsity, the spatio-temporal correlation,
+and the heterogeneity of time series. A different approach was followed by Yang
+et al. (2018); Dai et al. (2016) assigning the weights to the paths instead of the
+edges of the road network, avoiding splitting the trajectories in small fragments.
+The authors in Yang et al. (2014) explored the use of weighted PageRank values
+of edges for assigning appropriate weights to all edges. The authors in Id´e and
+Sugiyama (2011) proposed a weight propagation model able to capture neigh-
+boring road-link dependencies and embedded the model to the regression task.
+In the same direction in Zheng and Ni (2013) the authors provided a multi
+task learning framework that simultaneously captures spatial dependencies and
+temporal dynamics encouraging spatio-temporal smoothness.
+Learning Latent Features on the Road Network: Recent techniques have
+suggested more sophisticated methods for taking advantage of historical data
+for travel time prediction. In Hofleitner et al. (2012) a technique that estimates
+the arterial travel time distributions is proposed, introducing hidden random
+variables that represent the road segments’ state (congested and undersatu-
+rated). Then a dynamic bayesian network learns the travel time distributions.
+A technique that detects the time-varying distribution of travel time of road seg-
+ments using Graph Convolutional Neural Network was introduced in Hu et al.
+(2019). In Deng et al. (2016) the authors proposed a method that imputes the
+short future speeds for the road segments, utilizing latent topological and tem-
+poral features learned and updated incrementally through matrix factorization.
+In Wang et al. (2014) the travel times of different road segments, drivers and
+time slots are modeled as a 3D sparse tensor. The missing values were filled
+in using the geospatial features and the recent and historical traffic informa-
+tion. A dynamic programming technique optimally concatenated the path into
+subpaths.
+Deep Learning Approaches: The recent success of deep learning in a variety
+of learning problems, lead to the design of deep learning architectures for the
+travel time estimation task. In DeepTravel Zhang et al. (2018) a deep learn-
+ing architecture is proposed with two major components. The first handles the
+representation of the features (spatial, temporal, driving state) with an embed-
+dings layer while the second consists of a BiLSTM layer that performs the actual
+4
+
+regression. An origin-destination travel time estimation method is MURAT Li
+et al. (2018a) that employs a graph embedding method for extracting roads’
+embeddings and an embedding layer for capturing the spatial and temporal fea-
+tures. These embeddings layers transform and provide the input to a Residual
+network.
+The authors in Wang et al. (2018a) proposed an end-to-end Deep
+learning framework for travel time estimation of an entire path (DeepTTE). A
+geo-convolution operation is proposed that handles the GPS points of the tra-
+jectory followed by a recurrent component. A multi-task learning component is
+used in order to learn both the total travel time of the given path and the travel
+times of smaller parts of the path. Finally in Wang et al. (2018b) the authors
+proposed a deep learning model that estimates the time of arrival using wide,
+deep and recurrent components.
+In essence, in our work we exploit novel techniques for the discovery of latent
+features in the spatial and temporal traffic data and at the same time leverage
+the use of sparse real-time information.
+3
+Our Approach
+3.1
+Problem Setup
+In this work we propose an efficient algorithm for estimating the travel time that
+is required for a vehicle to traverse a path of the road network. The proposed
+framework receives firstly as input a set of vehicles’ trajectories. Each trajectory
+is a sequence of time ordered spatial points T : (p1, t1) → · · · → (pn, tn), where
+each point pi ∈ R2 is the sampled GPS position and ti is the corresponding
+timestamp of the measurement. Then the points of the trajectories are mapped
+on a Road Network.
+A Road Network is defined as a topological structure
+of a network captured by a graph G, where the nodes of G correspond to a
+collection of road segments ri that link different urban areas together and the
+set of edges represent the connections between these road segments. A map-
+matched trajectory TG is a projection of a trajectory T in the road network G.
+TG : (r1, tin,1, t1) → · · · → (rn′, tin,n′, tn′) is defined as a sequence of the visited
+road segments ri along with the timestamps that the vehicle entered tin,i and
+left ti each road segment. In this work we are computing the estimated travel
+time of a given query path, without maintaining profiles for each driver.
+Each vehcile that traverses a segment ri generates a Travel Time Report
+Ri = (ri, ti, TTi). The reports Ri are available when the vehicle exits the road
+segment. TTi = ti−tin,i is the travel time required for the traversal and ti is the
+time when the vehicle left the road segment. The travel time reports are stored
+in a collection RH that is incrementally updated as new reports are provided.
+Also, a Road Segment Embedding is a mapping E : ri → RD , that maps a road
+segment ri of the road network G to a D-dimensional latent space. Finally, a
+query path Pq : rq1 → · · · → rqm is an ordered sequence of m consecutive road
+segments of G.
+Problem Definition. Given a query q that consists of a query path Pq and
+5
+
+Query
+q
+Offline 
+Data
+Real Time
+Data
+Road 
+Network
+Trajectory
+Database
+Map 
+Matching
+Segments 
+TT - RH
+Matrix 
+Factorization
+Embeddings
+Gaussian 
+Processes 
+Training
+Hyperpar
+ameters
+Gaussian 
+Process
+Real Time/ 
+Offline Data
+Travel Time 
+Estimation
+TTq
+Figure 1: Framework of our approach.
+a departure time tdep,q, predict the travel time TTq� that is required for a vehicle
+to traverse all the road segments of Pq departing at tdep,q using the collection
+of historical travel time reports RH that have been received until the time of the
+query.
+3.2
+Overview of the Approach
+The overview of our framework for estimating the travel time of a given query
+path is illustrated in Figure 1. Our framework has two major tasks. Initially, it
+aims to model the historical data by examining the traffic behavior of the road
+segments. Then it makes real-time predictions that exploit both historical and
+real-time information. Our architecture consists of the following modules:
+Module 1: Road Network & Trajectory Partitioning. The first pro-
+cessing component receives as input raw GPS trajectories and maps them onto
+paths in a road network, such as OpenStreetMap (OSM). In this case, the GPS
+points of a trajectory are mapped to road segments (i.e. OSM road segments)
+using the Barefoot2 library. Also, in this work we consider more abstract models
+of the road network. Under this case, the input GPS points could be mapped
+to sequences of road segments, or to pairs of GPS locations that represent a
+transition from an origin to a destination.
+One way to obtain a compact set of road segment sequences for learning the
+traffic behaviour is using the concept of pathlet dictionary. Given a set of map
+matched trajectories S, the pathlet dictionary (PD) is a collection of paths
+(road segment sequences) on the road network that reconstructs all trajectories
+in S by concatenation. Entries in the pathlet dictionary are referred as pathlets.
+A pathlet dictionary is considered optimal if it satisfies the following criteria:
+(i) The number of pathlets in the dictionary, |PD| is minimized. (ii) For each
+map matched trajectory TG ∈ S, the number of pathlets used to reconstruct TG,
+|p(TG)| is minimized.
+Although computing the optimal pathlet dictionary from a trajectory col-
+lection is an NP-hard problem, Chen et al. (2013) proposed an efficient approx-
+imation algorithm to find solution in O(|S|·n2) time, where |S| is the size of the
+trajectory collection and n is the maximum number of road segments. When the
+2https://github.com/bmwcarit/barefoot
+6
+
+Figure 2: Example segments mapping into the embedding space. The t-SNE
+technique is used in order to project the embeddings in a 3-dimensional space.
+dictionary is computed, it’s easy to query the decomposition of map matched
+trajectories using graph search creating the travel time reports for the pathlets.
+After the mapping to the road segments is completed using the former or the
+latter approach the travel time reports RH are generated, containing the time
+required to travel the road segments or the pathlets. This processing module is
+common for both the historical and the real-time data. These reports are the
+fundamental element of the proposed travel time estimation technique.
+Module 2: A latent representation for road segments.
+Identifying
+segments with similar traffic patterns is crucial for tackling the data sparsity
+problem. This component receives as input timestamped travel time reports for
+the various road segments. A latent representation for each segment is learned
+capturing the correlations among the road segments. That is, segments with
+similar traffic behaviour are placed close in the latent space. On the other hand,
+segments with divergent traffic behaviour are placed far apart in the latent space.
+Figure 2 on the right illustrates on the map several road segments with similar
+traffic behavior (similar embeddings). On the left part of the figure the mapping
+of these road segments into the embedding space is presented.
+Module 3: Travel Time Estimation. The final module estimates the time
+needed to travel a given query path Pq. This module is comprised by an offline
+and a real-time stage. A Gaussian Process (Williams and Rasmussen, 2006)
+model is trained offline with a set of historical reports RH using a complex
+covariance function, able to capture a variety of data aspects such as the data
+periodicity and the magnitude of the most recent values.
+Then our system
+receives in real-time queries q and estimates the total travel time required to
+traverse the given query path Pq, estimating the travel times of all the individual
+road segments of Pq.
+4
+THE HTTE Algorithm
+We describe the HTTE travel time estimation algorithm. Initially, we describe
+a technique for modeling the traffic data and for identifying road segments with
+similar traffic behavior. Then we discuss the desirable properties of our data
+and present a covariance function that captures these properties. Finally, we
+describe a method for predicting in real-time the travel times of the query paths.
+7
+
+40
+30
+20
+10
+0
+10
+-20
+30
+20
+30
+0
+10
+10
+20
+-10
+0
+30
+-30
+-20A Latent Representation for Road Segments
+Data sparsity is one of the major obstacles in estimating the travel time of a road
+segment since traffic information is provided only by a few vehicles. Information
+for the recent traffic state for the majority of road segments is often missing.
+It is essential to ensure that the method will use the traffic information from
+the segments for which we have recent reports in order to infer the state for a
+segment with similar traffic behavior but without recent information.
+Here we propose a technique that detects a latent embedding representation
+for the segments. The main property of this mapping is that segments with sim-
+ilar traffic behaviour should be placed close in this embedding space. This latent
+representation is used by the proposed prediction model in order to address the
+data sparsity issue. For the purpose of constructing the embedding representa-
+tion for the segments we decided to employ a matrix factorization approach. A
+similar technique is also used in Deng et al. (2016); Wang et al. (2014). Our
+aim is to discover some latent features that describe the road segments traffic
+characteristics. Each road segment ri and time window w is associated with
+a D-dimensional embedding vector of latent features. Then, the actual travel
+time report for the segment ri in the time window w can be approximated by
+the product of these two latent vectors. In order to satisfy this condition, the
+learned embeddings of segments with similar traffic behaviour should be close
+in the latent space.
+In order to apply the matrix factorization method we need to convert the
+historical reports RH to a sparse matrix M ∈ RN×W , where the N rows corre-
+spond to the segments ri of the road network G and the W columns correspond
+to all the time windows of the historical data w. Each time window has a size of
+30 minutes and each cell corresponds to the average travel time for a segment ri
+in this window (i.e. from 10:00 till 10:30), considering all the vehicles that tra-
+versed ri. After applying the matrix factorization the matrix M is decomposed
+into two matrices P ∈ RN×D, Q ∈ RW ×D such that M ≈ P × QT = M ′. The
+rows of P are the D-dimensional embeddings of the road segments and the rows
+of Q are the D-dimensional embeddings of the time windows. For estimating
+the matrices P and Q we minimize the Mean Squared Error (MSE) between
+the original matrix M and the matrix reconstruction M ′ using the Stochastic
+Gradient Descent (SGD). The SGD starts with the random matrices P and Q
+and at each step alters them considering the direction of the gradient of the
+objective function. The algorithm terminates when the objective function does
+not significantly change.
+Constructing the Covariance Function: Here, we first introduce the prop-
+erties that characterise the road segments’ travel times. Then we describe the
+complex covariance function that fits our data. Our aim is to predict the travel
+time of several queried road segments considering multiple travel time reports
+which are transmitted by the moving vehicles. Multiple evolving time-series are
+generated, one for each road segment, and our aim is to make accurate forecasts
+for their future traffic condition.
+To accurately estimate travel time the following key properties of the time
+8
+
+series data should be considered: (i) Periodicity: the traffic condition of the road
+segments is periodic in a daily basis, since commuters tend to follow similar trips.
+(ii) Correlation among road segments: The information provided by multiple
+road segments can be used in order to make predictions jointly, exploiting the
+correlations among the road segments and allows us to ameliorate the effects of
+data sparsity. (iii) Short term irregularities: even if the time series are periodic
+the traffic condition can be affected by multiple factors (i.e. constructions in
+the road network, an accident or a social event). This could generate traffic
+congestion events that are impossible to detect without monitoring the real-
+time traffic reports. (iv) Noisiness: the travel time reports are extremely noisy,
+for instance a driver may be stopped by a traffic light spending 1 minute waiting
+while another driver may not.
+In this work we use Gaussian processes to tackle the travel time estima-
+tion problem. We construct appropriately the covariance function providing an
+excellent fit to the data and characterizing the correlations among the differ-
+ent travel time reports in the process. Here we consider Gaussian processes
+with a zero mean function. Our goal is to model the travel time (TT) of the
+road segments as a function of the input vector x �� RD+1. x contains the D-
+dimensional embedding representation of the road segment along with the time
+that the vehicle left the road segment.
+x =
+�t, e�T
+(1)
+e =
+�
+e1, . . . , eD�
+(2)
+We model the daily variation of the road segments’ travel times using a
+periodic covariance function on the timestamp of measurements t, modified by
+taking the product with a squared exponential component on (i) the timestamp
+measurement t reducing the impact of older reports and (ii) the embeddings in
+order to reduce the impact of irrelevant road segments.
+k1(x, x′) = θ2
+1exp(−(t − t′)2
+2θ2
+2
+− (e − e′)T (e − e′)
+2θ2
+3
+− 2sin2(π(t − t′))
+θ2
+4
+)
+(3)
+The next term of the covariance function, models the medium term irreg-
+ularities and the correlations among similar road segments. This term uses a
+rational quadratic component on the timestamp t and a squared exponential
+component on the road segments’ embeddings e. Using this term the travel
+time prediction of a road segment is affected by the recent reports of road seg-
+ments with similar traffic behaviour (close in the embedding space), treating
+the data sparsity problem.
+k2(x, x′) = θ2
+5(1 + (t − t′)2
+2θ6θ7
+)−θ6exp(−(e − e′)T (e − e′)
+2θ2
+8
+)
+(4)
+Finally, a noise model is introduced considering the timestamp and the em-
+beddings of the datapoints.
+k3(x, x′) = θ2
+9exp(−(e − e′)T (e − e′)
+2θ2
+10
+− (t − t′)2
+2θ2
+11
+)
+(5)
+9
+
+The final covariance function is the sum of the previously described covari-
+ance functions, k(x, x′) = k1(x, x′) + k2(x, x′) + k3(x, x′) , with hyperparam-
+eters θ = [θ1 . . . θ11]. We empirically initialize these hyperparameters based on
+our prior beliefs about the data. During the learning procedure θ is automati-
+cally adjusted in order to appropriately fit the training data.
+Hybrid Travel Time Estimation (HTTE): We now present in detail our
+travel time estimation algorithm for the received query paths. Our algorithm,
+presented in Algorithm 1, consists of an offline and a real-time stage. The offline
+stage is responsible to initialize the required variables and is executed only once.
+The real-time stage of the algorithm receives in a streaming manner query paths
+and a departure time for each path and estimates instantly the corresponding
+travel time considering the already received travel time reports RH.
+Offline stage: Here we perform a set of tasks that initialize our system. Firstly,
+we compute the average travel time for each road segment for different times of
+the day, referred as TTavg. These average travel times will provide us later an
+approximate estimation of the departure time for each road segment of the given
+query path. In order to compute TTavg the day is partitioned in time windows
+of 30 minutes. Additionally, we compute for each road segment the average
+travel time and the standard deviation of travel time, referred as segsStats.
+The segsStats variable does not consider the time of the day. These statistics
+will be used later in order to standardize the travel time reports of the various
+road segments. Then our algorithm computes an embedding representation E
+for each road segment as it was described in Section 4. In the next step of the
+algorithm, a Gaussian process model, with the covariance function described in
+Section 4 and zero-mean is trained and its hyperparmeters θ are learned.
+Having all the travel time reports that have been received until now and
+the road segments statistics and embeddings, our next step is to initialize the
+Gaussian process models, defined as gpModels. In order to avoid generating a
+large Gaussian process model that would consider the travel time reports for all
+the city and the whole day we perform spatial and temporal partitioning of RH.
+This results in multiple gpModels and expedits the estimation of travel times
+queries. Each model is affiliated with a particular spatial area and time of the
+day. When a gpModel is generated, a covariance matrix K is constructed using
+the covariance function of Section 4, and the hyperparameters θ. The covariance
+matrix, for each gpModel, describes the correlation between the different travel
+time reports that have been received till now for a particular time window and
+area.
+Since the road segments have different lengths and their travel times
+deviate significantly, we decided to standardize the travel time reports for each
+road segment using the corresponding statistics segsStats. Thus the targets y
+for each gpModel are the standardized, with the statistics, travel times and not
+the raw travel times.
+The travel time reports are partitioned into different gpModels considering
+the location of the road segments and their timestamp (Figure 3). More specifi-
+cally the whole city is decomposed in smaller areas applying a grid of uniformly
+sized cells. In order to feed the model we allow spatial overlaps with neighbor-
+ing spatial grid cells in order to improve the accuracy for the road segments
+10
+
+Overlapping Area
+Query area
+...
+00
+01
+02
+10
+12
+13
+14
+21
+22
+23
+...
+11
+Query time 
+window
+Overlapping 
+time window
+Figure 3: Spatial and temporal partitioning.
+that are near the edges of each cell. Also, each day is partitioned into smaller
+time windows. Here we allow temporal overlaps with the previous and the next
+time windows, providing traffic information at the beginning of each query time
+window.
+The travel time reports that belong in overlapping areas and time
+windows are inserted into multiple gpModels. Finally, each gpModel contains
+the recent and historical reports of each query and overlapping time window
+and area that is associated with.
+Real-time stage:
+Our system receives query paths in real-time from the
+QueriesStream and performs instantly the travel time estimation for each given
+query. Initially our algorithm updates the gpModels adding the newly received
+travel time reports. The update of the gpModels is performed in predetermined
+periods and not every time a travel time report or a query is received. Such
+frequent updates would be time consuming. In order to do this we identify the
+current time window of the day w, considering the current time. If the window w
+has changed from the window of the previous query prevW then the gpModels
+are updated. Each gpModel is updated extending its covariance matrix K with
+the travel time reports that have been received from the previous update of the
+gpModel for the investigated spatial area and time window.
+The next step of the algorithm is to decompose the given query path in a
+set of individual road segments (SubQueries) and estimate their travel times
+querying the corresponding gpModels. In order to query the gpModels it is
+required to estimate an approximate departure time ti for each road segment
+rqi of the given query path. In order to approximate the departure times we
+begin with the first road segment rq1 of the query path setting as departure
+time t1 for this road segment the trip’s departure time tdep,q. Then in order to
+estimate the departure time for the next road segment rqi we add to the previous
+road segment departure time ti−1 the average travel time TTavg of the previous
+road segment rqi−1, as it was computed in the offline stage of the algorithm.
+This procedure iterates till the last road segment of the query path. Having an
+approximate estimation for the departure time for each road segment will allow
+to perform batch queries to the affected gpModels of the query path, speeding
+up the execution time of the queries.
+Finally, individual queries for the road segments’ travel times are posed
+to the appropriate gpModel, considering the spatial and temporal partitioning.
+The gpModels return standardized travel times, thus the segsStats are required
+in order to get the actual travel time estimations. The total travel time of the
+query path TT�q is updated considering the estimates of the gpModels for the
+individual road segments.
+Finally, TT�q is the estimated travel time for the
+11
+
+DRUMCONDRA
+[R807]
+R803
+ad
+R131]
+FainviewPark
+R834
+CrokePark
+R101
+R803
+OADSTONE
+R131
+EastPointBusinessPark
+H
+The Mater Misericordiae
+University Hospital
+R802]
+NORTHSTRAND
+[R803]
+Mountjoy
+R105]
+Square Park
+M50
+Dublin City Gallery
+JamesJoyceCentre
+The Hugh Lane
+EASTWALL
+Rotunda Hospital C
+R803
+R101]
+R131
+R108
+NT
+St Mary'sProCathedral
+CA.
+国
+Dublin
+R101
+NORTH WALL
+R10S]
+R801
+EPICTheIrish
+3Arena
+rch
+EmigrationMuseum
+Ha'penny Bridge
+R802
+R814
+RiverLiffey
+R148
+Trinity College
+R131
+TEMPLEBAR
+Dublin
+Bord Gais
+Cathedral
+EnergyTheatre
+Dublin Castle
+R131
+R108]
+R802
+National Gallery
+oscarwilde House
+R802
+Ringsend
+of Ireland
+R118
+TheLittleMuseum
+R138
+R802
+Cathedral
+of Dublin
+Merrion
+R815]
+国
+Square
+STELLA GARDENS
+R131
+StStephen's
+TheFitzwilliam
+Technological
+UniversityDublin R110
+Green
+Casinoand Card Club
+R815
+R137
+R111
+Irishtown
+R110
+Dicey's Garden ClubO
+BEGGAR'SBUSH
+UNT
+AvivaStadium
+Hey
+R114
+Iveagh
+TheNational
+Gardens
+ConcertHall
+R118]
+Royal Victoria Eye H
+国
+R816M50
+R108]
+R104]
+R809]
+R107
+Finglas
+R103]
+RT
+Coolock
+NCHARDSTOWN
+R135
+Glasnevin
+WHITEHALL
+Artane
+Kilba
+RahenyR105
+M50
+R147
+R102
+[R103]
+R808)
+National
+R102
+M50
+R807
+Castleknock
+Botanic
+R108
+R107
+Gardens
+NT
+R808
+Saint
+R806
+Marino
+R105]
+AnnesPark
+ASHTOWN
+Cabra
+Clontarf
+Fairview
+Dollymount
+R806]
+DRUMCONDRA
+R147
+[R80S]
+BROADSTONE
+EastPointBusinessPark
+DublinZoo
+R101]
+nerstown
+R105
+PhoenixPark
+M50
+QDublin Port
+JamesonDistillery Bow St
+Dublin
+3Arena
+R109
+R112
+R109
+R801
+R111R148
+[R802]
+R131
+Ballyfermot
+R833
+DublinEhbourg
+GuinnessStorehouse
+AD
+StStephen's
+Rinasend
+R810
+StPatrick'sCathedral
+Green
+INCHICORE
+R812
+R111
+DolphinsBarn
+West
+Bluebell
+DRIMNAGH
+R111
+x&Geese
+Ballsbridge
+R131
+Ranelagh
+R112
+R114
+Rathmines
+R815
+R110
+R117
+.
+Ballymount
+R817
+Rathgar
+Donnybrook
+IndustrialEstate
+Greenhills
+R820
+TERENURE
+UCDInstitute
+forDiscovery
+Booterstown
+M5D
+R112]
+R118
+[R838]
+TymonPark
+R138
+R817]
+R112
+Blackrock
+[R137]
+Kilnamanagh
+R117
+R825
+N31
+R114
+R112
+DUNDRUM
+R825
+Monks
+R821]
+Goatstown
+R817
+N11
+Tallaght
+R115
+M50
+Ballyboden
+R826
+R826
+R113]
+R827
+Firhouse
+Knocklyon
+R822
+RE
+R113]Algorithm 1: Travel Time Estimation Algorithm
+Data: RH, QueriesStream = [q1, . . . , q∞]
+Result: TT�1, . . . , TT�∞)
+1 Offline Stage;
+2 TTavg ← computeAvgTravelTime(RH);
+3 segsStats ← computeRoadSegmentsStats(RH);
+4 E ← computeEmbeddings(RH, segsStats);
+5 θ ← computeHyperparametersGP(RH, E, segsStats);
+6 gpModels ← initializeMultipleGPs(RH, E, segsStats, θ);
+7 prevW ← None;
+8 Online Stage;
+9 foreach q =< Pq, tdep,q > in QueriesStream do
+10
+TT�q ← 0;
+11
+w ← getTimeWindow();
+12
+if w ̸= prevW then
+13
+gpModels.update(RH, segsStats, θ);
+14
+prevW ← w;
+15
+SubQueries ← decompose path(Pq, tdep,q, TTavg);
+16
+foreach < ri, ti >∈ SubQueries do
+17
+gpModel ← findGP(ri, ti);
+18
+TT�q ← TT�q + gpModel.query(ri, ti, segsStats)
+query q.
+5
+Conclusion
+We develop a novel hybrid technique for travel time estimation, that considers
+recent and historical traffic reports. An embedding representation for each road
+segment is learned based on its traffic behaviour. This representation is incor-
+porated by a regression technique, handling the data sparsity problem. This
+allows our technique to make accurate estimations even if there are no recent
+traffic reports available for a segment. Finally, our technique adapts different
+levels and types of abstraction that allow the real-time travel time estimation.
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+15
+

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+THE ONE-VISIBILITY LOCALIZATION GAME
+ANTHONY BONATO, TRENT G. MARBACH, MICHAEL MOLNAR, AND JD NIR
+Abstract. We introduce a variant of the Localization game in which the cops only
+have visibility one, along with the corresponding optimization parameter, the one-
+visibility localization number ζ1.
+By developing lower bounds using isoperimetric
+inequalities, we give upper and lower bounds for ζ1 on k-ary trees with k ≥ 2 that
+differ by a multiplicative constant, showing that the parameter is unbounded on k-
+ary trees. We provide a O(√n) bound for Kh-minor free graphs of order n, and we
+show Cartesian grids meet this bound by determining their one-visibility localization
+number up to four values. We present upper bounds on ζ1 using pathwidth and the
+domination number and give upper bounds on trees via their depth and order. We
+conclude with open problems.
+1. Introduction
+Pursuit-evasion games, such as the Localization game and the Cops and Robber
+game, are combinatorial models for detecting or neutralizing an adversary’s activity on
+a graph. In such models, pursuers attempt to capture an evader loose on the vertices
+of a graph. How the players move and the rules of capture depend on which variant is
+studied. Such games are motivated by foundational topics in computer science, discrete
+mathematics, and artificial intelligence, such as robotics and network security.
+For
+surveys of pursuit-evasion games, see the books [7, 11]; see Chapter 5 of [11] for more
+on the Localization game.
+Among the many variants of the game of Cops and Robbers, one theme is to limit the
+visibility of the robber. For a nonnegative integer k, in k-visibility Cops and Robbers,
+the robber is visible to the cops only when a cop is distance at most k. The case when
+k = 0 has been studied [17, 18, 23], as has the case when k = 1 [26, 27, 28], and a recent
+paper covers the cases k ≥ 1 [16].
+The Localization game was first introduced for one cop by Seager [15, 21]. The game
+in the present form was first considered in the paper [15], and subsequently studied in
+several papers such as [2, 8, 9, 10, 12, 13, 14]. We consider a novel analogue of one-
+visibility Cops and Robbers in the setting of the Localization game. In the one-visibility
+Localization game, there are two players playing on a graph, with one player controlling
+a set of k cops, where k is a positive integer, and the second controlling a single robber.
+The game is played over a sequence of discrete time-steps; a round of the game is a
+move by the cops and the subsequent move by the robber. The robber occupies a vertex
+of the graph, and when the robber is ready to move during a round, they may move to a
+neighboring vertex or remain on their current vertex. A move for the cops is a placement
+2020 Mathematics Subject Classification. 05C57,05C12.
+Key words and phrases. localization number, limited visibility, pursuit-evasion games, isoperimetric
+inequalities, graphs.
+1
+arXiv:2301.03534v1  [math.CO]  9 Jan 2023
+
+2
+A. BONATO, T.G. MARBACH, M. MOLNAR, AND JD NIR
+of cops on a set of vertices (note that the cops are not limited to moving to neighboring
+vertices). The players move on alternate time-steps, with the robber going first. In each
+round, the cops C1,C2,...,Ck occupy a set of vertices u1,u2,...,uk and each cop sends
+out a cop probe di, where 1 ≤ i ≤ k. If a cop Ci is on the vertex of the robber, then di = 0.
+If the cop Ci is adjacent to the robber, then di = 1. In all other cases, the cop probe
+returns no information, and we set di = ∗. Hence, in each round, the cops determine a
+distance vector D = (d1,d2,...,dk) of cop probes. Relative to the cops’ position, there
+may be more than one vertex x with the same distance vector. We refer to such a vertex
+x as a candidate of D or simply a candidate. The cops win if they have a strategy to
+determine, after a finite number of rounds, a unique candidate, at which time we say
+that the cops capture the robber. We assume the robber is omniscient, in the sense
+that they know the entire strategy for the cops. If the robber evades capture, then the
+robber wins. For a graph G, define the one-visibility localization number of G, written
+ζ1(G), to be the least positive integer k for which k cops have a winning strategy in
+the one-visibility Localization game. The standard Localization game is played in the
+same way, except that each cops’ probe returns di as the distance between this cop and
+the robber. The localization number is the minimum number of cops required for this
+game and is denoted ζ(G).
+For a graph G of order n, ζ(G) ≤ ζ1(G), as a winning cop strategy in one-visibility
+localization will also be winning in the Localization game. Further, ζ1(G) ≤ n − 1. If
+G has diameter at most 2, then a probe of ∗ by a one-visibility cop can only represent
+a distance of 2, and so ζ(G) = ζ1(G). We may define ζj for all integers j ≥ 0 in an
+analogous fashion, although we will only consider the case j = 1 in this paper. A recent
+work [3] introduced the so-called zero-visibility search game, which is equivalent to ζj
+in the case when j = 0.
+We illustrate briefly how the parameters ζ(G) and ζ1(G) may differ. Spiders are
+trees with exactly one vertex of degree at least 3. This vertex is referred to as the head,
+and the paths from the head to the leaves, not including the head, are referred to as
+arms. Let G be the spider consisting of head vertex r and three arms of length three.
+It is straightforward to see that ζ(G) = 1; however, ζ1(G) = 2. To see that ζ1(G) > 1,
+suppose one cop plays. The robber chooses one of the neighbors of r on an arm the cop
+will not probe first and passes until the cop is about to probe on that arm. When the
+cop moves to the robber’s arm, the robber moves to r. Anticipating the next cop probe,
+they move to a neighbor of r on an arm that will not be probed in the next round and
+the process repeats. To see that ζ1(G) ≤ 2, have one cop probe r in every round, and
+the other cop scans each of the arms until the robber is captured.
+The paper is organized as follows. We begin in Section 2 by considering a relaxation
+of the one-visibility Localization game to the one-proximity game, where the robber
+is captured if they occupy a neighbor of the cop. We consider bounds on ζ1 in terms
+of the corresponding one-proximity number prox1(G). In Section 3, we give several
+techniques for bounding ζ1(G) and prox1(G) for general graphs G. Upper bounds are
+given using pathwidth and the domination number, and are found for certain minor-free
+graphs. Lower bounds are derived by using isoperimetric inequalities and a new graph
+parameter we call the h-index. In Section 4, we show that ζ1 and prox1 differ on trees
+by at most 1. We derive upper bounds on trees via their depth and order and give lower
+bounds on k-ary trees with k ≥ 2 using their isoperimetric peaks. One consequence of
+
+THE ONE-VISIBILITY LOCALIZATION GAME
+3
+these results is that the one-visibility number is unbounded on the family of k-ary trees;
+this contrasts significantly from the Localization game, where trees have localization
+number at most 2. We consider Cartesian grid graphs in Section 5 and derive bounds
+there that differ by four values. We conclude with further directions and open problems.
+All graphs we consider are finite, undirected, reflexive, and do not contain multiple
+edges. We only consider connected graphs, unless otherwise stated. The set of vertices
+that share an edge with x is denoted N(x), and we refer to vertices in N(x) as neighbors
+of x. Although our graphs are reflexive, we insist that x ∉ N(x). We define N[x] =
+N(x)∪{x}. For a set S of vertices, N[S] = ⋃u∈S N[u]. For a graph G, let ∆(G) b e the
+maximum degree of a vertex in G. For further background on graph theory, see [25].
+2. The one-proximity game
+Before we present results on the one-visibility Localization game, we give a simpler
+version that will prove useful for bounding ζ1(G). In the one-proximity game, play is
+defined as in the one-visibility Localization game, except that the cops win immediately
+if any probe returns a distance other than ∗. We call the corresponding graph parameter
+the one-proximity number, written as prox1(G). This game corresponds to the probes
+returning perfect information about the neighborhood of a vertex, rather than merely
+whether or not the robber is adjacent to the probed vertex. Note that prox1 is the
+analogue of the one-visibility seeing cop-number c′
+1, where the robber is captured if they
+are in the neighborhood of a cop; see [16].
+Observe that prox1(G) ≤ ζ1(G), as the one-proximity game cops can use the same
+strategy as the Localization game cops until the final round when, having located the
+robber, the one-proximity game cop probes the robber’s last known location and must
+be within distance one from the robber. As noted for k-visibility Cops and Robber in
+[16], seeing the robber for the first time could be much more resource intensive than the
+subsequent capture. The extra expense cannot be too large, however.
+Theorem 1. For every graph G, we have
+ζ1(G) ≤ ∆(G)prox1(G).
+Proof. Suppose that when prox1(G) cops play the one-proximity game, and that if these
+cops move on the vertices Vt in round t, then the cops win. We play with ∆(G)prox1(G)
+cops in the one-visibility Localization game. In round t, for each u ∈ Vt, a cop is placed
+on u and on ∆(G)−1 of the at most ∆(G) vertices in N(u), chosen arbitrarily. We know
+that in some round t′, there is a v ∈ Vt′ such that the robber is in N[v]. (This was the
+requirement for the cops to win independent of the robber strategy in the one-proximity
+game.) Before round t′, every cop in the one-proximity game received a distance of ∗,
+so the cops in the one-visibility Localization game are playing with no less information.
+In round t′, we have either a cop on the same vertex as the robber, or the robber is on
+the unique vertex in N(v) that does not contain a cop. In the latter case, the robber’s
+exact location is now known, so they are captured.
+□
+Theorem 1 is tight on the complete graphs. We can say more if prox1(G) is large
+compared to the maximum degree of G. We do not claim the bound on prox1 in the
+hypothesis of the following theorem is optimal.
+
+4
+A. BONATO, T.G. MARBACH, M. MOLNAR, AND JD NIR
+Theorem 2. If G is a graph and prox1(G) ≥ ∆(G)2, then ζ1(G) = prox1(G).
+Proof. The prox1(G)-many cops play the one-visibility Localization game, following a
+winning strategy from the one-proximity game. At some point, since the strategy is
+winning, at least one probe returns a distance of 1 on vertex v. On the robber’s move,
+the robber moves to a vertex of distance 0, 1, or 2 from v.
+During the cops’ next move, a cop is placed on each vertex of distance 1 or 2 from
+v, which requires at most ∆(G)2 cops. Either a cop on some vertex u probes 0 and the
+robber is found on u, or no cop probes 0 and the robber is found on v.
+□
+One benefit of considering the one-proximity game instead of the one-visibility Local-
+ization game is the success or failure of the cops strategy is independent of the robber’s
+strategy. Let G be a graph, and for t ≥ 1, let Vt denote the set of vertices probed by the
+cops in round t. Define St = St(G,{V1,...,Vt}) to be the set of vertices on which the
+robber may reside immediately after the cops’ tth move without having been captured.
+Consider the following three properties:
+(1) the robber can start on any vertex;
+(2) if the robber was on some vertex v before the robber’s (t+1)th move, then they
+can move to any vertex in N[v] on their (t + 1)th move; and
+(3) the robber is captured on the cops’ (t + 1)th move if they are in a vertex of
+⋃v∈Vt+1 N[v].
+The following theorem translates these properties to statements about St. For a set
+of vertices S, let δ(S) be the set of vertices not in S that are adjacent to some vertex
+in S. To avoid conflicting notation, we do not use δ to denote the minimum degree of
+a graph.
+Lemma 3. When playing the one-proximity game on a graph G where the cops play
+on Vt in round t, the robber can be on a vertex u if and only if u ∈ St, where
+(1) S1 = V (G); and
+(2) St+1 = (St ∪ δ(St)) ∖ (⋃v∈Vt+1 N[v]).
+Proof. The robber may start on any vertex, so S1 = V (G). Immediately before the
+robber’s (t + 1)th move, the robber may be on a vertex v if and only if v ∈ St. The
+robber uses their (t + 1)th move to occupy some vertex in N[v]. Thus, the robber can
+be on vertex v if and only if v ∈ (St ∪ δ(St)) after the robber’s (t + 1)th move.
+The robber will be captured on the cops’ (t+1)th move if and only if it is in a vertex
+of ⋃v∈Vt+1 N[v].
+Therefore, the robber remains uncaptured after the cop’s (t + 1)th
+move if and only if it is on a vertex in (St ∪ δ(St)) ∖ (⋃v∈Vt+1 N[v]). This completes the
+proof.
+□
+There are a variety of different terminologies for the sets St. These can be called the
+robber territory, or the set of contaminated vertices. We use the term contaminated,
+denoting these vertices as red in the figures. The vertices not in St are usually called
+either clean or cleared. We use the term cleared and denote these vertices as white in
+any figures. A set of vertices is contaminated (respectively, cleared) if all of its contained
+vertices are contaminated (respectively, cleared). We say a cleared set S is fully cleared
+when the robber can never return to recontaminate the vertices of S under the given
+cop strategy.
+
+THE ONE-VISIBILITY LOCALIZATION GAME
+5
+Lemma 3 is the one-visibility Localization game equivalent of Proposition 10 of [3],
+which is a result about the zero-visibility Localization game. As a result, we can treat
+play in the one-proximity game as a single-player game where the cops clear vertices on
+their move and the contamination spreads between the cops’ moves. This will often be
+easier to analyze because the robber strategy is no longer necessary.
+3. Bounds on ζ1
+In the present section, we focus on several bounds for ζ1, including upper bounds
+using pathwidth, the domination number, and one using properties of certain minor-
+free graphs. We finish by giving lower bounds using isoperimetric inequalities.
+3.1. Upper bounds. We begin with an upper bound using pathwidth. In [12], the lo-
+calization number of a graph is bounded above by the graph’s pathwidth. An analogous
+result holds for the one-localization number.
+Given a graph G, a path-decomposition of G is a pair (X,P), where the set X =
+{B1,B2,...,Bn} consists of subsets of V (G) called bags, and P is a path whose vertices
+are the bags Bi, satisfying the following properties:
+(1) V (G) = ⋃n
+i=1 Bi;
+(2) for every edge (u,v) ∈ E(G), there exists a bag that contains both u and v; and
+(3) for all 1 ≤ i ≤ k ≤ j ≤ n, Bi ∩ Bj ⊆ Bk.
+The width of the path-decomposition is the cardinality of its largest bag minus 1, and
+the pathwidth of the graph G, denoted pw(G), is the minimum width among all possible
+path-decompositions of G. While the proof of the following theorem is analogous to the
+proof bounding the localization number by pathwidth given in [12], we include it for
+completeness.
+Theorem 4. For any graph G, ζ1(G) ≤ pw(G).
+Proof. Assume G has at least two vertices and let P be a path-decomposition of G.
+Without loss of generality, linearly order the bags B1,B2,...,Bk from left to right. For
+all 1 ≤ i ≤ k we assume Bi ∖Bi+1 is nonempty; otherwise, Bi can be eliminated from the
+path-decomposition. Furthermore, for every u ∈ Bi ∖ Bi+1, we assume u has a neighbor
+in Bi. If this were not the case, then we remove u from bag Bi without changing the
+path-decomposition.
+For each 1 ≤ i < k, let ui be a fixed vertex in Bi ∖Bi+1 and let vi be a neighbor of ui in
+Bi. Also let uk be a vertex in Bk ∖Bk−1 and vk be a neighbor of uk in Bk. Sequentially,
+for i = 1,2,...,k, the cops probe each vertex of Bi ∖ vi.
+Starting with B1, which is a leaf of P, cops probe B1 ∖ v1. Suppose the robber is in
+B1. If they are in B1 ∖v1, then they are captured since a cop will probe 0. If the robber
+is on v1, the cop at u1 probes 1. Since u1 must have a neighbor in B1 and no cop has
+probed 0, the robber is captured at v1. In this way, we can ensure the robber is not in
+B1. We then proceed inductively, probing the vertices in Bj ∖ vj, for j > 1, to ensure
+the robber is not in Bi, with i ≤ j. The robber is forced to move into Bk where they
+will be captured.
+□
+The bound in Theorem 4 is tight for complete graphs Kn, as pw(Kn) = ζ1(Kn) = n−1.
+We note that proof of Theorem 4 also holds for the zero-visibility Localization game,
+
+6
+A. BONATO, T.G. MARBACH, M. MOLNAR, AND JD NIR
+except that we must place a cop on all vertices of a bag as we sequentially probe the
+bags. Therefore, we also have the following.
+Lemma 5. For any graph G, ζ0(G) ≤ pw(G) + 1.
+For graphs which are C4-free (that is, do not contain the 4-cycle as a subgraph),
+we have the following bound for the one-visibility localization number in terms of two
+common graph parameters, including the domination number, written γ(G).
+Theorem 6. If G is C4-free, then ζ1(G) ≤ γ(G) + ∆(G).
+Proof. Let S = {v1,v2,...,vγ(G)} be a dominating set of G and have γ(G)-many cops
+probe each vertex of S in each round. On the first probe, either some cop probes 0 and
+the robber is captured, or the robber is on some vertex u ∈ V (G) ∖ S, and there is at
+least one cop who probes 1. Say this is the cop at v1.
+In the next round, we will use the additional ∆(G)-many cops to probe each neighbor
+of v1, and so the robber must move to avoid capture. The robber moves to w ∈ V (G)∖S,
+where w is not a neighbor of v1. On the next round of probes, since S is a dominating
+set, there will be some other cop, say the one at v2, who probes 1, while the cop at
+u also probes 1. If v2 were adjacent to a second neighbor of u, then G would contain
+a 4-cycle. Therefore, the cops can uniquely determine the robber’s location to be at
+w.
+□
+Our next result provides an upper bound on ζ1 for a large family of graphs. A graph
+H formed from G by first taking a subgraph and then contracting some of the remaining
+edges is said to be a minor of G. The family of Kh-minor free graphs includes planar
+graphs in the case h = 5. The following separator theorem for Kh-minor-free graphs will
+be useful for bounding the one-localization number.
+Theorem 7 ([1]). If h ≥ 1 is a fixed integer and G is a Kh-minor-free graph of order
+n, then there are sets of vertices A, B, and C so that no vertex in A is adjacent with a
+vertex in B, neither A nor B contains more than 2/3n vertices, and C contains no more
+than h3/2√n vertices.
+We refer to the set C in Theorem 7 as a separator, and the sets A and B of order at
+most 2/3n as parts. The following theorem gives an upper bound on the ζ1 number for
+several classes of graphs, including planar graphs. The proof uses a divide-and-conquer
+approach. Later in the paper, we will give bounds Ω(√n) in square grids. We do not
+attempt to optimize constants in the upper bound.
+Theorem 8. If h > 3 and n are integers, and G is a Kh-minor free graph of order n,
+then ζ1(G) = O(√n). In particular, if G is planar of order n, then ζ1(G) = O(√n).
+Proof. Define the function f(m) = h3/2√m(
+1
+1−
+√
+2/3) + √n. We write P(m) to be the
+statement that for each Kh-minor-free graph G of order m, there exists a strategy using
+at most f(m) cops to capture the robber on G. We apply induction, assuming that
+P(m) is true for 1 ≤ m ≤ n−1, and show that P(n) holds. Once that is established, the
+proof of the theorem follows.
+The base cases are when 1 ≤ m ≤ √n. In any such graph, we can place a cop on every
+vertex to capture the robber in one round. This uses at most √n ≤ f(m) cops, and so
+P(m) is true in these cases.
+
+THE ONE-VISIBILITY LOCALIZATION GAME
+7
+For the inductive step, for a graph G of order n, we apply Theorem 7 to give a
+separator C of G with ∣C∣ ≤ h3/2√n, with parts A and B of cardinalities at most 2n/3.
+We note that A and B are not necessarily connected; however, they are both Kh-minor-
+free graphs.
+The cops employ a strategy in two phases. During both phases (and hence, in all
+rounds), at most h3/2√n cops are played on C so that each vertex in C contains a cop;
+we label this set of cops by X. As such, the robber cannot move between A and B
+without being captured on a cop move.
+We know by the inductive hypothesis that a strategy exists to capture the robber on
+A using at most f(∣A∣) cops. Therefore in the first phase, we play this strategy on A
+using the cops while also playing the cops in X on C. After this, the robber will be
+captured if it was ever on a vertex in A or C, and so we may assume that the robber is
+now in B after the cops’ last move.
+There similarly exists a strategy to capture the robber on B using at most f(∣B∣)
+cops, and in the second phase, we play this strategy on B while also playing the cops in
+X on C. After this process, the robber will be captured either on a vertex of B or C.
+Assuming without loss of generality that ∣A∣ ≥ ∣B∣, we used at most
+max(h3/2√n + f(∣A∣),h3/2√n + f(∣B∣))
+≤
+h3/2√n + f(∣A∣)
+≤
+h3/2√n + h3/2√
+2n/3⎛
+⎝
+1
+1 −
+√
+2/3
+⎞
+⎠ + √n
+=
+h3/2√n⎛
+⎝
+1
+1 −
+√
+2/3
+⎞
+⎠ + √n
+cops to capture the robber on G. In the first inequality, we used the fact that ∣f(B)∣ ≤
+∣f(A)∣, while the second follows by inductive hypothesis. Hence, P(n) holds, and the
+proof follows.
+□
+Interestingly, we show that the bound in Theorem 8 is tight in the sense that there
+exist planar graphs G (in particular, Cartesian grid graphs) with ζ1(G) =
+√
+∣V (G)∣ +
+O(1).
+3.2. Lower bounds from isoperimetric inequalities. The isoperimetric problem
+of a graph G asks for the minimum cardinality of the boundary of a set of vertices,
+given the set of vertices has cardinality k. For a subset of vertices S, this border can
+be either the vertex border δ(S) = N[S] ∖ S, or the edge border
+∂(S) = ∣E(S,S)∣ = ∣{(u,v) ∈ E(G) ∶ u ∈ S,v ∉ S}∣,
+which is the set of edges that have exactly one endpoint in S.
+We consider the following two standard isoperimetric parameters for graphs:
+ΦE(G,k) =
+min
+S⊆V ∶∣S∣=k ∣∂(S)∣,
+ΦV (G,k) =
+min
+S⊆V ∶∣S∣=k ∣δ(S)∣.
+The isoperimetric problem for either of these two parameters asks for an exact eval-
+uation of ΦE(G,k) or ΦV (G,k), while an isoperimetric inequality is a bound on these
+
+8
+A. BONATO, T.G. MARBACH, M. MOLNAR, AND JD NIR
+values. The edge isoperimetric problem is also studied along with the problem of max-
+imizing the number of edges between vertices of a k-set of vertices, maxS∶∣S∣=k ∣E(S,S)∣.
+For a survey on edge isoperimetric inequalities, see [4].
+The two isoperimetric problems are closely related. We have that ΦV (G,k) ≤ ΦE(G,k).
+Since each vertex in δ(S) is incident to at most ∆(G) vertices in S, it follows that
+ΦE(G,k) ≤ ∆(G)ΦV (G,k). Therefore, these parameters differ at most by a factor of
+∆(G):
+ΦE(G,k)
+∆(G)
+≤ ΦV (G,k) ≤ ΦE(G,k).
+The isoperimetric peak is the maximum over the isoperimetric numbers on the graph:
+ΦE(G) = max
+k
+ΦE(G,k),
+ΦV (G) = max
+k
+ΦV (G,k).
+The vertex isoperimetric peak has been explicitly studied for trees [5, 20, 24], although
+further results for the isoperimetric peak problem appear implicitly within many works.
+We introduce a modification to this concept inspired by the h-index metric of citation
+metrics. For some function f ∶ Z → Z, define the h-index function on f, H(f), as follows:
+H(f) = max{h ∈ Z ∶ for some k1, we have f(k) ≥ h for k1 ≤ k ≤ k1 + h − 1}.
+In particular, there are H(f) consecutive integers k1 ≤ k ≤ k1 + h − 1 with f(k) ≥ H(f).
+The vertex-h-index of a graph is
+HV (G) = H(ΦV (G,k)),
+and similarly, the edge-h-index of a graph is
+HE(G) = H(ΦE(G,k)).
+See Figure 1 for an illustration of HV (G).
+The following lemma establishes inequalities for the HV and HE parameters.
+Lemma 9. For a graph G, we have that
+HE(G)
+∆(G) ≤ HV (G) ≤ HE(G).
+Proof. Let h = HV (G). By the definition of HV (G), there exists an integer ka such that
+each k ∈ [ka,...,ka + h − 1] satisfies ΦV (G,k) ≥ h. However, since ΦE(G,k) ≥ ΦV (G,k),
+this gives that ΦE(G,k) ≥ h for k ∈ [ka,...,ka + h − 1]. Therefore, HE(G) ≥ h = HV (G)
+by the definition of HE.
+Let h = HE(G). By the definition of HE(G), there exists an integer ka such that each
+k ∈ [ka,...,ka + h − 1] has ΦE(G,k) ≥ h. However, since ΦV (G,k) ≥ ΦE(G,k)/∆(G),
+this gives that ΦV (G,k) ≥ h/∆(G) for k ∈ [ka,...,ka+⌈h/∆(G)⌉−1] ⊆ [ka,...,ka+h−1].
+Therefore, HV (G) ≥ h/∆(G) = HE(G)/∆(G) by the definition of HV .
+□
+The following theorem gives a lower bound on prox1(G) in terms of HV (G), and
+hence, gives a lower bound for ζ1(G).
+Theorem 10. If G is a graph, then
+prox1(G) > HV (G)
+∆(G) + 1.
+
+THE ONE-VISIBILITY LOCALIZATION GAME
+9
+k
+ΦV (G,k)
+ka
+kb = ka + HV (G) − 1
+HV (G)
+k1
+k2
+k3
+cleared
+re-contaminated
+Figure 1. A graph of Φ(G,k) that illustrates HV (G), where a contigu-
+ous set of HV (G) integers each have Φ(G,k) ≥ HV (G). For Theorem 10,
+if the cops manage to reduce the number of contaminated vertices k be-
+low kb + 1 (in our example, moving from k = k1 to k = k2 that is in
+the gray region), then the contamination is guaranteed to grow to some
+cardinality at least kb + 1 (that is, moving from k = k2 to k = k3 in our
+example, which is to the right of the gray region).
+Proof. We may assume HV (G) ≥ (∆(G)+1), or else the proof is immediate. See Figure
+1 as an aid to this proof. As a high-level overview of the proof, p cops can clear at most
+p(∆(G) + 1) vertices per round (that is, a cop on u clears the at most ∆ + 1 vertices in
+N[u]) and the contamination spreads at a rate of at least ΦV (G,k), with k being the
+number of currently contaminated vertices. If ΦV (G,k) is larger than p(∆(G) + 1) for
+enough contiguous values k, then there will always be some point in the game where
+the contamination will grow faster than the cops can clear the contamination.
+Consider a game in which the cop player controls p = ⌊ HV (G)
+∆(G)+1⌋ cops. By the definition
+of HV (G), there must exist ka and kb = ka + HV (G) − 1 such that for any value k ∈
+{ka,ka + 1,...,kb} we have ΦV (G,k) ≥ HV (G). Note that ka > 1, since
+ΦV (G,1) = min
+v∈V (G)deg(v) < ∆(G) + 1 ≤ HV (G),
+so the inequality ΦV (G,1) ≥ HV (G) does not hold.
+Suppose there are at least kb+1 contaminated vertices just before the cops move. If a
+cop plays on vertex v, then the vertices on N[v] that were contaminated are no longer
+contaminated. As a consequence, after this cop round at most p(∆(G) + 1) ≤ HV (G)
+vertices have been cleared, which implies that at least
+kb + 1 − HV (G) = (ka + HV (G) − 1) + 1 − HV (G) = ka
+vertices remain contaminated.
+That is, in a round where the cops reduce the con-
+taminated vertices below kb + 1, there will always be at least ka contaminated vertices
+remaining. For the cops to win by eliminating all contaminated vertices, there must
+be some round where between ka and kb vertices are contaminated. Suppose we are in
+such a round after the cops move.
+
+10
+A. BONATO, T.G. MARBACH, M. MOLNAR, AND JD NIR
+Now ΦV (G,k) ≥ HV (G) for each k with ka ≤ k ≤ kb, so on the contamination round
+at least HV (G) clear vertices become re-contaminated, and so at least ka + HV (G) =
+kb + 1 vertices are now contaminated. That is, if the cops ever reduce the number of
+contaminated vertices to be kb or fewer, then there will be kb + 1 or more contaminated
+vertices after the subsequent contamination round. Therefore, we conclude that the
+cops can never reduce the number of contaminated vertices below ka.
+□
+An analogous bound for HE can be found using Lemma 9 on Theorem 10 is the
+following.
+Corollary 11. For a graph G,
+prox1(G) >
+HE(G)
+(∆(G) + 1)∆(G).
+Since ζ1(G) ≥ prox1(G), both Theorem 10 and Corollary 11 allow results in isoperi-
+metric bounds to be applied to yield lower bounds on the k-visibility Location number.
+The primary challenge remaining is that even when the isoperimetric parameters are
+known exactly, computing HV (G) and HE(G) is often a complex task. As an example,
+the vertex isometric peak of a binary tree of radius d is known to be asymptotically
+equal to d/2 [19]; it is also known that the number of vertices in the vertex border
+is small for many sporadic values of k. However, we can find a lower bound of HV
+and HE using the corresponding isoperimetric peak, showing that an h-index and its
+corresponding isoperimetric parameter differ by a small multiplicative constant without
+complicated direct analysis.
+Theorem 12. For a graph G,
+ΦV (G)
+2
+(1 +
+1
+2∆(G) + 1) ≤ HV (G) ≤ ΦV (G).
+Proof. The upper bound is clear by the definition of HV and ΦV . For the lower bound,
+we will show that two consecutive values cannot have their ΦV (G,k) values differ too
+much.
+Consider a set S of cardinality k + 1. If we remove any vertex u from S, then the
+only vertex that can be in δ(S ∖ {u}) that is not in δ(S) is the vertex u, and so
+∣δ(S)∣ ≥ ∣δ(S ∖ {u})∣ − 1. As a consequence, if ∣δ(S)∣ = ΦV (G,k + 1) and u ∈ S, then we
+have that
+ΦV (G,k + 1) = ∣δ(S)∣ ≥ ∣δ(S ∖ {u})∣ − 1 ≥
+min
+S′∶∣S′∣=k ∣δ(S′)∣ − 1 = ΦV (G,k) − 1.
+(1)
+Now consider a set S of cardinality k − 1. If we add a vertex v to S, then any vertex
+in δ(S ∪{v}) that is not in δ(S) must be a neighbor of v. As a consequence, δ(S ∪{v})
+contains at most ∆(G) more vertices than δ(S), and so similar to the previous case,
+ΦV (G,k − 1) ≥ ΦV (G,k) − ∆(G).
+(2)
+Let kp be a value such that ΦV (G,kp) = ΦV (G). By recursively applying inequality
+(1), we find htat
+ΦV (G,kp + i) ≥ ΦV (G,kp) − i ≥ ΦV (G) − ΦV (G)∆(G)/(2∆(G) + 1)
+
+THE ONE-VISIBILITY LOCALIZATION GAME
+11
+for each i ∈ {0,...,ΦV (G)∆(G)/(2∆(G) + 1)}. Similarly, by recursively applying in-
+equality (2),
+ΦV (G,kp − i) ≥ ΦV (G,kp) − i∆(G) ≥ ΦV (G) − ΦV (G)∆(G)/(2∆(G) + 1)
+for each i ∈ {1,...,ΦV (G)/(2∆(G) + 1)}. Therefore, there are at least
+ΦV (G)∆(G)
+2∆(G) + 1
++ 1 +
+ΦV (G)
+2∆(G) + 1 = ΦV (G) ∆(G) + 1
+2∆(G) + 1 + 1
+contiguous values of k such that
+ΦV (G,k) ≥ ΦV (G) − ΦV (G)∆(G)
+2∆(G) + 1
+= ΦV (G) ∆(G) + 1
+2∆(G) + 1.
+By the definition of HV , this yields that
+HV (G) ≥ ΦV (G) ∆(G) + 1
+2∆(G) + 1 = ΦV (G)
+2
+(1 +
+1
+2∆(G) + 1),
+as required.
+□
+We note that as ΦV (G) ≥ HV (G), the vertex-h-index and the vertex isoperimetric
+peak differ by a multiplicative factor of at most a little over 2. There is an analogous
+result for the edge-h-index, as follows.
+Theorem 13. For a graph G,
+HE(G) ≥
+2
+∆(G) + 2ΦE(G).
+Proof. Follows similarly to the proof of Theorem 12, except using
+ΦE(G,k + 1) ≥ ΦE(G,k) − ∆(G)
+in place of the inequality (1).
+□
+We have the following corollary as a consequence of Theorems 10 and 12.
+Corollary 14. For a graph G, we have that
+prox1(G) = Ω(ΦV (G)
+∆(G) ),
+and as ΦV (G) ≥ ΦE(G)/∆(G),
+prox1(G) = Ω(ΦE(G)
+∆(G)2 ).
+4. Trees
+As was proved first in [21] and later in [13], the localization number of trees is at
+most two. For the one-visibility Localization game, the situation is quite different. We
+explore bounds on ζ1 for trees and give upper and lower bounds on ζ1 for k-ary trees
+with k ≥ 2 that differ by a multiplicative constant; this family is shown as a result to
+have ζ1 unbounded.
+We begin by showing that ζ1 is monotone on subtrees of a tree.
+Lemma 15. If T is a tree and S is a subtree of T, then ζ1(S) ≤ ζ1(T).
+
+12
+A. BONATO, T.G. MARBACH, M. MOLNAR, AND JD NIR
+Proof. For each v ∈ T, let des(v) ∈ S be the unique vertex in S at the shortest distance
+(in T) from v. Using ζ1(T) cops, if a successful strategy on T calls for a cop to probe
+v ∈ T, they instead probe des(v). Since the robber cannot be in T ∖ S, the distance
+in T between the robber and des(v) is at most the distance between the robber and v.
+Thus, this strategy gives no less information than it would in T. As the cops win in T,
+they will win in S.
+□
+Our next result shows that the one-visibility localization number and one-proximity
+number differ by at most one for trees.
+Lemma 16. For any tree T,
+prox1(T) ≤ ζ1(T) ≤ prox1(T) + 1.
+Furthermore, if prox1(T) ≥ ∆(T), then ζ1(T) = prox1(T).
+Proof. Let m = prox1(T). Root T arbitrarily and let r be the root vertex. The cops
+of the one-visibility Localization game use m cops to follow a winning strategy in the
+one-proximity game and place one additional cop at r on each round. If the robber
+ever tries to cross from one subtree of T − r to another, then they must pass through
+r, at which point the cop on r would probe a distance of 0, and the robber would be
+captured. Therefore, the robber may only move in one subtree.
+As the cops’ strategy succeeds in the one-proximity game, they eventually probe a
+vertex and receive a distance of at most one. If the probe returns distance zero, the
+robber has been located, so assume it returns one. This uniquely determines on which
+subtree the robber is located. If the cop on r also returned a probe of distance 1 to
+the robber, then the robber would be captured as r has exactly one neighbor on each
+subtree. Therefore, we may assume the robber has distance at least 2 from r. The cop
+dedicated to probing r can now probe the root of the robber’s subtree while the other
+m cops start the winning strategy from the beginning. After repeating this process at
+most d times, where d is the depth of T, the subtree onto which the robber is forced
+will be a leaf. The cops can then win by probing each leaf until locating the robber.
+□
+Let T be a tree. Note that given any v ∈ T, T −v is a forest. Call v a midway vertex of
+T if each component T1,T2,...,Tk of T − v satisfies ∣V (Ti)∣ ≤ n/2. The following result
+is folklore.
+Lemma 17. Every tree has at least one midway vertex.
+Proof. Given u ∈ V (T), let T1,T2,...,Tk be the components of T − u and define
+s(u) = max
+1≤i≤k ∣V (Ti)∣.
+Assume for the sake of contradiction that minu∈V (T) s(u) > n/2. Let v be a vertex with
+s(v) minimal and let T1,T2,...,Tk be the components of T −v, where we have s(v) > n/2
+by our initial assumption. Note that at most one component satisfies ∣V (Ti)∣ > n/2 as
+∑∣V (Ti)∣ = n − 1; without loss of generality, let this component be T1.
+Let w ∈ T1 be the neighbor of v in T. We then have that T − w is a collection of
+components, say S1,S2,...,Sr. The largest of these components, say S1, cannot be a
+subtree of T1 as any such subset does not contain w and therefore has fewer than ∣V (T1)∣
+vertices, contradicting that s(v) was minimal. However, any Si intersecting T1 must,
+
+THE ONE-VISIBILITY LOCALIZATION GAME
+13
+in fact, be a subtree of T1 as the only path from elements in T1 to elements in T − T1
+contains w.
+Thus, the largest component S1 must be the subtree formed by combining the subtrees
+T2,...,Tk and v, and so we have that ∣V (S1)∣ ≤ n − ∣V (T1)∣. As ∣V (T1)∣ > n/2, we have
+that
+∣V (S1)∣ ≤ n − ∣V (T1)∣ < n − n/2 = n/2,
+which contradicts that s(w) ≥ s(v).
+□
+We now derive the following bound in terms of the order of the tree.
+Theorem 18. If T is a tree of order n ≥ 2, then ζ1(T) ≤ ⌈log2 n⌉.
+Proof. The proof is by induction on n. The base case with n = 2 is straightforward: the
+only tree on two vertices is an edge, for which log2 2 = 1 probe suffices.
+Now assume T is a tree on n vertices. Let x be a midway vertex of T and probe x
+every round. This prevents the robber from moving from one component of T − x to
+another. Each component of T −x contains at most n/2 vertices and thus, by induction,
+requires at most log2(n/2) = log2 n − 1 probes to search. As the robber is restricted
+to a single component of T − x, the cops can use these log2 n − 1 probes to clear each
+component of T −x before moving on to the next. The winning cop strategy we outlined
+uses at most 1 + log2 n − 1 = log2 n probes, and the proof follows.
+□
+The depth of a vertex in a rooted tree is the number of edges in a shortest path from
+the vertex to the tree’s root. The depth of a rooted tree T is the greatest depth in T.
+The depth of a tree is the smallest depth of a rooted tree over all ways of rooting T. We
+next turn to two bounds in terms of the depth of a tree.
+Theorem 19. For a tree T of depth d, we have that
+ζ1(T) ≤ ⌊d
+4⌋ + 2.
+Proof. We provide a strategy using m = ⌊d
+4⌋ + 1 cops to win the one-proximity game on
+T. The result then follows from Lemma 16. The idea of the proof is to clear paths
+sequentially, based on an ordering of the leaves. We clear all the paths to leaves from
+lower to higher index, using two cop moves to clear a given path. We ensure that the
+robber cannot reinfect previously infected paths to leaves with a lower index in each
+round.
+Let u0 be the root vertex of T, and let ℓ1,...,ℓp denote an ordering of the leaves of
+T, where the ordering is obtained by performing a depth-first search on T. Let i be the
+smallest index such that ℓi has not yet been chosen. Let Pi = u0u1,...,uq = ℓi denote
+the path from the root to ℓi. Define vi as the vertex in Pi that is not in Pi+1 but is as
+close to the root u0 as possible.
+By induction, we assume that an even number of rounds have occurred, and it is
+immediately before the cops’ move on a round of odd parity. Further, we assume that
+each subtree in the forest T − Pi either has:
+(1) all vertices cleared; or
+(2) all vertices infected, except perhaps the unique vertex with a neighbor in Pi
+(within the graph T).
+
+14
+A. BONATO, T.G. MARBACH, M. MOLNAR, AND JD NIR
+We call these subtrees cleared and infected, respectively. The base step for induction is
+follows as, on the first round, T − P1 is composed of infected trees. We note that under
+these assumptions and since the ℓi were defined using a depth-first search, directly before
+the cops take their move, the only infected descendants of vi are in Pi.
+Recall that q ≤ d is the length of the path Pi. The cop player places a cop on the
+vertex u4j for each 0 ≤ j ≤ ⌊q/4⌋ using at most ⌊q/4⌋ + 1 ≤ m cops. If the robber was
+on Pi and was not captured in this move, then they must have been on u4j+2 for some
+0 ≤ j ≤ ⌊(q − 2)/4⌋ or else on uq = ℓi if q ≡ 3 (mod 4). Therefore, after the robber moves,
+they are on either an infected subtree of T −Pi or on N[u4j+2] for some 0 ≤ j ≤ ⌊(q−2)/4⌋.
+On the next cop move, a cop is placed on vertex u4j+2 for each 0 ≤ j ≤ ⌊(q−2)/4⌋ using at
+most ⌊(q−2)/4⌋+1 ≤ m cops. The robber may now only be on an infected subtree and is
+not on Pi nor on a cleared tree. As we noted before these two rounds, the descendants of
+vi were infected only if they were in Pi. Consequently, each descendant of vi (including
+vi itself) is cleared after these two moves. The robber then takes their move, and after
+this move may be on an infected tree or on a vertex in V (Pi) ∩ V (Pi+1).
+We observe that the descendants of vi (including vi itself) form a cleared tree of
+T −Pi+1, which we label as S. In addition, a cleared tree in T −Pi is either a cleared tree
+in T −Pi+1 or is a subtree of S. Similarly, we can show that each subtree of T −Pi+1 that
+is not one of these cleared trees is an infected tree. Therefore, we have that an even
+number of rounds has occurred, it is now the cops’ move on an odd-parity round, and
+each subtree of T − Pi+1 is either cleared or infected, completing the inductive step.
+□
+We provide a third and final upper bound for ζ1 on trees. An example will follow,
+illustrating three graphs such that each bound is best on exactly one graph.
+Let T be a tree, rooted at vertex v, of depth d.
+Let Lv = {L1,L2,...,Ld} be a
+level decomposition of T rooted at v, where Li = {u ∈ V (T) ∶ d(u,v) = i}. Define the
+nonnegative integer Li = ∣{w ∈ Li ∶ deg(w) ≥ 2}∣, which counts the number of non-leaf
+vertices within each level. We have the following upper bound on prox1(T).
+Theorem 20. If Lv is the level decomposition of T rooted at v, then
+prox1(T) ≤ ⌈maxi{Li}
+3
+⌉ + 1.
+Proof. Let k = ⌈maxi{Li}
+3
+⌉. We give a strategy for k + 1 one-proximity cops to clear T
+starting from level Lm−1 and working up to the root v. Divide the non-leaf vertices of
+Lm−1 into disjoint groups of three vertices, assigning one cop to each set, and consider
+the first such group consisting of vertices x, y, and z. For x, let x1 denote the parent of
+x, x2 denote the parent of x1, and define y1,y2,z1 and z2 analogously. We refer to the
+cop initially assigned to these three vertices as C1. At the start of the game, the robber
+may be anywhere on T, so all vertices start contaminated.
+In the first three rounds, C1 will probe x, then y, then z. After these probes, imme-
+diately before the robber’s move, the robber may be at x (had they begun on x2, they
+could move to x1 prior to the cop probing y, and then to x prior to the cop probing z).
+The robber could also be at y1 or z2. We note that the robber cannot currently be on
+a leaf adjacent to x, y, or z without being previously detected by one of the first three
+probes. See Figure 2.
+
+THE ONE-VISIBILITY LOCALIZATION GAME
+15
+x
+⋯
+x1
+x2
+y
+⋯
+y1
+y2
+z
+⋯
+z1
+z2
+Lm
+Lm−1
+Lm−2
+Lm−3
+Figure 2. Possible robber locations are shown in red after C1’s third
+probe.
+The robber now takes their move. To keep the leaves adjacent to x clear, C1 will next
+probe x. They could then probe y, and then z, repeating the previous three probes.
+This would protect the leaves adjacent to x, y, and z, but would not allow the cop player
+to make progress. We now introduce our additional cop, C∗, who will help C1 shift from
+guarding x,y and z to guarding x1,y1, and z1. To do this, C∗ probes x instead of C1.
+Had the robber moved to x they will be detected by this next probe regardless of where
+they now move. This allows C1’s next three probes to be x1, then y, then z, while C∗
+continues to probe x. Following these three rounds of cop probes, x is fully cleared.
+In the next three rounds we will have C∗ probe y, while C1 probes x1, then y1, then
+z. Next, C∗ probes z while C1 probes x1, then y1, then z1. On the robber’s move
+following this sequence of probes, they may now be at x1, y2, or z3. Next, C∗ will move
+to the second trio of non-leaf vertices in Lm−1 and will perform this same strategy with
+the cop there, C2. At the same time, C1 will repeatedly probe x1, then y1, then z1,
+ensuring that the robber can only reach Lm−1 in this part of the graph, and will be
+detected doing so, thereby clearing x, y, and z, and the adjacent leaves in Lm. This is
+continued for each group of three non-leaf vertices in Lm−1, of which we have at most
+k.
+When C∗ has concluded with the last group of three non-leaves in Lm−1, the cops will
+probe in Lm−1 so that every non-leaf vertex there is probed every three rounds. This
+ensures that the robber will be captured if they ever move onto a non-leaf in Lm−1.
+The clearing strategy continues up the tree, one level at a time. When the cops move
+to Lm−2, we note that we may have non-leaf vertices in Lm−2 which have yet to be
+probed (those that have all their children as leaves in Lm−1). The tree T has at most
+k sets of three non-leaves in any level. For such vertices, divide them into groups of
+three and assign an unused cop to each. We have these cops repeatedly probe each of
+their three vertices (as C1 probed x, y, then z initially), clearing their adjacent leaves
+in Lm−1.
+The cop C∗ is then used to extend the cop territory up from Lm−2 into Lm−3. First,
+C∗ probes x1 while C1 probes x2, then y1, then z1. The process is repeated as in the
+level below. In each grouping of vertices, the cop C∗ extends the cop territory up the
+tree, one vertex at a time. Since each vertex in a level is probed every three terms
+while C∗ is probing that level, the robber cannot move down the tree into cop territory
+without being detected.
+
+16
+A. BONATO, T.G. MARBACH, M. MOLNAR, AND JD NIR
+In this way, C∗ can extend the cop territory eventually up to the root v, cleaning the
+tree and winning the one-proximity game. The proof follows.
+□
+For integers d,k ≥ 2, we use the notation T k
+d for the k-ary tree of depth d, where each
+non-leaf vertex has k children. We note that the bound in Theorem 20 is tight for T 3
+2 .
+Rooted at the midway vertex, the tree has maxi{Li} = ∣L1∣ = 3, and so Theorem 20
+provides that prox1(T 3
+2 ) ≤ 2. It is known [22] that the original localization number
+satisfies ζ(T 3
+2 ) = 2, so we have ζ1(T 3
+2 ) ≥ 2 as well.
+To compare the upper bounds on ζ1(T) provided by Theorems 18, 19, and 20, let
+T0 = T 3
+3 , which has 40 vertices and 39 edges. The corresponding upper bounds for T0
+are displayed in the first row of Table 1. Define the trees Ti to be T0 with each edge
+subdivided into i vertices. We then have that Ti will have depth 3(i + 1) and order
+40 + 39i. The best upper bound for each tree is shown in bold. Note that each of the
+theorems gives the best bound depending on the tree considered.
+Theorem 18
+Theorem 19
+Theorem 20
+T0
+6
+2
+4
+T10
+9
+10
+10
+T100
+12
+77
+10
+Table 1. Upper bounds for ζ1 from Theorems 18, 19, and 20.
+Finding lower bounds for the one-visibility localization number is challenging in most
+cases. We determined that the isoperimetric peak can give a lower bound on prox1,
+enabling us to utilize such isoperimetric results when they exist. We finish by applying
+such results to binary trees, where k = 2. We cite the following result, which gives
+asymptotically tight values for the isoperimetric peak of binary trees.
+Theorem 21 ([19]). If d ≥ 2 is an integer, then
+d
+2 − O(log d) ≤ ΦE(T 2
+d ) ≤ d
+2 + O(1).
+We therefore have that HE(T 2
+d ) ≥ d
+5 − O(log d) from Theorem 13, and so we have the
+following bounds.
+Corollary 22. If d ≥ 2 is an integer, then
+d
+60 − O(log d) < prox1(T 2
+d ) ≤ ζ1(T 2
+d ) ≤ d
+4 + 2.
+Proof. The upper bound follows from Theorem 19. The lower bound follows from using
+Corollary 11, then applying Theorem 13 to the result, and then finally using the lower
+bound of Theorem 21.
+□
+Similar lower bounds of the vertex isoperimetric peak on k-ary trees are also useful
+to us here.
+
+THE ONE-VISIBILITY LOCALIZATION GAME
+17
+Theorem 23 ([24]). If d,k ≥ 2 are integers, then
+ΦV (T k
+d ) ≥ 3
+40(d − 2).
+Theorem 23 provides the following bounds on prox1 and ζ1 for k-ary trees.
+Corollary 24. If d,k ≥ 2 are integers, then
+3
+80(d − 2)(
+2
+2k + 3) < prox1(T k
+d ) ≤ ζ1(T k
+d ) ≤ d
+4 + 2.
+Proof. The upper bound follows from Theorem 19. The lower bound follows from using
+Theorem 10, then applying Theorem 12, and then finally using Theorem 23.
+□
+Corollaries 22 and 24 provide families of trees with unbounded ζ1 number, in stark
+contrast to ζ being bounded by 2 for trees. The result of Corollary 24 is far from tight.
+We note that improvements to the isoperimetric value of k-ary trees would improve this
+result.
+5. Cartesian grid graphs
+We proved in Theorem 8 that for planar graphs of order n, ζ1(G) ≤ O(√n). In this
+section, we show that grid graphs make this bound tight, in the sense that such graphs
+have ζ1 numbers in
+√
+∣V (G)∣ + O(1).
+For a positive integer n, let Gn,n be the n × n Cartesian grid, which consists of the
+Cartesian product of the n-order path with itself, or Pn ◻ Pn. As we only consider
+Cartesian grids, we refer to them as grids.
+The lower bound for grids follows by using our earlier results with the h-index.
+Theorem 25. For a positive integer n > 1, prox1(Gn,n) ≥ n
+5 + 1.
+Proof. The vertex isoperimetric values are known for the grids [6]. In particular, for
+Gn,n, ΦV (Gn,n,k) = n for k ∈ {n2−3n+4
+2
+,..., n2+n−2
+2
+}, which are 2n−2 contiguous values of
+k. Thus, it follows that HV (Gn,n) ≥ n, and by Theorem 10 we have that prox1(Gn,n) >
+n/5, as required.
+□
+We next establish upper bounds for grid graphs that differ from the lower bound in
+Theorem 25 by an additive constant.
+Theorem 26. Let m be the odd integer such that n = 5m − i for some integer 0 ≤ i ≤ 9.
+We then have that
+prox1(Gn,n) ≤ m + 3.
+We prove Theorem 26 at the end of this section. As a consequence of Theorem 26,
+we know prox1(Gn,n) up to one of four values.
+This gives a class of graphs where
+Theorem 10 is close to being tight.
+Corollary 27. For n a positive integer,
+⌈n
+5 ⌉ + 1 ≤ prox1(Gn,n) ≤ ⌈n
+5 ⌉ + 4.
+Proof. Theorem 25 gives prox1(Gn,n) ≥ ⌈n
+5 ⌉+1. When we note that m = ⌈n
+5 ⌉+⌊ i
+5⌋ ≤ ⌈n
+5 ⌉+1
+in Theorem 26, it follows that prox1(Gn,n) ≤ ⌈n
+5 ⌉ + 4.
+□
+
+18
+A. BONATO, T.G. MARBACH, M. MOLNAR, AND JD NIR
+The following result determines the one-visibility localization number of Cartesian
+grids up to four values, assuming that n is sufficiently large.
+Corollary 28. If n ≥ 11, then
+⌈n
+5 ⌉ + 1 ≤ ζ1(Gn,n) ≤ ⌈n
+5 ⌉ + 4.
+Proof. Using prox1(Gn,n) cops in the one-visibility Localization game, the cops may
+initially follow the strategy guaranteed by Corollary 27 until some cop probes a distance
+of 1 from the robber, say the cop on vertex u. The robber moves and may now be on
+any vertex of distance 0, 1, or 2 from u. Since n ≥ 11, we have that ⌈n
+5 ⌉ + 1 ≥ 4, and so
+we know that at least four cops are playing. For the cops’ second move, play four cops
+on the vertices of N(u).
+If all four cops probe a distance of 1, then the robber is on u. If exactly two of these
+cops probe a distance of 1, then the robber must be on the unique vertex of distance
+2 from u that is adjacent to the two vertices containing these cops. If exactly one cop
+probes a distance of 1, then the robber must be on the unique vertex of distance 1 from
+this cop that is not adjacent to any of the other cops. Thus, the robber’s location is
+determined by the cops.
+□
+We finish the section with the proof of Theorem 26, but as the proof is quite technical,
+we motivate our technique by exploring some less efficient but more intuitive strategies.
+A first strategy one may think of for the cops is to clear the grid using n cops to probe
+an entire row and march upwards, clearing the grid from bottom to top. This approach
+is effective but inefficient.
+A natural improvement is to place a cop in every other column, alternating between
+the first and second row, so that two full rows of the grid are cleared using at most
+(n+1)/2 cops; see Figure 3. As the cops march upwards, two rows are cleared, and only
+c
+c
+c
+c
+c
+c
+Figure 3. Cops placed on every other column can clear two rows.
+one row is reinfected each round. This strategy avoids overlapping the neighborhoods of
+the cops’ probes, but only the “forward” edge of the cops’ line clears infected vertices.
+The “rear” of each probe is already cleared. Matching the isoperimetric lower bound
+requires most cops to clear the maximum ∆(G) + 1 = 5 new vertices with every probe.
+To improve on this second strategy, note that it would take the robber multiple rounds
+to cross the cops’ formation: three rounds on a column with a cop and two rounds on
+a column without a cop. This implies that cops only need to play on these positions
+every other round to prevent the robber from reinfecting the portion of the grid they’ve
+
+THE ONE-VISIBILITY LOCALIZATION GAME
+19
+fully cleared. By shifting the set of columns where the cops’ probe, it is possible for the
+cops to protect these rows by playing twice every five rounds, so that each column has
+two vertices cleared by one cop move and three vertices cleared by the other.
+We now give a high-level description of our strategy. We partition the grid into five
+rectangles of width approximately n/5 vertices. Using two sets of approximately n/10
+cops, each rectangle is probed twice every five rounds with a cop probing every other
+column in the rectangle. To prevent the robber from slipping from the infected portion
+of one rectangle to the cleared portion of the next, the cops clear a diagonal, rather
+than the two rows from Figure 3. Although it takes several rounds to do so, we show
+that the cops can move these diagonals up the rectangles, closing in on the robber until
+they are captured.
+We now turn to the proof of our main result in this section.
+Proof of Theorem 26. Let Gn,m′ denote the n×m′ grid and G denote the square lattice
+with vertices in Z×Z. It will be convenient to allow the cops to play on G and to restrict
+the robber’s position to a subgraph Gn,m′, where m′ = m for the first part of the proof
+and m′ = n in the second part of the proof. (Recall that m is the odd integer for which
+n = 5m − i for some 0 ≤ i ≤ 9.) We will then argue this relaxation did not benefit the
+cop player.
+We break the proof into three parts which prove the following three claims, respec-
+tively:
+(1) When the robber is restricted to a subgraph Gn,m within G, m+3
+2
+cops which
+only play on rounds t with t ≡ 0,3 (mod 5) can capture the robber.
+(2) When the robber is restricted to a subgraph Gn,5m of G, the robber can be
+captured by m + 3 cops.
+(3) At most m + 3 cops are required to capture the robber on Gn,n.
+Throughout the proof, let
+fi,j(c) =
+⎧⎪⎪⎨⎪⎪⎩
+i + 1 + ⌊c−j
+2 ⌋
+for c − j > 0
+i + ⌈c−j
+2 ⌉
+for c − j ≤ 0,
+and define Fi,j = {(r,c) ∶ 1 ≤ c ≤ m′,fi,j(c) ≤ r ≤ n}, which we call a forced region. These
+forced regions describe different subsets of vertices that the cop player will contain the
+robber within. The strategy we describe will reduce the cardinality of the forced region
+over time. It is important to note here that when we show that the robber must be
+in a given forced region, there may be some vertices the robber cannot occupy.
+In
+particular, it will be convenient to assume that the forced region will, in some rounds,
+contain several vertices (x,y) with x < 1.
+The (i,j) index of Fi,j refers to a cell along the lower edge of the region, which cuts
+diagonally from southwest to northeast across columns 1 through m′. In particular, the
+second index describes the column of focus of the forced region, which we pay special
+attention to in the proof. The function fi,j, given a column c, gives a certain key position
+in Fi,j related to where the cops will play. See Figure 4 for a visual reference.
+Let
+Si,j ={(fi,j(c) + 1,c) ∶ c > j and c − j is odd}∪
+{(fi,j(c) + 1,c) ∶ c ≤ j and c − j is even}.
+
+20
+A. BONATO, T.G. MARBACH, M. MOLNAR, AND JD NIR
+The set Si,j contains points spaced apart in an L-shape, similar to a knight move
+in chess. These vertices are chosen so that the neighborhoods of the points cover a
+diagonal stripe along the bottom of Fi,j with minimal overlap.
+Claim 1: When the robber is restricted to a subgraph Gn,m within G, m+3
+2
+cops which
+only play on rounds t with t ≡ 0,3 (mod 5) can capture the robber.
+The cops��� strategy consists of two moves. If the cops have restricted the robber to
+Fi,j immediately before the cops’ move, then the cops will play on the vertices Si,j that
+are within the columns [0,m+1]. By the definition of Si,j, the cops will only be playing
+on every other column in [0,j] and every other column in [j + 1,m + 1], for a total of
+⌈j
+2⌉ + ⌈(m + 1) − (j + 1) + 1
+2
+⌉ ≤ j + 1
+2
++ (m + 1) − (j + 1) + 1 + 1
+2
+= m + 3
+2
+cops.
+Immediately after this cop’s move, the robber may only be on a vertex in Fi,j ∖
+N[Si,j] = Fi+2,j−1. To see this fact, we consider the cases c > j and c ≤ j. For the
+first case, when c − j is odd, N[(fi,j(c) + 1,c)] is a superset of {(fi,j(c),c),(fi,j(c) +
+1,c),(fi,j(c) + 2,c)}; if the robber is in column c, then it must be on a vertex (r,c)
+with r ≥ fi,j(c) + 3 = fi+2,j−1(c).
+When c − j is even, {(fi,j(c),c),(fi,j(c) + 1,c)} ⊆
+N[(fi,j(c)+1,c)], and the robber must be in a row r ≥ fi,j(c)+2 = fi+2,j−1(c). The case
+when c ≤ j follows similarly to find that if the robber is in column c, then the robber
+must be on row r ≥ fi+2,j−1(c), which defines the set Fi+2,j−1. We call this the natural
+cop move and label this as P1 for future reference.
+In the case that the robber is known to be on a vertex of Fi+2,0, the cops instead think
+of the robber as being on a vertex of Fi+2+ m+1
+2
+,m. These two forced regions describe the
+same set of vertices, but as Si,j is determined by the index (i,j), this change of index
+describes a different cop move. We label this replacement as P2 for future reference.
+Observe that when the robber is on a vertex in Fi,j but the cops do not play during
+their next move (as in rounds 1,2,4 (mod 5)), then the robber moves to a vertex in
+Fi−1,j = N[Fi,j]. We label this as P3 for future reference.
+We prove Claim 1 recursively. As a base case, we can assume the robber is contained
+within Fi,m for any i ≤ 1, where we note that we will take i to be negative in some cases.
+Note that for such i, Fi,m contains the subset of vertices [1,n] × [1,m], so this initial
+assumption is always true.
+For the recursive step, assume the robber is contained within some Fi,m in round t
+with t ≡ 0 (mod 5). Repeating the natural move P1, the column of focus j will shift
+from m down to 1, with rounds 0 (mod 5) focusing on an odd column and rounds 3
+(mod 5) focusing on an even column. We refer to these rounds collectively as the first
+sweep. Once the column of focus is 1, we continue to play, and the column of focus
+again becomes m. For this second sweep, the column of focus will shift from m all the
+way down to 1, but with the parity reversed: rounds 3 (mod 5) focus on an odd column
+and rounds 0 (mod 5) focus on an even column. After both sweeps, the cops will have
+successfully moved the robber from Fi,m to Fi+1,m.
+We refer the reader to Figure 4, which depicts the following cop moves.
+First sweep: The cops play the natural move so that the robber must be contained
+within Fi+2,m−1 (see P1). The robber moves three times, first to a vertex in Fi+1,m−1,
+
+THE ONE-VISIBILITY LOCALIZATION GAME
+21
+then to a vertex in Fi,m−1, and finally to a vertex in Fi−1,m−1 (see P3). The cops then play
+the natural move in round t + 3, so that the robber must be contained within Fi+1,m−2
+(see P1). Then the robber moves twice, first to a vertex in Fi,m−2, then to a vertex in
+Fi−1,m−2 (see P3). This shows that every five moves, the indices of the forced region
+decrease by one in the first coordinate and two in the second coordinate. This process
+repeats (m − 1)/2 times until the robber is known to reside on Fi− m−1
+2
+,1 immediately
+before the cops’ (t + 5m−1
+2 )th move, where we note that t + 5m−1
+2
+≡ 0 (mod 5). The
+cops again play the natural move, and so the robber is known to reside in Fi− m−1
+2
++2,0,
+immediately after the cops’ move, which is then replaced with Fi+3,m by P2. The robber
+takes three moves and is on a vertex in Fi,m.
+Second sweep: The cops again play the natural move, and so the robber is known
+to reside in Fi+2,m−1. The robber takes two moves and is on a vertex in Fi,m−1. Once
+again, every five moves the first index of the forced region is decreased by one and
+the second index by two. Play continues in this fashion (the cops playing the natural
+move on round 0,3 (mod 5)) for (m − 1)/2 rounds, until the robber is known to reside
+in Fi− m−1
+2
+,1 immediately before the cops’ move in the (t + 5(m − 1))th round, where
+t + 5(m − 1) ≡ 0 (mod 5). The cops play the natural move, so the robber is known
+to reside in Fi− m−1
+2
++2,0 immediately after the cops’ move, which is then replaced with
+Fi+3,m by P2. The robber takes two moves and is on a vertex in Fi+1,m immediately
+before the cops’ (t + 5m)th move.
+We note that we started with the robber being on a vertex of Fi,m in some round 0
+(mod 5), and now have that Fi+1,m in some round 0 (mod 5). This completes the recur-
+sive step, and so we can conclude that after sufficiently many rounds, as Fn+ 3m
+2 ,m = ∅,
+the robber is captured, and the proof of Claim 1 follows.
+For the next claim, it will be useful to note that if this process is initialized with Fi,m,
+then immediately before the cops’ move in round t = 5mα + 1, the robber must be in
+the forced region Fi+α,m.
+Claim 2: When the robber is restricted to a subgraph Gn,5m of G, the robber can be
+captured by m + 3 cops.
+We split the subgraph Gn,5m into five subgraphs A1,A2,A3,A4, and A5, where Aj is
+on the vertices of Gn,5m in columns [(j − 1)m + 1,jm]. We use a set of m+3
+2
+cops on
+Aj on rounds j,j + 3 (mod 3). This requires two sets of m+3
+2
+cops, and so m + 3 cops
+are used in total. The technique described in Claim 1 is applied to each Ai, shifting
+the rounds on which the cops play appropriately, and with the additional condition
+that the process in Claim 1 is started in Aj in round j with forced region Fi,m where
+i = −2m + (j − 1)m−1
+2 .
+To illustrate how the process in Claim 1 is extended to Gn,5m, we describe the first
+six cops moves of the m + 3 cops on the infinite square grid G, focused on the subgraph
+Gn,5m.
+Cop move 1: The first set of m+3
+2
+cops play on S−2m,m on the columns [0,m + 1] (this
+is the first move for the cops in A1).
+Cop move 2: The first set of m+3
+2
+cops play on S−2m+ m−1
+2
+,m on the columns [m,2m + 1]
+(this is the first move for the cops in A2).
+
+22
+A. BONATO, T.G. MARBACH, M. MOLNAR, AND JD NIR
+Figure 4. The first sweep of claim 1 where m = 7. Red and pink dots
+indicate the forced region before the cops’ move, and the square indicates
+the i,j such that this forced region is Fi,j. Red dots indicate the forced
+region after the cops’ move. Black circles indicate the locations of the
+cops their move. From left to right, top to bottom, the images indicate
+play just before and after the cops’ 1st, 4th, 6th, 9th, 11th, 14th, 16th,
+and 19th move.
+
+THE ONE-VISIBILITY LOCALIZATION GAME
+23
+Cop move 3: The first set of m+3
+2
+cops play on S−2m+2 m−1
+2
+,m on the columns [2m,3m+1]
+(this is the first move for the cops in A3).
+Cop move 4: The first set of m+3
+2
+cops play on S−2m+3 m−1
+2
+,m on the columns [3m,4m+1]
+(this is the first move for cops in A4) and the second set of m+3
+2
+cops play on S−2m−1,m−1
+on the columns [0,m + 1] (this is the fourth round in A1).
+Cop move 5: The first set of m+3
+2
+cops play on S−2m+4 m−1
+2
+,m on the columns [4m,5m +
+1] (this is the first move for cops in A5) and the second set of
+m+3
+2
+cops play on
+S−2m−1+ m−1
+2
+,m−1 on the columns [m,2m + 1] (this is the fourth round in A2).
+Cop move 6: The first set of m+3
+2
+cops play on S−2m−1,m−2 on the columns [0,m + 1]
+(this is the sixth round in A1) and the second set of m+3
+2
+cops play on S−2m−1+2 m−1
+2
+,m−1
+on the columns [2m,3m + 1] (this is the fourth round in A3).
+If the robber stays within a subgraph Aj, then by Claim 1 they will eventually be
+captured. Suppose the robber moves from one subgraph to another, say from Aa to
+Ab. The robber must have been on a vertex of the cops’ current forced region in Aa in
+round t. If the robber moved to the cops’ current forced region in Ab, then the robber
+has not made progress as they may as well have started in Ab and stayed there until
+the current move. Therefore, we can assume that the robber moves to a vertex of Ab
+outside of the cops’ forced region.
+Assume without loss of generality that a,b ∈ {1,2}. We analyze the moves that occur
+on the border of A1 and A2, which affect where the robber can be on either column m
+or m + 1. Before we begin, we analyze each of the cops’ moves of A1 to find which cops
+played on either the left-most columns 0, 1, and 2, or the right-most columns m − 1, m,
+and m + 1. This will require a deeper analysis of the moves in Claim 1. We note the
+following properties.
+Property 1: The first sweep of Claim 1 utilized 5m−1
+2
++ 3 rounds and the second
+sweep of Claim 1 utilized 5m−1
+2
++ 2 rounds. Together, this is 5m−1
+2
++ 3 + 5m−1
+2
++ 2 = 5m
+rounds needed to perform both sweeps. Therefore, if the cops were on Si,m in round t,
+then the cops are on Si+1,m in round t+5m. Since the cops play on S−2m,m in the round
+with t = 1, we conclude that the cops play on S−2m+α,m during round t = 1 + (5m)α.
+Property 2: If the cops played on Si,j in round t, then the cops play on Si−1,j−2
+in round t + 5, unless j ∈ {1,2}, in which case the cops play on Si+ m−1
+2
+,j+m−2. Since
+the cops play on S−2m+α,m in round t = 1 + (5m)α, we conclude that the cops play on
+S−2m+α−β,m−2β in round t = 1 + (5m)α + 5β when 0 ≤ β ≤ m−1
+2 , and the cops play on
+S−2m+α−β+ m−1
+2
+,m−2β+(m−2) in round t = 1 + (5m)α + 5β when m+1
+2
+≤ β ≤ m − 1.
+Property 3: If the cops play on Si,j in round t where t ≡ 1 (mod 5), then in the
+round t + 3 the cops play on Si−1,j−1 if j ≠ 1, and on Si+ m−1
+2
+,j+(m−1) if j = 1.
+We next consider the situation where the cops play near the left and right edges.
+For each t ≡ 1 (mod 5), we describe which of these cops in A1 played on a column in
+{0,1,2,m − 1,m,m + 1}.
+(1) If we are playing in round t = 1+(5m)α+5β where β = 0, then the cops in these
+columns were played on vertices {(−2m + α + 1,m),(−2m + α + 2,m + 1),(−2m +
+α + 1 − m−1
+2 ,1)}.
+
+24
+A. BONATO, T.G. MARBACH, M. MOLNAR, AND JD NIR
+(2) If we are playing in round t = 1 + (5m)α + 5β where 1 ≤ β ≤ m−1
+2 , then the cops
+in these columns were played on vertices {(−2m+α+1,m−1),(−2m+α+2,m+
+1),(−2m + α + 1 − m−1
+2 ,1)}.
+(3) If we are playing in round t = 1 + (5m)α + 5β where m+1
+2
+≤ β ≤ m − 1, then the
+cops in these columns were played on vertices {(−2m + α + 2,m),(−2m + α + 1 −
+m−1
+2 ,0),(−2m + α + 2 − m−1
+2 ,2)}.
+For each t ≡ 4 (mod 5), we describe which of these cops in A1 played on a column in
+{0,1,2,m − 1,m,m + 1}.
+(1) If we are playing in round t = 4 + (5m)α + 5β where 0 ≤ β ≤ m−1
+2 , then the
+cops in these columns were played on vertices {(−2m + α + 1,m),(−2m + α −
+m−1
+2 ,0),(−2m + α + 1 − m−1
+2 ,2)}.
+(2) If we are playing in round t = 4 + (5m)α + 5β where β = m+1
+2 , then the cops
+in these columns were played on vertices {(−2m + α + 1,m),(−2m + α + 2,m +
+1),(−2m + α + 1 − m−1
+2 ,1)}.
+(3) If we are playing in round t = 4 + (5m)α + 5β where m+1
+2
+≤ β ≤ m − 1, then the
+cops in these columns were played on vertices {(−2m − 1 + α + 1,m − 1),(−2m +
+α + 2,m + 1),(−2m + α + 1 − m−1
+2 ,1)}.
+Property 4: A cop on A1 plays on the vertex (i,j) during round t if and only if a
+cop on A2 plays on the vertex (i + m−1
+2 ,j + m) in round t + 1. As a consequence, for
+each of the vertices (i,j) with j ∈ {0,1,2} that were visited by a cop in A1 in round t as
+described above, the corresponding vertex (i′,j′) = (i + m−1
+2 ,j + m) in A2 was visited in
+round t + 1, where j′ ∈ {m,m + 1,m + 2}. Therefore, for every round, we can now derive
+which cops probed a vertex in column {m − 1,m,m + 1,m + 2}.
+This is relevant as only the cops playing in columns {m − 1,m,m + 1,m + 2} will
+impact the robber’s location on the border of A1 and A2. To simplify, take α = 2m. A
+similar argument follows for all other α. In Table 2, we describe exactly which vertices
+are probed by the cops on columns {m − 1,m,m + 1,m + 2} in rounds i + 5β + 5m(2m),
+where 0 ≤ β ≤ m − 1 and 1 ≤ i ≤ 5.
+i
+β = 0
+1 ≤ β ≤ m−1
+2
+β = m+1
+2
+m+1
+2
+≤ β ≤ m − 1
+1
+(1,m) (2,m + 1)
+(1,m − 1) (2,m + 1)
+(2,m)
+(2,m)
+2
+(1,m + 1)
+(1,m + 1)
+(1,m),(2,m + 2)
+(1,m) (2,m + 2)
+3
+4
+(1,m)
+(1,m)
+(1,m) (2,m + 1)
+(1,m − 1) (2,m + 1)
+5
+(0,m) (1,m + 2)
+(0,m) (1,m + 2)
+(1,m + 1)
+(1,m + 1)
+Table 2. The vertices in columns {m − 1,m,m + 1,m + 2} where a cop
+plays in round t = i + 5β + 5mα, where α = 2m.
+We may also analyze Claim 1 to find that if the robber is on a vertex (i,m) in
+column m that is in the forced region of A1 immediately after the cops move in round
+t = 1+5β +(5m)α, then i ≥ α−2m+3 if 0 ≤ β ≤ m−1
+2 , and i ≥ α−2m+4 if m+1
+2
+≤ β ≤ m−1.
+Similarly, after the cops move in round t = 4 + 5β + (5m)α, then i ≥ α − 2m + 3 for
+0 ≤ β ≤ m − 1.
+
+THE ONE-VISIBILITY LOCALIZATION GAME
+25
+If the robber is on a vertex (i,m + 1) in column m + 1 that is in the forced region of
+A2 immediately after the cops move in round t = 2 + 5β + (5m)α, then i ≥ α − 2m + 3.
+Similarly, after the cops move in round t = 5 + 5β + (5m)α, then i ≥ α − 2m + 2 if
+0 ≤ β ≤ m−1
+2 , and i ≥ α − 2m + 3 if m+1
+2
+≤ β ≤ m − 1.
+Let xt
+1 denote the smallest value of x such that a cop may be on (x,m) in round t in
+the forced region of A1, and let xt
+2 denote the smallest value of x such that a cop may be
+on (x,m+1) in round t in the forced region of A2. The robber may then move from the
+forced region of A1 onto a vertex not in the forced region of A2 only when xt
+1 ≤ xt
+2 + 2.
+Similarly, the robber may move from the forced region of A2 onto a vertex not in the
+forced region of A1 only when xt
+2 ≤ xt
+1 +2. We note that by the above analysis, this only
+occurs when t ≡ 1,4 (mod 5). In each of these rounds and for every possible move of
+the robber from a vertex of a forced region onto a vertex not in a forced region, there is
+a cop that prevents it by either being adjacent to the robber before or after their move.
+The complete list of such events is presented in Table 3 for the case α = 2m. The proof
+of Claim 2 follows.
+t ≡
+robber at t
+robber at t + 1
+capturing cop
+0 ≤ β ≤ m−1
+2
+1
+(1,m+1)
+(1,m)
+(2,m + 1) on round t
+4
+(1,m + 1)
+(1,m)
+(1,m) on round t
+β = m+1
+2
+1
+(1,m + 1)
+(1,m)
+(1,m) on round t + 1
+1
+(2,m+1)
+(2,m)
+(1,m) on round t + 1
+4
+(1,m + 1)
+(1,m)
+(1,m) on round t
+m+3
+2
+≤ β ≤ m − 1
+1
+(2,m + 1)
+(2,m)
+(2,m) on round t
+4
+(2,m + 1)
+(1,m + 1)
+(2,m + 1) on round t
+Table 3. In round t = i+5β +(5m)α with α = 2m, each possible robber
+move from the forced region of one Aj to the unforced region of the other
+Aj′ is represented as a row, with the corresponding cop that captures
+the robber if it performs this move.
+Claim 3: At most m + 3 cops are required to capture the robber on Gn,n.
+We now show that at most m + 3 cops are required to capture the robber on Gn,n,
+which proves Claim 3 and will complete the proof of the theorem. Recall that n = 5m−i
+for some 0 ≤ i ≤ 9. To capture the robber on Gn,n, the cops’ will observe and modify
+the strategy to capture the robber on the subgraph Gn,5m of G given in Claim 2.
+That is, suppose that the m + 3 cops play on vertices St in round t in G where the
+robber is restricted to Gn,5m. We further restrict the robber so that it can only be on
+Gn,n ⊆ Gn,5m. A simple modification to the cop moves St also ensures that the cops
+only play on the subset Gn,n of G. However, this game is identical to just playing on
+the graph Gn,n, and so is a winning strategy for m + 3 cops to capture the robber on
+Gn,n.
+We note that each cop outside of [0,n + 1] × [0,n + 1] will not affect the robber, since
+the robber is contained within the vertices [1,n]×[1,n] in all rounds. Delete all vertices
+in St that are not within [0,n + 1] × [0,n + 1]. This has no impact on capturing the
+robber.
+
+26
+A. BONATO, T.G. MARBACH, M. MOLNAR, AND JD NIR
+Suppose (0,x) ∈ St. This cop clears only the vertex (1,x) in round t, and so it is a
+strictly better move for the cop to play on (1,x). We therefore, replace (0,x) ∈ St with
+(1,x) ∈ St. Similarly, we replace (n + 1,x) ∈ St with (n,x) ∈ St, replace (x,0) ∈ St with
+(x,1) ∈ St, and replace (x,n + 1) ∈ St with (x,n) ∈ St.
+The resulting cop moves are, therefore, strictly better at capturing the robber on
+Gn,n, but also have the robber contained within Gn,n. This completes the proof.
+□
+6. Conclusion and future directions
+We introduced the one-visibility localization number and proved asymptotically tight
+bounds on Cartesian grids and bounds on k-ary trees. We gave bounds for trees in terms
+of their order and depth. Determining a tree’s exact one-visibility localization number
+based on its structural features remains an open problem.
+The one-visibility localization number may be investigated in various graph families
+where the localization number has been studied, such as Kneser graphs, Latin square
+graphs, or the incidence graphs of projective planes and combinatorial designs. Our
+approach using isoperimetric inequalities should apply to the families of hypercubes,
+higher dimensional Cartesian grids, and strong grids.
+Another natural direction would be to consider the k-limited visibility Localization
+game for k > 1, with corresponding optimization parameter ζk. It would be interesting
+to find graphs G such ζi(G) ≠ ζj(G) for all distinct values of i and j that are at most
+the radius of G.
+7. Acknowledgements
+The authors were supported by NSERC.
+References
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+1–18.
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+THE ONE-VISIBILITY LOCALIZATION GAME
+27
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+(A1,A2,A3,A4) Toronto Metropolitan University, Toronto, Canada
+Email address, A1: (A1) [email protected]
+Email address, A2: (A2) [email protected]
+Email address, A3: (A3) [email protected]
+Email address, A4: (A4) [email protected]
+

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+arXiv:2301.05303v1  [eess.SY]  12 Jan 2023
+1
+Probabilistic Constraint Construction for
+Network-safe Load Coordination
+Sunho Jang, Student Member, IEEE, Necmiye Ozay, Senior Member, IEEE,
+Johanna L. Mathieu, Senior Member, IEEE
+Abstract—Distributed Energy Resources (DERs) can provide
+balancing services to the grid, but their power variations might
+cause voltage and current constraint violations in the distribution
+network, compromising network safety. This could be avoided by
+including network constraints within DER control formulations,
+but the entities coordinating DERs (e.g., aggregators) may not
+have access to network information, which typically is known only
+to the utility. Therefore, it is challenging to develop network-safe
+DER control algorithms when the aggregator is not the utility;
+it requires these entities to coordinate with each other. In this
+paper, we develop an aggregator-utility coordination framework
+that enables network-safe control of thermostatically-controlled
+loads to provide frequency regulation. In our framework, the
+utility sends a network-safe constraint set on the aggregator’s
+command without directly sharing any network information. We
+propose a constraint set construction algorithm that guarantees
+satisfaction of a chance constraint on network safety. Assuming
+monotonicity of the probability of network safety with respect
+to the aggregator’s command, we leverage the bisection method
+to find the largest possible constraint set, providing maximum
+flexibility to the aggregator. Simulations show that, compared to
+two benchmark algorithms, the proposed approach provides a
+good balance between service quality and network safety.
+Index Terms—chance constraints, distributed energy resources,
+load control, network safety, thermostatically-controlled loads
+I. INTRODUCTION
+A
+S the amount of intermittent renewable generation is
+rapidly growing, it is becoming more difficult to rely
+solely on the conventional ways of balancing power systems.
+One emerging solution is to leverage Distributed Energy
+Resources (DERs), such as thermostatically-controlled loads
+(TCLs), batteries, and electric vehicles, to provide grid ser-
+vices. By doing so, they can improve the reliability, and
+reduce the operating cost and environmental impact of power
+systems. However, DERs coordinated to provide balancing
+services might cause issues in the distribution network, such as
+under/over-voltages, over-current violations, and transformer
+overheating, compromising network safety.
+When the distribution network operator (i.e., the utility)
+coordinates DERs to provide grid services it can adopt a cen-
+tralized algorithm that explicitly manages distribution network
+constraints, e.g., the algorithms provided in [1]–[3]. However,
+in competitive U.S. electricity markets it is becoming more
+likely that third-party (i.e., non-utility) DER aggregators will
+take on this role. Unfortunately, the aggregator does not have
+This work was supported by U.S. National Science Foundation Award
+CNS-1837680. The authors are with the Department of Electrical Engineering
+and Computer Science, University of Michigan, Ann Arbor, MI 48109 USA
+{sunhoj,necmiye,jlmath}@umich.edu.
+access to detailed distribution network information typically
+known only to the utility, and so it is unable to directly de-
+termine how its actions would affect the distribution network.
+This challenge has already been recognized by the US Federal
+Energy Regulatory Commission (FERC) [4].
+Thus, there is a need for coordination between the aggrega-
+tor and the utility to ensure network-safe operation of DERs.
+The recent FERC Order No. 2222 [5] provided some guidance
+on the development of operational coordination architectures
+between DER aggregators, utilities, and market coordinators;
+however, it is still unclear how these architectures will evolve
+and which architecture is “best.” Beyond ensuring network
+safety, coordination architectures should also 1) ensure that
+each entity’s private information (e.g., sensitive network in-
+formation held by the utility, proprietary DER coordination
+strategies held by the aggregator, and private DER state
+information held by the DERs’ end-users) is not shared with
+the other entities and 2) communication between the entities
+is minimal for compatibility with current communications
+infrastructure and/or to reduce the cost of any newly required
+infrastructure. Furthermore, architectures need to specify coor-
+dination protocols on different timescales, for example, 1) for
+operational planning such that the aggregator can determine
+its offer for balancing services, and 2) for real-time control
+in case network conditions differ significantly from forecasts
+and aggregator actions need to be curtailed.
+In this paper, we propose an aggregator-utility coordination
+framework for a collection of TCLs to provide balancing
+services like frequency regulation while ensuring distribution
+network-safety with high probability. We focus on real-time
+coordination, specifically, a setting in which an aggregator is
+already committed to provide a certain amount of balancing
+services, but real-time distribution network conditions require
+curtailment of those services. In our framework, the utility
+sends the aggregator a one-step ahead constraint set on the
+aggregator’s control input, which guarantees the satisfaction of
+a chance constraint on network safety with a certain confidence
+level. This method leverages estimation from Monte Carlo
+simulation and the bisection method to provide the largest
+possible constraint set to maximize the network-safe TCL
+flexibility. To achieve light communication requirements, the
+aggregator control algorithm assumes the TCLs all respond to
+the same scalar control input. This constrains the aggregator’s
+degrees-of-freedom but also makes it possible for the utility
+to define a simple constraint set on the control input.
+Previous work, e.g. [6]–[9], has proposed strategies to con-
+trol aggregations of TCLs, such as air conditioners and water
+
+2
+heaters, to provide balancing services in ways that are non-
+disruptive to end-users. TCLs have inherent thermal energy
+storage capacity and non-disruptiveness can be achieved, e.g.,
+by keeping internal temperatures inside a narrow temperature
+dead-band. However, network safety was not considered in the
+above papers. Some work has developed network-safe control
+algorithms for TCLs coordinated by third-party aggregators.
+Ref. [10] proposes both a utility-centric and an aggregator-
+centric coordination framework, differentiated by which entity
+ultimately sends control commands to the TCLs. That paper
+and [11] develop utility-centric strategies wherein the utility
+blocks aggregator’s commands that would otherwise cause net-
+work constraint violations. In contrast, our proposed approach
+would be considered aggregator-centric.
+Aggregator-centric network-safe DER coordination could
+be achieved through (convex) inner approximation of safe
+operating regions [12]–[14], which could be computed by the
+utility and sent to the aggregator as constraints on the net
+DER power deviations at each node. Research from Australia
+refers to these nodal constraints as operating envelopes [15]–
+[17]. Ref. [18] proposes an optimization problem to obtain a
+hyper-rectangular constraint set on the net power consumption
+of controllable DERs at each node in order to satisfy chance
+constraints on the voltage at each node. However, these
+approaches all require constraints to be applied at each node,
+rather than applying a constraint on aggregate power devia-
+tions by DERs located across a network. Ref. [19] proposes a
+method to constrain the norm of the power deviations across
+all nodes in the network, but requires significant computation
+to compute the constraint. Assuming an aggregate power
+deviation constraint exists, our previous work [20] develops an
+aggregator-centric TCL coordination algorithm using formal
+methods, but does not develop an approach to obtain the
+constraint, and the solutions are very conservative.
+In contrast to this previous work, this paper makes the
+following contributions: 1) we develop a new aggregator-
+centric approach to enable network-safe control of TCLs for
+balancing services; 2) assuming a simple control scheme that
+leverages a scalar control input to coordinate TCLs to provide
+balancing services (the aggregator’s algorithm), we develop
+an approach to constrain the control input to satisfy a chance
+constraint on network safety (the utility’s algorithm); and
+3) we demonstrate our approach in simulation and compare
+its performance to two benchmark approaches. In contrast to
+past work on network-safe control that assumes the system is
+deterministic, e.g., [19], here we consider uncertainty in the
+power consumption of non-participating loads. Furthermore,
+in contrast to [20], we assume the aggregator has incomplete
+information about the TCLs to reduce communication require-
+ments and preserve some level of privacy. Lastly, though some
+past work leveraged chance constraints to develop network-
+safe DER coordination approaches, e.g., [18], [21]–[26], these
+papers all assume that the controller has detailed distribution
+network information (enabling the formulation of a chance-
+constrained optimal power flow problem), which is inconsis-
+tent with our utility-aggregator coordination framework.
+The organization of the paper is as follows. Section II
+introduces the coordination framework and problem of in-
+terest. Section III explains the aggregator’s control approach
+and Section IV details the proposed constraint construction
+algorithm used by the utility to achieve network safety at
+a high level of probability. Section V presents the results
+of a case study comparing the proposed approach to two
+benchmarks. The appendix includes proofs of two of the
+theorems.
+Notation: N, [N], [N]0 are the set of natural num-
+bers, {1, . . ., N}, and {0, 1, . . ., N}, respectively. The n-
+dimensional Euclidean space is Rn. The jth element of the
+vector y is yj. Binomial distribution B(ns, ν) has ns trials,
+each with success probability ν, and cumulative density func-
+tion (cdf) FB(x; ns, ν). N(µ, σ2) is the normal distribution
+with mean µ and variance σ2. Function
+1(A) is 1 if A is
+true, and 0 otherwise. All random variables are capitalized
+English letters, e.g., X, with realizations denoted ˜x and esti-
+mates/approximates denoted ˆx. All other variables are denoted
+by symbols other than capitalized English letters. Vectors and
+matrices are bolded.
+II. FRAMEWORK & PROBLEM OF INTEREST
+We consider a framework in which a utility and aggregator
+coordinate to provide network-safe grid balancing services,
+e.g., frequency regulation, by aggregations of TCLs. TCLs
+switch ON/OFF to maintain temperature within a dead-band.
+We focus on real-time coordination, i.e., we assume that the
+aggregator has already participated in the ancillary services
+market and committed balancing service capacity to the in-
+dependent system operator (ISO). The amount of balancing
+service capacity offered by the aggregator was based on
+forecasts of the capabilities of the TCLs and the network state.
+However, the real-time network state differs significantly from
+its forecasts and so the committed balancing service capacity
+must be curtailed to avoid distribution network constraint
+violations. This could happen when load consumption and/or
+renewable power injections are significantly different from
+forecasts and the network is operating close to its limits.
+We assume that the following coordination steps occur at
+each discrete time step t, where the length of each time step
+is ∆t. The coordination scheme is shown in Fig. 1.
+1) The aggregator receives a constraint set U(t) from the
+utility and a reference signal pref(t) (e.g., a scaled and
+biased frequency regulation signal) from the ISO.
+2) The aggregator determines the control command u(t) ∈
+U(t) and broadcasts the same command to all TCLs.
+3) Each TCL maintains or switches its ON/OFF mode based
+on its temperature and the aggregator’s command u(t).
+4) The utility observes the real and reactive power consump-
+tion at each network node p(t) and q(t), and obtains
+some information from the aggregator (described below).
+Then, it constructs a one-step ahead constraint set U(t+1)
+and sends it to the aggregator. (And go back to step 1.)
+The aggregator’s goal is to select u(t) to maximize the
+quality of grid balancing services. This means that the aggre-
+gator should choose a command u(t) that is likely to adjust
+the aggregate power of the TCLs to match the reference
+signal pref(t) as closely as possible. Here, we assume the
+
+3
+Fig. 1. Coordination between the aggregator, utility, and the TCLs.
+aggregator’s command u(t) is a real scalar in the range [−1, 1]
+and is interpreted by each TCL as the probability it should
+switch modes; the details of how it switches are given in
+Section III. TCL coordination through probabilistic switching
+has been considered in previous work e.g., [6], [10]. An
+advantage of this type of command is that it only needs simple
+broadcast communication infrastructure. However, it does not
+allow the aggregator to directly adjust the power consumption
+of individual TCLs, which means that the aggregator has a
+low degree-of-freedom in control.
+Since the aggregator does not have detailed distribution
+network information and cannot evaluate how its command
+would affect the network, the utility sends a one-step ahead
+constraint set U(t+1) on the aggregator’s command u(t+1).
+This set U(t+1) is designed such that, if u(t+1) ∈ U(t+1),
+then probability of network safety is over a desired value 1−ǫ.
+We propose a method for the utility to compute U(t + 1) in
+Section IV, which is the main contribution of this work. To
+do this, the utility leverages:
+1) Real-time data from household smart meters to obtain
+the real and reactive power consumption at each node, p(t)
+and q(t). We recognize that in practice most utilities do not
+currently gather smart meter data in real-time, but this is
+possible with most existing smart meters and could be enabled
+through reconfiguration of their settings.
+2) Forecasts of the probability distributions of the one-
+step ahead real and reactive power consumption of non-
+participating loads at each node, P L(t+1) and QL(t+1). We
+assume that these distributions are estimated using historical
+and real-time data from household smart meters, and lever-
+aging a disaggregation technique [27] to separate the power
+consumption of the TCLs from that of the non-participating
+loads. We assume that P L(t) and QL(t) are correlated and
+fP L,QL(t) is their joint probability density function (pdf).
+3) Some real-time TCL information from the aggregator
+that is necessary for constraint set computation. This should
+be minimal to protect end-user privacy. In our framework, the
+aggregator provides the one-step ahead estimated fractions of
+TCLs that will be outside of their temperature dead-band and
+switched OFF-to-ON and ON-to-OFF by their thermostats,
+ˆ
+wON(t + 1) and ˆ
+wOFF(t + 1). Details on how this information
+is used are provided in Section IV-A.
+In this paper, for the sake of simplicity, we define network
+safety in terms of under-voltage violations. Specifically, we
+say that the network is safe if there are no under-voltage
+violations, and unsafe if there are any violations. The approach
+can be easily extended to include over-voltage violations and
+other distribution network constraint violations. The formal
+statement problem is as follows.
+Problem 1. Given the desired safety probability 1−ǫ, the real-
+time real and reactive power consumption at each node p(t)
+and q(t), the joint pdfs of the uncontrollable loads fP L,QL(t),
+fP L,QL(t + 1), and the fractions of TCLs that are outside of
+their dead-band wON(t + 1), wOFF(t + 1), find a one-step
+ahead constraint set U(t + 1) such that the following chance
+constraint holds if u(t + 1) ∈ U(t + 1),
+Pr
+�
+min
+j∈[n] Vj(t + 1) ≥ v
+�
+≥ 1 − ǫ,
+(1)
+where v is the lower bound on each of the nodal voltages Vj
+and n is the number of nodes other than the substation.
+To solve this problem, we define the one-time step ahead
+voltage at each node Vj(t + 1) as a random variable whose
+distribution depends on the command u(t + 1); the details are
+explained in Section IV. It is difficult to obtain a closed-form
+expression for the probability distribution of each Vj(t + 1).
+Therefore, our approach leverages Monte Carlo simulation to
+estimate the left side of (1) given a one-step ahead command
+u(t + 1). Since estimation from sampling leads to error, we
+find a constraint set U(t + 1) with a confidence level over a
+desired level 1 − β rather than giving an exact solution.
+III. AGGREGATOR’S CONTROL APPROACH
+In this section, we explain how the TCLs operate under the
+aggregator’s command u(t). For simplicity, we assume that all
+participating TCLs are cooling TCLs (e.g., air conditioners),
+though the approach also applies to heating TCLs. We denote
+by nTCL the vector whose element nTCL
+j
+is the number of
+participating TCLs at node j, and by nTCL := 1⊤nTCL the total
+number of participating TCLs, which satisfies �n
+j=1 nTCL
+j
+=
+nTCL. The internal temperature of the ith TCL at time t is
+denoted by θi(t) and its mode is denoted by mi(t), which
+is 0 when it is OFF, and 1 when it is ON. The temperature
+dynamics of the ith TCL follow the affine model from [28],
+θi(t + 1) = ai
+thθi(t) +
+�
+1 − ai
+th
+� �
+θi
+a(t) + ri
+thpi
+trmi(t)
+�
+,
+(2)
+where
+θi
+a(t)
+is
+the
+ambient
+temperature
+and
+ai
+th
+=
+exp(−∆t/(ri
+thci
+th)) is a parameter computed from the thermal
+resistance ri
+th and capacitance ci
+th of the ith TCL. Also, pi
+tr is
+the energy transfer rate of the ith TCL, which is negative for
+a cooling TCL. The power consumption of the ith TCL in the
+ON mode is pi := pi
+tr/ζi where ζi is the coefficient of per-
+formance; the power consumption in the OFF mode is 0. We
+assume that the reactive power consumption of the ith TCL is
+qi := ωipi, where ωi is a positive constant. The aggregate real
+power consumption of the TCLs is pagg(t) := �nTCL
+i=1 pimi(t).
+Each TCL has a temperature range [θi, θ
+i] within which its
+internal temperature should always be; this range is called the
+temperature dead-band. The temperature set-point, which is
+set by its end-user, θi
+s := (θi + θ
+i)/2 is the middle point of
+the dead-band. Whenever a TCL’s internal temperature reaches
+
+Input
+Reference
+Constraints
+Signal
+Utility
+Aggregator
+Network
+Aggregate
+Probabilistic
+information
+Power
+Input
+TCL
+TCL
+TCL
+Distribution
+1
+2
+NT
+Network
+Aggregate TCLs4
+or goes beyond the boundary of its dead-band it switches its
+mode to go back into the dead-band.
+At each time step t, the aggregator determines its command
+u(t) and broadcasts it to all participating TCLs. TCLs within
+their dead-bands interpret this command as the desired prob-
+ability of OFF TCLs to switch ON when u(t) > 0, and the
+desired probability of ON TCLs to switch OFF when u(t) < 0.
+To determine whether or not to switch, each TCL draws a
+random number zi(t) from the uniform distribution on the
+interval [0, 1) and compares it to the command u(t). If it is
+OFF and zi(t) ≤ u(t), then it switches ON. If it is ON and
+zi(t) ≤ −u(t), then it switches OFF.
+In summary, the mode of the ith TCL is
+mi(t) =
+
+
+
+
+
+1
+if θi(t) ≥ θ
+i
+0
+if θi(t) ≤ θi
+mc(zi(t), u(t))
+otherwise,
+(3)
+where mc(zi(t), u(t)) is equal to
+
+
+
+
+
+1
+if mi(t − 1) = 0 and zi(t) ≤ u(t)
+0
+if mi(t − 1) = 1 and zi(t) ≤ −u(t)
+mi(t − 1)
+otherwise.
+Note that, when positive (negative) u(t) is broadcast to
+the TCLs, the fraction of the OFF (ON) TCLs within their
+dead-bands that are switched is approximately u(t) (−u(t)).
+Thus, |u(t)| can be interpreted by the aggregator as the ratio
+of the power consumption increase (decrease) compared to
+the maximal increase (decrease). Therefore, even though the
+power consumption of each TCL is not directly controlled
+by the aggregator, the aggregator can manipulate pagg(t) by
+selecting the u(t) ∈ U(t) that is likely to adjust pagg(t) to
+match the reference signal pref(t) as closely as possible, i.e.,
+uopt(t) = arg min
+u∈U(t)
+|E [Pagg(t)] − pref(t)| ,
+(4)
+where U(t) is provided by the utility.
+IV. UTILITY’S CONSTRAINT CONSTRUCTION METHOD
+As mentioned in Section II, the utility computes a one-step
+ahead constraint set U(t + 1), which should be a solution to
+Problem 1. This requires the utility to be able to evaluate
+how the command u(t + 1) would affect the probability of
+network safety. In this section, we first show how the voltage
+at each node is modeled as a random variable. For ease
+of exposition, we consider only balanced radial distribution
+networks. Then, we derive the probability of network safety
+(i.e., the probability that no under-voltage violations happen)
+as a function of the command u(t + 1) = u.
+Next, we show how to verify whether or not the chance
+constraint (1) is satisfied under u(t + 1) = u with a desired
+confidence level, and how the utility can construct U(t+1) to
+ensure (1) is satisfied. We introduce a theorem establishing a
+confidence interval for the success probability of a Bernoulli
+random variable using Monte Carlo simulations. Using this
+result, we leverage the bisection method to find the largest
+upper bound on u(t + 1) that guarantees (1) with a desired
+confidence level. The largest upper bound gives the aggregator
+the greatest possible flexibility in determining its command.
+A. Modeling the probability of network safety
+We denote the real and reactive power consumption of
+participating TCLs across all nodes by P T(t) and QT(t) ∈ Rn.
+The utility approximates the nodal values as
+P T
+j (t) ≈ pjN ON
+j
+(t), QT
+j (t) ≈ qjN ON
+j
+(t)
+∀j ∈ [n],
+(5)
+where N ON
+j
+(t) and N OFF
+j
+(t) are the number of ON and OFF
+TCLs at node j, and pj and qj are the average real and reactive
+power rating (i.e., the ON-mode consumption) of the TCLs at
+node j. We additionally define diagonal matrices Ξp and Ξq ∈
+Rn×n whose jth diagonal elements are pj and qj, respectively.
+Then, P T(t) = ΞpN ON(t) and QT(t) = ΞqN ON(t), and the
+total real and reactive power consumption across all nodes is
+P (t) = ΞpN ON(t) + P L(t) and Q(t) = ΞqN ON(t) + QL(t).
+We first show how the one-step ahead number of ON TCLs
+N ON(t + 1) ∈ Rn is modeled as a random variable under the
+command u(t+1) = u. The number N ON(t+1) depends upon
+how many TCLs are switched both by their thermostat (i.e., the
+first and second cases of (3)) and by the aggregator’s command
+(i.e., the third case of (3)). The number of TCLs at each node
+j that will be switched ON, OFF by their thermostats is
+SON
+j (t + 1) = wON
+j (t + 1)N OFF
+j
+(t),
+SOFF
+j
+(t + 1) = wOFF
+j
+(t + 1)N ON
+j
+(t),
+(6)
+where, as defined in Section II, wON
+j
+(t + 1) is the one-step
+ahead fraction of OFF TCLs that will be switched ON and
+wOFF
+j
+(t + 1) is the one-step ahead fraction of ON TCLs that
+will be switched OFF by their thermostats at bus j. We assume
+that the aggregator estimates wON
+j (t + 1) and wOFF
+j
+(t + 1)
+using a model of the aggregate TCL dynamics and sends
+the estimated values ˆwON
+j (t + 1) and ˆwOFF
+j
+(t + 1) to the
+utility, which corresponds to the TCL information illustrated
+in Fig. 1. The utility uses these estimates to obtain realizations
+of SON
+j (t + 1) and SOFF
+j
+(t + 1) via Monte Carlo simulation,
+which will be explained in Section IV-B.
+According to (3), the numbers of TCLs at each node j that
+will be switched ON and OFF by the aggregator’s command
+follow binomial distributions,
+CON
+u,j(t + 1) ∼ B
+�
+N OFF
+j
+(t) − SON
+j (t + 1), u+�
+,
+COFF
+u,j (t + 1) ∼ B
+�
+N ON
+j
+(t) − SOFF
+j
+(t + 1), u−�
+,
+(7)
+where u+ := max(u, 0) and u− = max(−u, 0). Therefore,
+the number of ON TCLs given the command u(t + 1) = u is
+N ON
+u (t + 1) = N ON(t) + SON(t + 1)
+−SOFF(t + 1) + CON
+u (t + 1) − COFF
+u
+(t + 1). (8)
+Since the distributions of CON
+u,j(t + 1) and COFF
+u,j (t + 1) depend
+on u, the real and reactive power consumption across all nodes
+P (t+1) and Q(t+1) also depend on u. Therefore, from now
+on, we denote these random variables under the one-step ahead
+command u(t + 1) = u as Pu(t + 1) and Qu(t + 1).
+The next step is to model the one-step ahead voltage
+Vj(t + 1) at each node j as a random variable. Suppose that
+vj is the voltage magnitude at node j; pb
+j and qb
+j are the
+real and reactive power flowing through the branch whose
+receiving end is node j; and the resistance and reactance
+
+5
+of the branch are rj > 0 and xj > 0, respectively. Then,
+the DistFlow equations [29] corresponding to a single-phase
+equivalent model of a radial three-phase balanced network are
+pb
+j =
+�
+k∈c(j)
+pb
+k + pj + rj|ib
+j|
+qb
+j =
+�
+k∈c(j)
+qb
+k + qj + xj|ib
+j|
+v2
+j = v2
+e(j) − 2(rjpb
+j + xjqb
+j) + (r2
+j + x2
+j)|ib
+j|,
+(9)
+where e(j) and c(j) are the parent node and set of child
+nodes of node j, respectively, and |ib
+j|= ((pb
+j)2 + (qb
+j)2)/v2
+e(j)
+is the magnitude of the current flowing through the branch
+whose receiving end is node j. Given real and reactive power
+consumption p and q ∈ Rn and substation voltage v0, we
+let fvj(p, q, v0) be the voltage solution of (9), which can
+be obtained by various algorithms such as Backward-Forward
+Sweep [30]. Then, the one-step ahead voltage at node j under
+the command u(t + 1) = u is Vu,j(t + 1) = fvj(Pu(t +
+1), Qu(t + 1), v0). Note that we cannot obtain an explicit pdf
+of Vu,j(t + 1) since there is no closed-form solution of fvj.
+Instead, we can obtain a realization of Vu,j(t + 1) by solving
+(9) for a set of realizations ˜p and ˜q of Pu(t+1) and Qu(t+1).
+Finally, we define a Bernoulli random variable that indicates
+whether or not an under-voltage violation exists,
+Xu(t + 1) =
+1
+�
+minj∈[n]Vu,j(t + 1) ≥ v
+�
+,
+(10)
+whose success probability νu(t + 1) = Pr (Xu(t + 1) = 1)
+corresponds to the one-step ahead probability of network
+safety under command u(t + 1) = u. Thus, the utility’s
+problem is to find a set U(t+1) such that, for any u ∈ U(t+1),
+νu(t+1) is larger than 1−ǫ with confidence level over 1−β.
+The solution to this problem is explained in the next section.
+B. Probabilistically-safe set construction
+In this section, we first present a theorem on computing a
+confidence interval for the success probability of a Bernoulli
+random variable via a Monte Carlo simulation. Based on this
+theorem, we then show how the utility can test whether a
+command u(t + 1) = u is probabilistically safe and how this
+test procedure can be used to construct the set U(t + 1) of all
+commands that satisfy the chance constraint.
+Theorem 1. Suppose that X(1), . . . , X(ns) are i.i.d. samples of
+a random variable X following Bernoulli distribution B(1, ν)
+for a positive ν (i.e. Pr(X(i) = 1) = ν, Pr(X(i) = 0) = 1−ν
+for any i ∈ [ns]). Let Mns := �ns
+i=1 X(i)/ns be the estimator
+of ν, and ˜mns a realization of Mns. If the following inequalities
+hold,
+˜mns > 1 − ǫ
+(11)
+ns > ln
+� 1
+β
+�
+1
+( ˜mns + ǫ) ln( ˜mns + ǫ) − ( ˜mns + ǫ − 1), (12)
+then [1 − ǫ, 1] is a confidence interval for the success proba-
+bility ν of X with the confidence level over 1 − β.
+The proof is given in Appendix A. In our problem, ˜mns is a
+realization of an estimator of the success probability νu(t+1)
+obtained from realizations of Xu(t+1). This theorem implies
+that, if both ˜mns and the number of samples ns are sufficiently
+large, then νu(t+1) is larger than 1−ǫ. Thus, to verify whether
+or not νu(t + 1) is larger than 1 − ǫ, the utility can obtain a
+number of realizations of Xu(t + 1) and check if inequalities
+(11) and (12) hold.
+Now, we introduce the procedure the utility uses to obtain
+realizations of Xu(t + 1) given some u ∈ [−1, 1]. The utility
+first computes the probability mass function (pmf) of N ON(t)
+given the observed p(t) and q(t) as follows,
+Pr
+�
+N ON(t) = nON | (P (t) = p(t)) ∩ (Q(t) = q(t))
+�
+=
+Pr
+� �
+P L(t) = p(t) − ΞpnON�
+∩
+�
+QL(t) = q(t) − ΞqnON�
+| (P (t) = p(t)) ∩ (Q(t) = q(t))
+�
+=
+fP L,QL
+�
+p(t) − ΞpnON, q(t) − ΞqnON�
+�
+n∈NON fP L,QL (p(t) − Ξpn, q(t) − Ξqn),
+(13)
+where NON :=
+�
+nON | nON
+j
+∈ [nTCL
+j
+]0 ∀j ∈ [n]
+�
+is the set of
+all possible vectors for N ON(t). Then, the utility obtains a
+realization ˜xu(t + 1) of Xu(t + 1) through the following
+sampling procedure, illustrated in Fig. 2.
+Step 1) a. Obtain a realization ˜nON(t) of N ON(t) by sampling
+from its pmf derived through (13), and compute
+˜nOFF(t) = nTCL − ˜nON(t).
+b. Obtain realizations
+˜pL(t + 1) and ˜qL(t + 1) of P L(t + 1) and QL(t + 1)
+by sampling from fP L,QL(t + 1).
+Step 2) Obtain realizations ˜sON and ˜sOFF of SON(t + 1) and
+SOFF(t + 1) by computing their elements per (6) as
+˜sON
+j (t + 1) = ˆwON
+j (t + 1)˜nOFF
+j
+(t) ∀j ∈ [n]
+˜sOFF
+j
+(t + 1) = ˆwOFF
+j
+(t + 1)˜nON
+j (t) ∀j ∈ [n].
+Step 3) Obtain
+realizations
+˜cON
+u (t + 1)
+and
+˜cOFF
+u
+(t +
+1) of CON
+u (t + 1) and COFF
+u
+(t + 1) by sam-
+pling
+their
+elements
+per
+(7)
+from
+the
+bino-
+mial distributions B
+�
+˜nOFF
+j
+(t) − ˜sON
+j (t + 1), u+�
+and
+B
+�
+˜nON
+j (t) − ˜sOFF
+j
+(t + 1), u−�
+.
+Step 4) Obtain realizations of N ON
+u (t+1), Pu(t+1), Qu(t+
+1), Vu(t + 1), and Xu(t + 1) as
+˜nON
+u (t + 1) = ˜nON(t) + ˜sON(t + 1) − ˜sOFF(t + 1)
++ ˜cON
+u (t + 1) − ˜cOFF
+u
+(t + 1)
+˜pu(t + 1) = ˜pL(t + 1) + Ξp ˜nON
+u (t + 1)
+˜qu(t + 1) = ˜qL(t + 1) + Ξq ˜nON
+u (t + 1)
+˜vu,j(t + 1) = fvj(˜pu(t + 1), ˜qu(t + 1), v0) ∀j ∈ [n]
+˜xu(t + 1) =
+1(minj∈[n]˜vu,j(t + 1) ≥ v).
+The utility can obtain multiple realizations of Xu(t + 1)
+by iterating this sampling procedure. Denote each realization
+i of Xu(t + 1) as ˜x(i)
+u (t + 1), where i ∈ [ns]. In each
+iteration, the utility updates the realization of the estimator
+˜mns = �ns
+i=1 ˜x(i)
+u (t + 1)/ns and checks if the inequalities
+(11), (12) hold. If they do, u(t + 1) = u satisfies the
+chance constraint with confidence level over 1−β; otherwise,
+the utility continues to iterate until ns reaches some pre-
+determined upper bound ns, as shown in Fig. 2.
+
+6
+Fig. 2. Flowchart of the test procedure to check if a one-step ahead command
+u(t + 1) = u satisfies the chance constraint. The information required for
+each step is in orange.
+Next, we construct a one-step ahead constraint set U(t+1).
+We first make an assumption on the monotonicity of νu(t+1).
+Assumption 1. The one-step ahead probability of network
+safety νu(t + 1) monotonically decreases with respect to u.
+The intuition behind this assumption is that the real and
+reactive power consumption at each node is likely to increase
+as u increases, which is also likely to lead to a voltage
+decrease at every node. This assumption will be justified in
+Section IV-C. Under this assumption, the following holds.
+Theorem 2. Suppose that Assumption 1 holds and let ˜x(1)
+u (t+
+1), . . . , ˜x(ns)
+u
+(t + 1) be ns realizations of Xu(t + 1) for a
+command u ∈ [−1, 1]. If ns and ˜mns = �ns
+i=1 ˜x(i)
+u (t + 1)/ns
+satisfy (11) and (12), then U(t + 1) = [−1, u] is a solution to
+the Problem 1 with confidence level over 1 − β.
+Proof. By Theorem 1, the interval [1 − ǫ, 1] is a confidence
+interval for νu(t + 1) with confidence level over 1 − β. Also,
+under Assumption 1, νu(t + 1) ≥ νu(t + 1) holds for any
+u ∈ U(t + 1) = [−1, u]. Thus, νu(t + 1) is greater than or
+equal to 1−ǫ for any u ∈ U(t+1) with confidence level over
+1 − β.
+This theorem means that, if the one-step ahead probability
+of network safety νu(t+1) under the command u(t+1) = u is
+greater than or equal to the desired safety probability, then any
+less aggressive command in the range [−1, u] also satisfies the
+chance constraint. Therefore, a solution to Problem 1 is the
+interval [−1, u], where u passes the test procedure in Fig. 2.
+The choice of probabilistically-safe set U(t + 1) is not
+unique. A larger U(t + 1) gives more flexibility to the aggre-
+gator, potentially improves the quality of balancing services,
+and reduces the conservativeness of our approach. Therefore,
+the utility should find the largest possible u that passes the test
+procedure. This can be achieved using the bisection method
+[31], starting with u = 1.
+Remark 1. To restrict the probability of over-voltage vi-
+olations, we can also apply the monotonicity assumption;
+the probability of over-voltage violations increases as the
+command u decreases. In this case, we can use the bisection
+method to obtain a lower bound on u(t + 1). Then, the utility
+can send both a lower and upper bound on u(t+1) to restrict
+the probability of over- and under-voltage violations.
+Remark 2. Since the utility approximates P T and QT in
+(5) and uses estimates of wON and wOFF in Step 2 of the
+sampling procedure, Theorem 2 holds only if those approxi-
+mations/estimates are accurate. We will justify the use of these
+approximations/estimations through simulation in Section V.
+C. Justification of Assumption 1
+In this section, we justify Assumption 1 by showing that
+an approximation of νu(t + 1) is a monotonically decreasing
+function with respect to u. We consider the LinDistFlow
+equations [32], which drop the nonlinear terms of (9), i.e.,
+ˆpb
+j =
+�
+k∈c(j)
+ˆpb
+k + pj,
+ˆqb
+j =
+�
+k∈c(j)
+ˆqb
+k + qj
+ˆv2
+j = ˆv2
+e(j) − 2(rj ˆpb
+j + xj ˆqb
+j),
+(14)
+where variables with hats correspond to approximations of the
+original DistFlow variables. Let ˆfvj(p, q, v0) be the voltage
+solution of (14), i.e., ˆVu,j(t + 1) := ˆfvj(Pu(t + 1), Qu(t +
+1), v0) is the approximate voltage at node j. Also, let
+ˆXu(t + 1) =
+1
+�
+minj∈[n] ˆVu,j(t + 1) ≥ v
+�
+,
+(15)
+whose success probability ˆνu(t + 1) := Pr( ˆXu(t + 1) = 1)
+approximates νu(t + 1). To show that ˆνu(t + 1) is decreasing
+with respect to u, we start with a proposition.
+Proposition 1. Suppose that p(1), p(2) ∈ Rn and q(1), q(2) ∈
+Rn are different instances of real and reactive power con-
+sumption where p(1)
+j
+≤ p(2)
+j
+and q(1)
+j
+≤ q(2)
+j
+∀ j ∈ [n]. Then,
+ˆfvj(p(1), q(1), v0) ≥ ˆfvj(p(2), q(2), v0) for all j ∈ [n].
+Proof. First let ˆfpb
+j(p, q, v0) and ˆfqb
+j(p, q, v0) be the solutions
+of (14) corresponding to pb
+j and qb
+j when the real and reactive
+power consumption at each node are p and q, and the
+substation voltage is v0. Then, for all j ∈ [n] [32]
+ˆfpb
+j(p, q, v0) =
+�
+k∈d(j)
+pk,
+ˆfqb
+j(p, q, v0) =
+�
+k∈d(j)
+qk
+ˆf 2
+vj(p, q, v0) = v2
+0 − 2
+�
+k∈a(j)
+�
+rk ˆfpb
+k(p, q, v0)
++ xk ˆfqb
+k(p, q, v0)
+�
+,
+(16)
+where d(j) := c(j)∪{j} is the set of indices of all descendants
+of node j including itself, and a(j) is the set of indices of
+all ancestors of node j including itself. Hence, ˆfpb
+j(p, q, v0)
+and ˆfqb
+j(p, q, v0) are increasing as pk and qk increase for
+all k ∈ [n], and ˆfpb
+j(p(1), q(1), v0) ≤ ˆfpb
+j(p(2), q(2), v0) and
+ˆfqb
+j(p(1), q(1), v0) ≤ ˆfqb
+j(p(2), q(2), v0) for all j ∈ [n]. Also,
+since all rk and xk are positive, ˆfvj(p, q, v0) is decreasing as
+
+Initialize
+ns ←1
+fpLou (t),p(t),q(t)
+Stepla
+Step1b
+fpLo
+L(t+1)
+Sample NON (t)
+Sample
+pl(t + 1), Q(t + 1)
+WON(t + 1)
+Step2
+WOFF(t + 1)
+Sample
+sON(t + 1), SOFF(t + 1)
+ns - ns +1
+u(t+1)=u
+Step3
+Sample
+CON(t +1),COFF(t+1)
+Step 4
+Step4
+SampleNoN(t+1),Pu(t+1),Qu(t+1)
+V,(t + 1), X,(t + 1) and obtain x(ns)(t + 1)
+No
+Do
+x(t+1)
+No
+rns
+Is
+m
+ns
+ns
+ns≥n,?
+satisfy(10),(11)?
+Yes
+Yes
+Safe
+Unsafe7
+Fig. 3. Demonstration of the monotonicity of νu(t + 1) with respect to u.
+ˆfpb
+k(p, q, v0) and ˆfqb
+k(p, q, v0) increase for all k ∈ [n]. There-
+fore, ˆfvj(p(1), q(1), v0) ≥ ˆfvj(p(2), q(2), v0) ∀ j ∈ [n].
+This proposition states that ˆfvj(p, q, v0) monotonically de-
+creases as the real and reactive power consumption pj and
+qj at every node increase for all j ∈ [n]. Since the one-step
+ahead real and reactive power consumption of the TCLs at
+each node are likely to increase as u increases (recall that
+in Section III we made the realistic assumption that TCLs
+have constant lagging power factors, and so their real and
+reactive power consumption change in the same direction),
+this proposition implies that the probability of under-voltage
+violations increases as u increases. This is stated in the
+following theorem.
+Theorem 3. The approximate probability of network safety
+ˆνu(t + 1) under the one-step ahead command u is a mono-
+tonically decreasing function of u.
+The proof is given in Appendix B. While Theorem 3
+justifies Assumption 1 for the approximation ˆνu(t + 1), we
+also empirically validate that νu(t + 1) is a monotonically
+decreasing function of u in Fig. 3. To create this plot, we
+generated ns = 106 realizations of Xu(t + 1) for each of 101
+uniformly spaced points u from -1 to 1.
+V. CASE STUDY
+We next present the result of a case study in which we com-
+pare the proposed approach with two benchmark approaches,
+a tracking controller benchmark and an Optimal Power Flow
+(OPF) benchmark. We first describe our simulation setup and
+detail the benchmark approaches. Then, we present our results.
+We use the 56-bus balanced distribution network from [33]
+where the nominal real and reactive power consumption at
+node j are denoted by pLn
+j
+and qLn
+j , respectively. We set
+the safe lower bound on the voltage to v
+= 0.95 pu.
+TCL parameters are randomly sampled1, and the TCLs are
+distributed throughout the network so that the aggregate
+TCLs’ nominal real power consumption at node j is ap-
+proximately 0.25pLn
+j . For simplicity, we assume that the
+real and reactive power consumption of the non-participating
+loads at each node P L
+j (t) and QL
+j (t) follow normal distribu-
+tions N(pLn
+j (t), (0.15pLn
+j )2) and N(qLn
+j (t), (0.15qLn
+j )2) trun-
+cated by the intervals [pLmin
+j
+, pLmax
+j
+] = [−0.25pLn
+j , 0.675pLn
+j ]
+and [qLmin
+j
+, qLmax
+j
+] = [−0.25qLn
+j , 0.675qLn
+j ], respectively. We
+1Each parameter is sampled from uniform distributions with intervals:
+θi
+a ∈ [29, 31] °C, ci
+th ∈ [1.5, 2.5]kWh/°C, ri
+th = [1.2, 2.5]°C/kW, pi
+tr ∈
+[−18, −14] kW, ζi ∈ [2.3, 2.7], θi
+s ∈ [20, 25]°C, θ
+i − θi ∈ [1.5, 2]°C, and
+ωi = tan(arccos(φi)), where φi ∈ [0.95, 0.99].
+13
+13.5
+14
+14.5
+15
+Time (h)
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+Fraction of TCLs
+Actual Fraction
+Estimated Fraction
+13
+13.5
+14
+14.5
+15
+Time (h)
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+Fraction of TCLs
+Actual Fraction
+Estimated Fraction
+Fig. 4. Actual and estimated fractions of TCLs switched ON (left) and OFF
+(right) by their thermostats, in the proposed approach (ǫ = 0.02).
+conduct 2h simulations (13h-15h) and let pLn
+j (t) and qLn
+j (t)
+linearly increase from 0.5 to 0.65 of their nominal values
+from 13.0h to 13.9h, stay constant from 13.9h to 14.1h, and
+linearly decrease to 0.5 of their nominal values from 14.1h to
+15.0h. The reference signal pref(t) is a scaled and shifted 2h
+segment of a PJM RegD signal [34]. We use the desired safety
+probabilities 1−ǫ = 0.95 and 0.98 and the desired confidence
+level 1 − β = 0.999.
+The aggregator obtains the estimates
+ˆwON
+j (t + 1) and
+ˆwOFF
+j
+(t + 1) for each node leveraging an approximate model
+of the dynamics of the TCL aggregation. The model was
+developed in past work, e.g., [6], and so not detailed here.
+While we could identify different models for each node,
+here we use the same model for each node j ∈ [n] and so
+ˆwON
+j (t + 1) and ˆwOFF
+j
+(t + 1) are identical across nodes. Fig. 4
+demonstrates the model’s estimation performance, showing the
+actual and estimated fractions of TCLs outside of their dead-
+bands. Although the estimates do not perfectly track the actual
+values, they capture the overall trends.
+The tracking controller benchmark does not take into ac-
+count network safety. It chooses the optimal command uopt(t)
+using (4) with U(t) = [−1, 1], where E [Pagg(t)] is the
+expected aggregate power of the TCLs under u(t) = u, which
+is computed with the same approximate aggregate TCL model.
+The OPF benchmark approximately enforces network safety
+assuming linearized power flow. It solves the following mixed
+integer linear program at each time step to compute the optimal
+one-step ahead mode of each TCL,
+min
+mi |pagg − pref|
+(17a)
+s.t. pagg =
+nTCL
+�
+i=1
+pimi
+(17b)
+pT
+j =
+�
+i∈Ij
+pimi, qT
+j =
+�
+i∈Ij
+qimi,
+∀j ∈ [n]
+(17c)
+TCL temperature dynamics (2),
+∀i ∈ [nTCL]
+(17d)
+θi ∈ [θi, θi],
+∀i ∈ [nTCL]
+(17e)
+v = Φp(pT + pLmax) + Φq(qT + qLmax) + Φc
+(17f)
+v ≤ v,
+(17g)
+where Ij is the set of indices of TCLs connected to j and
+(17f) is the linearized power flow developed in [1]. The OPF
+benchmark is different from the proposed approach and opti-
+mal tracking controller in that it can observe the TCLs’ internal
+temperatures and directly control the TCLs’ modes. In contrast
+to the proposed approach, it has a deterministic constraint
+(17g) on network safety rather than a chance constraint.
+
+1.00
+0.75
+十
++0.50
+0.25
+0.00
+1.0
+-0.5
+0.0
+0.5
+1.0
+u8
+0
+0.5
+1
+1.5
+2
+Power (MW)
+Ref Signal
+Agg Power
+Nom Power
+13
+13.5
+14
+14.5
+15
+Time (h)
+0.94
+0.95
+0.96
+0.97
+0.98
+Voltage (V)
+Minimum Voltage
+Lower Bound
+13.5
+14
+14.5
+15
+Time (h)
+13.5
+14
+14.5
+15
+Time (h)
+13.5
+14
+14.5
+15
+Time (h)
+Fig. 5. The reference signal and the TCLs’ aggregate power (top), and the minimum network voltage and the safe lower bound (bottom) for each algorithm.
+TABLE I
+TRACKING AND SAFETY PERFORMANCE OF EACH ALGORITHM
+Track Ctrl
+OPF
+Proposed Approach
+Benchmark
+Benchmark
+ǫ = 0.05
+ǫ = 0.02
+RMSE (kW)
+77.05
+168.3
+102.8
+118.8
+Safety Probability
+0.908
+1.00
+0.981
+0.986
+Fig. 5 illustrates the results of the comparison between the
+two benchmarks and our proposed approach with ǫ = 0.05 and
+0.02. Table I shows the root mean squared error (RMSE) of
+the aggregate power from the reference signal, along with the
+empirical safety probability computed as the fraction of time
+steps in which under-voltage violations (computed with the
+nonlinear power flow equations) do not happen. The tracking
+controller benchmark has the best tracking performance, but
+frequently causes under-voltage violations. This demonstrates
+the need to employ network-safe DER control strategies. In
+contrast, the OPF benchmark avoids under-voltage violations,
+but has the worst tracking performance, demonstrating that
+approaches that (approximately) enforce network safety will
+at times have poor balancing service performance.
+Our approach achieves a better trade-off between tracking
+performance and network safety; specifically, it achieves better
+tracking performance than the OPF benchmark and satisfies
+the chance constraint on network safety, resulting in fewer
+under-voltage violations than the tracking controller bench-
+mark. As shown in Table I, the empirical safety probabilities
+are over the target values 1 − ǫ. The RMSE increases as ǫ
+decreases, which is expected since higher 1−ǫ results in more
+conservative bounds on the input commands.
+VI. CONCLUSION
+This paper proposed an approach to coordinate a collection
+of TCLs to provide balancing services while guaranteeing
+network safety with high probability. In particular, we pro-
+posed a constraint construction method that would allow the
+utility to constrain the input commands of an aggregator
+providing balancing services like frequency regulation. The
+approach imposes a chance constraint on network safety,
+wherein both the violation probability and confidence level are
+design parameters that can be selected by the utility. We used
+the bisection method to compute the largest possible constraint
+set, which provides the most flexibility to the aggregator.
+Future work will extend the proposed approach to incor-
+porate different types of DERs, such as stationary batteries,
+electric vehicles, and curtailable solar photovoltaics, into the
+framework; we already have some preliminary work along this
+direction [35].
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+APPENDIX
+A. Proof of Theorem 1
+By Theorem 4.1 in [36], the following inequality is derived
+from the Chernoff bound for any 0 < δ ≤ 1−ν
+ν ,
+Pr(Mns ≥ (1 + δ)ν) ≤
+�
+1
+1 + δ
+�(1+δ)nsν
+eδnsν
+= ensν(δ−(1+δ) ln(1+δ)).
+(18)
+We substitute c/ν, with c ∈ [0, 1 − ν], for δ and obtain
+Pr (Mns − ν ≥ c) ≤ ens(c−(ν+c) ln(1+ c
+ν))
+⇐⇒ Pr (ν ≥ Mns − c) ≥ 1 − ens(c−(ν+c) ln(1+ c
+ν)).
+(19)
+Hence, [ ˜mns − c, 1] is a confidence interval for ν with con-
+fidence level over 1 − ens(c−(ν+c) ln(1+ c
+ν )). Thus, if there
+exists c > 0 that satisfies ˜mns − c ≥ 1 − ǫ and 1 −
+ens(c−(ν+c) ln(1+ c
+ν )) > 1 − β, then [1 − ǫ, 1] is a confidence
+interval for ν with confidence level over 1 − β. Next, we
+show that such a c exists. First, we derive a lower bound on
+1 − ens(c−(ν+c) ln(1+ c
+ν)). Focusing on the exponent, observe
+that
+∂
+∂ν
+�
+c − (ν + c) ln
+�
+1 + c
+ν
+��
+= − ln
+�
+1 + c
+ν
+�
++ c
+ν .
+(20)
+If we let h1(x) := − ln(1+x)+x, the right side of (20) is equal
+to h1(c/ν). From h1(0) = 0 and ∂h1(x)/∂x = −1/(1 + x)+
+1 ≥ 0
+∀x ∈ [0, ∞), we have h1(x) ≥ 0 for all x ∈ [0, ∞),
+which means h1(c/ν) is non-negative. Hence, the exponent is
+increasing with respect to ν, and thus achieves its maximum
+at ν = 1. Therefore,
+Pr (ν ≥ Mns − c) ≥ 1 − ens(c−(1+c) ln(1+c)).
+(21)
+Since ν ≤ 1, (21) implies that [ ˜mns − c, 1] is a confidence in-
+terval for ν with confidence level over 1−ens(c−(1+c) ln(1+c)).
+Now, suppose that (11), (12) hold and define h2(x) :=
+x − (1 + x) ln(1 + x); the exponent on the right side of (21)
+is nsh2(c). From h2(0) = 0 and ∂h2(x)/∂x < 0 for all x ∈
+(0, ∞), we have h2(x) < 0 for all x ∈ (0, ∞). Since ˜mns −
+(1 − ǫ) > 0 by (11), ( ˜mns + ǫ − 1) − ( ˜mns + ǫ) ln( ˜mns +
+ǫ)) = h2( ˜mns − (1 − ǫ)) is negative. Also, substituting
+c with
+˜mns − (1 − ǫ), the right side of (21) becomes
+1 − ens(( ˜mns+ǫ−1)−( ˜mns+ǫ) ln( ˜mns+ǫ)) which is less than 1 −
+e− ln( 1
+β) = 1 − β, per (12). Hence, 1 − ens(c−(1+c) ln(1+c)) ≥
+1 − β and, thus, the interval [1 − ǫ, 1] = [ ˜mns − c, 1] is a
+confidence interval for ν with confidence level over 1−β.
+B. Proof of Theorem 3 and supporting lemmas
+We first introduce and prove Lemma 1, which is required
+for the proof of Lemma 2. Then, we prove Lemma 2, which is
+used in the proof of Theorem 3. Finally, we prove Theorem 3.
+Lemma 1. Suppose that aw(x), bw(x) : X → R+ are non-
+negative functions with parameter w ∈ R, and {˜x1, . . . , ˜xN}
+(˜x1 ≤ . . . ≤ ˜xN) is a finite subset of the domain X. Also, as-
+sume that the following two conditions hold: 1) �j
+k=1 aw(˜xk)
+is a decreasing function with respect to w for any j ∈
+{1, . . ., N}, and 2) bw(x) is decreasing function with respect
+to both x and w. Then, g(w) := �N
+k=1 aw(˜xk)bw(˜xk) is a
+decreasing function with respect to w.
+Proof. We prove the lemma by showing that, for w ≤ w,
+�j
+k=1 aw(˜xk)bw(˜xk) ≥ �j
+k=1 aw(˜xk)bw(˜xk) for any j ∈ [N]
+and w1, w2 ∈ R as follows:
+j
+�
+k=1
+aw(˜xk)bw(˜xk) ≥
+j
+�
+k=1
+aw(˜xk)bw(˜xk)
+(22a)
+= bw(˜xj)
+j
+�
+k=1
+aw(˜xk) +
+j−1
+�
+k=1
+∆bw(˜xk)
+k
+�
+l=1
+aw(˜xl)
+(22b)
+≥ bw(˜xj)
+j
+�
+k=1
+aw(˜xk) +
+j−1
+�
+k=1
+∆bw(˜xk)
+k
+�
+l=1
+aw(˜xl)
+(22c)
+=
+j
+�
+k=1
+aw(˜xk)bw(˜xk)
+(22d)
+
+10
+where ∆bw(˜xk) := (bw(˜xk) − bw(˜xk+1)), (22a) holds by
+condition 2 and (22c) holds by condition 1.
+Lemma 2. Suppose that Y (j)
+w
+(j ∈ [n]) is a discrete random
+variable with the finite sample space Y(j) = {´yj
+1, . . . , ´yj
+κj}
+(´yj
+1
+≤ . . . ≤
+´yj
+κj) with parameter w ∈ R having the
+following properties: 1) Y (1)
+w , · · · , Y (n)
+w
+are independent of
+each other, and 2) the cdf FY (j)(y; w) of Y (j)
+w
+is a decreasing
+function with respect to w for any y ∈ Y(j). Then, for any
+z(i) ∈ R (i ∈ [nc]) and non-negative coefficients aij ∈ R+,
+Pr
+��nc
+i=1
+��n
+j=1 aijY (j)
+w
+≤ z(i)��
+monotonically decreases
+as w increases.
+Proof. Let Yw = (Y (1)
+w , . . . , Y (n)
+w
+)⊤ be a multivariate random
+variable with elements Y (j)
+w
+and P = {y | Ay ≤ z} be a
+polyhedron with elements aij. Then,
+Pr
+
+
+nc
+�
+i=1
+
+
+n
+�
+j=1
+aijY (j)
+w
+≤ z(i)
+
+
+
+ = Pr (Yw ∈ P) .
+Note that P is a lower polyhedron in Πn
+j=1[´yj
+1, ´yj
+κj]; if y ∈ P,
+then y′ ∈ P also holds for any y′ ≤ y. Thus, it is sufficient
+to show that Pr (Yw1 ∈ P′) ≥ Pr (Yw2 ∈ P′) ∀ w1 ≥ w2 and
+any lower polyhedron P′, which we do as follows:
+1) Let n = 1 and P′
+1 ⊂ [´y1
+1, ´y1
+κ1] be a 1-dimensional lower
+polyhedron. Then, there exists y such that P′
+1 = [´y1
+1, y],
+and Pr(Y (1)
+w1
+∈ P′
+1) = FY (1)(y; w1) ≥ FY (1)(y; w2) =
+Pr(Y (1)
+w2 ∈ P′
+1), which proves the statement for n = 1.
+2) Let n
+=
+k and suppose Pr(Y (1:k)
+w1
+∈
+P′
+k)
+≥
+Pr(Y (1:k)
+w2
+∈ P′
+k) holds ∀ w1 ≥ w2 and for any k-
+dimensional lower polyhedron P′
+k ⊂ Πk
+j=1[´yj
+1, ´yj
+κj]. De-
+fine P−
+k (yk+1) = {(y1, . . . , yk)⊤ | (y1, . . . , yk, yk+1)⊤ ∈
+P′
+k+1}. Then, P−
+k (yk+1) is a lower polyhedron for
+any yk+1 ∈ [´yk+1
+1
+, ´yk+1
+κk+1]. Therefore, Pr(Y (1:k+1)
+w1
+∈
+P′
+k+1)
+=
+�κk+1
+j=1 Pr(Y (k+1)
+w1
+=
+´yk+1
+j
+)Pr(Y (1:k)
+w1
+∈
+P−
+k (´yk+1
+j
+)) for any k + 1-dimensional lower polyhedron
+P′
+k+1 ⊂ Πk+1
+j=1[´yj
+1, ´yj
+κj]. This is greater than or equal to
+�κk+1
+j=1 Pr(Y (k+1)
+w2
+= ´yk+1
+j
+)Pr(Y (1:k)
+w2
+∈ P−
+k (´yk+1
+j
+)) by
+Lemma 1, which in turn equals Pr(Y (1:k+1)
+w2
+∈ P′
+k+1).
+This proves the statement for n = k + 1.
+Therefore, by mathematical induction, Pr (Yw1 ∈ P′)
+≥
+Pr (Yw2 ∈ P′) holds for any lower polyhedron P′.
+Proof of Theorem 3. From (16), we obtain
+ˆV 2
+u,j(t + 1) = ˆf 2
+vj(Pu(t + 1), Qu(t + 1), v0)
+= v2
+0 − 2
+�
+k∈a(j)
+�
+rk ˆfpb
+k(Pu(t + 1), Qu(t + 1), v0)
++ xk ˆfqb
+k(Pu(t + 1), Qu(t + 1), v0)
+�
+= v2
+0 − 2
+�
+k∈a(j)
+�
+l∈d(k)
+(rkPu,l(t + 1) + xkQu,l(t + 1)) .
+(23)
+Substituting Pu,l(t+1) with P L
+l (t+1)+plN ON
+u,l (t+1), Qu,l(t+
+1) with QL
+l (t + 1) + qlN ON
+u,l (t + 1), N ON
+u,l (t + 1) with the right
+side of (8), and leveraging (23) we obtain
+ˆVu,j(t + 1) ≥ v ⇐⇒ ˆV 2
+u,j(t + 1) ≥ v2
+⇐⇒ gj(Cu(t + 1)) ≤ hj(R),
+where
+vector
+R
+:=
+(N ON(t)⊤, P L(t + 1)⊤, QL(t +
+1)⊤, SON(t + 1)⊤, SOFF(t + 1)⊤)⊤ collects random variables,
+Cu(t + 1) := CON
+u (t + 1) − COFF
+u
+(t + 1) is the net number of
+TCL OFF to ON switches by the aggregator’s command, and
+the functions gj and hj are
+gj (Cu(t + 1)) := 2
+�
+k∈a(j)
+�
+l∈d(k)
+(rkpl + xkql)Cu,l(t + 1),
+hj(R) := v2
+0 − v2 − 2
+�
+k∈a(j)
+�
+l∈d(k)
+�
+rkP L
+l (t + 1) + xkQL
+l (t + 1)
++ (rkpl + xkql)
+�
+N ON
+l
+(t) + SON
+l
+(t + 1) − SOFF
+l
+(t + 1)
+� �
+.
+Note that gj is a non-negative linear combination of Cu,l(t+1)
+for all j ∈ [n], i.e., there exist ajl ≥ 0 for any j, l ∈ [n] such
+that gj(Cu(t + 1)) is equal to �n
+l=1 ajlCu,l(t + 1).
+Let R be the sample space of R and fR be the joint
+probability density function of R. Then, we have
+ˆνu(t + 1) = Pr
+
+
+n
+�
+j=1
+�
+ˆVu,j(t + 1) ≥ v
+�
+
+
+=
+�
+˜r∈R
+Pr
+
+
+n�
+j=1
+(gj(Cu(t + 1)) ≤ hj(˜r))
+����R = ˜r
+
+ fR(˜r)d˜r.
+(24)
+For any realization, ˜r := (˜nON(t)⊤, ˜pL(t + 1)⊤, ˜qL(t +
+1)⊤, ˜sON(t+1)⊤, ˜sOFF(t+1)⊤)⊤ ∈ R, Cu,l(t+1) = CON
+u (t+
+1) when u ≥ 0, and Cu,l(t+ 1) = −COFF
+u
+(t+ 1) when u < 0.
+Thus, by (7), the conditional cdf of Cu,l(t+1) is computed as
+Pr(Cu,l(t + 1) ≤ k|R = ˜r) = FB(k; ˜nOFF
+l
+(t) − ˜sON
+l
+(t + 1), u)
+when u ≥ 0, and Pr(Cu,l(t + 1) ≤ k|R = ˜r) = 1 −
+FB(−k; ˜nON
+l
+(t) − ˜sOFF
+l
+(t + 1), −u) when u < 0. In addition,
+from [37], the cdf of a binomial random variable B(n; ν) is
+FB(k; n, ν) = (n − k)
+�n
+k
+� � 1−ν
+0
+tn−k−1(1 − t)kdt,
+which is a monotonically decreasing function with respect to
+ν. Thus, Pr(Cu,l(t + 1) ≤ k|R = ˜r) monotonically decreases
+as u increases, and Cu,1(t+1)|˜r, . . . , Cu,n(t+1)|˜r for any ˜r ∈
+R satisfies the conditions on the random variables in Lemma 2.
+Thus, Pr
+��n
+j=1
+�
+gj(Cu(t + 1)) ≤ hj(˜r)
+����R = ˜r
+��
+is a de-
+creasing function with respect to u. Therefore, by (24),
+ˆνu(t + 1) is also a decreasing function with respect to u.
+

8tA0T4oBgHgl3EQfOv9y/content/tmp_files/2301.02165v1.pdf.txt

@@ -0,0 +1,1045 @@
+Towards the Resolution of a Quantized Chaotic Phase Space: The Interplay of
+Dynamics with Noise
+Domenico Lippolis1 and Akira Shudo2
+1
+Institute for Applied Systems Analysis, Jiangsu University, Zhenjiang 212013, China; [email protected]
+2
+Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji, Tokyo 192-0397, Japan
+Abstract—We outline formal and physical similarities between the quantum dynamics of open systems, and the
+mesoscopic description of classical systems affected by weak noise. The main tool of our interest is the dissipative
+Wigner equation, that, for suitable timescales, becomes analogous to the Fokker-Planck equation describing classical
+advection and diffusion. This correspondence allows in principle to surmise a a finite resolution, other than the Planck
+scale, for the quantized state space of the open system, particularly meaningful when the latter underlies chaotic classical
+dynamics. We provide representative examples of the quantum-stochastic parallel with noisy Hopf cycles and Van der Pol
+oscillators.
+1. Introduction
+Efforts to reconcile classical and quantum mechanics are just about as old as quantum mechanics itself. While the
+formulation in Hilbert space makes it difficult to establish a direct correspondence between the two, a projection of the
+wave function to phase space may reveal some formal affinities between the quantum evolution of probability density and
+the traditional Liouville formalism of classical mechanics. The closest one can get to relate the two is by projecting the
+Liouville-von Neumann equation onto a suitable state space. For example, choosing the traditional phase space, we may
+obtain the so-called Wigner representation, that shares similarities with the aforementioned classical density evolution.
+Yet, there are also notable differences, as it stands to reason. The Wigner function, that is the projection of the density
+operator onto the phase-space, may also take on negative values, its evolution is governed by an equation plagued with
+an infinity of derivatives, and, as an indirect consequence of that, it may attain scales smaller than Planck’s constant [1].
+This is especially true in systems whose underlying classical dynamics exhibits chaotic behavior.
+In reality, however, no system is perfectly and eternally isolated, and exchange of matter or energy with the surround-
+ing environment is inevitable, whether due to measurements, thermal interactions, or shot noise [2, 3, 4]. That brings
+dissipation into the picture, and with that, decoherence.
+The effect of the environment on the evolution of a density matrix in a phase-space representation was first studied by
+Feynman and Vernon [5], who extended the path-integral formalism to dissipative quantum dynamics. Later, Caldeira and
+Leggett [6] derived an equivalent partial differential equation for the density matrix, that bears diffusive and dissipative
+terms, similarly to a Fokker-Planck equation. The latter analogy was then thought to hold but in the semiclassical limit,
+until a new wave of contributions [7, 8, 9, 10] reexamined the problem in a quantum chaotic setting. A most remarkable
+outcome of those works is the identification of a decoherence time, beyond which the Wigner equation is in all a Fokker-
+Planck equation since the higher-order derivatives may be safely neglected, and the quantized phase space may be resolved
+only up to a finite scale. Such resolution does not depend on the Planck constant, but rather emerges from the balance of
+the phase-space contraction rate (Lyapunov exponent) with the coupling of the system with an Ohmic environment.
+More recent contributions have focussed on the efficiency of Wigner evolution for general types of dissipation [11],
+and on obtaining a Lindblad-based dissipative Wigner equation to tackle quantum friction [12, 13].
+Once it is established that, under suitable conditions and after a sufficiently long time of evolution, the Wigner equation
+has the form of a Fokker-Planck equation, the quantum dissipative problem is cast into a classical stochastic process.
+Moreover, if the underlying classical dynamics of the quantum system in exam is chaotic, the limiting resolution of the
+phase space postulated in refs. [8, 10] is not expected to be uniform, but it will depend on the local interplay of the
+stretching/contraction with the dissipation. In the equivalent classical noisy problem, it is the ‘Brownian’ diffusion that
+plays the role of the dissipation.
+In the past decade, significant steps [14, 15, 17, 18, 16] were taken to determine the resolution of a chaotic state space
+in the presence of weak noise, and so reduce the dynamics to a Markov process of finite degrees of freedom, in the form
+of a matrix. Low-dimensional discrete-time dynamical systems such as logistic- or Hénon-type maps have been treated in
+a non-Hamiltonian setting, whose quantum analogs are in principle difficult to identify. The optimal resolution hypothesis
+arXiv:2301.02165v1  [quant-ph]  4 Jan 2023
+
+should be extended to continuous-time flows as well, and the starting point of that roadmap is a thorough comprehension
+of the steady-state solutions of the Fokker-Planck equation around the building blocs of chaos: periodic orbits.
+Here, we intend to lay the foundation of that understanding, by solving the Fokker-Planck equation of nonlinear
+paradigmatic dynamical systems, classical and with weak noise. We examine two-dimensional flows featuring nonlinear-
+ities but not yet chaos, where the competition between contraction and noise around a limit cycle results in a stationary
+density, which characterizes the steady state, and, as shown at the very end of the manuscript, shares common traits with
+the steady-state Wigner function of a case study in quantum dissipative dynamics.
+The article is structured as follows: in section 2 we review the basic tools of the phase-space representation of quantum
+dynamics, both in closed and open systems. We follow up in section 3 by discussing the main issues related to the evolution
+of the Wigner function in a quantum chaotic setting, the effects of dissipation, and the correspondence of the Wigner- with
+the Fokker-Planck equation. In section 4 a novel methodology is introduced to evaluate the steady-state solution of the
+Fokker-Planck equation around a periodic orbit, which casts the partial differential equation into an ordinary differential
+equation for the covariance matrix, known as Lyapunov equation. We first present a proof of concept on the simplest limit
+cycle, of circular shape as from a Hopf bifurcation, to be followed in section 5 by the treatment of the nonlinear oscillators
+that are the main object of the current study. At the end of the section the results on the Fokker-Planck steady-state
+densities are paralleled to those obtained for the Wigner function in a recent study of a quantum-dissipative model of the
+same oscillators. Summary and discussion close the paper.
+2. Density matrix, Wigner function, and dissipation
+Given a collection of physical systems, the ensemble average of an observable A is given by
+⟨A⟩ =
+�
+i
+ρi⟨ψi|A|ψi⟩ ,
+(1)
+or, using
+ρ =
+�
+i
+ρi|ψi⟩⟨ψi| ,
+(2)
+one can simply write
+⟨A⟩ = Tr �ρ A� ,
+(3)
+so that, if the observable A is time-independent, knowing ρ at all times means solving the problem of dynamics. That is
+the motivation for studying the density matrix ρ in the first place.
+2.1. Quantum dynamics in the phase space
+Now, the density ρ evolves according to the Liouville-von Neumann equation
+iℏρt = �ρ, H� ,
+(4)
+the quantum analog of the well-known Liouville equation
+ρt = {ρ, H}
+(5)
+of classical dynamics, which we spell out in phase space:
+∂tρ = − p
+m∂xρ +
+�
+∂xV(x)∂p
+�
+p ,
+(6)
+assuming a Hamiltonian of the form H = p2
+2m + V(x).
+In order to integrate the Liouville-von Neumann equation (4), we need to project it onto some basis, and several
+representations are already available to us, for instance, the P- or the Q-representation (a.k.a. Husimi’s)
+Q(α, α∗) = 1
+π⟨α|ρ|α⟩ ,
+(7)
+with |α⟩ a coherent state. Studying quantum-to-classical correspondence, especially of a system that exhibits chaotic
+behavior, is generally best achieved by using the Wigner representation [19]
+W(x, p) =
+1
+2πℏ
+�
+e−ipy/ℏψ
+�
+x + y
+2
+�
+ψ∗ �
+x − y
+2
+�
+dy ,
+(8)
+
+which can also be expressed in terms of the density matrix, as
+W(x, p) = 1
+πℏ
+�
+e−2ipy/ℏ⟨x − y|ρ|x + y⟩dy ,
+(9)
+of which ⟨x − y|ρ|x + y⟩ is called Weyl transform. An operator A may also be projected onto phase space, by applying a
+Weyl transform:
+˜A(x, p) =
+�
+e−ipy/ℏ⟨x + y/2|A|x − y/2⟩dy ,
+(10)
+which can prove handy in the evaluation of expectation values, that is
+⟨A⟩ = Tr �ρ A� =
+�
+W(x, p) ˜A(x, p)dxdp ,
+(11)
+since, in general,
+Tr [AB] =
+�
+˜A(x, p) ˜B(x, p)dxdp .
+(12)
+Thus, expectation values of observables are determined by means of phase-space averages, and the problem of quantum
+mechanics boils down to that of the time evolution of the Wigner function. It has been shown [19] that W(x, p) obeys the
+Wigner equation
+∂tW(x, p) = − p
+m∂xW(x, p) +
+∞
+�
+s=0
+cs(−ℏ2)s∂2s+1
+x
+V(x)∂2s+1
+p
+W(x, p) ,
+(13)
+that, in general, bears an infinite number of terms. In reality, integrating equation (13) can already be impractical if there
+are just a few nontrivial terms in the summation [20]. If the potential V(x) is at most quadratic, the Wigner equation
+reduces to Liouville’s, as in (6). Otherwise, Eq. (13) is still not easy to deal with, and, importantly, it may not be truncated
+in the semiclassical limit, since the terms ∂2s+1
+p
+W(x, p) bring down powers of ℏ−1−2s, so that
+ℏ2s ·
+1
+ℏ2s+1 ∼ ℏ−1 ,
+(14)
+and O
+�
+ℏ−1�
+does grow in the limit ℏ → 0, making no terms in the Wigner equation negligible, in principle.
+2.2. Open systems
+On the other hand, let us suppose the system is connected to an environment, whose interaction produces two addi-
+tional terms in the right-hand side of the Wigner equation (13), that is [8]
+2γ∂p
+�pW(x, p)� + D∂2
+ppW(x, p) .
+(15)
+The first term produces relaxation, due to the exchange of energy with the environment, and γ is the relaxation rate. The
+second term means diffusion, responsible for the so-called decoherence process, where one sets D = 2γMkBT, with M
+mass of the system, and T temperature of the environment. The dissipation and diffusion terms are obtained from a path-
+integral formulation of the system-environment interaction, that traces back to the works of Feynman and Vernon [5], and,
+later, of Caldeira and Leggett [6].
+If the potential V(x) is at most quadratic, one recovers the Fokker-Planck equation, that describes the classical evolu-
+tion of the density of trajectories produced by a particle subject to Brownian motion:
+∂tW(x, p) = − p
+m∂xW(x, p) + ∂xV(x)∂pW(x, p) + 2γ∂p
+�pW(x, p)� + D∂2
+ppW(x, p) .
+(16)
+This equation is fully quantum mechanical, and W(x, p) may take on negative values, unlike the classical phase-space
+density of a Brownian particle.
+Yet, for a general potential V(x), the evolution of the dissipative system is ruled by the full-fledged Wigner equa-
+tion (13) plus the terms (15) due to the environment:
+∂tW(x, p) = − p
+m∂xW(x, p) +
+∞
+�
+s=0
+cs(−ℏ2)s∂2s+1
+x
+V(x)∂2s+1
+p
+W(x, p) + 2γ∂p
+�pW(x, p)� + D∂2
+ppW(x, p) .
+(17)
+The resulting equation is still plagued with an infinite number of derivatives, and is thus of impractical integration. In the
+next section, we discuss whether and how it is safe to neglect the higher-order terms in Eq. (17), in the context of quantum
+chaos.
+
+3. Stretching, contracting, and Zaslavsky’s time
+Let us examine some aspects of the evolution of the Wigner function, when the underlying classical dynamics of the
+system is chaotic. By established knowledge [21], the two main features of chaos are
+1. Nearby trajectories diverge exponentially fast, meaning that, letting x = (x, p),
+λ = lim
+t→∞ ln
+�����
+δx(t)
+δx(0)
+����� > 0 ,
+(18)
+in other words, the difference δx(t) between any two nearby trajectories grows exponentially fast for any initial
+conditions. This feature is also described as extreme sensitivity of the system to initial conditions.
+2. The number M of qualitatively distinct orbits (‘configurations’, tagged by symbolic sequences) scales exponentially
+with their length, so that the topological entropy is positive:
+S = lim
+t→∞
+1
+t ln M(t) > 0 .
+(19)
+3.1. Chaos and the Wigner function
+Chaos is the result of a stretching and folding process mainly due to nonlinearities. For a Hamiltonian system, volumes
+in the phase space are conserved (by Liouville’s theorem), so that the amount of stretching (diverging trajectories) in some
+directions must be compensated by an equal amount of contraction in others.
+As a result, the inconvenient higher-order terms in the Wigner equation (13) can be estimated to evolve as
+∂2s+1
+p
+W(x, p) ∝ W(x, p)
+δp2s+1(t) ∼
+W(x, p)
+δp(0) e−(2s+1)λt ,
+(20)
+for smooth enough W(x, p). Thus, the inherent problem is in principle not with the smoothness of the density, but rather
+with the fact that the phase space contracts at an exponential rate, and therefore the contribution of higher-order derivatives
+in the equation (13) is more and more important, as time proceeds. To better illustrate that, let us compare the terms in
+the Poisson brackets of the Liouville equation (6) (also present in the full-fledged Wigner equation), with the higher-order
+terms in Eq. (13):
+∂xV(x)∂pW(x, p)
+cs∂2s+1
+x
+V(x)∂2s+1
+p
+W(x, p) ∼ 1
+cs
+∂xV(x)
+∂2s+1
+x
+V(x)
+δp2s
+W2s(x, p) ∼ 1
+cs
+∂xV(x)
+∂2s+1
+x
+V(x)
+δp(0) e−2sλt
+W2s(x, p) ≫ 1
+(21)
+is a condition for the higher-order terms to be negligible with respect to the lower-order, ‘Liouville’ terms. The above
+inequality can be inverted, and turned into a condition for the time t:
+t ≪ 1
+λ ln
+� ∂xV(x)
+∂2s+1
+x
+V(x)
+δp(0)
+csW2s(x, p)
+�−1/2s
+.
+(22)
+Identifying the quantity XV δp(0) =
+∂xV(x)δp(0)
+∂2s+1
+x
+V(x)W2s(x,p) with the typical action of the system, we can now understand
+t∗ = 1
+λ ln XV δp(0)
+ℏ
+(23)
+as the time scale within which the inconvenient higher-order terms of the Wigner equation may be neglected. Some
+literature refers to t∗ as Zaslavsky’s time [22]. Its meaning is somehow related to the more commonly mentioned Ehrenfest
+time, as in fact t∗ is longer the larger the ratio of the typical action to ℏ, and longest in the semiclassical limit. Still, the
+basic idea of this correspondence time does not relate directly with interference or need ‘semiclassical’ dynamics, but
+rather implies a finite resolution for the quantized phase space within a certain time scale, irrespective of the scale of the
+action. In general, the smoother the potential V(x), the longer t∗, the larger the Lyapunov exponent λ, the shorter t∗.
+3.2. A resolution for the quantized phase space
+In a simplified but physically meaningful description, that will then prove more accurate as a local model, we may rec-
+ognize and estimate the competing effects of dynamical contraction on the one hand, and of dissipation-induced diffusion
+on the other hand, by quantizing the Hamiltonian H = λxp. A wave packet of the form
+W(x, p) ∼ e−x2/σ2−σ2p2
+(24)
+
+evolves separately along the stretching x−direction, and the contracting p−direction. In momentum space, we have the
+Schrödinger equation
+∂tu(p, t) = λp∂pu(p, t) ⇒ u(p, t) = u0
+�
+peλt�
+,
+(25)
+that maps the wave packet in the p−direction as
+e−σ2p2/2 → ee2λtσ2p2/2 ,
+(26)
+and thus the width σ−2 shrinks by a factor of e−2λt after time t. Identifying σ−1 with the uncertainty δp(t) of the momentum,
+we may say that
+δp(t) ∼ δp(0)e−λt
+(27)
+along the contracting direction. On the other hand, connecting the system to an environment brings about diffusion, and
+∂tu(p, t) = D∂ppu(p, t) ⇒ u(p, t) ∼ e−p2/2(δp(0)+Dt) ,
+(28)
+whose variance evolves as
+√
+Dt:
+δp(t) ∼ �δp(0) + Dt�1/2
+(29)
+Then, intuitively, there must be some minimal scale in the contracting direction, set by
+δpmin ∼
+� D
+2λ
+�1/2
+.
+(30)
+The full picture is called Ornstein-Uhlenbeck problem [23]
+∂tu(p, t) = D∂ppu(p, t) − λ∂pu(p, t) .
+(31)
+In particular, the larger δpmin, the closer the evolution of the Wigner function to a stochastic process. More precisely, the
+regime where we may neglect the higher-order derivatives in the Wigner equation is deduced from Eq. (21) as
+XV δpmin
+ℏ
+≫ 1 ,
+(32)
+and that requires a relatively smooth potential, and the coefficient of the decoherence term in Eq. (16), D, to be comparable
+to the Lyapunov exponent λ. Now, if δpmin is an ‘equilibrium’ value as argued above, the chaotic contraction is no
+longer shrinking the scale of phase-space probability exponentially and indefinitely as in the non-dissipative setting (
+recall δp(t) ∼ δp(0)e−λt). Hence, in principle there would be no Zaslavsky’s time t∗, but rather, the quantum dissipative
+evolution may be well described by the Fokker-Planck type of equation (16) at all times, provided that the initial condition
+is smooth enough. Importantly, the semiclassical limit is not required for this approximation to work.
+4. Contraction vs. diffusion in stochastic dynamics
+Equation (16) and the discussion from the previous section suggest that, under suitable conditions, the problem of
+the dynamics of a quantum system connected to an environment may be cast into the classical evolution of a density
+according to a Fokker-Planck equation. As a consequence, studying the interplay of stretching/contracting dynamics with
+weak noise may also help shed light on quantum dissipation. Particularly interesting scenarios arise when the deterministic
+dynamics exhibits chaotic behavior. It is in fact well known that the phase space of a chaotic system has a self-similar
+(fractal) structure of infinite resolution. However, in reality, every system experiences noise, coming from experimental
+uncertainties, neglected degrees of freedom, or roundoff errors, for example. No matter how weak, noise smoothens out
+fractals, giving the system a finite resolution. The consequences are dramatic for the computation of long-time dynamical
+averages, such as diffusion coefficients or escape rates, since infinite-dimensional operators describing the evolution of
+the system (such as Fokker-Planck) effectively become finite matrices. With the aim of efficiently estimating long-time
+averages of observables for a chaotic dynamical system affected by background noise, a recent endeavor carried on over
+the past decade has achieved a technique to partition the chaotic phase space up to its optimal resolution, using unstable
+periodic orbits. The benchmark models already treated range from one-dimensional discrete-time repellers [14], and
+general unimodal maps [15], to two-dimensional chaotic attractors [18, 17]. Most importantly, a finite resolution for
+the state space of these models has effectively changed the dimensionality of the Fokker-Planck operator from infinite
+to inherently finite. Consequently, computations of the desired long-time averages become simpler and more efficient.
+On a more intuitive note, the present results also bear physical significance because, even when the external noise is
+
+uncorrelated, additive, isotropic, and homogenous, the interplay of noise and nonlinear dynamics always results in a
+local stochastic neighborhood, whose covariance depends on both the past and the future noise integrated and nonlinearly
+convolved with deterministic evolution along the trajectory. In that sense, noise is effectively never ‘white’ in nonlinearity,
+and thus the optimal resolution varies from neighborhood to neighborhood and has to be computed locally.
+As stated in the introduction, here we attack continuous-time dynamical systems, and begin by studying the evolution
+of noisy neighborhoods of periodic orbits. The simplest yet meaningful models are two-dimensional limit cycles, that can
+serve as testbed for parsing the interaction of deterministic dynamics with noise.
+4.1. The Lyapunov equation around a cycle
+Consider the Fokker-Planck equation
+∂tρ(x, t) = −∂x (v(x)ρ(x, t)) + ∆∂xxρ(x, t) ,
+(33)
+where ∆ is the diffusion tensor, whose entries are the noise amplitudes along each direction (∆ is diagonal with identical
+entries for isotropic noise). If we look at the dynamics in the neighborhood of a particular deterministic trajectory, we
+may linearize the velocity field v(x) locally, and replace it with Aa(x − xa), where Aa = ∂v(x)
+∂x
+����x=xa is the so-called matrix of
+variations. Moreover, we may switch to a co-moving reference frame in the desired neighborhood, say za = x − xa .
+Suppose we start off with an initial density of trajectories of Gaussian shape, that is ρa(za) =
+1
+Ca exp
+�
+−z⊤
+a
+1
+Qa za
+�
+. The
+short-time solution to (33) can then be written in the path-integral form
+ρa+1(za+1)
+=
+1
+Ca
+�
+[dza] exp
+�
+−1
+2(za+1 − (1 + Aaδt)za)⊤ 1
+∆δt (za+1 − (1 + Aaδt)za) − 1
+2z⊤
+a
+1
+Qa
+za
+�
+=
+1
+Ca+1
+exp
+�
+−1
+2z⊤
+a+1
+1
+(1 + Aaδt)Qa(1 + Aaδt)T + ∆δt za+1
+�
+.
+(34)
+One can then infer the relation between input and output quadratic forms in the exponential
+Qa+1 = ∆δt + (1 + Aaδt)Qa(1 + Aaδt)⊤ ,
+(35)
+and, neglecting terms of order δt2, recover the time-dependent Lyapunov equation
+˙Q = A(t)Q + QA⊤(t) + ∆ ,
+Q(t0) = Q0 .
+(36)
+Following the theory of time-dependent ordinary differential equations [24], we may write the solution of (36) as
+Q(t) = J(t, t0)
+�
+Q(t0) +
+� t
+t0
+J−1(s, t0)∆
+�
+J−1(s, t0)
+�⊤ds
+�
+J⊤(t, t0) .
+(37)
+Here J(t, t0) is the Jacobian along a flow x = x(t):
+d
+dt J(t, t0) = A(x)J(t, t0),
+J(t0, t0) = 1 .
+(38)
+One can verify this by just plugging the solution above into the equation. Alternatively, one can write Eq. (37) in the
+simpler form
+Q(t) = J(t, t0)Q(t0)J⊤(t, t0) +
+� t
+t0
+J(t, s)∆J⊤(t, s)ds ,
+(39)
+where the notation J(t, s) means that the Jacobian is computed following a trajectory that starts at time s and ends at time
+t, consistently with Eq. (38).
+4.2. Noisy circle
+Next, consider one of the simplest 2-dimensional dynamical systems, a pair of ODEs with a circular limit cycle of
+radius rc, together with additive isotropic white noise of strength 2D:
+˙x
+=
+λ(rc −
+�
+x2 + y2)x − ωy +
+√
+2Dξx
+˙y
+=
+λ(rc −
+�
+x2 + y2)y + ωx +
+√
+2Dξy
+(40)
+
+�1.0
+�0.5
+0.5
+1.0
+x
+�1.0
+�0.5
+0.5
+1.0
+y
+Figure 1: Solution of the numerically integrated Eq. (40) without noise. Any initial condition converges to the circular
+limit cycle.
+where
+< ξx(t)ξx(τ) >= δ(t − τ),
+< ξx(t)ξy(τ) >= 0.
+(41)
+In polar coordinates, this is written
+˙r
+=
+λ(rc − r)r +
+√
+2Dξx cos θ +
+√
+2Dξy sin θ
+˙θ
+=
+ω −
+√
+2Dξx
+sin θ
+r
++
+√
+2Dξy
+cos θ
+r
+(42)
+This Langevin-type equation produces the drift and diffusion coefficients [23]
+Dr
+=
+λ(rc − r)r + 2D
+r
+Dθ
+=
+ω
+Drr
+=
+2D
+Dθθ
+=
+2D
+r2
+(43)
+which then determine the Fokker-Planck equation for the system:
+∂tP + 1
+r ∂r[λ(rc − r)rP] + ∂θωP − D
+r ∂r(r∂rP) − D
+r2 ∂θθP = 0
+(44)
+The limit cycle r = rc can be either stable or unstable depending on the sign of λ. Let us consider the stable case.
+The first thing to look for is a stationary solution to the asymptotic form of (44):
+∂r[λ(rc − r)rP∞] − D∂r(r∂rP∞) = 0
+(45)
+A solution is
+P∞(r) = Ce− λ
+2D (r−rc)2 ,
+(46)
+which implies that P∞ is a Gaussian of width 2 √D/λ in the neighborhood of the limit cycle. The general solution to (44)
+is [23]
+P(r, θ, t) = e− λ
+2D (r−rc)2
+∞
+�
+n=0
+∞
+�
+ν=−∞
+Aν
+ne−sν
+nt(r − rc)|ν|L|ν|
+n (r − rc)eiνθ
+(47)
+where L|ν|
+n (r − rc) are generalized Laguerre polynomials and both the eigenvalues sν
+n and the coefficients Aν
+n can be found
+numerically.
+
+4.2.1. Neighborhood and coordinates
+This problem has an obvious symmetry, which allows us to guess the right (nonlinear!) change of coordinates, as well
+as the stationary solution, that is independent of the angular coordinate. The result is that the noisy neighborhood of the
+limit cycle is determined by the variance of the stationary solution (46). In general, however, we might not be so lucky,
+and guessing a suitable, possibly nonlinear, change of coordinates is probably beyond our reach. One way to identify a
+neighborhood for a periodic or any other orbit is to integrate the time-dependent Lyapunov equation (36) in the original
+(Cartesian) coordinates, but in a co-moving frame defined by the local coordinates za = x − xa introduced in section 4.1.
+Figure 2 illustrates this second approach: the forward Lyapunov equation (36) is numerically solved along an orbit that
+converges to the circular limit cycle 1, and its solution (39) is sampled along the trajectory and inverted to obtain Q−1(t),
+the covariance matrix of the Gaussian density, that produces a tube (in the figure in light yellow) along the orbit. The
+eigenvalues of Q−1(t) are found to converge to Λ1 =
+λ
+2D, consistently with the result for the width of the stationary-state
+solution (46) of the full-blown radial Fokker-Planck equation (45): that determines the width of the tube, σ = √1/2Λ1.
+The second eigenvalue of Q−1 is Λ2 = 0, as it appears from fig. 2(b). The latter eigenvalue is to be read as follows: while
+the forward Lyapunov equation converges to a finite limit in the stable (radial) direction, where noise balances contracting
+dynamics, it diverges along the marginal (tangent) direction, and therefore its inverse converges to zero, asymptotically.
+As shown in Fig. 2(c)-(d), the exact solution (46) of the Fokker-Planck equation is well reproduced by piecing together
+Gaussian tubes of covariance Q−1(t), each computed around a definite point of the noiseless limit cycle (the spurious
+lines orthogonal to the circle in Fig. 2(d) are due to the finite sampling of the Gaussian tubes, that should ideally be a
+continuum).
+As we may mostly be interested in the solution of the Lyapunov equation near unstable periodic orbits (like in a chaotic
+system), we then need to solve the same problem backwards in time, otherwise said by studying adjoint evolution, or the
+adjoint Lyapunov equation.
+4.3. Adjoint Lyapunov equation
+The backward evolution is described by the adjoint Fokker-Planck equation
+∂tρ(x, t) = v(x)∂xρ(x, t) + ∆∂xxρ(x, t) .
+(48)
+Following the line of thought of section 4.1, we can write the path-integral evolution of a Gaussian density in the neigh-
+borhood of an orbit
+ρa(za)
+=
+1
+Ca+1
+�
+[dza+1] exp [−1
+2(za+1 − (1 + Aaδt)za)⊤ 1
+∆δt (za+1 − (1 + Aaδt)za) − 1
+2z⊤
+a+1
+1
+Qa+1
+za+1]
+=
+1
+Ca
+exp
+�
+−1
+2z⊤
+a
+1
+Qa
+za
+�
+,
+(49)
+where
+Qa = (1 + Aaδt)−1 (Qa+1 + ∆δt)
+�
+(1 + Aaδt)⊤�−1 .
+(50)
+Analogously to the forward evolution, we can take the limit of infinitesimal time intervals and get the differential equation
+˙Q = ∆ − A(t)Q − QA⊤(t) ,
+(51)
+the adjoint (or backward-) Lyapunov equation. Compared to the forward Lyapunov equation (36), the adjoint evolu-
+tion (51) features the ‘time reversal’ operation A(t) → −A(t), and therefore we can still use Eq. (39) as a solution, as long
+as the Jacobian along the orbit is computed as
+d
+dt J(t, t0) = −A(x)J(t, t0),
+J(t0, t0) = 1 ,
+(52)
+and its computation follows the time reversed flow, that is the solution to the dynamical system ˙x = −v(x) .
+1Here we take the diffusion tensor ∆ =
+� 2D
+0
+0
+2D
+�
+
+(a)
+�1.0
+�0.5
+0.5
+1.0
+x
+�1.0
+�0.5
+0.5
+1.0
+y
+(b)
+0
+5
+10
+15
+20
+25
+30t
+0.2
+0.4
+0.6
+0.8
+1.0
+Σ, �
+(c)
+(d)
+Figure 2: (a) Solution of the numerically integrated Eq. (40), together with the eigenvectors (arrows) of the covariant
+matrix Q−1, as given by the solution (39) of the forward Lyapunov equation, for noise amplitude 2D = 0.1. The light
+yellow stripe is a pictorial representation of the Gaussian tube along the limit cycle (plan view); (b) The eigenvalues
+Λ1 (blue dots), Λ2 (red dots) of Q−1, and the width σ (solid line) of the evolved density versus time t; (c) The exact
+steady-state solution (46) to the Fokker-Planck equation for a noisy circular limit cycle; (d) The approximation to the
+same steady-state, obtained by piecing together solutions (39) to the Lyapunov equation around the limit cycle.
+5. Non-circular limit cycles: Classical noise vs. quantum dissipation
+We now turn our attention to non-circular limit cycles with background noise, and determine the steady-state density
+distribution yielded by the Fokker-Planck equation. We do so by integrating the Lyapunov equation in the neighborhood
+of a trajectory that eventually converges to the limit cycle.
+The paradigmatic models of our choice both come from the nonlinear oscillator
+¨x + ω2
+0x + µ
+�
+a ˙x
+3 − b˙x
+�
+= 0 .
+(53)
+In what follows, we shall set a = 1, b = 3, ω0 = 1, and we will tweak µ. Equation (53) may be reduced to a dynamical
+system as the Van der Pol oscillator
+˙x
+=
+y
+˙y
+=
+−µ
+�
+x2 − 3
+�
+y − x ,
+(54)
+
+yo
+0
+xyo
+0
+xor as the Rayleigh model
+˙x
+=
+y − µ
+�1
+3 x3 − 3x
+�
+˙y
+=
+−x .
+(55)
+�6
+�4
+�2
+2
+4
+6 x
+�6
+�4
+�2
+2
+4
+6
+y
+�4
+�2
+2
+4
+x
+�4
+�2
+2
+4
+y
+Figure 3: Solution of the numerically integrated (a) Eq. (55), and (b) Eq. (54), without noise. Any initial condition
+converges to a limit cycle.
+In both classical systems, the dynamics converges to a limit-cycle of non-circular shape, which depends on the pa-
+rameters, and it is characterized by fast and slow motion. One therefore expects the interplay of noise with the inherent
+nonlinear contraction to be non-uniform, unlike in the circular limit cycle examined in the previous section, and to give
+rise to a stationary density distribution of varying covariance along the cycle. We consider both models (54) and (55) for
+different values of the parameter µ, so as to gradually increase the eccentricity of the limit cycle, from a deformed circle
+[fig. 4(a)] to a nearly rectangular orbit [fig. 5(b)], where the deterministic stretching/contraction are most inhomogeneous
+along the cycle. The most notable feature is the oscillation along the direction orthogonal to that of noiseless motion of the
+covariance of the Gaussian solution of the linearized Fokker-Planck equation, denoted by σ in figs. 4(c)-(d) and 5(c)-(d):
+the more eccentric the limit cycle, the more widely and rapidly σ oscillates. That translates to a Gaussian steady-state
+density featuring a width that increasingly depends on the position along the orbit with the parameter µ, as we can see in
+the three-dimensional/density plots of figs. 4(e)-(f), and 5(e)-(f). As anticipated, these monodromic Gaussian distributions
+computed by means of the Lyapunov equation and portrayed in the figures would be, in a chaotic setting, the building
+blocs of a partition of the noisy phase space, whose non-uniform resolution is determined by their widths.
+The Gaussian solutions of the Lyapunov equation computed and illustrated here share common traits with the steady-
+state Wigner function of the same two oscillators (54) and (55), as obtained from a fully quantum mechanical compu-
+tation that has recently appeared in the literature [25]. In that work, the quantization is performed by means of cre-
+ation/annihilation operators
+ˆa = 1
+2 (ˆx + iˆy) ,
+(56)
+and its adjoint ˆa†, while dissipative terms are added to the Liouville-von Neumann equation (4), in the spirit of Lindblad’s
+formalism:
+iℏρt =
+�
+ρ, ˆa†ˆa
+�
+− γ1D
+�
+ˆa†�
+− γ2D
+�
+ˆa2�
+,
+(57)
+where
+D [ˆc] ρ = ˆcρˆc† − 1
+2 ˆc†ˆcρ − 1
+2ρˆc†ˆc .
+(58)
+The above equation (57) was numerically integrated, and the Wigner function was then found to eventually concentrate
+around the classical limit cycles, that feature similar eccentricities to the ones considered in the present work and plotted in
+figs. 4 and 5. In particular (Fig. 6), the steady-state Wigner distribution is enhanced along ‘tubes’ of varying width, as it can
+be noticed in the more eccentric density plots of the Rayleigh model [Fig. 6(c)-(d)]. This feature is especially apparent in
+Fig. 6(d), where the high-density region is narrower along the vertical segments of the limit cycle (faster classical motion),
+
+(a)
+�4
+�2
+2
+4
+x
+�4
+�2
+2
+4
+y
+(b)
+�4
+�2
+2
+4
+x
+�4
+�2
+2
+4
+y
+(c)
+6
+8
+10
+12
+14
+16
+18t
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Σ
+(d)
+6
+8
+10
+12
+14
+16t
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7Σ
+(e)
+(f)
+Figure 4: Top: solution of the numerically integrated Eq. (54), together with the eigenvectors (arrows) of the covariant
+matrix Q−1, as given by the solution (39) of the forward Lyapunov equation. In Eq. (54), we take (a)-(c)-(e) µ = 0.03, and
+(b)-(d)-(f) µ = 0.2, while the amplitude of the noise is set to 2D = 0.1. The light yellow stripe represents the width σ of
+the Gaussian density around the limit cycle; Middle: width of the Lyapunov tube, determined by the nonzero eigenvalue
+of Q−1(t), vs. time t. The cycle period is tp ≈ 7 time units; Bottom: Gaussian solutions of the linearized Fokker-Planck
+equation along the limit cycle.
+
+yo
+S
+0
+5
+xyo
+-5
+0
+5
+x(a)
+�4
+�2
+2
+4 x
+�4
+�2
+2
+4
+y
+(b)
+�6
+�4
+�2
+2
+4
+6 x
+�6
+�4
+�2
+2
+4
+6
+y
+(c)
+0
+2
+4
+6
+8
+10 12 14
+t
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Σ
+(d)
+0
+2
+4
+6
+8
+t
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Σ
+(e)
+(f)
+Figure 5: Top: solution of the numerically integrated Eq. (55), together with the eigenvectors (arrows) of the covariant
+matrix Q−1, as given by the solution (39) of the forward Lyapunov equation. In Eq. (55), we take (a)-(c)-(e) µ = 0.3, and
+(b)-(d)-(f) µ = 0.8, while the amplitude of the noise is set to 2D = 0.1. The light yellow stripe represents the width σ of
+the Gaussian density around the limit cycle; Middle: width of the Lyapunov tube, determined by the nonzero eigenvalue
+of Q−1(t), vs. time t. The cycle period is tp ≈ 7 time units; Bottom: Gaussian solutions of the linearized Fokker-Planck
+equation along the limit cycle.
+
+yo
+.5
+0
+5
+x5
+yo
+0
+5
+xand wider along its horizontal segments (slower classical motion). It compares directly with the Gaussian solution of the
+Lyapunov equation portrayed in Fig. 5(f).
+It is noted that in the cited work, the authors did not integrate the Wigner equation, but the Lindblad equation with
+a full-fledged quantum-mechanical algorithm. In particular, the localization of the Wigner function around the classical
+limit cycles is not to be taken for granted, and it legitimates the parallel between their steady-state solutions and the local
+Gaussian tubes obtained in the present work from the noisy classical system.
+On the other hand, the numerical steady-state solutions to Eq. (58) obtained by the authors of [25] are clearly not
+Gaussians centered at the limit cycles (except in Fig. 6(a), the case of least eccentricity), as demonstrated by their varying
+intensities along the periodic orbits. In that sense, the Gaussian Ansatz that turns the Fokker-Planck- into the Lyapunov
+equation carries but limited information on the phase-space density distributions at equilibrium. Therefore, the analogy
+proposed here should be taken with a grain of salt, and only considered as a hint for the noisy-classical to quantum-
+dissipative correspondence in a particular system with nontrivial interplay of contraction and diffusion.
+Finally, we would like to briefly comment on the difference between classical and quantum dissipation in the present
+models featuring limit cycles. In the classical system the dissipation produces damping, which is then balanced by the
+noise-induced diffusion. Instead, the quantum dissipation, generated by the characteristic Lindblad terms in Eq. (57), is
+responsible for both the ‘friction’ that drives densities to localize along the classical limit cycles, and the diffusion that
+spreads out the steady-state Wigner density distribution in the same region of the attracting orbits. This is consistent with
+the more general picture of section 2.2, where the quantum dissipation brings about both a damping- and a diffusive term
+in the Wigner equation.
+Figure 6: Quantum Van der Pol- and Rayleigh oscillators from [25]. In the figures the steady-state Wigner function is
+portrayed for increasing eccentricity parameter µ from Eqs. (54) [in (a)-(b)] and (55) [in (c)-(d)]. Reproduced with the
+permission of the American Physical Society from Ref. [25].
+6. Summary and discussion
+Having reviewed the parallels between the problem of dynamical evolution of a quantum system subject to dissipation
+and that of a stochastic process ruled by Fokker-Planck’s equation, we have narrowed our attention down to chaos, and,
+in particular, to the problem of an inherent scale resolution of the phase space. The issue is measuring the conjugated
+variables down to a certain precision, which may be set by the balance of the contraction rate of the classical chaos with
+the coupling to the environment, the source of dissipation.
+Using the analogy with the problem of classical chaotic dynamics with background noise, we consider the Fokker-
+Planck equation, and study its local solutions in the neighborhood of a periodic orbit, that effectively give the latter a
+finite width, in the phase space. Solving the problem for two-dimensional limit cycles, as done here, is the starting
+point: in a chaotic setting, a number of periodic tubes of finite width that proliferate exponentially with their length must
+end up overlapping, and thus determine the finest resolution for the noisy/quantized state space, that is expected to be
+non-uniform, as chaos interacts differently with diffusion/dissipation everywhere, in general.
+The analysis performed in this venue shows that the problem is tractable, and it provides the basic technology to attack
+it. Complications and obstacles are ahead for higher-dimensional systems, where stable, unstable, and marginal directions
+coexist along the same orbit, and where the solution to the adjoint Fokker-Planck operator introduced here will almost
+certainly be instrumental to the method. Still, the progress already achieved by periodic orbit theory in such complex
+models as the Kuramoto-Sivashinsky or the Navier-Stokes equation give us confidence in the feasibility of the optimal
+partition hypothesis in higher-dimensional chaos.
+
+(a)
+5 (b)
+p
+0
+P o
+5
+-5
+-5
+0
+5
+-5
+0
+5
+Q
+Q5 (c)
+5 (d)
+P o
+p 0
+-5
+-5
+0
+5
+-5
+0
+5
+Q
+QReferences
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+

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+Posterior Collapse and
+Latent Variable Non-identifiability
+Yixin Wang
+University of Michigan
+[email protected]
+David M. Blei
+Columbia University
+[email protected]
+John P. Cunningham
+Columbia University
+[email protected]
+Abstract
+Variational
+autoencoders
+model
+high-dimensional
+data
+by
+positing
+low-
+dimensional latent variables that are mapped through a flexible distribution
+parametrized by a neural network. Unfortunately, variational autoencoders often
+suffer from posterior collapse: the posterior of the latent variables is equal to its
+prior, rendering the variational autoencoder useless as a means to produce mean-
+ingful representations. Existing approaches to posterior collapse often attribute it
+to the use of neural networks or optimization issues due to variational approxima-
+tion. In this paper, we consider posterior collapse as a problem of latent variable
+non-identifiability. We prove that the posterior collapses if and only if the latent
+variables are non-identifiable in the generative model. This fact implies that pos-
+terior collapse is not a phenomenon specific to the use of flexible distributions or
+approximate inference. Rather, it can occur in classical probabilistic models even
+with exact inference, which we also demonstrate. Based on these results, we pro-
+pose a class of latent-identifiable variational autoencoders, deep generative mod-
+els which enforce identifiability without sacrificing flexibility. This model class
+resolves the problem of latent variable non-identifiability by leveraging bijective
+Brenier maps and parameterizing them with input convex neural networks, with-
+out special variational inference objectives or optimization tricks. Across syn-
+thetic and real datasets, latent-identifiable variational autoencoders outperform ex-
+isting methods in mitigating posterior collapse and providing meaningful repre-
+sentations of the data.
+1
+Introduction
+Variational autoencoders (VAE) are powerful generative models for high-dimensional data [28, 46].
+Their key idea is to combine the inference principles of probabilistic modeling with the flexibility of
+neural networks. In a VAE, each datapoint is independently generated by a low-dimensional latent
+variable drawn from a prior, then mapped to a flexible distribution parametrized by a neural network.
+Unfortunately, VAE often suffer from posterior collapse, an important and widely studied phe-
+nomenon where the posterior of the latent variables is equal to prior [6, 8, 38, 62]. This phenomenon
+is also known as latent variable collapse, KL vanishing, and over-pruning. Posterior collapse ren-
+ders the VAE useless to produce meaningful representations, in so much as its per-datapoint latent
+variables all have the exact same posterior.
+Posterior collapse is commonly observed in the VAE whose generative model is highly flexible,
+leading to the common speculation that posterior collapse occurs because VAE involve flexible
+neural networks in the generative model [11], or because it uses variational inference [59]. Based
+on these hypotheses, many of the proposed strategies for mitigating posterior collapse thus focus on
+modifying the variational inference objective (e.g. [44]), designing special optimization schemes
+35th Conference on Neural Information Processing Systems (NeurIPS 2021).
+arXiv:2301.00537v1  [stat.ML]  2 Jan 2023
+
+for variational inference in VAE (e.g. [18, 25, 32]), or limiting the capacity of the generative model
+(e.g. [6, 16, 60].)
+In this paper, we consider posterior collapse as a problem of latent variable non-identifiability. We
+prove that posterior collapse occurs if and only if the latent variable is non-identifiable in the gen-
+erative model, which loosely means the likelihood function does not depend on the latent vari-
+able [40, 42, 56]. Below, we formally establish this equivalence by appealing to recent results in
+Bayesian non-identifiability [40, 42, 43, 49, 58].
+More broadly, the relationship between posterior collapse and latent variable non-identifiability im-
+plies that posterior collapse is not a phenomenon specific to the use of neural networks or varia-
+tional inference. Rather, it can also occur in classical probabilistic models fitted with exact inference
+methods, such as Gaussian mixture models and probabilistic principal component analysis (PPCA).
+This relationship also leads to a new perspective on existing methods for avoiding posterior collapse,
+such as the delta-VAE [44] or the β-VAE [19]. These methods heuristically adjust the approximate
+inference procedure embedded in the optimization of the model parameters. Though originally mo-
+tivated by the goal of patching the variational objective, the results here suggest that these adjust-
+ments are useful because they help avoid parameters at which the latent variable is non-identifiable
+and, consequently, avoid posterior collapse.
+The relationship between posterior collapse and non-identifiability points to a direct solution to the
+problem: we must make the latent variable identifiable. To this end, we propose latent-identifiable
+VAE, a class of VAE that is as flexible as classical VAE while also being identifiable. Latent-
+identifiable VAE resolves the latent variable non-identifiability by leveraging Brenier maps [36,
+39] and parameterizing them with input-convex neural networks [2, 35]. Inference on identifiable
+VAE uses the standard variational inference objective, without special modifications or optimization
+tricks. Across synthetic and real datasets, we show that identifiable VAE mitigates posterior collapse
+without sacrificing fidelity to the data.
+Related work. Existing approaches to avoiding posterior collapse often modify the variational in-
+ference objective, design new initialization or optimization schemes for VAE, or add neural network
+links between each data point and their latent variables [1, 3, 6, 8, 12, 15, 16, 17, 18, 21, 25, 27, 32,
+34, 38, 44, 50, 51, 52, 55, 61, 62, 63]. Several recent papers also attempt to provide explanations
+for posterior collapse. Chen et al. [8] explains how the inexact variational approximation can lead
+to inefficiency of coding in VAE, which could lead to posterior collapse due to a form of informa-
+tion preference. Dai et al. [11] argues that posterior collapse can be partially attributed to the local
+optima in training VAE with deep neural networks. Lucas et al. [33] shows that posterior collapse
+is not specific to the variational inference training objective; absent a variational approximation, the
+log marginal likelihood of PPCA has bad local optima that can lead to posterior collapse. Yacoby
+et al. [59] discusses how variational approximation can select an undesirable generative model when
+the generative model parameters are non-identifiable. In contrast to these works, we consider poste-
+rior collapse solely as a problem of latent variable non-identifiability, and not of optimization, varia-
+tional approximations, or neural networks per se. We use this result to propose the identifiable VAE
+as a way to directly avoid posterior collapse.
+Outside VAE, latent variable identifiability in probabilistic models has long been studied in the
+statistics literature [40, 42, 42, 43, 49, 56, 58]. More recently, Betancourt [5] studies the effect of
+latent variable identifiability on Bayesian computation for Gaussian mixtures. Khemakhem et al.
+[23, 24] propose to resolve the non-identifiability in deep generative models by appealing to auxiliary
+data. Kumar & Poole [29] study how the variational family can help resolve the non-identifiability
+of VAE. These works address the identifiability issue for a different goal: they develop identifia-
+bility conditions for different subsets of VAE, aiming for recovering true causal factors of the data
+and improving disentanglement or out-of-distribution generalization. Related to these papers, we
+demonstrate posterior collapse as an additional way that the concept of identifiability, though classi-
+cal, can be instrumental in modern probabilistic modeling. Considering identifiability leads to new
+solutions to posterior collapse.
+Contributions.
+We prove that posterior collapse occurs if and only if the latent variable in the
+generative model is non-identifiable. We then propose latent-identifiable VAE, a class of VAE
+that are as flexible as classical VAE but have latent variables that are provably identifiable. Across
+synthetic and real datasets, we demonstrate that latent-identifiable VAE mitigates posterior collapse
+without modifying VAE objectives or applying special optimization tricks.
+2
+
+2
+Posterior collapse and latent variable non-identifiability
+Consider a dataset x = (x1,...,xn); each datapoint is m-dimensional. Positing n latent variables
+z = (z1,..., zn), a variational autoencoder (VAE) assumes that each datapoint xi is generated by a
+K-dimensional latent variable zi:
+zi ∼ p(zi),
+xi | zi ∼ p(xi | zi ; θ) = EF(xi | fθ(zi)),
+(1)
+where xi follows an exponential family distribution with parameters fθ(zi); fθ parameterizes the
+conditional likelihood. In a deep generative model fθ is a parameterized neural network. Classical
+probabilistic models like Gaussian mixture model [45] and probabilistic PCA [10, 47, 48, 54] are
+also special cases of Eq. 1.
+To fit the model, VAE optimizes the parameters θ by maximizing a variational approximation of the
+log marginal likelihood. After finding an optimal ˆθ, we can form a representation of the data using
+the approximate posterior q ˆφ(z|x) with variational parameters ˆφ or its expectation Eq ˆφ(z|x) [z|x].
+Note that here we abstract away computational considerations and consider the ideal case where the
+variational approximation is exact. This choice is sensible: if the exact posterior suffers from pos-
+terior collapse then so will the approximate posterior (a variational approximation cannot “uncol-
+lapse” a collapsed posterior). That said we also note that there exist in practice situations where
+variational inference alone can lead to posterior collapse. A notable example is when the variational
+approximating family is overly restrictive: it is then possible to have non-collapsing exact posteriors
+but collapsing approximate posteriors.
+2.1
+Posterior collapse ⇔ Latent variable non-identifiability
+We first define posterior collapse and latent variable non-identifiability, then proving their connec-
+tion.
+Definition 1 (Posterior collapse [6, 8, 38, 62]). Given a probability model p(x, z; θ), a parameter
+value θ = ˆθ, and a dataset x = (x1,...,xn), the posterior of the latent variables z collapses if
+p(z|x; ˆθ) = p(z).
+(2)
+The posterior collapse phenomenon can occur in a variety of probabilistic models and with dif-
+ferent latent variables. When the probability model is a VAE, it only has local latent variables
+z = (z1,..., zn), and Eq. 2 is equivalent to the common definition of posterior collapse p(zi |xi ; ˆθ) =
+p(zi) for all i [12, 17, 33, 44]. Posterior collapse has also been observed in Gaussian mixture mod-
+els [5]; the posterior of the latent mixture weights resembles their prior when the number of mixture
+components in the model is larger than that of the data generating process. Regardless of the model,
+when posterior collapse occurs, it prevents the latent variable from providing meaningful summary
+of the dataset.
+Definition 2 (Latent variable non-identifiability [42, 56]). Given a likelihood function p(x| z; θ), a
+parameter value θ = ˆθ, and a dataset x = (x1,...,xn), the latent variable z is non-identifiable if
+p(x| z = ˜z′ ; ˆθ) = p(x| z = ˜z; ˆθ)
+∀˜z′, ˜z ∈ Z ,
+(3)
+where Z denotes the domain of z, and ˜z′, ˜z refer to two arbitrary values the latent variable z can
+take. As a consequence, for any prior p(z) on z, we have the conditional likelihood equal to the
+marginal p(x| z = ˜z; ˆθ) =
+�
+p(x| z; ˆθ)p(z)dz = p(x; ˆθ)
+∀˜z ∈ Z .
+Definition 2 says a latent variable z is non-identifiable when the likelihood of the dataset x does
+not depend on z. It is also known as practical non-identifiability [42, 56] and is closely related to
+the definition of z being conditionally non-identifiable (or conditionally uninformative) given ˆθ [40,
+42, 43, 49, 58]. To enforce latent variable identifiability, it is sufficient to ensure that the likelihood
+p(x| z,θ) is an injective (a.k.a. one-to-one) function of z for all θ. If this condition holds then
+˜z′ ̸= ˜z
+⇒
+p(x| z = ˜z′ ; ˆθ) ̸= p(x| z = ˜z; ˆθ).
+(4)
+Note that latent variable non-identifiability only requires Eq. 3 be true for a given dataset x and
+parameter value ˆθ. Thus a latent variable may be identifiable in a model given one dataset but not
+another, and at one θ but not another. See examples in Appendix A.
+3
+
+Latent variable identifiability (Definition 2) [42, 56] differs from model identifiability [41], a related
+notion that has also been cited as a contributing factor to posterior collapse [59]. Latent variable
+identifiability is a weaker requirement: it only requires the latent variable z be identifiable at a
+particular parameter value θ = ˆθ, while model identifiability requires both z and θ be identifiable.
+We now establish the equivalence between posterior collapse and latent variable non-identifiability.
+Theorem 1 (Latent variable non-identifiability ⇔ Posterior collapse). Consider a probability model
+p(x, z; θ), a dataset x, and a parameter value θ = ˆθ. The local latent variables z are non-identifiable
+at ˆθ if and only if the posterior of the latent variable z collapses, p(z|x) = p(z).
+Proof. To prove that non-identifiability implies posterior collapse, note that, by Bayes rule,
+p(z|x; ˆθ) ∝ p(z)p(x| z; ˆθ) = p(z)p(x; ˆθ) ∝ p(z),
+(5)
+where the middle equality is due to the definition of latent variable non-identifiability. It implies
+p(z|x; ˆθ) = p(z) as both are densities. To prove that posterior collapse implies latent variable non-
+identifiability, we again invoke Bayes rule. Posterior collapse implies that p(z) = p(z|x; ˆθ) ∝ p(z)·
+p(x| z; ˆθ), which further implies that p(x| z; ˆθ) is constant in z. If p(x| z; ˆθ) nontrivially depends
+on z, then p(z) must be different from p(z)p(x| z; ˆθ) as a function of z.
+The proof of Theorem 1 is straightforward, but Theorem 1 has an important implication. It shows
+that the problem of posterior collapse mainly arises from the model and the data, rather than from
+inference or optimization. If the maximum likelihood parameters ˆθ of the VAE renders the latent
+variable z non-identifiable, then we will observe posterior collapse. Theorem 1 also clarifies why
+posteriors may change from non-collapsed to collapsed (and back) while fitting a VAE. When fitting
+a VAE, Some parameter iterates may lead to posterior collapse; others may not.
+Theorem 1 points to why existing approaches can help mitigate posterior collapse. Consider the β-
+VAE [19], the VAE lagging encoder [18], and the semi-amortized VAE [25]. Though motivated by
+other perspectives, these methods modify the optimization objectives or algorithms of VAE to avoid
+parameter values θ at which the latent variable is non-identifiable. The resulting posterior may not
+collapse, though the optimal parameters for these algorithms no longer approximates the maximum
+likelihood estimate.
+Theorem 1 can also help us understand posterior collapse observed in practice, which manifests
+as the phenomenon that the posterior is approximately (as opposed to exactly) equal to the prior,
+p(z|x; ˆθ) ≈ p(z). In several empirical studies of VAE (e.g. [12, 18, 25]), we observe that the
+Kullback-Leibler (KL) divergence between the prior and posterior is close to zero but not exactly
+zero, a property that stems from the likelihood p(x| z) being nearly constant in the latents z. In
+these cases, Theorem 1 provides the intuition that the latent variable is nearly non-identifiable ,
+p(x| ˜z′) ≈ p(x| ˜z),∀˜z, ˜z′ and so Eq. 2 holds approximately.
+2.2
+Examples of latent variable non-identifiability and posterior collapse
+We illustrate Theorem 1 with three examples. Here we discuss the example of Gaussian mixture
+VAE (GMVAE). See Appendix A for probabilistic principal component analysis (PPCA) and
+Gaussian mixture model (GMM).
+The GMVAE [13, 51] is the following model:
+p(zi) = Categorical(1/K),
+p(wi | zi ; µ,Σ) = N (µzi,Σzi),
+p(xi |wi ; f ,σ) = N (f (wi),σ2 · Im),
+where µk’s are d-dimensional, Σk are d × d-dimensional, and the parameters are θ = (µ,Σ, f ,σ2).
+Suppose the function f is fully flexible; thus f (wi) can capture any distribution of the data. The
+latent variable of interest is the categorical z = (z1,..., zn). If its posterior collapses, then p(zi =
+k|x) = 1/K for all k = 1,...,K.
+Consider fitting a GMVAE model with K = 2 to a dataset of 5,000 samples. This dataset is drawn
+from a GMVAE also with K = 2 well-separated clusters; there is no model misspecification. A GM-
+VAE is typically fit by optimizing the maximum log marginal likelihood ˆθ = argmaxθ log p(x|θ).
+Note there may be multiple values of θ that achieve the global optimum of this function.
+We focus on two likelihood maximizers. One provides latent variable identifiability and the posterior
+of zi does not collapse. The other does not provide identifiablity; the posterior collapses.
+4
+
+1. The first likelihood-maximizing parameter ˆθ1 is the truth; the distribution of the K fitted clusters
+correspond to the K data-generating clusters. Given this parameter, the latent variable zi is
+identifiable because the K data-generating clusters are different; different cluster memberships
+zi must result in different likelihoods p(xi | zi ; ˆθ1). The posterior of zi does not collapse.
+2. In the second likelihood-maximizing parameter ˆθ2, however, all K fitted clusters share the
+same distribution, each of which is equal to the marginal distribution of the data. Specifically,
+(µ∗
+k,Σ∗
+k) = (0, Id) for all k, and each fitted cluster is a mixture of the K original data generating
+clusters, i.e., the marginal. At this parameter value, the model is still able to fully capture the
+mixture distribution of the data. However, all the K mixture components are the same, and thus
+the latent variable zi is non-identifiable; different cluster membership zi do not result in differ-
+ent likelihoods p(xi | zi ; ˆθ2), and hence the posterior of zi collapses. Figure 1a illustrates a fit of
+this (non-identifiable) GMVAE to the pinwheel data [22]. In Section 3, we construct an latent-
+identifiable VAE (LIDVAE) that avoids this collapse.
+Latent variable identifiability is a function of the both the model and the true data-generating distri-
+bution. Consider fitting the same GMVAE with K = 2 but to a different dataset of 5,000 samples,
+this one drawn from a GMVAE with only one cluster. (There is model misspecification.) One max-
+imizing parameter value ˆθ3 is where both of the fitted clusters correspond to the true data generating
+cluster. While this parameter value resembles that of the first maximizer ˆθ1 above—both correspond
+to the true data generating cluster—this dataset leads to a different situation for latent variable iden-
+tifiability. The two fitted clusters are the same and so different cluster memberships do not result in
+different likelihoods of p(xi | zi ; ˆθ3). The latent variable zi is not identifiable and its posterior col-
+lapses.
+Takeaways.
+The GMVAE example in this section (and the PPCA and GMM examples in Ap-
+pendix A) illustrate different ways that a latent variable can be non-identifiable in a model and suffer
+from posterior collapse. They show that even the true posterior—without variational inference—can
+collapse in non-identifiable models. They also illustrate that whether a latent variable is identifiable
+can depend on both the model and the data. Posterior collapse is an intrinsic problem of the model
+and the data, rather than specific to the use of neural networks or variational inference.
+The equivalence between posterior collapse and latent variable non-identifiability in Theorem 1 also
+implies that, to mitigate posterior collapse, we should try to resolve latent variable non-identifiability.
+In the next section, we develop such a class of latent-identifiable VAE.
+3
+Latent-identifiable VAE via Brenier maps
+We now construct latent-identifiable VAE, a class of VAE whose latent variables are guaranteed to
+be identifiable, and thus the posteriors cannot collapse.
+3.1
+The latent-identifiable VAE
+To construct the latent-identifiable VAE, we rely on a key observation that, to guarantee latent
+variable identifiability, it is sufficient to make the likelihood function P(xi | zi ; θ) injective for all
+values of θ. If the likelihood is injective, then, for any θ, each value of zi will lead to a different
+distribution P(xi | zi ; θ). In particular, this fact will be true for any optimized ˆθ and so the latent zi
+must be identifiable, regardless of the data. By Theorem 1, its posterior cannot collapse.
+Constructing latent-identifiable VAE thus amounts to constructing an injective likelihood function
+for VAE. The construction is based on a few building blocks of linear and nonlinear injective
+functions, then composed into an injective likelihood p(xi | zi ; θ) mapping from Z d to X m, where
+Z and X indicate the set of values zi and xi can take. For example, if xi is an m-dimensional binary
+vector, then X = {0,1}m; if zi is a K-dimensional real-valued vector, then Z = Rd.
+The building blocks of LIDVAE: Injective functions. For linear mappings from Rd1 to Rd2 (d2 ≥
+d1), we consider matrix multiplication by a d1 × d2-dimensional matrix β. For a d1-dimensional
+variable z, left multiplication by a matrix β⊤ is injective when β has full column rank [53]. For
+example, a matrix with all ones in the diagonal and all other entries being zero has full column rank.
+For nonlinear injective functions, we focus on Brenier maps [4, 37]. A d-dimensional Brenier map
+is is the gradient of a convex function from Rd to R. That is, a Brenier map satisfies g = ∇T for
+5
+
+some convex function T : Rd → R. Brenier maps are also known as a monotone transport map. They
+are guaranteed to be bijective [4, 37] because their derivative is the Hessian of a convex T, which
+must be positive semidefinite and has a nonnegative determinant [4].
+To build a VAE with Brenier maps, we require a neural network parametrization of the Brenier map.
+As Brenier maps are gradients of convex functions, we begin with the neural network parametrizaton
+of convex functions, namely the input convex neural network (ICNN) [2, 35]. This parameterization
+of convex functions will enable Brenier maps to be paramterized as the gradient of ICNN.
+An L-layer ICNN is a neural network mapping from Rd to R. Given an input u ∈ Rd, its lth layer is
+z0 = u,
+zl+1 = hl(Wlzl +Alu+bl),
+(l = 0,...,L −1),
+(6)
+where the last layer zL must be a scalar, {Wl} are non-negative weight matrices with W0 = 0. The
+functions {hl : R → R} are convex and non-decreasing entry-wise activation functions for layer l;
+they are applied element-wise to the vector (Wlzl + Alu + bl). A common choice of h0 : R → R
+is the square of a leaky RELU, h0(x) = (max(α · x,x))2 with α = 0.2; the remaining hl’s are set to
+be a leaky RELU, hl(x) = max(α· x,x). This neural network is called “input convex” because it is
+guaranteed to be a convex function.
+Input convex neural networks can approximate any convex function on a compact domain in sup
+norm (Theorem 1 of Chen et al. [9].) Given the neural network parameterization of convex functions,
+we can parametrize the Brenier map gθ(·) as its gradient with respect to the input gθ(u) = ∂zL/∂u.
+This neural network parameterization of Brenier map is a universal approxiamtor of all Brenier maps
+on a compact domain, because input convex neural networks are universal approximators of convex
+functions [9].
+The latent-identifiable VAE (LIDVAE). We construct injective likelihoods for LIDVAE by com-
+posing two bijective Brenier maps with an injective matrix multiplication. As the composition of in-
+jective and bijective mappings must be injective, the resulting composition must be injective. Sup-
+pose g1,θ : RK → RK and g2,θ : RD → RD are two Brenier maps, and β is a K ×D-dimensional matrix
+(D ≥ K) with all the main diagonal entries being one and all other entries being zero. The matrix
+β⊤ has full column rank, so multiplication by β⊤ is injective. Thus the composition g2,θ(β⊤ g1,θ(·))
+must be an injective function from a low-dimensional space RK to a high-dimensional space RD.
+Definition 3 (Latent-identifiable VAE (LIDVAE) via Brenier maps). An LIDVAE via Brenier maps
+generates a D-dimensional datapoint xi,∈ {1,...,n} by:
+zi ∼ p(zi),
+xi | zi ∼ EF(xi | g2,θ(β⊤ g1,θ(zi))),
+(7)
+where EF stands for exponential family distributions; zi is a K-dimensional latent variable, discrete
+or continuous. The parameters of the model are θ = (g1,θ, g2,θ), where g1,θ : RK → RK and g2,θ :
+RD → RD are two continuous Brenier maps. The matrix β is a K × D-dimensional matrix (D ≥ K)
+with all the main diagonal entries being one and all other entries being zero.
+Contrasting LIDVAE (Eq. 7) with the classical VAE (Eq. 1), the LIDVAE replaces the function
+fθ : Z K → X D with the injective mapping g2,θ(β⊤ g1,θ(·)), composed by bijective Brenier maps
+g1,θ, g2,θ and a zero-one matrix β⊤ with full column rank. As the likelihood functions of exponential
+family are injective, the likelihood function p(xi | zi ; θ) = EF(g2,θ(β⊤ g1,θ(zi))) of LIDVAE must
+be injective. Therefore, replacing an arbitrary function fθ : Z K → X D with the injective mapping
+g2,θ(β⊤ g1,θ(·)) plays a crucial role in enforcing identifiability for latent variable zi and avoiding
+posterior collapse in LIDVAE. As the latent zi must be identifiable in LIDVAE, its posterior does
+not collapse.
+Despite its injective likelihood, LIDVAE are as flexible as VAE; the use of Brenier maps and ICNN
+does not limit the capacity of the generative model. Loosely, LIDVAE can model any distributions
+in RD because Brenier maps can map any given non-atomic distribution in Rd to any other one in
+Rd [37]. Moreover, the ICNN parametrization is a universal approximator of Brenier maps [2]. We
+summarize the key properties of LIDVAE in the following proposition.
+Proposition 2. The latent variable zi is identifiable in LIDVAE, i.e. for all i ∈ {1,...,n}, we have
+p(xi | zi = ˜z′ ; θ) = p(xi | zi = ˜z; θ)
+⇒
+˜z′ = ˜z,
+∀ ˜z′, ˜z,θ.
+(8)
+Moreover, for any VAE-generated data distribution, there exists an LIDVAE that can generate the
+same distribution. (The proof is in Appendix B.)
+6
+
+15
+10
+5
+0
+5
+10
+15
+20
+15
+10
+5
+0
+5
+10
+15
+(a) Non-ID GMVAE
+15
+10
+5
+0
+5
+10
+15
+20
+15
+10
+5
+0
+5
+10
+15
+(b) IDGMVAE
+0
+100
+200
+300
+400
+500
+Epoch
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Accuracy
+(c) Accuracy
+0
+200
+400
+Epoch
+400
+350
+300
+250
+Test Log Likelihood
+3-layer ID-GMVAE
+6-layer ID-GMVAE
+9-layer ID-GMVAE
+3-layer GMVAE
+6-layer GMVAE
+9-layer GMVAE
+(d) Log-likelihood
+Figure 1: (a)-(b): The posterior of the classical GMVAE [13, 26, 51] collapses when fit to the
+pinwheel dataset; the latents predict the same value for all datapoints. The posteriors of latent-
+identifiable Gaussian mixture VAE (LIDGMVAE), however, do not collapse and provide meaning-
+ful representations.
+(c)-(d) The latent-identifiable GMVAE produces posteriors that are substan-
+tially more informative than GMVAE when fit to fashion MNIST. It also achieves higher test log
+likelihood.
+3.2
+Inference in LIDVAE
+Performing inference in LIDVAE is identical to the classical VAE, as the two VAE differ only in
+their parameter constraints. To fit an LIDVAE, we use the classical amortized inference algorithm
+of VAE; we maximize the evidence lower bound (ELBO) of the log marginal likelihood [28].
+In general, LIDVAE are a drop-in replacement for VAE. Both have the same capacity (Proposi-
+tion 2) and share the same inference algorithm, but LIDVAE is identifiable and does not suffer from
+posterior collapse. The price we pay for LIDVAE is computational: the generative model (i.e. de-
+coder) is parametrized using the gradient of a neural network; its optimization thus requires calcu-
+lating gradients of the gradient of a neural network, which increases the computational complex-
+ity of VAE inference and can sometimes challenge optimization. While fitting classical VAE using
+stochastic gradient descent has O(k · p) computational complexity, where k is the number of itera-
+tions and p is the number of parameters, fitting latent-identifiable VAE may require O(k · p2) com-
+putational complexity.
+3.3
+Extensions of LIDVAE
+The construction of LIDVAE reveals a general strategy to make the latent variables of generative
+models identifiable: replacing nonlinear mappings with injective nonlinear mappings. We can em-
+ploy this strategy to make the latent variables of many other VAE variants identifiable. Below we
+give two examples, mixture VAE and sequential VAE.
+The mixture VAE, with GMVAE as a special case, models the data with an exponential family
+mixture and mapped through a flexible neural network to generate the data. We develop its latent-
+identifiable counterpart using Brenier maps.
+Example 1 (Latent-identifiable mixture VAE (LIDMVAE)). An LIDMVAE generates a D-
+dimensional datapoint xi, i ∈ {1,...,n} by
+zi ∼ Categorical(1/K),
+wi | zi ∼ EF(wi |β⊤
+1 zi),
+xi |wi ∼ EF(xi | g2,θ(β⊤
+2 g1,θ(wi))),
+(9)
+where Wi is a K-dimensional one-hot vector that indicates the cluster assignment. The parameters
+of the model are θ = (g1,θ, g2,θ), where the functions g1,θ : RM → RM and g2,θ : RD → RD are two
+continuous Brenier maps. The matrices β1 and β2 are a K × M-dimensional matrix (M ≥ K) and
+a M × D-dimensional matrix (D ≥ M) respectively, both having all the main diagonal entries being
+one and all other entries being zero.
+The LIDMVAE differs from the classical mixture VAE in p(xi | zi), where we replace its neural
+network mapping with its injective counterpart, i.e. a composition of two Brenier maps and a matrix
+multiplication g2,θ(β⊤
+2 g1,θ(·)). As a special case, setting both exponential families in Example 1 as
+Gaussian gives us LIDGMVAE, which we will use to model images in Section 4.
+Next we derive the identifiable counterpart of sequential VAE, which models the data with an au-
+toregressive model conditional on the latents.
+7
+
+Fashion-MNIST
+Omniglot
+AU
+KL
+MI
+LL
+AU
+KL
+MI
+LL
+VAE [28]
+0.1
+0.2
+0.9
+-258.8
+0.02
+0.0
+0.1
+-862.1
+SA-VAE [25]
+0.2
+0.3
+1.3
+-252.2
+0.1
+0.2
+1.0
+-853.4
+Lagging VAE [18]
+0.4
+0.6
+1.6
+-248.5
+0.5
+1.0
+3.6
+-849.4
+β-VAE [19] (β=0.2)
+0.6
+1.2
+2.4
+-245.3
+0.7
+1.4
+5.9
+-842.6
+LIDGMVAE (this work)
+1.0
+1.6
+2.6
+-242.3
+1.0
+1.7
+7.5
+-820.3
+Synthetic
+Yahoo
+Yelp
+AU
+KL
+MI
+LL
+AU
+KL
+MI
+LL
+AU
+KL
+MI
+LL
+VAE [28]
+0.0
+0.0
+0.0
+-46.5
+0.0
+0.0
+0.0
+-519.7
+0.0
+0.0
+0.0
+-635.9
+SA-VAE [25]
+0.4
+0.1
+0.1
+-40.2
+0.2
+1.0
+0.2
+-520.2
+0.1
+1.9
+0.2
+-631.5
+Lagging VAE [18]
+0.5
+0.1
+0.1
+-40.0
+0.3
+1.6
+0.4
+-518.6
+0.2
+3.6
+0.1
+-631.0
+β-VAE [19] (β=0.2)
+1.0
+0.1
+0.1
+-39.9
+0.5
+4.7
+0.9
+-524.4
+0.3
+10.0
+0.1
+-637.3
+LIDSVAE
+1.0
+0.5
+0.6
+-40.3
+0.8
+7.2
+1.1
+-519.5
+0.7
+9.1
+0.9
+-634.2
+Table 1: Across image and text datasets, LIDVAE outperforms existing VAE variants in preventing
+posterior collapse while achieving similar goodness-of-fit to the data.
+Example 2 (Latent-identifiable sequential VAE (LIDSVAE)). An LIDSVAE generates a D-
+dimensional datapoint xi, i ∈ {1,...,n} by
+zi ∼ p(zi),
+xi | zi,x<i ∼ EF(g2,θ(β⊤
+2 g1,θ([zi, fθ(x<i)]))),
+where x<i = (x1,...,xi−1) represents the history of x before the ith dimension. The function fθ :
+X<i → RH maps the history X<i into an H-dimensional vector. Finally, [zi, fθ(x<i)] is an (K+H)×1
+vector that represents a row-stack of the vectors (zi)K×1 and (fθ(x<i))H×1.
+Similar with mixture VAE, the LIDSVAE also differs from sequential VAE only in its use of
+g2,θ(β⊤
+2 g1,θ(·)) function in p(xi | zi,x<i). We will use LIDSVAE to model text in Section 4.
+4
+Empirical studies
+We study LIDVAE on images and text datasets, finding that LIDVAE do not suffer from posterior
+collapse as we increase the capacity of the generative model, while achieving similar fits to the data.
+We further study PPCA, showing how likelihood functions nearly constant in latent variables lead
+to collapsing posterior even with Markov chain Monte Carlo (MCMC).
+4.1
+LIDVAE on images and text
+We consider three metrics for evaluating posterior collapse:
+(1) KL divergence between
+the posterior and the prior, KL(q(z|x)||p(z)); (2) Percentange of active units (AU):AU =
+�D
+d=1 1{Covp(x)(Eq(z|x) [zd]) ≥ ϵ}, where zd = (z1d,..., znd) is the dth dimension of the latent vari-
+able z for all the n data points. In calculating AU, we follow Burda et al. [7] to calculate the
+posterior mean, (E [z1d |x1],...,E [znd |xn])] for all data points, and calculate the sample variance
+of E [zid |xi] across i’s from this vector. The threshold ϵ is chosen to be 0.01 [7]; the theoretical
+maximum of %AU is one; (3) Approximate Mutual information (MI) between xi and zi, I(x, z) =
+Ex
+�
+Eq(z|x) [log(q(z|x))]
+�
+− Ex
+�
+Eq(z|x) [log(q(z))]
+�. We also evaluate the model fit using the impor-
+tance weighted estimate of log-likelihood on a held-out test set [7]. For mixture VAE, we also eval-
+uate the predictive accuracy of the categorical latents against ground truth labels to quantify their
+informativeness.
+Competing methods. We compare LIDVAE with the classical VAE [28], the β-VAE (β=0.2) [19],
+the semi-amortized VAE [25], and the lagging VAE [18]. Throughout the empirical studies, we use
+flexible variational approximating families (RealNVPs [14] for image and LSTMs [20] for text).
+Results: Images. We first study LIDGMVAE on four subsampled image datasets drawn from pin-
+wheel [22], MNIST [31], Fashion MNIST [57], and Omniglot [30]. Figures 1a and 1b illustrate a
+fit of the GMVAE and the LIDGMVAE to the pinwheel data [22]. The posterior of the GMVAE
+8
+
+z1
+2
+0
+2
+z2
+2
+0
+2
+LL
+1e4
+8
+6
+4
+2
+0
+4
+2
+0
+2
+4
+0
+1
+2
+3
+4
+5
+6
+Density
+(a) σ = 0.2
+z1
+2
+0
+2
+z2
+2
+0
+2
+LL
+1e4
+8
+6
+4
+2
+0
+4
+2
+0
+2
+4
+0
+1
+2
+3
+4
+5
+6
+Density
+(b) σ = 0.5
+z1
+2
+0
+2
+z2
+2
+0
+2
+LL
+1e4
+8
+6
+4
+2
+0
+2.5
+0.0
+2.5
+0
+2
+4
+6
+Density
+(c) σ = 1.0
+z1
+2
+0
+2
+z2
+2
+0
+2
+LL
+1e4
+8
+6
+4
+2
+0
+2.5
+0.0
+2.5
+0
+2
+4
+6
+Density
+posterior
+prior
+(d) σ = 1.5
+Figure 2: As the noise level increases in PPCA, the latent variable becomes closer to non-
+identifiable because the likelihood and more susceptible to posterior collapse. Its likelihood sur-
+face becomes flatter and its posterior becomes closer to the prior. Top panel: Likelihood surface of
+PPCA as a function of the two latents z1, z2. When σ increase, the likelihood surface becomes flat-
+ter and the latent variables z1, z2 are closer to non-identifiable. Bottom panel: Posterior of z1 under
+different σ values. When σ increase, the posterior becomes closer to the prior.
+latents collapse, attributing all datapoints to the same latent cluster. In contrast, LIDGMVAE pro-
+duces categorical latents faithful to the clustering structure. Figure 1 examines the LIDGMVAE
+as we increase the flexibility of the generative model. Figure 1c shows that the categorical latents
+of the LIDGMVAE are substantially more predictive of the true labels than their classical coun-
+terparts. Moreover, its performance does not degrade as the generative model becomes more flexi-
+ble. Figure 1d shows that the LIDGMVAE consistently achieve higher test log-likelihood. Table 1
+compares different variants of VAE in a 9-layer generative model. Across four datasets, LIDGM-
+VAE mitigates posterior collapse. It achieves higher AU, KL and MI than other variants of VAE.
+It also achieves a higher test log-likelihood.
+Results: Text.
+We apply LIDSVAE to three subsampled text datasets drawn from a synthetic
+text dataset, the Yahoo dataset, and the Yelp dataset [60]. The synthetic dataset is generated from a
+classical two-layer sequential VAE with a five-dimensional latent. Table 1 compares the LIDSVAE
+with the sequential VAE. Across the three text datasets, the LIDSVAE outperforms other variants
+of VAE in mitigating posterior collapse, generally achieving a higher AU, KL, and MI.
+4.2
+Latent variable non-identifiability and posterior collapse in PPCA
+Here we show that the PPCA posterior becomes close to the prior when the latent variable becomes
+close to be non-identifiable. We perform inference using Hamiltonian Monte Carlo (HMC), avoid-
+ing the effect of variational approximation on posterior collapse.
+Consider a PPCA with two latent dimensions, p(zi) = N (zi ; 0, I2), p(xi | zi ; θ) = N (xi ; z⊤
+i w,σ2 ·
+I5), where the value of σ2 is known, zi’s are the latent variables of interest, and w is the only
+parameter of interest. When the noise σ2 is set to a large value, the latent variable zi may become
+nearly non-identifiable. The reason is that the likelihood function p(xi | zi) becomes slower-varying
+as σ2 increases. For example, Figure 2 shows that the likelihood surface becomes flatter as σ2
+increases. Accordingly, the posterior becomes closer to the prior as σ2 increases. When σ = 1.5, the
+posterior collapses. This non-identifiability argument provides an explanation to the closely related
+phenomenon described in Section 6.2 of [33].
+5
+Discussion
+In this work, we show that the posterior collapse phenomenon is a problem of latent variable non-
+identifiability. It is not specific to the use of neural networks or particular inference algorithms in
+9
+
+VAE. Rather, it is an intrinsic issue of the model and the dataset. To this end, we propose a class of
+LIDVAE via Brenier maps to resolve latent variable non-identifiability and mitigate posterior col-
+lapse. Across empirical studies, we find that LIDVAE outperforms existing methods in mitigating
+posterior collapse.
+The latent variables of LIDVAE are guaranteed to be identifiable. However, it does not guarantee
+that the latent variables and the parameters of LIDVAE are jointly identifiable. In other words, the
+LIDVAE model may not be identifiable even though its latents are identifiable. This difference be-
+tween latent variable identifiability and model identifiability may appear minor. But the tractability
+of resolving latent variable identifiability plays a key role in making non-identifiability a fruitful one
+perspective of posterior collapse. To enforce latent variable identifiability, it is sufficient to ensure
+that the likelihood p(x| z, ˆθ) is an injective function of z. In contrast, resolving model identifiability
+for the general class of VAE remains a long standing open problem, with some recent progress re-
+lying on auxiliary variables [23, 24]. The tractability of resolving latent variable identifiability is a
+key catalyst of a principled solution to mitigating posterior collapse.
+There are a few limitations of this work. One is that the theoretical argument focuses on the collapse
+of the exact posterior. The rationale is that, if the exact posterior collapses, then its variational ap-
+proximation must also collapse because variational approximation of posteriors cannot “uncollapse”
+a posterior. That said, variational approximation may “collapse” a posterior, i.e. the exact posterior
+does not collapse but the variational approximate posterior collapses. The theoretical argument and
+algorithmic approaches developed in this work does not apply to this setting, which remains an in-
+teresting venue of future work.
+A second limitation is that the latent-identifiable VAE developed in this work bear a higher compu-
+tational cost than classical VAE. While the latent-identifiable VAE ensures the identifiability of its
+latent variables and mitigates posterior collapse, it does come with a price in computation because
+its generative model (i.e. decoder) is parametrized using gradients of a neural network. Fitting the
+latent-identifiable VAE thus requires calculating gradients of gradients of a neural network, leading
+to much higher computational complexity than fitting the classifical VAE. Developing computation-
+ally efficient variants of the latent-identifiable VAE is another interesting direction for future work.
+Acknowledgments.
+We thank Taiga Abe and Gemma Moran for helpful discussions, and anony-
+mous reviewers for constructive feedback that improved the manuscript. David Blei is supported by
+ONR N00014-17-1-2131, ONR N00014-15-1-2209, NSF CCF-1740833, DARPA SD2 FA8750-18-
+C-0130, Amazon, and the Simons Foundation. John Cunningham is supported by the Simons Foun-
+dation, McKnight Foundation, Zuckerman Institute, Grossman Center, and Gatsby Charitable Trust.
+10
+
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+
+Supplementary Materials
+Posterior Collapse and Latent Variable Non-identifiability
+A
+Examples of posterior collapse continued
+We present two additional examples of posterior collapse, probabilistic principal component analysis
+and Gaussian mixture model.
+A.1
+Probabilistic principal component analysis
+We consider classical probabilistic principal component analysis (PPCA) and show that its local
+latent variables can suffer from posterior collapse at maximum likelihood parameter values (i.e.
+global maxima of log marginal likelihood). This example refines the perspective of Lucas et al. [7],
+which demonstrated that posterior collapse can occur in PPCA absent any variational approximation
+but due to local maxima in the log marginal likelihood. Here we show that posterior collapse can
+occur even with global maxima, absent optimization issues due to local maxima.
+Consider a PPCA with two latent dimensions,
+p(zi) = N (zi |0, I2),
+p(xi | zi ; θ) = N (xi | z⊤
+i w,σ2 · I5),
+where zi’s are the latent variables of interest and others θ = (w,σ2) are parameters of the model.
+Consider fitting this model to two datasets, each with 500 samples, focusing on maximum likelihood
+parameter values. Depending on the true distribution of the dataset, PPCA may or may not suffer
+from posterior collapse.
+1. Sample the data from a one-dimensional PPCA,
+xi ∼ N (xi |N (0, I1)· ¯w1, ¯σ1 · I5).
+(10)
+(The model remains two dimensional.) The latent variables zi’s are not (fully) identifiable in this
+case. The reason is that one set of maximum likelihood parameters is ˆθ = ( ˆw, ˆσ) = ([0, ¯w1], ¯σ1),
+i.e. setting one latent dimension as zero and the other equal to the true data generating direction.
+Under this ˆθ, the likelihood function is constant in the first dimension of the latent variable, i.e.
+zi1; see Figure 3a. The posterior of zi1 thus collapses, matching the prior, while the posterior of
+zi2 stays peaked (Figure 3b).
+2. Sample the data from from a two-dimensional PPCA,
+xi ∼ N (xi |N (0, I2)· ¯w2, ¯σ2 · I5).
+(11)
+The latent variables zi are identifiable. The likelihood function varies against both zi1 and zi2;
+the posteriors of both zi1 and zi2 are peaked (Figures 3c and 3d).
+A.2
+Gaussian mixture model
+Though we have focused on the posterior collapse of local latent variables, a model can also suffer
+from posterior collapse of its global latent variables. Consider a simple Gaussian mixture model
+(GMM) with two clusters,
+p(α) = Beta(α|5,5),
+p(xi |α; θ) = α·N (xi |µ1,σ2
+1)+(1−α)·N (xi |µ2,σ2
+2).
+Here α is a global latent variable and θ = (µ1,µ2,σ1,σ2) are the parameters of the model. Fit this
+model to three datasets, each with 105 samples.
+1. Sample the data from two non-overlapping clusters,
+xi ∼ 0.15·N (−10,1)+0.85·N (10,1).
+(12)
+The latent variable α is identifiable. The two data generating clusters are substantially different,
+so the likelihood function varies across α ∈ [0,1] under the maximum likelihood (ML) parameters
+(Figure 4a). The posterior of α is also peaked (Figure 4b) and differs much from the prior.
+1
+
+2. Sample the data from two overlapping clusters,
+xi ∼ 0.15·N (−0.5,1)+0.85·N (0.5,1).
+(13)
+The latent variable α is identifiable.
+However, it is nearly non-identifiable.
+While the two
+data generating clusters are different, they are very similar to each other because they over-
+lap. Therefore, the likelihood function p(xi |α; θ∗) is slowly varying under ML parameters
+θ∗ = (µ∗
+1,µ∗
+2,σ∗
+1,σ∗
+2) = (−0.5,0.5,1,1); see Figure 4a. Consequently, the posterior of α remains
+very close to the prior; see Figure 4b.
+3. Sample the data from a single Gaussian distribution, xi ∼ N (−1,1). The latent variable α is non-
+identifiable. The reason is that one set of ML parameters is θ∗ = (µ∗
+1,µ∗
+2,σ∗
+1,σ∗
+2) = (−1,−1,1,1),
+i.e. setting both of the two mixture components equal to the true data generating Gaussian distri-
+bution.
+Under this θ∗, the latent variable α is non-identifiable and its likelihood function p({xi}n
+i=1 |α; θ∗)
+is constant in α because the two mixture components are equal; Figure 4a illustrates this fact.
+Moreover, the posterior of α collapses, p(α|{xi}n
+i=1 ; θ∗) = p(α). Figure 4b illustrates this fact:
+The HMC samples of the α posterior closely match those drawn from the prior. (Exact inference
+is intractable in this case, so we use HMC as a close approximation to exact inference.) This
+example demonstrates the connection between non-identifiability and posterior collapse; it also
+shows that posterior collapse is not specific to variational inference but is an issue of the model
+and the data.
+As for PPCA, these GMM examples demonstrate that whether a latent variable is identifiable in a
+probabilistic model not only depends on the model but also the data. While all three examples were
+fitted with the same GMM model, their identifiability situation differs as the samples are generated
+in different ways.
+B
+Proof of Proposition 2
+We prove a general version of Proposition 2 by establishing the latent variable identifiability and
+flexibility of the most general form of the LIDVAE. The LIDVAE, LIDMVAE, and LIDSVAE
+(Definition 4 and examples 1 and 2) will all be its special cases. Then Proposition 2 will also be a
+special case of the more general result stated below (Proposition 3).
+We first define the most general form of LIDVAE.
+Definition 4 (General LIDVAE via Brenier maps). A general LIDVAE via Brenier maps generates
+an D-dimensional data-point xi,∈ {1,...,n} by:
+(zi)K×1 ∼ p(zi),
+(14)
+(wi)M×1 | zi ∼ EF(wi |β⊤
+1 zi),
+(15)
+(xi)D×1 |wi,x<i ∼ EF(xi |h◦ g2,θ(β⊤
+2 g1,θ([wi, fθ(x<i)]))),
+(16)
+where EF stands for exponential family distributions; zi is a K-dimensional latent variable, discrete
+or continuous. The parameters of the model are θ = (g1,θ, g2,θ, fθ), where fθ : X<i → RH is a
+function that maps all previous data points x<i to an H-dimensional vector, g1,θ : RM+H → RM+H
+and g2,θ : RD → RD are two continuous monotone transport maps. The function h(·) is a bijective
+link function for the exponential family, e.g. the sigmoid function. The matrix β1 is a K × M-
+dimensional matrix (M ≥ K) all the main diagonal entries being one and all other entries being
+zero, and thus with full row rank. Similarly, β2 is a (M + H)× D-dimensional matrix (D ≥ M + H)
+with all the main diagonal entries being one and all other entries being zero, also with full row rank.
+Finally, [wi, fθ(x<i)] is an (M+H)×1 vector that represents a row-stack of the vectors (wi)M×1 and
+(fθ(x<i))H×1.
+The general LIDVAE differs from the classical VAE whose general form is
+(zi)K×1 ∼ p(zi),
+(17)
+(wi)M×1 | zi ∼ EF(wi |β⊤
+1 zi),
+(18)
+(xi)D×1 |wi,x<i ∼ EF(xi |h◦ gθ([wi, fθ(x<i)])),
+(19)
+2
+
+The key difference is in Eq. 19, where the classical VAE uses an arbitrary function g : RM+H → RD
+in Eq. 19. In contrast, LIDVAE uses a composition g2,θ(β⊤
+2 g1,θ(·)) with additional constraints in
+Eq. 16.
+General LIDVAE can handle both i.i.d. and sequential data. For i.i.d data (e.g. images), we can set
+fθ(·) to be a zero function, which implies P(xi |wi,x<i) = P(xi |wi). For sequential data (e.g. text),
+we can set fθ(·) to be an LSTM that embeds the history x<i into an H-dimensional vector.
+General LIDVAE emulate many existing VAE. Letting zi be categorical (one-hot) vectors, the dis-
+tribution EF(z⊤
+i βθ) is an exponential family mixture. The identifiable VAE then maps this mixture
+model through a flexible function gθ. When zi is real-valued, it mimics classical VAE by mapping
+an exponential family PCA through flexible functions.
+LIDGMVAE is a special case of the general LIDVAE when we set zi be categorical (one-hot)
+vectors, set the exponential family distribution EF to be Gaussian in Eqs. 15 and 16. In this case,
+wi ∼ Gaussian(z⊤
+i βθ,γθ) is a Gaussian mixture. Then, we set fθ(·) to be a zero function, which
+implies P(xi |wi,x<i) = P(xi |wi), and finally set h as the identity function.
+This general LIDVAE also subsumes the Bernoulli mixture model, which is a common variant
+of LIDGMVAE for the MNIST data. Specifically, we can set zi be categorical (one-hot) vec-
+tors, and then set the exponential family distribution EF to be Gaussian in Eq. 15, making wi ∼
+Gaussian(z⊤
+i βθ,γθ) to be a Gaussian mixture. Next we set fθ(·) to be a zero function, which im-
+plies P(xi |wi,x<i) = P(xi |wi), then set h to be the sigmoid function, and finally set the EF to be
+Bernoulli in Eq. 16.
+LIDSVAE is another special case of the general LIDVAE when we set the EF to be a point mass
+and β1,θ to be identity matrix in Eq. 15, which implies wi = zi. Then setting the EF to be a categor-
+ical distribution and h to be identity in Eq. 16 leads to a configuration that is the same as Example 2.
+LIDVAE can be made deeper with more layers by introducing additional full row-rank matrices βk
+(e.g. ones with all the main diagonal entries being one and all other entries being zero) and additional
+Brenier maps gk,θ. For example, we can expand Eq. 16 with an additional layer by setting
+(xi)D×1 |wi,x<i ∼ EF(g3,θ(β⊤
+3 g2,θ(β⊤
+2 g1,θ([wi, fθ(x<i)])))).
+Next we establish the latent variable identifiability and flexibility of this general class of LIDVAE,
+which will imply the identifiability and flexibility of all the special cases above.
+Proposition 3. The latent variable zi is identifiable in LIDVAE, i.e. for all i ∈ {1,...,n}, we have
+p(xi | zi = ˜z′,x<i ; θ) = p(xi | zi = ˜z,x<i ; θ)
+⇒
+˜z′ = ˜z,
+∀ ˜z′, ˜z,θ.
+(20)
+Moreover, for any data distribution generated by the classical VAE (Eqs. 17 to 19), there exists an
+LIDVAE that can generate the same distribution.
+Proof. We first establish the latent variable identifiability. To show that the latent variable zi is
+identifiable, it is sufficient to show that the mapping from zi to p(xi | zi ; θ) is injective for all θ.
+The injectivity holds because all the transformations (β1,β2, g1,θ, g2,θ) involved in the mapping is
+injective, and their composition must be injective: the linear transformations (β1,β2) have full row
+rank and hence are injective; the nonlinear transformations (g1,θ, g2,θ) are monotone transport maps
+and are guaranteed to be bijective [1, 9]; finally, the exponential family likelihood is injective.
+We next establish the flexibility of the LIDVAE, by proving that any VAE-generated p(x) can be
+generated by an LIDVAE. The proof proceeds in two steps: (1) we show any VAE-generated p(x)
+can be generated by a VAE with injective likelihood p(xi | zi ; θ); (2) we show any p(x) generated
+by an injective VAE can be generated by an LIDVAE.
+To prove (1), suppose β1 does not have full row rank and gθ is not injective. Then there exists some
+Z′ ∈ Rd, d < K, and injective β′
+1,θ, g′
+θ such that the new VAE can represent the same p(x). The
+reason is that we can always turn an non-injective function into an injective one by considering its
+quotient space. In particular, we consider the quotient space with the equivalence relation between
+z, z′ defined as p(x| z; θ) = p(x| z′ ; θ)}, which induces a bijection into Rd. When p(z′) is no longer
+standard Gaussian, there must exist a bijective Brenier map ˜z = ft(z′) such that p(˜x) is standard
+Gaussian (Theorem 6 of McCann et al. [8]).
+3
+
+Pinwheel
+MNIST
+AU
+KL
+MI
+LL
+AU
+KL
+MI
+LL
+VAE [6]
+0.2
+1.4e-6
+2.0e-3
+-6.2 (5e-2)
+0.1
+0.1
+0.2
+-108.2 (5e-1)
+SA-VAE [5]
+0.2
+1.6e-5
+2.0e-2
+-6.5 (5e-2)
+0.4
+0.4
+0.6
+-106.3 (7e-1)
+Lagging VAE [3]
+0.6
+0.7e-3
+1.5e0
+-6.5 (4e-2)
+0.5
+0.8
+1.7
+-105.2 (5e-1)
+β-VAE [4] (β=0.2)
+1.0
+1.2e-3
+2.3e0
+-6.6 (6e-2)
+0.8
+1.5
+2.8
+-100.4 (6e-1)
+LIDGMVAE (this work)
+1.0
+1.2e-3
+2.2e0
+-6.5 (5e-2)
+1.0
+1.8
+3.9
+-95.4 (7e-1)
+Table 2: LIDGMVAE do not suffer from posterior collapse and achieves better fit than its classical
+counterpart in a 9-layer generative model. The reported number is mean (sd) over ten different
+random seeds. (Higher is better.)
+To prove (2), we show that any VAE with injective mapping can be reparameterized as a LIDVAE.
+To prove this claim, it is sufficient to show that any injective function lθ : RM+H → RD can be
+reparametrized as g2,θ(β⊤
+2 g1,θ(·)). Below we provide such a reparametrization by solving for g1, g2
+and β in lθ(z) = g2,θ(β⊤
+2 g1,θ(z)). We set g1,θ as an identity map, β2 as an (M + H)× D matrix with
+all the main diagonal entries being one and all other entries being zero, and g2,θ as an invertible
+Rd → Rd mapping which coincides with lθ on the (M + H)-dimensional subspace of z.
+Finally, we note that the same argument applies to the variant of VAE where wi = zi. It coincides
+with the classical VAE in Kingma & Welling [6]. Applying the same argument as above establishes
+Proposition 2.
+C
+Experiment details
+For image experiments, all hidden layers of the neural networks have 512 units. We choose the
+number of continuous latent variables as 64 and the dimensionality of categorical variables as the
+number of ground truth labels. Then we use two-layer RealNVP ([2]) as an approximating family
+to tease out the effect of variational inference.
+For text experiments, all hidden layers of the neural networks have 1024 units. We choose the
+dimensionality of the embedding as 1024. Then we use two-layer LSTM as an approximating family
+following common practice of fitting sequential VAE.
+D
+Additional experimental results
+Table 2 includes additional experimental results of LIDVAE on image datasets (Pinwheel and
+MNIST).
+4
+
+z1
+2
+1
+0
+1
+2
+z2
+2
+1
+0
+1
+2
+likelihood
+3000
+2500
+2000
+1500
+1000
+500
+(a) Likelihood (1D PPCA)
+2
+0
+2
+4
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+prior of z1 and z2
+posterior of z1
+posterior of z2
+(b) Posterior (1D PPCA)
+z1
+2
+1
+0
+1
+2
+z2
+2
+1
+0
+1
+2
+likelihood
+8000
+6000
+4000
+2000
+(c) Likelihood (2D PPCA)
+2
+0
+2
+0.0
+0.5
+1.0
+1.5
+prior of z1 and z2
+posterior of z1
+posterior of z2
+(d) Posterior (2D PPCA)
+Figure 3: Fitting PPCA with more latent dimensions than enough leads to non-identifiable local
+latent variables and collapsed posteriors. (a)-(b) Fit a two-dimensional PPCA to data drawn from a
+one-dimensional PPCA. The likelihood surface is constant in one dimension of the latent variable,
+i.e. this latent variable is non-identifiable. Hence its corresponding posterior collapses. (c)-(d) Fit
+a two-dimensional PPCA to data from a two-dimensional PPCA does not suffer from posterior
+collapse; its likelihood surface varies in all dimensions.
+0.00
+0.25
+0.50
+0.75
+1.00
+mixture weight
+6
+4
+2
+likelihood
+data
+2 clusters (non-overlap) : ID
+2 clusters (overlap): 
+non-ID
+1 cluster: non-ID
+(a) Likelihood function
+0.00
+0.25
+0.50
+0.75
+1.00
+mixture weight
+0
+250
+500
+750
+1000
+2 clusters (non-overlap)
+2 clusters (overlap)
+1 cluster
+prior
+(b) Posterior histogram
+Figure 4: When a latent variable is non-identifiable (non-ID) in a model, its likelihood function
+is a constant function and its posterior is equal to the prior, i.e. its posterior collapses. Consider
+a Gaussian mixture model with two clusters x ∼ α · N (µ1,σ2
+1) + (1 − α) · N (µ2,σ2
+2), treating the
+mixture weight α as the latent variable and others as parameters. Fit the model to datasets generated
+respectively by one Gaussian cluster (α non-identifiable), two overlapping Gaussian clusters (α
+nearly non-identifiable), and two non-overlapping Gaussian clusters (α identifiable). Under optimal
+parameters, the likelihood function p(x|α) is (close to) a constant when the latent variable α is (close
+to) non-identifiable; its posterior is also (close to) the prior. Otherwise, the likelihood function is
+non-constant and the posterior is peaked.
+5
+
+References
+[1] Ball, K. (2004). An elementary introduction to monotone transportation. In Geometric aspects
+of functional analysis (pp. 41–52). Springer.
+[2] Dinh, L., Sohl-Dickstein, J., & Bengio, S. (2016). Density estimation using real NVP. arXiv
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+séminaire de Mathématiques Supérieure (SMS) Montréal, (pp. 145–180).
+6
+

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+arXiv:2301.03336v1  [math.FA]  8 Dec 2022
+On existence theorems of a functional differential
+equations in partially ordered Banach algebras
+Amor Fahem(1), Aref Jeribi (1) and Najib Kaddachi (2)
+(1) Department of Mathematics. University of Sfax. Faculty of Sciences of Sfax.
+Soukra Road Km 3.5 B.P. 1171, 3000, Sfax, Tunisia
+E-mail: [email protected]
+E-mail: [email protected]
+(2) University of Kairouan. Faculty of Science and Technology of Sidi Bouzid. Agricultural
+University City Campus − 9100, Sidi Bouzid, Tunisia
+E-mail: [email protected]
+Abstract. In this paper we are concerned with existence results for a coupled system of
+quadratic functional differential equations. This system is reduced to a fixed point problem
+for a 2 × 2 block operator matrix with nonlinear inputs. To prove the existence we are
+established some fixed point theorem of Dhage’s type for the block matrix operator acting
+in partially ordered Banach algebras.
+Keywords:
+Partially
+ordered
+Banach
+algebra,
+Fixed
+point
+theory,
+Partial
+measure
+of
+noncompactness, differential equations, block Operator matrix.
+Mathematics Subject Classification: 47H10, 47H08, 47H09.
+1
+Introduction
+The theory of fixed point is one of the most powerful and most fruitful tools of modern
+mathematics and can consider a fundamental material of non-linear analysis. In recent years a
+number of excellent monographs and surveys by distinguished authors about fixed point theory
+have appeared such as, [1, 2, 5, 6, 9, 20]. Based on the fact that the Banach spaces are the
+fundamental underlying spaces on linear and nonlinear analysis, it leads us to consider the
+following problem: if a Banach space is equipped with an ordering structure, partial order or
+lattice, this Banach space becomes a partially ordered Banach space. Then, when we solve
+some problems on this Banach space, in addition to the topological structure and the algebraic
+structure, the ordering structure will provide a new powerful tool. This important idea has
+been widely used in solving integral equations [4, 7, 12, 18, 19, 21, 24, 26]. In this work, we are
+1
+
+mainly concerned with the existence results of solutions of the following system of Quadratic
+nonlinear functional differential equation (in short QFDE)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+�x(t) − f1(t, x(t))
+f2(t, y(t))
+�′
++ λ
+�x(t) − f1(t, x(t))
+f2(t, y(t))
+�
+= g(t, y(t))
+y(t) =
+1
+1 − b(t)|x(t)| − p
+�
+t,
+1
+1 − b(t)|x(t)|
+�
++ p(t, y(t))
+�
+x(0), y(0)
+�
+= (x0, y0) ∈ R2,
+(1.1)
+where λ ∈ R+, b : J −→ R and f1, f2, g, p : J × R −→ R are continuous with f2 not ever
+vanishing. The first equation of FDE (1.1) is general in the sense that it includes some im-
+portant classes of functional differential equations. If we take x = y, f1(t, x) = φ(t, x) = 0
+and f2(t, x) = f(t, x, y) and g(t, x) = g(t, x, y), then the FDE (1.1) reduces to the following
+functional differential equation with a delay:
+
+
+
+
+
+
+
+�
+x(t)
+f(t, x(t), X(t))
+�′
++ λ
+�
+x(t)
+f(t, x(t), X(t))
+�
+= g(t, x(t), xt),
+t ∈ I
+x0 = φ ∈ C([−r, 0], R),
+r > 0
+whenever I = [0, T ], T > 0. The equation was examined in the paper [15, 17] and some special
+cases of this equation were considered in [25].
+Again, when f1(t, x) = φ(t, x) = 0, f2(t, x) = 1 and g(t, x) = g(t, x, y), the QFDE (1.1) reduces
+to the following known nonlinear differential equation with maxima
+
+
+
+x′(t) + λx(t) = g(t, x(t), X(t),
+t ∈ I
+x(0) = α0 ∈ R+.
+The above nonlinear differential equation with maxima has already been discussed in [3] for
+existence and uniqueness of the solutions via classical methods of Schauder and Banach fixed
+point principles.
+Furthermore, if f1(t, x) = f(t, x, y), f2(t, x) = 1, φ(t, x) = 0 and g(t, x) = g(t, x, y), the QFDE
+(1.1) reduces to the following HDE without maxima,
+
+
+
+(x(t) − f(t, x(t), X(t))′ + λ(x(t) − f1(t, x(t))) = g(t, x(t), X(t))
+x(0)) = α0 ∈ R+,
+(1.2)
+which is discussed in [14] via Dhage iteration method and established the existence and approx-
+imation result under some mixed partial Lipschitz and partial compactness conditions.
+Note that the system (1.1) may be transformed into the following fixed point problem of
+the 2 × 2 block operator matrix
+� A
+B · B′
+C
+D
+�
+,
+(1.3)
+where the entries of the matrix are, in general, nonlinear operators defined on partially ordered
+Banach algebras. The operators occurring in the representation (1.3) are nonlinear, and our
+assumptions are as follows: A a maps nondecreasing in a partially ordered Banach algebra E
+into E and B, C, D and B′ are nondecreasing and positive operators from E into E.
+In this direction, the authors A. Jeribi, B. Krichen and N. Kaddachi in [22, 23] have es-
+tablished some fixed point for a 2 × 2 operator matrix (1.3), when X is a Banach algebra
+2
+
+satisfying certain condition. It is important to mention that the theoretical study was based
+on the existence of a solution of the following equation
+x = Ax · Bx + Cx
+(1.4)
+and obtained a lot of valuable results ([10, 12, 13] and the references therein). These studies
+were mainly based on the closure of the bounded domain, and properties of the operators A,
+B and C (cf. partially completely continuous, partially nonlinear k-set contractive, partially
+condensing, and the potential tool of the axiomatic partially measure of noncompactness,...).
+This paper is organized as follows. In the next section, we give some preliminary results
+needed in the sequel. In Section 3, we present existence results for Equation (1.4). In Section 4,
+we will deal with some fixed point results for 2 × 2 block operator matrices in partially ordered
+Banach algebra. The main results of this section are Theorems 4.1 and 4.2. In Section 5, we
+give an application showing the existence of solutions of the system (1.1) in partially ordered
+Banach algebra.
+2
+Auxiliary Results
+Throughout this paper, let (E, ⪯, ∥·∥) be a partially ordered Banach algebra with zero element
+θ. Two elements x and y in E are called comparable if either the relation x ⪯ y or y ⪯ x
+holds. A non-empty subset C of E is called a chain or totally ordered set if all elements of
+C are comparable. It is know that E is regular if {xn} a nondecreasing (resp. nonincreasing)
+sequence in E and xn → x∗ as n → ∞, then xn ⪯ x∗ (resp. xn ⪰ x∗) for all n ∈ N. The
+conditions guaranteeing the regularity of E many be found in Guo and Lakshmikentham [20]
+and Nieto and Lopez [26] and the references therein.
+At the beginning of this section, we present some basic facts concerning the partially mea-
+sures of noncompactness in E. If C is a chain in E, then C′ denotes the set of all limit points
+of C in E. The symbol C stands for the closure of C in E defined by C = C ∪ C′. The set C
+is called a closed chain in E. Thus, C is the intersection of all closed chains containing C. In
+what follows, we denote by Pcl(E), Pbd(E), Prcp(E), Pch(E), Pbd,ch(E), Prcp,ch(E), the class
+of all nonempty and closed, bounded, relatively compact, chains, bounded chains and relatively
+compact chains of E respectively. Recall that the notion of the partial Kuratowski measure of
+noncompactness αp(.) on E by the formula:
+αp(C) = inf
+�
+r > 0, C =
+n�
+i=1
+Ci, diam(Ci) ≤ r ∀i
+�
+where diam(Ci) = sup{∥x − y∥ : x, y ∈ Ci}. For convenience we recall some basic properties of
+αp(.) needed below [11, 12, 13, 15].
+Definition 2.1 A mapping αp : Pbd,ch(E) −→ R+ is said to be a partially measure of noncompactness
+in E if it satisfies the following conditions:
+1. ∅ ̸= (αp)−1({0}) ⊂ Prcp,ch(E),
+2. αp(C) = αp(C),
+3. αp is nondecreasing,
+4. If {Cn} is a sequence of nondecreasing closed chains from Pbd,ch(E) with lim
+n→∞ αp(Cn) = 0,
+then C∞ = ∩∞
+n=0Cn is a nonempty set and αp(C∞) = 0,
+3
+
+The family of sets described in 1. is said to be the kernel of the measure of noncompactness
+αp and is defined as
+ker αp = {C ∈ Pbd,ch(E)|αp(C) = 0}
+Clearly, ker αp ⊂ Prcp,ch(E). Observe that the intersection set C∞ from condition 4. is
+a member of the family kerαp. In fact, since αp(C∞) ≤ αp(Cn) for any n, we infer that
+αp(C∞) = 0. This yields that αp(C∞) ∈ ker αp. This simple observation will be essential
+in our further investigations.
+The partially measure αp of noncompactness is called sublinear if it satisfies
+5. αp(C1 + C2) ≤ αp(C1) + αp(C2), for all C1, C2 ∈ Pbd,ch,
+6. αp(λC1) = |λ|α(C1), for all λ ∈ R,
+Again, αp is said to satisfy maximum property if
+7. αp(C1 ∪ C2) = max{αp(C1), αp(C2)}.
+Finally, αp is said to be full or complete if
+8. ker αp = Prcp,ch(E)
+♦
+The following definitions (see [11, 12, 13] and the references therein) are frequently used in the
+subsequent part of this paper.
+Definition 2.2 A mapping T : E −→ E is called monotone nondecreasing if it preserves the
+order relation ⪯, that is, if x ⪯ y implies T x ⪯ T y for all x, y ∈ E. Similarly, T is called
+monotone nonincreasing if x ⪯ y implies T x ⪰ T y for all x, y ∈ E. A monotone mapping T is
+one which is either monotone nondecreasing or monotone nonincreasing on E.
+♦
+Definition 2.3 A mapping ψ : R+ −→ R+ is called a dominating function or, in short,
+D-function if it is an upper semi-continuous and monotonic nondecreasing function satisfying
+ψ(0) = 0
+♦
+Definition 2.4 A mapping T : E −→ E is called partially nonlinear D-Lipschitzian if there
+exist a D-function ψ : R+ −→ R+ satisfying
+∥T x − T y∥ ≤ ψ(∥x − y∥),
+for all comparable elements x, y ∈ E where ψ(0) = 0. The function ψ is called a D-function of
+T on E. If ψ(r) = kr, k > 0, then T is called partially Lipschitzian with the Lipschitz constant
+k. In particular, if k < 1, then T is called a partially contraction on E with the contraction
+constant k. Finally, T is called a partially nonlinear D-contraction if it is a partially nonlinear
+D-Lipschitzian with ψ(r) < r for r > 0.
+♦
+Remark 2.1 Obviously, every partially Lipshitzian mapping is partially nonlinear D-Lipshitizian.
+the converse may be not true.
+Definition 2.5 A nondecreasing mapping T : E −→ E is called partially nonlinear D-set-Lipschitzian
+if there exists a D-function ψ such that
+αp(T C) ≤ ψ(αp(C)),
+for all bounded chain C in E. T is called partially k-set-Lipschitzian if ψ(r) = kr, k > 0. T is
+called partially k-set-contraction if it is a partially k-set-Lipschitzian with k < 1. Finally, T is
+called a partially nonlinear D-set-contraction if it is a partially nonlinear D-Lipschitzian with
+ψ(r) < r for r > 0.
+♦
+4
+
+Definition 2.6 A mapping T : E −→ E is called partially continuous at a point a ∈ E if
+for ε > 0 there exist a δ > 0 such that ∥T x − T a∥ < ε whenever x is comparable to a and
+∥x − a∥ < δ. T called partially continuous on E if it is partially continuous at every point of
+it. It is clear that if T is partially continuous on E, then it is continuous on every chain C
+contained in E.
+♦
+Definition 2.7 A mapping T : E −→ E is called partially bounded if T (C) is bounded for every
+chain C in E. T is called uniformly partially bounded if all chains T (C) in E are bounded by
+a unique constant. T is called bounded if T (E) is a bounded subset of E.
+♦
+Definition 2.8 A mapping T : E −→ E is called partially compact if T (C) is a relatively
+compact subset of E for all totally ordered sets or chains C in E. T is called uniformly partially
+compact if T (C) is a uniformly partially bounded and partially compact on E.
+T is called
+partially totally bounded if for any totally ordered and bounded subset C of E, T (C) is a relatively
+compact subset of E. If T is partially continuous and partially totally bounded, then it is called
+partially completely continuous on E.
+♦
+Remark 2.2 Note that every compact mapping on a partially normed linear space is partially
+compact and every partially compactmapping is partially totally bounded, however the reverse
+implications do not hold. Again, every completely continuous mapping is partially completely
+continuous and every partially completely continuous mapping is partially continuous and partially
+totally bounded, but the converse may not be true.
+♦
+Definition 2.9 The order relation ⪯ and the metric d on a non-empty set E are said to be
+compatible if {xn} is a monotone, that is, monotone nondecreasing or monotone nondecreasing
+sequence in E and if a subsequence {xnk} of {xn} converges to x∗ implies that the whole
+sequence {xn} to x∗. Similarly, given a partially ordered normed linear space (E, ⪯, ∥ · ∥) the
+order relation ⪯ and the norm ∥ · ∥ are said to be compatible if ⪯ and the metric d defined
+through the norm ∥ · ∥ are compatible.
+♦
+Definition 2.10 A map T : E −→ E is called T -orbitally continuous on E if for any sequence
+{xn} ⊆ O(x; T ) = {x, T x, T 2x, . . . , T nx, . . .} we have that xn → x∗ implies that T xn → T x∗
+for each x ∈ E. The metric space E is called T -orbitally complete if every cauchy sequence
+{xn} ⊆ O(x; T ) converses to a point x∗ ∈ E.
+We need the following results in the sequel.
+Let (E, ⪯, ∥ · ∥) be partially ordered Banach
+algebra. Denote
+E+ = {x ∈ E/x ⪰ θ} and K = {E+ ⊂ E/uv ∈ E+ for all u, v ∈ E+},
+where θ is the zero element of E. The members K are called positive vectors in E.
+Lemma 2.1 [15] If u1, u2, v1, v2 ∈ K are such that u1 ⪯ v1 and u2 ⪯ v2, then u1u2 ⪯ v1v2. ♦
+Definition 2.11 An operator T : E −→ E is said to be positive if the range R(T ) of T is such
+that R(T ) ⊆ K.
+♦
+Lemma 2.2 [12] If C1 and C2 are two bounded chains in a partially ordered Banach algebra
+E, then
+αp(C1 · C2) ≤ ∥C2∥αp(C1) + ∥C1∥αp(C2)
+where ∥C∥ = sup{∥c∥, c ∈ C}.
+♦
+Theorem 2.1 [11] Let (E, ⪯, ∥.∥) be a partially ordered set and let T : E → E be a nondecreasing
+mapping. Suppose that there is a metric d in X such that (E, d) is a T -orbitally complete metric
+space. Assume that there exists a D-function ψ such that
+d(T x, T y) ≤ ψ(d(x, y))
+5
+
+for all comparable elements x, y ∈ E satisfying ψ(r) < r for r > 0.Further assume that either
+T is T -orbitally continuous on E or E is such that if {xn} is a nondecreasing sequence with
+xn → x ∈ E, then x ⪯ x for all n ∈ N. If there is an element x0 ∈ E satisfying x0 ⪯ T x0 or
+x0 ⪰ T x0, then T has a fixed point which is further unique if "every pair of elements in E has
+a lower and an upper bound".
+♦
+Theorem 2.2 [11] Let (E, ⪯, ∥·∥) be a partially ordered Banach algebra. Let T : E −→ E be a
+nondecreasing, partially compact and continuous mapping. Further if the order relation ⪯ and
+the norm ∥ · ∥ in E are compatible and if there is an element x0 ∈ E satisfying x0 ⪯ T x0 or
+x0 ⪰ T x0, then T has a fixed point.
+♦
+Theorem 2.3 [12] Let (E, ⪯, ∥ · ∥) be a regular partially ordered complete normed linear space
+such that the order relation and ⪯ the norm ∥ · ∥ are compatible.
+Let T : E −→ E be a
+nondecreasing, partially continuous and partially bounded mapping. If T is partially nonlinear
+D-set-contraction and if there exists an element x0 ∈ E such that x0 ⪯ T x0 or x0 ⪰ T x0, then
+T has a fixed point x∗ and the sequence {T nx0} of successive iterations converges monotonically
+to x∗.
+♦
+3
+Reformulation of Dhage’s fixed point theorems
+In this section, we prove some fixed point theorems in partially ordered Banach algebras. Our
+results are formulated using some Dhage’s type fixed point. Now, we establish our first fixed
+point theorem by modifying some assumptions of Theorem 4.1 in [12].
+Theorem 3.1 Let S be non-empty closed and partially bounded subset of a regular partially
+ordered Banach algebra (E, ⪯, ∥ · ∥) such that the order relation ⪯ and the norm ∥ · ∥ in E are
+compatible. Let A : E −→ K, B : S −→ K and C : E −→ E be three nondecreasing operators
+satisfying the following conditions:
+(i) A and C are partially nonlinear D-Lipschitzians with D-functions ψA and ψC,
+(ii) B is continuous and partially compact,
+(iii) There exists x0 ∈ S such that x0 ⪯ Ax0 ·By +Cx0 or x0 ⪰ Ax0 ·By +Cx0 for each y ∈ S,
+(iv) Ax · By + Cx ∈ S for all y ∈ S
+(v) Every pair of elements x, y ∈ E has a lower and an upper bound in E.
+Then the equation (1.4) has a fixed point in Eas soon as MψA(r) + ψC(r) < r if r > 0, where
+M = ∥B(E)∥.
+♦
+Proof.
+Suppose that there exists x0 ∈ S such that x0 ⪯ Ax0 ·By +Cx0. Let y ∈ S and define
+a mapping define a mapping Ay : S −→ S by the formula
+Ay(x) = Ax · By + Cx.
+Since A and B are positive and A, B and C are nondecreasing, Ay is nondecreasing Now, let
+x1, x2 ∈ E be two comparable elements. Then
+∥Ay(x1) − Ay(x2)∥
+=
+∥Ax1 · By + Cx1 − Ax2 · By − Cx2∥
+≤
+∥Ax1 · By − Ax2 · By∥ + ∥Cx1 − Cx2∥
+≤
+∥By∥∥Ax1 − Ax2∥ + ∥Cx1 − Cx2∥
+≤
+(MψA + ψC) (∥x1 − x2∥).
+This implies that the operator Ay is a partially nonlinear D-contraction. Hence, by Theorem
+2.1 there exist a unique point x∗ ∈ E such that
+x∗ = Ax∗ · By + Cx∗.
+6
+
+because the hypothesis (iv) for all y ∈ S we have x∗ ∈ S. Define a mapping
+T :
+S
+−→ S
+y
+�−→ x∗,
+where x∗ is the unique solution of the equation Ax∗ · By + Cx∗. Since A, B and B′ are nonde-
+creasing and B and B′ are positive , T is nondecreasing. Now we show that T is continuous.
+Let {yn}∞
+n=0 be any sequence in B(E) converging to a point y, Since S is closed, y ∈ S. Then,
+∥T yn − T y∥
+=
+∥Ax∗
+n · yn + Cx∗
+n − Ax · y − Cx∥
+≤
+∥Ax∗
+n · yn − Ax · y∥ + ∥Cx∗
+n − Cx∥
+≤
+∥Ax∗
+n · yn − Ax · yn∥ + ∥Ax · yn − Ax · y∥ + ∥Cx∗
+n − Cx∥
+≤
+(MψA + ψc)(∥xn − x∥) + ∥Ax∥∥yn − y∥.
+Hence
+lim sup
+n ∥T yn − T y∥ ≤ (MψA + ψc)(lim sup
+n ∥xn − x∥) + ∥Ax∥ lim sup
+n ∥yn − y∥.
+This show that
+lim
+n→+∞ ∥T yn − T y∥ = 0 and the claim is approved. Next, we shows that T is
+partially compact. In fact, let C be a chain in S, for any z ∈ C we have
+∥Az∥
+≤
+∥Aa∥ + ψA(∥z − a∥)
+≤
+∥Aa∥ + ∥z − a∥
+M
+≤
+c,
+where c = ∥Aa∥ + diamC
+M
+for some fixed point a in C.
+Let ε > 0 be given. By assumption (ii), we infer that B(C) is partially totally bounded, then
+there exist a chain Y = {y1, . . . , yn} of point in C such that
+B(C) ⊆
+n�
+i=1
+Bδ(wi),
+where wi = Byi and δ = 1
+c(ε − (MφA(ε) + φC(ε))) and Bδ(wi) is an open ball in E centered at
+wi of radius δ. Therefore, for any y ∈ C, we have yk ∈ Y such that
+c∥Byi − By∥ ≤ δ.
+Also, we have
+∥T yk − T y∥
+≤
+∥Ax∗
+k · Byk − Ax · By∥ − ∥Cx∗
+k − Cx∥
+≤
+M(ψA + ψC)∥x∗
+k − x∥ + ∥Ax∥∥Byk − By∥
+≤
+M(ψA + ψC)∥x∗
+k − x∥ + c∥Byk − By∥.
+Then
+(I − M(ψA + ψC))(∥T yk − T y∥) ≤ c∥Byk − By∥.
+So,
+∥T (yk) − T (y)∥ < ε,
+because y ∈ C is arbitrary,
+T (S) ⊆
+n
+�
+i=1
+Bε(ki),
+7
+
+where ki = T (yi). As a result, T (S) is partially totally bounded in E. Hence, T is is partially
+compact. The order relation ⪯ and the norm ∥.∥ are compatible, so the desired conclusion
+follows of Theorem 2.2 we have T has a fixed point in S.
+□
+Now, modifying same assumptions of Theorem 4.8 in [13], we have the following result.
+Theorem 3.2 Let S be closed and partially bounded subset of a regular partially ordered Banach
+algebra (E, ⪯, ∥ · ∥) such that the order relation ⪯ and the norm ∥ · ∥ are compatible in every
+compact chain C of S. Let A, B : S → K and C : S −→ E be three nondecreasing operators
+satisfying the following conditions
+(i) A and C are partially nonlinear D-Lipschitzians with D-functions ψA and ψC respectively,
+(ii) B is partially completely continuous ,
+(iii)
+�I − C
+A
+�−1
+exist on B(S) and is nondecreasing,
+(iv) There exist x0 ∈ S such that x0 ⪯ Ax0 · By + Cx0 or x0 ⪰ Ax0 · By + Cx0, for each y ∈ S,
+(v) Ax · By + Cx ∈ S for all y ∈ S,
+Then the equation (1.4) has a fixed point in S as soon as MψA(r) + ψC(r) < r if r > 0 where
+M = ∥B(S)∥.
+Proof.
+Suppose that there exist x0 ⪯ Ax0 · By + Cx0. It is easy to check that the vector
+x ∈ E is a solution for the equation (1.4), if and only if x is a fixed point for the operator
+T :=
+�I − C
+A
+�−1
+B. From assumption (iii) it follows that, for each y ∈ S there is a unique
+xy ∈ E such that
+�I − C
+A
+�
+xy = By
+or, in an equivalently way
+Axy · By + Cxy = xy.
+Since the assumption (v) holds, then xy ∈ S. Hence the map T : S −→ S is well define. from
+assumption (iii), it follows that T is nondecreasing. Now, in view of Theorem 2.2, it suffices
+to prove that T is continuous and partially compact. Indeed let {yn}∞
+n=0 be any sequence in S
+converging to a point y. Because S is closed, y ∈ S. Now
+∥T yn − T y∥
+=
+∥Ax∗
+n · Byn + Cx∗
+n − Ax∗ · By − Cx∗∥
+≤
+∥Ax∗
+n · Byn − Ax∗ · By∥ + ∥Cx∗
+n − Cx∗∥
+≤
+∥Ax∗
+n · Byn − Ax∗ · Byn∥ + ∥Ax∗ · Byn − Ax∗ · By∥ + ∥Cx∗
+n − Cx∗∥
+≤
+(MψA + ψC)(∥x∗
+n − x∗∥) + ∥AT y∥∥Byn − By∥.
+Hence,
+lim
+n sup ∥T yn − T y∥ ≤ (MψA + ψC) lim
+n sup ∥T yn − T y∥ + ∥AT y∥ lim
+n sup ∥Byn − By∥.
+This shows that lim
+n→∞ ∥Nyn − Ny∥ = 0 and the claim is approved. Next, we shows that T
+is paritally compact. Since B is partially compact and
+�I − C
+A
+�−1
+is continuous, then the
+composition mapping T =
+�I − C
+A
+�−1
+B is continuous and partially compact on E. Next, the
+order relation ⪯ and the norm ∥ · ∥ in E are compatible. Hence, an application of Theorem 2.2
+infer that T has, at least, one fixed point in S.
+□
+8
+
+4
+Application of Dhage’s fixed point to block matrix operator
+In what follows, we will study the existence of a fixed point for the block matrix operators.
+Theorem 4.1 Let (E, ⪯, ∥·∥) be a regular partially ordered Banach algebra such that the order
+relation ⪯ and the norm ∥ · ∥ in E are compatible. Let A, C, D : E −→ E and B, B′ : E −→ K
+be five nondecreasing operators satisfying the following assumptions:
+(i) A, B and C are partially bounded and partially nonlinear D-Lipschitzians with D-functions
+ψA, ψB and ψC respectively,
+(ii) (I − D)−1 exist and partially nonlinear D-Lipschitz with D-function ψφ and (I − D)−1C is
+nondecreasing,
+(iii) B′ is partially continuous and C is compact,
+(iv) There exist x0 ∈ E, such that x0 ⪯ Ax0 + T x0 · T ′x0 or x0 ⪰ Ax0 + T x0 · T ′x0, where
+T = B(I − D)−1C and T ′ = B′(I − D)−1C.
+Then the operator matrix (1.3) has a fixed point in E × E, whenever Mψ(r) ≤ r if r > 0 with
+ψ(r) = ψB ◦ ψφ ◦ ψC(r) + ψA(r) and M = ∥T ′(E)∥.
+Proof.
+Suppose that there is an element x0 ∈ E such that x0 ⪯ Ax0 + T x0 · T ′x0. Define a
+mapping F : E −→ E by the formula
+Fx = Ax + T x · T ′x.
+Since B and B′ are positive and A, B, B′ and (I − D)−1C are nondecreasing we infer that F
+is nondecreasing. Next, we claim that F is partially continuous. To do us, let {xn; n ∈ N}
+be a sequence in E which converge to a point x such that xn and x are comparable. From
+assumption (i), it follows that:
+∥Fxn − Fx∥
+=
+∥Axn + T xn · T ′xn − Ax − T x · T ′x∥
+≤
+∥Axn − Ax∥ + ∥T xn · T ′xn − T x · T ′x∥
+≤
+∥Axn − Ax∥ + ∥T xn∥∥T ′xn − T ′
+x∥ + ∥T ′x∥∥T xn − T x∥
+≤
+ψA + MψB ◦ ψφ ◦ ψC(∥xn − x∥) + ∥T xn∥∥T ′xn − T ′x∥.
+From the partially continuity of B′, and taking limit supremum in the aforementioned inequality
+yields that
+lim
+n→∞ ∥Fxn − Fx∥ = 0.
+This proves that F is a partially continuous operators on E. Again by assumption (iv), it is
+clear that x0 ⪯ Fx0. Moreover, it easy to show that F is partially bounded. Furthermore, we
+show that F is a partially nonlinear D-set-contraction on E. Let C be a bounded chain in E.
+Then by definition of F, we have
+F(C) ⊆ A(C) + T (C)T ′(C).
+Since F is nondecreasing and partially continuous F(C) is again a bounded chain in E. Keeping
+in mind the relatively compactness of T ′(C) and making use of Lemma 2.2 together with the
+subadditivity of the partially Kuratowski measure of noncompactness αp, enables us to have
+αp(F(C))
+≤
+∥T (C))∥αp(T ′(C)) + ∥T ′(C)∥αp(T (C)) + αp(A(C))
+≤
+Mαp(T (C)) + αp(A(C))
+≤
+MψB ◦ ψφ ◦ ψC(αp(C)) + ψA(αp(C))
+=
+ψ(αp(C)),
+where ψ(r) = ψA(r) + MψB ◦ ψφ ◦ ψC(r) < r for all r > 0. This shows that F is a partially
+nonlinear D-set-contraction on E. Finally the relation ⪯ and the norm ∥ · ∥ are compatible, so
+the desired conclusion follows by an application of Theorem 2.3 we have x = Ax + T x · T ′x has
+a solution in E. Now, use the vector y = (I − D)−1Cx to achieve the proof.
+□
+9
+
+Remark 4.1 If A is a contraction on S into itself with constant contraction k and B a partially
+nonlinear D-lipshitzian with a D-function ψB, and C(S) ⊂ (I−D)(E), then the operator inverse
+(I − A)−1B exist and partially nonlinear D-lipshiztian with a D-function ψ(r) =
+1
+1−kψB(r).
+Next, we will combine Theorem 4.1 and Remark 4.1 in order to to obtain the following fixed
+point theorem:
+Corollary 4.1 Let (E, ⪯, ∥·∥) be a regular partially ordered Banach algebra such that the order
+relation ⪯ and the norm ∥ · ∥ in E are compatible. Let A, C, D : E −→ E and B, B′ : E −→ K
+are five nondecreasing operators satisfying the following assumptions:
+(i) A, B and C are partially bounded and partially nonlinear D-Lipschitzians with D-functions
+ψA, ψB and ψC respectively,
+(ii) D is a contraction with a contraction constant k and (I − D)−1C is nondecreasing,
+(iii) B′ is partially continuous and C is partially compact, and C(E) ⊂ (I − D)(E)
+(iv) x0 ⪯ Ax0 +T x0 ·T ′x0 or x0 ⪰ Ax0 +T x0 ·T ′x0, for some x0 ∈ E, where T = B(I −D)−1C
+and T ′ = B′(I − D)−1C.
+Then the operator matrix (1.3) has a fixed point in E × E, whenever Mψ(r) ≤ r if r > 0 with
+ψ(r) = ψB ◦ (
+1
+1−k)ψC(r) + ψA(r) and M = ∥T ′(E)∥.
+Based on the conditions C ≡ 0E and D = 1E in which 0E and 1E represents respectively the
+zero and the unit element of the partially ordered Banach algebra E, we infer from Theorem
+4.1 the following result:
+Corollary 4.2 [12] Let (E, ⪯, ∥ · ∥) be a regular partially ordered Banach algebra such that the
+order relation ⪯ and the norm ∥ · ∥ in E are compatible. Let A : E −→ E and B, B′ : E −→ K
+be five nondecreasing operators satisfying the following assumptions:
+(i) A, B and are partially bounded and partially nonlinear D-Lipschitzians with D-functions
+ψA, ψB respectively,
+(ii) B′ is partially continuous and compact,
+(ii) There exist x0 ∈ E, such that x0 ⪯ Ax0 + Bx0 · B′x0 or x0 ⪰ Ax0 + Bx0 · B′x0.
+Then the operator matrix (1.3) has a fixed point in E × E, whenever Mψ(r) ≤ r if r > 0 with
+ψ(r) = ψB + ψA(r) and M = ∥B′(E)∥.
+Next, we can modify some assumptions of Theorem 3.2 in order to study the problem in block
+operator matrix.
+Theorem 4.2 Let S be a non-empty closed partially bounded subset of a regular partially
+ordered Banach algebra (E, ⪯, ∥ · ∥) such that the order relation ⪯ and the norm ∥ · ∥ are
+compatible in every compact chain C of S. Let A, C : S −→ E and B, B′, D : E −→ K be five
+nondecreasing operators satisfying the following assumptions:
+(i) A, B and C are partially nonlinear D-Lipschitzians with D-functions ψA, ψB and ψC
+respectively,
+(ii) (I − D)−1 is nondecreasing and partially nonlinear D-Lipschitzian with D-functions ψφ,
+(iii) B′ is continuous and C is a partially compact operator such that C(S) ⊆ (I − D)(S)
+(iv) Ax + T x · T ′y ∈ S for all y ∈ S where T = B(I − D)−1C and T ′ = B′(I − D)−1C,
+(v)
+�I − A
+T
+�−1
+exist on T ′(S) and nondecreasing,
+(vi) x0 ⪯ Ax0 + T x0 · T ′y or x0 ⪰ Ax0 + T x0 · T ′y, for each y ∈ S,
+(vii) every pair of elements x, y ∈ E has a lower and an upper bound in E.
+Then the operator matrix (1.3) has a fixed point in S × S, whenever Mψ(r) ≤ r if r > 0 with
+ψ(r) = ψB ◦ ψφ ◦ ψC(r) + ψA(r) and M = ∥T ′(S)∥.
+Proof.
+Suppose that there is an element x0 ∈ E such that x0 ⪯ Ax0 + T x0 · T ′y. From
+assumption (iv), it follows that for each y ∈ S, there exist a unique point x ∈ S such that
+�I − A
+T
+�
+x = T ′y
+10
+
+or, equivalently x = Ax + T x · T ′y. Because assumption (iv) hold, then x ∈ S. Therefore, we
+can define F : S −→ S by the formula
+F(x) =
+�I − A
+T
+�−1
+T ′x.
+(4.1)
+In view of assumption (v), it follows that, F is nondecreasing on E. Next, we will prove that
+F is continuous. To see that, let {zn}∞
+n=0 be any sequence on E such that zn → z, and let
+
+
+
+yn = T ′(zn) and y = T ′(z)
+xn =
+�I − A
+T
+�−1
+(yn) and x =
+�I − A
+T
+�−1
+(y).
+Then, it easy to show that yn → y, and we have
+�
+xn
+=
+Axn + T xn · T ′yn
+x
+=
+Ax + T x · T ′y.
+So,
+∥xn − x∥
+=
+∥Axn + T xn · T ′yn − Ax + T x · T ′y∥
+≤
+∥Axn − Ax∥ + ∥T xn · T ′yn − T x · T ′y∥
+≤
+∥Axn.yn − Ax.yn∥ + ∥Ax.yn − Ax.y∥ + ∥Cxn − Cx∥
+≤
+ψA + MψB ◦ ψφ ◦ ψC(∥xn − x∥) + ∥T x∥∥yn − y∥,
+where ψ(r) = ψA(r) + (MψB ◦ ψφ ◦ ψC)(r) < r for all r > 0 and the constant M exists
+in view of the fact T ′ is partially bounded operator on S.
+Taking limit supermum in the
+aforementioned inequality yields that F is continuous. Further, from assumption (iii), we show
+that B′(I − D)−1C as well as F is partially compact. Finally the relation ⪯ and the norm ∥ · ∥
+are compatible. Hence, an application of Theorem 2.2 infer that F has at least, one solution in
+S × S. Now, use the vector y = (I − D)−1Cx to achieve the proof.
+□
+As a consequence we have the following fixed point result.
+Corollary 4.3 Let S be a non-empty closed partially bounded subset of a regular partially
+ordered Banach algebra (E, ⪯, ∥ · ∥) such that the order relation ⪯ and the norm ∥ · ∥ in E
+are compatible. Let A, C : S −→ E and B, B′, D : E −→ K be five nondecreasing operators
+satisfying the following assumptions:
+(i) A, B and C are partially bounded and partially nonlinear D-Lipschitzians with D-functions
+ψA, ψB and ψC respectively,
+(ii) D is contraction with a contraction constant k and C(S) ⊂ (I − D)(S),
+(iii) B′ is continuous and C is a partially compact operator,
+(iv) x = Ax + T x · T ′y, y ∈ S ⇒ x ∈ S where T = B(I − D)−1C and T ′ = B′(I − D)−1C,
+(v)
+�I − A
+T
+�−1
+exists on T ′(S) and nondecreasing,
+(vi) x0 ⪯ Ax0 + T x0 · T ′x0 or x0 ⪰ Ax0 + T x0 · T ′x0, for some x0 ∈ E,
+(vii) every pair of elements x, y ∈ E has a lower and an upper bound in E.
+Then the operator matrix (1.3) has a fixed point in S × S, whenever Mψ(r) ≤ r if r > 0 with
+ψ(r) = ψB ◦ (
+1
+1−k)ψC(r) + ψA(r) and M = ∥T ′(S)∥.
+From Theorem 4.2 and corollary 4.3 , without any hurdle we can derive the following corollary.
+Corollary 4.4 Let S be a non-empty closed partially bounded subset of a regular partially
+ordered Banach algebra (E, ⪯, ∥ · ∥) such that the order relation ⪯ and the norm ∥ · ∥ in E
+are compatible. Let A, C : S −→ E and B, B′, D : E −→ K be five nondecreasing operators
+satisfying the following assumptions:
+11
+
+(i) A, B and C are partially bounded and partially nonlinear D-Lipschitzians with D-functions
+ψA, ψB and ψC respectively,
+(ii) D is contraction with a contraction constant k and C(S) ⊂ (I − D)(E),
+(iii) B′ is partially completely continuous
+(iv) x = Ax + T x · T ′y, y ∈ S ⇒ x ∈ S where T = B(I − D)−1C and T ′ = B′(I − D)−1C,
+(v)
+�I − A
+T
+�−1
+exist on B′(E) and nondecreasing,
+(vi) x0 ⪯ Ax0 + T x0 · T ′x0 or x0 ⪰ Ax0 + T x0 · T ′x0, for some x0 ∈ E,
+(vii) every pair of elements x, y ∈ E has a lower and an upper bound in E.
+Then the operator matrix (1.3) has a fixed point in S × E, whenever Mψ(r) ≤ r if r > 0 with
+ψ(r) = ψB ◦ (
+1
+1−k)ψC(r) + ψA(r) and M = ∥T ′(E)∥.
+5
+Existence solutions for a system of functional differential
+equation
+The FDE (1.1) is considered in the function space E = C(J, R) of continuous real-valued
+functions defined on J. We define a norm ∥ · ∥ and the order relation ⪯ in C(J, R) by
+∥x∥∞ = sup
+t∈J
+|x(t)|
+(5.1)
+and
+x ≤ y ⇐⇒ x(t) ≤ y(t)
+(5.2)
+for all t ∈ J. Clearly, C(J, R) is a Banach space with respect to above supremum norm and
+also partially ordered with respect to the above partially order relation ≤. It is known that
+the partially ordered Banach space C(J, R) has some nice properties with respect to the above
+order relation in it. The following lemma follows by an application of Arzela-Ascolli theorem.
+Lemma 5.1 [16] Let (C(J, R), ≤, ∥ · ∥) be a partially ordered Banach space with the norm ∥ · ∥
+and the order relation ≤ defined by (5.1) and (5.2) respectively. Then ∥·∥ and ≤ are compatible
+in every partially compact subset of C(J, R).
+The purpose of this section is to apply theorem 4.1 to discuss the existence of solutions for the
+following nonlinear quadratic functional differential equations QFDE (1.1).
+We need the next definition in what follows.
+Definition 5.1 A functions u, v ∈ C(J, R) is said to be a lower solution of the (1.1), if it
+satisfies
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+�u(t) − f1(t, u(t))
+f2(t, v(t))
+�′
++ λ
+�u(t) − f1(t, u(t))
+f2(t, v(t))
+�
+≤ g(t, v(t))
+v(t) ≤
+1
+1 − b(t)|u(t)| − p
+�
+t,
+1
+1 − b(t)|u(t)|
+�
++ p(t, v(t))
+�
+u(0), v(0)
+�
+≤ (u0, v0) ∈ R2.
+for all t ∈ J. Similarly, a functions u′, v′ ∈ C(J, R) is said to be an upper solution of the FDE
+(1.1) if it satisfies the conditions above with the inequality reversed.
+We consider the following set of assumptions in what follows:
+(H0) The functions b : J −→ R is continuous.
+(H1) The function t �−→ f1(t, 0) is bounded on J with bound F0.
+12
+
+(H2) f2 define a function f2 : J × R −→ R+ are nondecreasing in x for all t ∈ J.
+(H3) There exist two constants L, K ∈ R∗
++ such that
+0 < fi(t, x(t)) − fi(t, y(t)) ≤
+L(x − y)
+K + (x − y)
+for all t ∈ J , i = 1, 2 and x, y ∈ R with x ≥ y. Moreover, L ≤ K.
+(H4) The function p : J × R −→ R is nondecreasing in x for all t ∈ J and contraction with a
+constant k.
+(H5) The function g : J × R −→ R+ is nondecreasing and there exist a function h ∈ L1(J) such
+that
+∥g(t, x)∥ ≤ ∥h∥L1 for all t ∈ J and x ∈ R.
+(H6) The QFDE (1.1) has a lower solution in C(J, R) × C(J, R)
+Remark 5.1 Assume that the assumption (H6) holds.
+Then a functions x ∈ C(J, R) is a
+solution of (1.1) if and only if is a solution of the functional integral equation
+
+
+
+
+
+
+
+x(t)
+=
+f1(t, x(t)) + f2(t, y(t))
+�
+ce−λt + e−λt
+� t
+0
+eλsg(s, y(s))ds
+�
+y(t)
+=
+1
+1 − b(t)|x(t)| − p
+�
+t,
+1
+1 − b(t)|x(t)|
+�
++ p(t, y(t))
+(5.3)
+where c = x0 − f1(0, x0)
+f2(0, y0)
+.
+Indeed, let g ∈ C(J, R). Assume first that x is a solution of the QFDE (1.1) define on J. By
+definition, the function t �−→ x(t) − f1(t, x(t))
+f2(t, y(t))
+is continuous on J, and so, differentiable there,
+whence
+�x(t) − f1(t, x(t))
+f2(t, y(t))
+�′
+is integrable on J. Applying integration to (1.1) from 0 to t, we
+obtain the HIE (5.3) on J.
+Conversely, assume that x satisfies Eq. (5.3). Then by direct differentiation we obtain the
+first equation in QFDE (1.1). Again, substituting t = 0 in Eq. (5.3) yields
+x(0) − f1(0, x(0))
+f2(0, y(0))
+= x0 − f1(0, x0)
+f2(0, y0)
+whence, (x(0), y(0)) = (x0, y0). Thus, the QFDE (1.1) holds.
+Now, we are in a position to prove the following existence theorem for QFDE (1.1).
+Theorem 5.1 Assume that the assumption (H0)) through (H7) hold. Furthermore, if
+
+
+
+
+
+
+
+L
+�����
+x0 − f1(0, x0)
+f2(0, y0)
+���� + ∥h∥L1
+�
+≤ K
+M1∥γ∥ + K ≤ 1 − k.
+(5.4)
+Then the system of the functional differential equations (1.1) has, at least, one solution in
+C(J, R) × C(J, R).
+♦
+13
+
+Proof.
+Observe that the above problem (1.1) may be written in the following form
+� x(t) = (Ax)(t) + (By)(t) · (B′y)(t)
+y(t) = (Cx)(t) + (Dy)(t)
+.
+where A, B, C, D and B′ on C(J, R) defined by:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+(Ax)(t) = f1(t, x(t))
+(By)(t) = f2(t, y(t))
+(Cx)(t) =
+1
+1 − b(t)|x(t)| − p
+�
+t,
+1
+1 − b(t)|x(t)|
+�
+(Dy)(t) = p(t, y(t))
+(B′y)(t) = ce−λt + e−λt
+� t
+0
+eλsg(s, y(s))ds.
+From the above assumptions and the continuity of the integral, it follows that the operators
+B, B′ : E −→ K. In order to apply Theorem 4.1, we have to verify the following steps.
+Step 1: A, B, C, D and B′ are nondecreasing on E.
+Let x, y ∈ E be such that x ≥ y. Then x(t) ≥ y(t) for all t ∈ J. Then by assumption (H3), we
+obtain
+Ax(t)
+=
+f1(t, x(t)
+≥
+f1(t, y(t)
+=
+Ay(t),
+for all t ∈ J. This shows that the operator that the operator A is nondecreasing on E. Similarly,
+by assumption (H3), we get
+Bx(t)
+=
+f2(t, x(t)
+≥
+f2(t, y(t))
+=
+By(t),
+for all t ∈ J. This shows that the operator B is also nondecreasing on E. Furthermore, by
+assumption (H5), we get C is nondecreasing operator on E. Indeed, let x, y ∈ E such that
+x(t) ≥ y(t)
+Cx(t)
+=
+1
+1 − b(t)|x(t)| − p
+�
+t,
+1
+1 − b(t)|x(t)|
+�
+≥
+1
+1 − b(t)|y(t)| − p
+�
+t,
+1
+1 − b(t)|y(t)|
+�
+=
+Cy(t),
+for all t ∈ J. This shows that the operator C is nondecreasing on E. Again, by assumption
+(H4) , we obtain
+Dx(t)
+=
+p(t, x(t)
+≥
+p(t, y(t))
+=
+Dy(t),
+for all t ∈ J. This shows that the operator D is nondecreasing on E. Finally, by assumption
+(H5) , we get
+B′x(t)
+=
+ce−λt + e−λt
+� t
+0
+e−λsg(s, x(s))ds
+≥
+ce−λt + e−λt
+� t
+0
+e−λsg(s, y(s))ds
+=
+B′y(t),
+14
+
+for all t ∈ J. This shows that the operator B′ is nondecreasing on E.
+Step 2: A, B and C are partially bounded and partially D-Lipschitzians on E.
+Let x ∈ E be arbitrary. Without loss of generality we assume that x ≥ 0. Then by assumptions
+(H1) and (H2), we have
+|Ax(t)|
+=
+|f1(t, x(t))|
+≤
+|f1(t, x(t)) − f1(t, 0)| + |f1(t, 0)|
+≤
+L∥x∥
+K + ∥x∥ + F0
+≤
+L + F0,
+for all t ∈ J. Taking the supremum over t in the above inequality, we obtain
+∥Ax∥ ≤ L + F0
+for all x ∈ E. So, A is bounded. This further implies that A is partially bounded on E.
+Next, let x, y ∈ E be such that x ≥ y. Then, we have
+|Ax(t) − Ay(t)|
+=
+|f1(t, x(t)) − f1(t, y(t))|
+≤
+L∥x − y∥
+K + ∥x − y ∥
+=
+ψA(∥x − y∥),
+for all t ∈ J, where ψA(r) =
+Lr
+K + r. Taking the supremum over t, we obtain
+∥Ax − Ay∥ ≤ ψA(∥x − y∥),
+for all x, y ∈ E with x ≥ y.
+Hence, A is a partial nonlinear D-Lipschitzian on E with a
+D-function ψA. By using the same argument, we conclude that B is partially bounded and
+partially D-Lipschitzian on E, where ψB(r) =
+Lr
+K + r . Also, we shall show that C is partially
+bounded and partially D-Lipschitzian. Indeed, for all t ∈ J, we get
+|Cx(t)|
+=
+����
+1
+1 − b(t)|x(t)| − p
+�
+t,
+1
+1 − b(t)|x(t)|
+�����
+≤
+1 +
+����p
+�
+t,
+1
+1 − b(t)|x(t)|
+�����
+≤
+1 + k
+This means that the operator C is partially bounded. Moreover, let x, y ∈ E such that x ≥ y.
+Then we get
+|Cx(t) − C(y)(t)|
+≤
+����
+1
+1 − b(t)|x(t)| − p
+�
+t,
+1
+1 ��� b(t)|x(t)|
+�
+−
+1
+1 − b(t)|y(t)| − p
+�
+t,
+1
+1 − b(t)|y(t)|
+�����
+≤
+����
+1
+1 − b(t)|x(t)| −
+1
+1 − b(t)|y(t)|
+���� +
+����p
+�
+t,
+1
+1 − b(t)|x(t)|
+�
+− p
+�
+t,
+1
+1 − b(t)|y(t)|
+�����
+≤
+(1 + k)
+����
+1
+1 − b(t)|x(t)| −
+1
+1 − b(t)|y(t)|
+����
+≤
+(1 + k)∥b∥|x(t) − y(t)|.
+Taking the supremum over t, we obtain that C is partially nonlinear D-Lipshitzian with D-
+function ψC(r) = (1 + k)∥b∥r
+15
+
+Step 3 : (I − D)−1 is partially nonlinear D-Lipshitzian and (I − D)−1C is nondecreasing.
+Since D is contraction then (I − D)−1 exist and is a contraction with constant
+1
+1 − k . Conse-
+quently, (I − D)−1 is partially D-lipshitzian with D-function ψφ(r) =
+1
+1 − k r
+Now, we shaw that (I − D)−1C is nondecreasing. Since (I − D)−1Cx =
+1
+1 − b|x|, for all x ∈ E.
+Then for all x, y ∈ E such that x ≤ y we have
+(I − D)−1Cx =
+1
+1 − b|x| ≤
+1
+1 − b|y| = (I − D)−1Cy.
+Step 4 : B′ est partially continuous and C is compact.
+Let {xn}n∈N be a sequence in a chain C such that xn → x as n → ∞. Then by the dominated
+convergence theorem for integration, we obtain
+lim
+n−→∞ Bxn(t)
+=
+ce−λt + e−λt
+� t
+0
+lim
+n−→∞ eλsg(s, xn(s))ds
+=
+ce−λt + e−λt
+� t
+0
+eλsg(s, x(s))ds
+=
+Bx(t),
+for all t ∈ J. This shows that {Bxn} converges to Bx pointwise on J.
+Now we show that {Bxn}n∈N is an equicontinuous sequence of functions in E. Let t1, t2 ∈ J
+with t1 > t2 > 0. Then we have
+|Bxn(t2) − Bxn(t1)|
+≤
+��ce−λt2 − ce−λt1�� +
+����e−λt2
+� t2
+0
+eλsg(s, xn(s))ds
+−
+e−λt1 � t1
+0 eλsg(s, xn(s))ds
+���
+≤
+|ce−λt2 − ce−λt1| +
+����e−λt2
+� t2
+0
+eλsg(s, xn(s)) ds
+−
+e−λt2
+� t1
+0
+eλsg(s, xn(s))ds +e−λt2
+� t1
+0
+eλsg(s, xn(s))ds
+−
+e−λt1
+� t1
+0
+eλsg(s, xn(s))ds
+����
+≤
+|ce−λt2 − ce−λt1| +
+����e−λt2
+� t2
+t1
+eλsg(s, xn(s))ds
+����
++
+����(e−λt2 − e−λt1)
+� t1
+0
+eλsg(s, xn(s))ds
+���� .
+This implies that lim
+t1→t2 |Bxn(t2) − Bxn(t1)| = 0 and consequently B is partially continuous.
+Next, we show that C is compact. To do this, for any x ∈ J
+|Cx(t)|
+=
+����
+1
+1 + b(t)|x(t)| − p
+�
+t,
+1
+1 + b(t)|x(t)|
+�����
+≤
+1 +
+����p
+�
+t,
+1
+1 + b(t)|x(t)|
+�����
+≤
+1 + k
+this show that C is bounded. Now we show that C is equicontinuous on E. For each t1, t2 ∈ J
+16
+
+such that t2 > t1, we get
+|Cx(t2) − Cx(t1)|
+=
+����
+1
+1 + b(t)|x(t2)| − p
+�
+t2,
+1
+1 + b(t2)|x(t2)|
+�
+−
+1
+1 + b(t1)|x(t1)| − p
+�
+t1,
+1
+1 + b(t1)|x(t1)|
+�����
+≤
+����
+1
+1 + b(t)|x(t2)| −
+1
+1 + b(t1)|x(t1)|
+����
++
+����p
+�
+t2,
+1
+1 + b(t2)|x(t2)|
+�
+− p
+�
+t1,
+1
+1 + b(t1)|x(t1)|
+�����
+≤
+∥b∥∞
+����
+x(t2) − x(t1)
+(1 + b(t2)|x(t2)|)(1 + b(t1)|x(t1)|)
+����
++k∥b∥∞
+����
+x(t2) − x(t1)
+(1 + b(t2)|x(t2)|)(1 + b(t1)|x(t1)|)
+����
+≤
+(1 + k)∥b∥∞|x(t2) − x(t1)|.
+This implies that
+lim
+t2→t1 |Cx(t2) − Cx(t1)| = 0
+This means that C is equicontinuous on J. Then by the Arzela-Ascoli’s theorem [8] the closure
+of C(E) is relatively compact, consequently C is compact operator.
+Step 5: u satisfies the operator inequality u ≤ Au + B(I − D)−1Cu · B′(I − D)−1Cu.
+By assumption (H6), we have
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+�u(t) − f1(t, u(t))
+f2(t, v(t))
+�′
++ λ
+�u(t) − f1(t, u(t))
+f2(t, v(t))
+�
+≤ g(t, v(t))
+v(t) ≤
+1
+1 − b(t)|u(t)| − p
+�
+t,
+1
+1 − b(t)|u(t)|
+�
++ p(t, v(t))
+�
+u(0), v(0)
+�
+≤ (u0, v0) ∈ R2
+for all t ∈ J.
+Then, by Remark 2.1 we get
+
+
+
+
+
+
+
+u(t)
+≤
+f1(t, u(t)) + f2(t, v(t))
+�
+ce−λt + e−λt
+� t
+0
+eλsg(s, v(s))ds
+�
+v(t)
+≤
+1
+1 − b(t)|u(t)| − p
+�
+t,
+1
+1 − b(t)|u(t)|
+�
++ p(t, v(t))
+Hence, from definitions of the operators A, B, C, D and B′ it follows that
+u(t) ≤ Au(t) + B(I − D)−1Cu(t) · B′(I − D)−1Cu(t),
+for all t ∈ J. Taking the suprumum we have
+u ≤ Au + B(I − D)−1Cu · B′(I − D)−1Cu.
+Finally By using the assumption (H5), we have
+M
+=
+∥T ′(E)∥
+=
+����ce−λt + e−λt
+� t
+0
+eλtg(s, x(s))ds
+����
+≤
+|c| + ∥h∥L1.
+17
+
+From equation (5.4), we infer that MψB ◦ ψφ ◦ ψC(r) + ψA(r) < r.
+Thus, the operators A, B, C, D and B′ satisfy all the requirement of Theorem 4.1 and so the
+QFDE (1.1) has a positive solution in C(J × R) × C(J × R).
+□
+Remark 5.2 The conclusion of Theorem 5.1 also remains true if we replace the assumption
+(H7) with the following one:
+(H′
+7) The QFDE (1.1) has a upper solution in C(J, R) × C(J, R)
+The proof under this new assumption is similar to Theorem 5.1 and the conclusion again follows
+by an application of Theorem 4.1.
+References
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+Integral Equations, Kluwer Acad. Publ., New York (2001).
+[3] D. D. Bainov and S. Hristova, Differential equations with maxima, Chapman and Hall/CRC
+Pure and Applied Mathematics, (2011).
+[4] J. Banas and M. Lecko, Fixed points of the product of operators in Banach algebras,
+Panamer. Math. J., 12(2002), no. 2, 101-109.
+[5] F. E. Browder, Fixed point theory and nonlinear problems, Bull. AMS, 9(1983), no. 1, 1-39.
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+solutions of an integral equation with linear modifcation of the argument, Acta Math. Sin.
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+(2011).
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+[10] B. C. Dhage, On some nonlinear alternatives of Leray-Schauder type and functional integral
+equations, Arch. Math. (Brno), 42(2006), no. 1, 11-23.
+[11] B. C. Dhage, Hybrid fixed point theory in partially ordered normed linear spaces and
+applications to fractional integral equations, Diff. Equa. Appl., 5(2013), no. 2, 155-184.
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+applications to functional integral equations, Tamkang J. Math., 45(2014), no. 4, 397-426.
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+spaces and applications to functional hybrid integral equations, Malaya J. Mat., 3(2015),
+no. 1, 62-85.
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+Existence of Solutions for a System of
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+New York, (1988)
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+19
+

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@@ -0,0 +1,2027 @@
+Families of bosonic suppression laws beyond the permutation symmetry principle
+M. E. O. Bezerra and V. S. Shchesnovich
+Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC, Santo Andr´e, SP, 09210-170 Brazil
+(Dated: January 6, 2023)
+Exact cancellation of quantum amplitudes in multiphoton interferences with Fock states at input,
+the so-called suppression or zero transmission laws generalizing the Hong-Ou-Mandel dip, are useful
+tool in quantum information and computation. It was recently suggested that all bosonic suppression
+laws follow from a common permutation symmetry in the input quantum state and the unitary
+matrix of interferometer.
+By using the recurrence relations for interference of Fock states, we
+find a wealth of suppression laws on the beamsplitter and tritter which are not explained by the
+permutation symmetry principle. Our results reveal that in interference with Fock states on unitary
+multiports there are whole families of suppression laws for arbitrary total number of bosons even on
+asymmetric unitary multiports, beyond the previously formulated permutation symmetry principle.
+Introduction.– One of the most distinctive features of
+quantum theory is the superposition principle which, un-
+der appropriate conditions, leads to the existence of to-
+tally destructive interference in multi-path scenario, with
+the probability of some outcomes being exactly zero.
+When two single photons become indistinguishable they
+bunch at the output of a balanced beamsplitter [1], which
+is the consequence of destructive interferences in the co-
+incidence outcomes.
+This is the well-known Hong-Ou-
+Mandel dip, which has found numerous applications such
+as characterization of photon indistinguishability [2, 3],
+generation and detection of entanglement [4–6] and de-
+sign of efficient quantum gates [7] for all-optical compu-
+tations. The exact cancellation be understood as the con-
+sequence of a symmetry in the setup: the beamsplitter is
+balanced and the Fock state of indistinguishable photons
+is symmetric under the transposition of the input modes.
+The totally destructive multiphoton interference for more
+than two photons was found in many subsequent works,
+including the even-odd number suppression events and
+four-photon enhancement on a beamsplitter [8, 10], the
+Hong-Ou-Mandel type effect in the coincidence counting
+on the symmetric Bell (a.k.a. Fourier) multiports [13],
+for which the conditions for all possible zero transmis-
+sion laws were formulated [14] and generalized to both
+bosons and fermions [16], and a series of experiments
+with various numbers of photons [17, 18, 20–23]. These
+works pointed on a connection between the suppression
+laws and some underlying symmetry in the setup. Such
+a connection was formulated as one common symmetry
+principle [24, 25], which seemed to explain all previously
+known suppression laws, for bosons and fermions, and
+generalize them to a wide class of and unitary multiports
+and input states.
+In this work we reveal the existence of whole families
+of suppression laws in quantum interferences with Fock
+states on unitary multiports, which are not explained by
+the common permutation symmetry principle.
+Recurrence relation for quantum amplitudes.– We start
+by demonstrating that the quantum amplitudes from
+multi-photon interference on a linear unitary multiport
+can be calculated by using the generating functions
+method. Let ˆa†
+k be the creation operator of optical mode
+in input port k of a unitary multiport of size M and
+consider the M-mode unnormalized coherent state in
+y = (y1, . . . , yM)
+|y⟩a = exp
+� M
+�
+k=1
+ykˆa†
+k
+�
+|0⟩ =
+�
+m
+M
+�
+k=1
+ymk
+k
+√mk!|m⟩a (1)
+where |m⟩a is a Fock state in M input modes m =
+(m1, . . . , mM).
+Introduce also analogous state |x⟩b,
+where x = (x1, . . . , xM), for the output modes ˆb†
+k, k =
+1, . . . , M, related to the input modes by an unitary mul-
+tiport U as follows
+a†
+k =
+M
+�
+l=1
+Uklb†
+l .
+(2)
+The N-photon quantum amplitude b⟨n|m⟩a between the
+input and output Fock states can be found from the gen-
+erating function G(x, y) ≡ b⟨x|y⟩a:
+b⟨n|m⟩a =
+M
+�
+k=1
+1
+√mk!nk!
+∂mk
+∂ymk
+k
+∂nk
+∂xnk
+k
+G(x, y)
+�����
+x=y=0
+. (3)
+Using the Baker-Campbell-Hausdorff formula for the
+product of two exponents as in Eq.(1) we obtain
+G(x, y) = exp
+�
+�
+M
+�
+k,l=1
+ykUklxl
+�
+� .
+(4)
+The expression in Eq.
+(3) admits many recurrence
+relations between the quantum amplitudes for different
+total number of photons N. For instance, by fixing the
+input state (computing the derivatives over y in Eq.(3))
+and taking one derivative over xl we get
+b⟨n|m⟩a =
+M
+�
+k=1
+�mk
+nl
+Ukl b⟨n − 1l|m − 1k⟩a
+(5)
+arXiv:2301.02192v1  [quant-ph]  3 Jan 2023
+
+2
+FIG. 1: Representation of the two interferometers that are
+considered to exemplify our method: a) Beamsplitter, that
+transforms two input modes into two output modes; b) Trit-
+ter, that is a composition of three different beamsplitters B1,
+B2, B3 and a control phase shifter θ. Here, each mk denotes
+the number of photons in the input mode k and nl denotes
+the number of photons in the output mode l.
+where 1k ≡ (0, . . . , 0, 1, 0, . . . , 0), etc.
+Eq.
+(5) is also
+a consequence of the fact that the quantum amplitude
+is given by the symmetrization of a product of the cor-
+responding N matrix elements of U, called the matrix
+permanent, and its properties [26].
+Let us now focus on a single output port l = 1, setting
+n = (n1, nS), where nS = (n2, ..., nM).
+Reusing the
+recurrence relation of Eq. (5) repeatedly nl times for the
+output modes l = 2, . . . , M we get the amplitude b⟨n|m⟩a
+as a linear combination of the amplitudes b⟨n1, 0S|m′⟩a,
+where m′ is the input configuration with fewer photons
+that appears in each term of the expansion due to the
+use of the recurrence relation. The latter can be easily
+calculated directly. In the end we get the amplitude in
+the form (see details in [28])
+b⟨n|m⟩a =
+�
+n1!
+nS!m!
+� M
+�
+k=1
+U mk−|nS|
+k1
+�
+f n
+m(U),
+(6)
+where f n
+m(U) is some polynomial in the matrix elements
+of U, which we call suppression function. This polyno-
+mial allows to find zero transmission laws, which are their
+roots.
+Below we restrict ourselves to small numbers of pho-
+tons in M −1 output ports (i.e., the power of the polyno-
+mial f n
+m(U) in Eq. (6)), setting |nS| = 1, 2 and illustrate
+our method on beamsplitter and tritter, given in Fig. 1.
+We will say that there is a “family of suppression
+laws” on the M-dimensional interferometer if for the in-
+put m and output n configurations of a given form, e.g.,
+m = (m, m) and n = (n1, 1) for a beamsplitter, and an
+arbitrary compatible total number of bosons there is a
+suppression law for the interferometer matrix with the
+elements given by the input and output configurations.
+Families of suppression laws for beamsplitter.– Let us
+first test the method using a beamsplitter, illustrated in
+Fig.1(a), with matrix
+B =
+�
+�
+�
+�
+√τ
+−√ρe−iϕ
+√ρeiϕ
+√τ
+�
+�
+�
+�
+(7)
+where τ = 1−ρ. In this case nS = n2. When considering
+beamsplitter we can neglect the arbitrary phase ϕ as it
+can be scaled out (however, when considering the tritter
+decomposition, as in Fig. 1(b), this phase is an important
+parameter).
+First, for n2 = 1 the recurrence in Eq. (6) has the
+following function
+f (n1,1)
+(m1,m2)(B) = (m1 + m2)τ − m1
+(8)
+which predicts that the following quantum amplitude
+b⟨n1, 1|m1, m2⟩a = 0 for an arbitrary n1 ≥ 1 and the
+transmission
+τ (1) =
+m1
+m1 + m2
+.
+(9)
+This suppression law coincides with the previous result
+[27], obtained by another method. This family reduces
+to the HOM effect [1] for the symmetric beamsplitter for
+m1 = m2 = 1.
+For n2 = 2 we get the suppression function
+f (n1,2)
+(m1,m2)(B) = (m1 + m2 − 1)(m1 + m2)
+�
+τ 2
+−
+2m1
+m1 + m2
+τ +
+m1(m1 − 1)
+(m1 + m2)(m1 + m2 − 1)
+�
+, (10)
+which predicts another (previously unknown) suppres-
+sion law ⟨n1, 2|m1, m2⟩ = 0 for the transmission
+τ (2) =
+m1
+m1 + m2
+�
+�1 ±
+�
+m2/m1
+m1 + m2 − 1
+�
+� .
+(11)
+This family of suppression laws contains also to the sym-
+metric beamsplitter τ (2) = 1/2 for specific inputs, e.g.,
+for four input photons b⟨2, 2|1, 3⟩a = 0 [8, 10]. Only such
+cases can be explained by the permutation symmetry ap-
+proach [14, 16, 24, 25] (in the above case transposition
+symmetry of the output ports with n1 = n2 = 2).
+The above approach allows one to derive all possible
+the suppression laws for the beamsplitter. The compu-
+tations, however, become quite involved as the minimum
+number of bosons in the input and output ports scales up
+(see [28] for more details). Nevertheless, general conclu-
+sions are allowed by the fact that quantum amplitudes
+b⟨n1, n2|m1, m2⟩a on a beamsplitter can be made real-
+valued functions of its transmission τ by removing the
+overall phase. Numerical simulations with various dis-
+tributions of bosons (i.e., Fock states) reveal that the
+number of zeros in such a quantum amplitude is given
+
+a)
+b)
+m1
+n2
+B1
+B3
+0
+m1
+n3
+B
+B2
+m2
+n1
+m3
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+-1
+-0.5
+0
+0.5
+n1,n2|m1m2
+FIG. 2:
+Typical behavior of the quantum amplitudes on a
+beamsplitter and the interlaced zeros (the suppression laws).
+Here we plot b⟨n1, n2|9, 4⟩a as functions of the beamsplitter
+transmission τ for n1 = 3 (solid line), n1 = 4 (dash-dotted
+line), and n1 = 5 (dashed line).
+by the minimum number of bosons min(nl, mk) in the
+four ports. Moreover, two quantum amplitudes related
+by the exchange of a single boson have interlaced zeros:
+between two zeros of one of them there is one zero of the
+other, see also Fig. 2 (at the end points, τ = 0 and τ = 1,
+a real-valued quantum amplitude is either equal to zero
+or to ±1, which allows for the above bound on the total
+number of zeros).
+Families of suppression laws for tritter.– We now con-
+sider the suppression laws on the tritter obtained by
+an arrangement of three beamsplitters according to the
+setup in Fig. 1(b) [9, 11]. Here each beamsplitter has
+matrix Bj similar to that of Eq. (7) and has the trans-
+mitivity τj and the phase ϕj. Moreover, an additional
+phase plate θ is inserted in one of the optical paths. Our
+tritter has in total seven free parameters, which is hard
+to analyze in general. We will therefore focus on two spe-
+cific families each having only two free parameters. For
+the first family we set: τ2 = 2/3, τ3 = 1/2, ϕj = π/2,
+leaving us with the free parameters τ1 and θ. It has the
+following matrix
+T (1) =
+=
+1
+√
+6
+�
+�
+�
+�
+�
+�
+�
+�
+2√τ1, −√τ1eiθ − i√3ρ1, −√τ1eiθ + i√3ρ1
+2√ρ1, −√ρ1eiθ + i√3τ1, −√ρ1eiθ − i√3τ1
+√
+2,
+√
+2eiθ,
+√
+2eiθ
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(12)
+For the second family we set: τ1 = τ3 = 1/2 and ϕj =
+π/2 leaving us with the free parameters τ2 and θ. It has
+the following matrix
+T (2) = 1
+2
+�
+�
+�
+�
+�
+�
+�
+�
+√2τ2, −i − √ρ2eiθ,
+i − √ρ2eiθ
+√2τ2,
+i − √ρ2eiθ,
+−i − √ρ2eiθ
+2√ρ2,
+√2τ2eiθ,
+√2τ2eiθ
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(13)
+The above tritter families reduce to the well-known sym-
+metric tritter (i.e., Bell multiport) when θ = 0 and, in
+the first case, τ1 = 1/2 or, in the second case, τ2 = 2/3.
+For tritter, in contrast to beamsplitter, two input
+mode occupations can vary for a given total number
+of bosons.
+We will focus below on the following two
+particular families of input states m(I) = (n1, 1, 1) and
+m(II) = (m, m, m), where n1 ≥ 1 and m ≥ 1 and oth-
+erwise arbitrary.
+This choice of specific inputs is dic-
+tated by the need to compare with the suppression laws
+due to permutation symmetry.
+For |nS| = 1 we have
+found suppression laws for the outputs n = (n1, 1, 0) and
+n = (n1, 0, 1), considering only the inputs m(II). In ad-
+dition, for |nS| = 2 we have found suppression laws for
+the outputs n = (n1, 1, 1) and n = (n1, 2, 0), consider-
+ing both of the previous inputs m(I) and m(II).
+The
+expressions for the corresponding suppression function
+f n
+m(T) are too complicated to be presented here (see de-
+tails in [28]). Instead we give the results in Fig. 3 with
+the explicit expressions for the tritter parameters given
+in Table I. Note that Table I contains only some of all
+possible suppression laws for the chosen inputs/outputs,
+e.g., m = (m, 0, 1) or m = (m, 1, 0) also correspond to
+other two families of suppression laws.
+Suppression laws from the permutation symmetry.–
+Only a fraction of the suppression laws discussed above
+(given by the red circles on the dashed line in Fig. 3),
+corresponding to the input m = (m, m, m) and out-
+put n = (n1, 2, 0) (with n1 = 3m − 2), is explained by
+the “general permutation symmetry principle” of Refs.
+[24, 25] (see for more details Ref. [28]). These appear for
+the symmetric tritter, with the three-dimensional Fourier
+matrix
+Ts =
+1
+√
+3
+�
+�
+�
+�
+�
+�
+�
+�
+1 − 1+i
+√
+3
+2
+−1+i
+√
+3
+2
+1
+−1+i
+√
+3
+2
+− 1+i
+√
+3
+2
+1
+1
+1
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(14)
+which results by setting either τ1 = 1/2 in Eq. (12) or
+τ2 = 2/3 in Eq. (13) and θ = 0, see also Fig. 1(b). Such
+suppression laws also correspond to some symmetry of
+the suppression function f n
+m(U) in Eq. (6): the roots do
+not depend on n1 and m. Interestingly, we have found
+a new symmetric tritter �Ts with the suppression laws
+obeying the same property. This new tritter corresponds
+
+4
+TABLE I: Suppression laws for tritter
+θ = 0, π
+θ = ± π
+2
+θ = 0, π
+θ = ± π
+2
+b⟨n|m⟩a
+τ1
+τ1
+τ2
+τ2
+b⟨n1, 1, 0|m, m, m⟩a
+1
+2
+1
+2
+2
+3
+2
+3
+b⟨n1, 1, 1|n1, 1, 1⟩a
+3n1(n1−1)
+2(n1+1)(n1+2)
+3n1
+4(n1+1) , n1 ̸= 1 a
+2n1(n1−1)
+(n1+1)(n1+2)
+4n1
+(n1+1)(n1+2)
+b⟨n1, 2, 0|n1, 1, 1⟩a
+1
+2 , n1 = 1, 2
+(too long, see [28])
+2
+3 , n1 = 1, 2
+(too long, see [28])
+b⟨n1, 1, 1|m, m, m⟩a
+1
+2
+�
+1 ±
+1
+√m
+�
+1
+2
+2m−1
+3m−1 ±
+�
+12(4m−1)
+6(3m−1)
+2
+3,
+2m
+3m−1
+b⟨n1, 2, 0|m, m, m⟩a
+1
+2
+(too long, see [28])
+2
+3 ,
+2m
+3m−1
+2m−1
+3m−1 ±
+�
+12(4m−1)
+6(3m−1)
+aFor n1 = 1 and θ = ±π/2 there is a suppression law for the
+tritter T (1) with an arbitrary τ1.
+FIG. 3: Non-trivial suppression laws for outputs n = (n1, 1, 1)
+and n = (n1, 2, 0).
+Here the suppression laws for τj = 0
+or τj = 1 are called trivial and then, are removed from the
+graph. For the tritter T (1) we have the suppression laws for
+inputs: a) m(I) = (n1, 1, 1) and b) m(II) = (m, m, m). For
+the tritter T (2) we have the suppression laws for inputs: c)
+m(I) = (n1, 1, 1) and d) m(II) = (m, m, m). The dashed line
+corresponds to the symmetric tritter τ1 = 1/2 and τ2 = 2/3
+for θ = 0.
+to an orthogonal matrix, which has the form of Ts in Eq.
+(14), as follows
+�Ts =
+1
+√
+3
+�
+�
+�
+�
+�
+�
+�
+�
+1 − 1+
+√
+3
+2
+−1+
+√
+3
+2
+1
+−1+
+√
+3
+2
+− 1+
+√
+3
+2
+1
+1
+1
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(15)
+The tritter �Ts is obtained by setting either τ1 = 1/2 in
+Eq. (12) or τ2 = 2/3 in Eq. (13) and θ = π/2 (factoring
+out the unimportant total phases in the output modes,
+i.e., a diagonal unitary from the right with (1, i, i) on
+the main diagonal). The tritter of Eq. (15) shares one
+of the symmetries with that of Eq. (14): it is invariant
+under the simultaneous permutation of rows 1 and 2 and
+columns 2 and 3 (a different type of symmetry used in
+the formulation of the “general permutation symmetry
+principle” of Refs. [24, 25]). The suppression laws on the
+symmetric tritter of Eq. (15) corresponding to the input
+m(II) = (m, m, m) and output n = (n1, 1, 1) are due to
+the roots of the suppression function f n
+m(U) in Eq. (6)
+that do not dependent on n1 and m (given by the blue
+points on the dashed line in Fig. 3).
+The symmetric tritter in Eq.
+(15) can also be real-
+ized by successive application of the transposition oper-
+ation of the first and the third inputs (P13), followed by
+a balanced beamsplitter on the second and third inputs
+(B(τs)), and by the inverse of the symmetric tritter Ts,
+i.e., we have
+�Ts = P13 (1 � B(τs)) T †
+s , where the beam-
+splitter is given by Eq. (7) with τs = (
+√
+3 + i)/4. The
+suppression laws for �Ts cannot be explained by the “gen-
+eral permutation symmetry principle” of Refs. [24, 25]
+which is applicable only to the standard symmetric trit-
+ter Ts (see details in Ref. [28]).
+In addition to the results discussed above, we also
+have found the suppression functions for the amplitudes
+
+1.0
+0.9
+0.8
+0.7
+0.6
+F0.5
+000000000
+0.4
+0.3
+0.2
+a)
+b)
+0.1
+0.0
+9 10
+8
+910
+8
+1
+2
+m
+n1
+1.0
+0.9
+0.8
+0.7
+0.6
+ 0.5
+0.4
+0.3
+0.2
+c)
+d)
+0.1
+0.0
+910
+9 10
+3
+4.
+5
+6
+8 
+1
+5
+6
+n1
+m
+n=(nl,l,l),θ=0,π
+n=(nl,2,0),θ= ±π/2
+n=(n1,1,1),θ= ±π/2
+n=(n1,2,0),θ= +π/2
+n=(n1,2,0),θ=0,π
+n =(n1,2, 0),θ= π/25
+b⟨n1, 1, 0|m, m, m⟩a and b⟨n1, 0, 1|m, m, m⟩a. These am-
+plitudes are zero only for the case of symmetric tritters
+τ1 = 1/2 or τ1 = 2/3, as shown in the first row of Table I.
+Considering the permutation symmetry of Refs. [24, 25],
+this result is already known only for the tritter Ts.
+Conclusion.– We have found several families of sup-
+pression laws on beamsplitter and on tritter for arbitrary
+total number of photons, which are not explained by
+the permutation symmetry principle advanced in Refs.
+[14, 16, 24, 25]. We have given only a fraction of new
+suppression laws on tritter for specific input/output con-
+figurations, numerical simulations reveal the existence of
+further families of the suppression laws not related to
+the permutation symmetry principle.
+Similar suppres-
+sion laws, not explained by the permutation symmetry
+principle, are expected to appear for multiports of any
+size, since by using our generation function approach
+one can, in principle, obtain all the suppression laws
+for a multiport of any size (though this is impractical
+by the complexity of the calculations which involve find-
+ing roots of higher-order polynomials). One can, on the
+other hand, explore the suppression laws experimentally,
+since recently there has been breakthrough in controlled
+production of Fock states with specified number of pho-
+tons. The recent methods that are able to generate these
+states include using heralded Fock states from a SPDC
+process [33], the interaction of a coherent state with two-
+level atoms [34], and by converting a coherent state into
+a Fock state inside a resonator by radiation losses [35].
+Our results also beg the important general question: Can
+the discovered families of suppression laws follow from a
+more general common symmetry principle? This could
+be the direction for future work.
+Acknowledgements.–
+M.E.O.B.
+was
+supported
+by
+the S˜ao Paulo Research Foundation (FAPESP), grant
+2021/03251-0 and V.S. was supported by the National
+Council for Scientific and Technological Development
+(CNPq) of Brazil, grant 307813/2019-3.
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+Supplementar material for
+“Families of bosonic suppression laws beyond the permutation symmetry principle”
+M. E. O. Bezerra and V. S. Shchesnovich
+Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC, Santo Andr´e, SP, 09210-170 Brazil
+THE MATRIX OF THE TRITTER
+The tritter is a three-mode interferometer that can be built in the triangular arrangement of three beamsplitters,
+one mirror, and in our case, an additional phase plate. Let us consider Bj the matrices of each beamsplitter acting on
+the input modes defined in Fig. 1(b) in the main text. We set the reflection phases of each beamsplitter as ϕj = π/2,
+then the matrices of these beamsplitters are
+B1 =
+�
+�
+�
+�
+�
+�
+�
+�
+√τ1
+i√ρ1 0
+i√ρ1
+√τ1
+0
+0
+0
+1
+�
+�
+�
+�
+�
+�
+�
+�
+, B2 =
+�
+�
+�
+�
+�
+�
+�
+�
+√τ2
+0 i√ρ2
+0
+1
+0
+i√ρ2 0
+√τ2
+�
+�
+�
+�
+�
+�
+�
+�
+, B3 =
+�
+�
+�
+�
+�
+�
+�
+�
+1
+0
+0
+0
+√τ3
+i√ρ3
+0 i√ρ3
+√τ3
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(1)
+where ρk = 1 − τk for each k = 1, 2, 3. In addition, we need to define the matrices related to the additional phase
+shifter Pθ = diag(1, 1, eiθ) and the ones related to the reflection phase in the first mode M1 = diag(−1, 1, 1), second
+mode M2 = diag(1, −1, 1) and third mode M3 = diag(1, 1, −1).
+Finally, the matrix of the tritter, denoted by T, is built by the sequence action of these matrices
+T = M1B1M2B2PθB3M3
+=
+�
+�
+�
+�
+�
+�
+�
+�
+−√τ1τ2
+i√ρ1τ3 + √τ1ρ2ρ3eiθ
+√ρ1ρ3 + i√τ1ρ2τ3eiθ
+i√ρ1τ2
+−√τ1τ3 − i√ρ1ρ2ρ3eiθ
+i√τ1ρ3 + √ρ1ρ2τ3eiθ
+i√ρ2
+i√τ2ρ3eiθ
+−√τ2τ3eiθ
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(2)
+where in our notation, the rows are related to the input modes and the columns with the output modes, in contrast
+to Ref. [1]. From Eq.(2), we arrive in the matrix T (1) of the main text by taking τ2 = 2/3 and τ3 = 1/2, preserving τ1
+and θ as free parameters, and factoring a diagonal matrix diag(i, 1, 1) from the left and diag(i, i, −1) from the right.
+Moreover, we arrive in the matrix T (2) by taking τ1 = τ3 = 1/2, preserving τ2 and θ as free parameters, and factoring
+the same diagonal matrices of the previous one.
+In the main text, we defined two types of symmetric tritters, denoted by Ts and �Ts. These tritters are obtained by
+setting τ1 = τ3 = 1/2 and τ2 = 2/3 in Eq.(2), and are built explicitly from the following construction:
+Ts(θ) =
+�
+�
+�
+�
+�
+�
+�
+�
+− 1
+√
+2
+i
+√
+2
+0
+i
+√
+2
+− 1
+√
+2 0
+0
+0
+1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+2
+3 0
+i
+√
+3
+0
+1
+0
+i
+√
+3
+0
+�
+2
+3
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1 0
+0
+0 1
+0
+0 0 eiθ
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+0
+0
+0
+1
+√
+2 − i
+√
+2
+0
+i
+√
+2 − 1
+√
+2
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(3)
+where we have Ts(0) = Ts and Ts(π/2) = �Ts, after factoring the appropriate diagonal matrices that do not contribute
+to the interference.
+SUPPRESSION FUNCTIONS
+It was proved in the main text that, when N photons enter a linear interferometer of dimension M in the M-mode
+input Fock state |m⟩a and are detected in the M-mode output Fock state |n⟩b, the corresponding amplitude is given
+arXiv:2301.02192v1  [quant-ph]  3 Jan 2023
+
+2
+by
+b⟨n|m⟩a =
+M
+�
+k=1
+1
+√mk!nk!
+∂mk
+∂ymk
+k
+∂nk
+∂xnk
+k
+G(x, y)
+�����
+x=y=0
+, G(x, y) = exp
+�
+�
+M
+�
+k,l=1
+ykUklxl
+�
+� ,
+(4)
+where U is a unitary matrix related to the interferometer. For simplicity, since the number of photons is fixed at
+the input, we can take all the derivatives over the input variables yk in the generating function G(x, y) obtaining the
+following expression
+b⟨n|m⟩a =
+M
+�
+k=1
+1
+√nk!
+∂nk
+∂xnk
+k
+Gm(x)
+�����
+x=0
+, Gm(x) =
+M
+�
+k=1
+1
+√mk!
+� M
+�
+l=1
+Uklxl
+�mk
+,
+(5)
+where here, we have defined a generating function for each input configuration m.
+Taking the derivative over the output variable xl we obtain the following recurrence relation for this new generating
+function
+∂
+∂xl
+Gm(x) =
+M
+�
+k=1
+√mkUklGm−1k(x) ,
+(6)
+where 1k is a vector of dimension M with the k-th element being 1 and the others being zero, i.e.
+1k ≡
+(0, . . . , 0, 1, 0, . . . , 0). Then, replacing Eq.(6) in Eq.(5), we arrive in the recurrence for the amplitudes
+b⟨n|m⟩a =
+M
+�
+k=1
+�mk
+nl
+Ukl b⟨n − 1l|m − 1k⟩a .
+(7)
+As assumed in the main text, we focus on the mode l = 1, which can have an arbitrary number of photons
+n1 ≥ 1, and considered that the other modes have a small number of photons. Denoting the output configurations
+as n = (n1, nS), with nS = (n2, ..., nM), we can remove the photons in each mode of nS by using the recurrence
+relation of Eq. (7) repeatedly nl times for each output modes l = 2, . . . , M. Following this procedure, we obtain the
+amplitude b⟨n|m⟩a as a linear combination of the simple ones
+b⟨n1, 0S|m′⟩a =
+1
+√n1!
+∂n1
+∂xn1
+1
+Gm′(x1, 0, ..., 0)
+����
+x1=0
+=
+�
+n1!
+m′! U m′
+k
+k1 ,
+(8)
+where m′ is the input configuration with fewer photons that appears in each term of the expansion due to the use of
+the recurrence relation. Finally, factoring m! = m1!...mM! and the smallest order of Ukl, i.e. mk − |nS|, we obtain
+the amplitude in the form
+b⟨n|m⟩a =
+�
+n1!
+nS!m!
+� M
+�
+k=1
+U mk−|nS|
+k1
+�
+f n
+m(U),
+(9)
+where the function f n
+m(U) is called suppression function and is obtained by collecting the matrix elements that appear
+from the Eqs. (7),(8) and the terms remaining in the factorization. This function is a polynomial in the parameters
+of the interferometers √ρj and √τ j.
+Beamsplitter
+Outputs |n1, 1⟩b and |n1, 2⟩b
+First of all, we want to find suppression laws for the amplitudes with less photons in the output mode l = 2 by
+using the recurrence relation given by Eq.(7) to remove all the photons in this mode. Then, using this method for
+
+3
+the output mode l = 2 one and two times, respectively, we arrive in the recurrence relations
+b⟨n|m⟩a =
+�m1
+n2
+U12 b⟨n − 12|m − 11⟩a +
+�m2
+n2
+U22 b⟨n − 12|m − 12⟩a,
+(10)
+b⟨n|m⟩a =
+�
+m1(m1 − 1)
+n2(n2 − 1) U 2
+12 b⟨n − 212|m − 211⟩a +
+�
+m2(m2 − 1)
+n2(n2 − 1) U 2
+22 b⟨n − 212|m − 212⟩a +
++2
+�
+m1m2
+n2(n2 − 1)U12U22 b⟨n − 212|m − 11 − 12⟩a,
+(11)
+The first recurrence removes the photons n2 of the amplitudes b⟨n1, 1|m1, m2⟩a and the second of the amplitudes
+b⟨n1, 2|m1, m2⟩a. Then, we can use Eq.(8) obtaining Eq.(9) with the suppression functions
+f (n1,1)
+(m1,m1)(B) = m1B12B21 + m2B11B22
+= (m1 + m2)τ − m1,
+(12)
+f (n1,2)
+(m1,m2)(B) = m1(m1 − 1)B2
+12B2
+21 + 2m1m2B11B12B21B22 + m2(m2 − 1)B2
+11B2
+22
+= (m1 + m2)(m1 + m2 − 1)
+�
+τ 2 −
+2m1
+m1 + m2
+τ +
+m1(m1 − 1)
+(m1 + m2)(m1 + m2 − 1)
+�
+.
+(13)
+where we have considered the matrix of the beamsplitter B as our unitary matrix U. The roots of the these functions
+are the suppression laws found for the beamsplitter and are shown in Eqs.(9),(11) in the main text.
+Outputs |1, n2⟩b and |2, n2⟩b
+In addition to the considered in the main text, we also can find suppression laws for the amplitude that have fewer
+photons in the mode l = 1. For this case, we need to use Eq.(7) for this mode one and two times, respectively,
+obtaining the recurrence relations:
+b⟨n|m⟩a =
+�m1
+n1
+U11 b⟨n − 11|m − 11⟩a +
+�m2
+n1
+U21 b⟨n − 11|m − 12⟩a,
+(14)
+b⟨n|m⟩a =
+�
+m1(m1 − 1)
+n1(n1 − 1) U 2
+11 b⟨n − 211|m − 211⟩a +
+�
+m2(m2 − 1)
+n1(n1 − 1) U 2
+21 b⟨n − 211|m − 212⟩a +
++2
+�
+m1m2
+n1(n1 − 1)U11U21 b⟨n − 211|m − 11 − 12⟩a,
+(15)
+and similarly to the previous case, these equations remove the photons in n1 of the amplitudes b⟨1, n2|m1, m2⟩a and
+b⟨2, n2|m1, m2⟩a respectively. Finally, we can use Eq.(8) obtaining Eq.(9) with the suppression functions
+f (1,n2)
+(m1,m1)(B) = m1B11B22 + m2B21B12
+= (m1 + m2)τ − m2,
+(16)
+f (2,n2)
+(m1,m2)(B) = m1(m1 − 1)B2
+11B2
+22 + 2m1m2B11B12B21B22 + m2(m2 − 1)B2
+21B2
+12
+= (m1 + m2)(m1 + m2 − 1)
+�
+τ 2 −
+2m2
+m1 + m2
+τ +
+m2(m2 − 1)
+(m1 + m2)(m1 + m2 − 1)
+�
+.
+(17)
+Note that, these functions have the same form as those of Eqs.(12),(13), however with m1 and m2 interchanged.
+Tritter
+Outputs |n1, 1, 0⟩b and |n1, 0, 1⟩b
+For the tritter, the simplest suppression laws are those with |nS| = 1, corresponding to the output configurations
+with nS = (1, 0) and nS = (0, 1). Then, for the first one, we need to use the recurrence once for l = 1, and for the
+
+4
+second one, once for l = 2, obtaining respectively
+b⟨n|m⟩a =
+�m1
+n2
+U12 b⟨n − 12|m − 11⟩a +
+�m2
+n2
+U22 b⟨n − 12|m − 12⟩a +
+�m3
+n2
+U32 b⟨n − 12|m − 13⟩a,
+(18)
+b⟨n|m⟩a =
+�m1
+n3
+U13 b⟨n − 13|m − 11⟩a +
+�m2
+n3
+U23 b⟨n − 13|m − 12⟩a +
+�m3
+n3
+U33 b⟨n − 13|m − 13⟩a
+(19)
+where the first removes the photons in n2 of the amplitudes b⟨n1, 1, 0|m1, m2, m3⟩a and the second, the photons n3 of
+the amplitudes b⟨n1, 0, 1|m1, m2, m3⟩a. Then, from Eqs.(8),(9) we obtain the general expression for the suppression
+functions
+f (n1,1,0)
+m
+(U) = m1U12U21U31 + m2U11U22U31 + m3U11U21U32,
+(20)
+f (n1,0,1)
+m
+(U) = m1U13U21U31 + m2U11U23U31 + m3U11U21U33
+(21)
+Finally, considering our families of tritters T (1) and T (2) as the unitary transformation U of the previous equation,
+we have
+f (n1,1,0)
+m,m,m (T (1)) = −f (n1,0,1)
+m,m,m (T (1)) = m
+3 (2τ1 − 1),
+(22)
+f (n1,1,0)
+m,m,m (T (2)) = f (n1,0,1)
+m,m,m (T (2)) = m
+√
+2
+4
+(3τ2 − 2)√τ2eiθ
+(23)
+whose non-trivial roots are τ1 = 1/2 or τ2 = 2/3, which correspond to the symmetric tritters. These suppression laws
+are shown in the first row of Table I in the main text.
+Outputs |n1, 1, 1⟩b
+Now, for |nS| = 2 we will first consider the outputs with nS = (1, 1). Using Eq.(7) for the modes l = 2 and l = 3
+symultaneously we obtain the recurrence relation
+b⟨n|m⟩a =
+=
+�m1m2
+n2n3
+(U12U23 + U22U13) b⟨n − 11 − 12|m − 11 − 12⟩a +
+�
+m3(m3 − 1)
+n2n3
+U32U33 b⟨n − 11 − 12|m − 213⟩a +
++
+�m1m3
+n2n3
+(U12U33 + U32U13) b⟨n − 11 − 12|m − 11 − 13⟩a +
+�
+m2(m2 − 1)
+n2n3
+U22U23 b⟨n − 11 − 12|m − 212⟩a +
++
+�m2m3
+n2n3
+(U22U33 + U32U23) b⟨n − 11 − 12|m − 12 − 13⟩a +
+�
+m1(m1 − 1)
+n2n3
+U12U13 b⟨n − 11 − 12|m − 211⟩a,
+(24)
+which removes the photons in n2 and n3 of the amplitudes b⟨n1, 1, 1|m1, m2, m3⟩a. Then, using Eqs.(8),(9), we found
+the corresponding suppression function
+f (n1,1,1)
+m
+(U) =
+= U11U21U31
+�
+m1m2 (U12U23 + U22U13) U31 + m1m3 (U12U33 + U32U13) U21 + m2m3 (U22U33 + U32U23) U11
+�
++
++
+�
+m1(m1 − 1)U12U13U 2
+21U 2
+31 + m2(m2 − 1)U22U23U 2
+11U 2
+31 + m3(m3 − 1)U32U33U 2
+11U 2
+21
+�
+.
+(25)
+The previous equation has too many parameters: the input configurations mk, the tritter parameters ρj and θ. To
+find suppression laws it is convenient to consider inputs with only one parameter, in our case m(I) = (n1, 1, 1) and
+m(II) = (m, m, m), and our families of tritters T (1) and T (2) as the unitary transformation U. Then, for which one
+of these cases, the suppression functions of Eq.(25) are given by:
+f (n1,1,1)
+(n1,1,1) (T (1)) =
+√
+2
+18
+� �
+4ei2θ + 3(1 + ei2θ)n1 + (3 − ei2θ)n2
+1
+�
+τ1 − 3n1(n1 − 1)
+�√
+1 − τ1,
+(26)
+
+5
+f (n1,1,1)
+(n1,1,1) (T (2)) =
+√
+2
+4
+�
+ei2θ �
+2 + 3n1 + n2
+1
+�
+τ2 + (3 − ei2θ)n1 − (1 + ei2θ)n2
+1
+��
+τ2(1 − τ2),
+(27)
+f (n1,1,1)
+(m,m,m)(T (1)) = m
+9
+�
+2(2m + ei2θ − 1)(τ1 − 1)τ1 + m − 1
+�
+,
+(28)
+f (n1,1,1)
+(m,m,m)(T (2)) = m
+8
+�
+3(3m − 1)ei2θτ 2
+2 − 2
+�
+(6m − 2)ei2θ − 1
+�
+τ2 + (4m − 2)ei2θ − 2
+�
+τ2,
+(29)
+The roots of the four previous equations given the suppression laws for the amplitudes b⟨n1, 1, 1|n1, 1, 1⟩a and
+b⟨n1, 1, 1|m, m, m⟩a. These results are shown in blue in Fig. 3 in the main text, where the non-trivial suppression
+laws are ignored (i.e. those that τ1, τ2 = 0, 1).
+Outputs |n1, 2, 0⟩b
+Furthermore, for |nS| = 2 we also considered the outputs with nS = (2, 0). In this case, we need to use Eq.(7) two
+times for l = 2, obtaining the recurrence relation
+b⟨n|m⟩a =
+=
+�
+m1(m1 − 1)
+n2(n2 − 1) U 2
+12 b⟨n − 212|m − 211⟩a +
+�
+m2(m2 − 1)
+n2(n2 − 1) U 2
+22 b⟨n − 212|m − 212⟩a +
++
+�
+m3(m3 − 1)
+n2(n2 − 1) U 2
+32 b⟨n − 212|m − 213⟩a + 2
+�
+m1m2
+n2(n2 − 1)U12U22 b⟨n − 212|m − 11 − 12⟩a
++2
+�
+m1m3
+n2(n2 − 1)U12U32 b⟨n − 212|m − 11 − 13⟩a + 2
+�
+m2m3
+n2(n2 − 1)U22U32 b⟨n − 212|m − 12 − 13⟩a,
+(30)
+which removes the photons in n2 of the amplitudes b⟨n1, 2, 0|m1, m2, m3⟩a. Then, using Eqs.(8),(9), we found the
+corresponding suppression function
+f (n1,2,0)
+m
+(U) = 2U11U21U31
+�
+m1m2U12U22U31 + m1m3U12U32U21 + m2m3U22U32U11
+�
++
++m1(m1 − 1)U 2
+12U 2
+21U 2
+31 + m2(m2 − 1)U 2
+22U 2
+11U 2
+31 + m3(m3 − 1)U 2
+32U 2
+11U 2
+21,
+(31)
+Finally, keeping only the parameters of interest, we have
+f (n1,2,0)
+(n1,1,1) (T (1)) =
+√
+2
+18
+�
+(4ei2θ − 3(ei2θ − 1)n1 − (3 + ei2θ)n2
+1)τ1 − 3n1(1 − n1)
+� √
+1 − τ1 +
++
+√
+6
+9 ieiθ �
+n1(2 − n1) − (2 + n1 − n2
+1)τ1
+� √τ1,
+(32)
+f (n1,2,0)
+(n1,1,1) (T (2)) =
+√
+2
+4
+�
+(2 + 3n1 + n2
+1)ei2θτ2 −
+�
+3 + ei2θ + (ei2θ − 1)n1
+�
+n1
+� �
+τ2(1 − τ2) +
++ 1
+√
+2ieiθ(1 − n1) [n1 − (1 + n1)τ2] √τ2,
+(33)
+f (n1,2,0)
+(m,m,m)(T (1)) = m
+9
+�
+(4m − 2 − 2ei2θ)(1 − τ1)τ1 − m + 1
+�
++ 2m
+27 ieiθ(2τ1 − 1)
+�
+3τ1(1 − τ1),
+(34)
+f (n1,2,0)
+(m,m,m)(T (2)) = m
+8
+�
+(9m − 3)ei2θτ 2
+2 − 2
+�
+(6m − 2)ei2θ + 1
+�
+τ2 + 2
+�
+(2m − 1)ei2θ + 1
+��
+τ2,
+(35)
+Now, the roots of the four previous equations give the suppression laws for the amplitudes b⟨n1, 2, 0|n1, 1, 1⟩a and
+b⟨n1, 2, 0|m, m, m⟩a. These results are shown in red in Fig. 3, where the non-trivial suppression laws are also ignored.
+
+6
+Suppression laws with constant solutions
+In addition, in Fig. 3 in the main text, we note four constant suppression laws for the reflectivities ρ2 = 1/3 and
+ρ1 = 1/2. It occurs because for these values the corresponding suppression functions are factorized in such a way that
+one of the terms does not depend on m, which corresponds to these constant solutions, as follows:
+f (n1,1,1)
+(m,m,m)(T (1))
+θ=±π/2
+=
+m(m − 1)
+9
+(2τ1 − 1)2,
+(36)
+f (n1,1,1)
+(m,m,m)(T (2))
+θ=±π/2
+=
+m
+8
+�
+(3m − 1)τ2 − 2m
+�
+(3τ2 − 2)τ2,
+(37)
+f (n1,2,0)
+(m,m,m)(T (1))
+θ=0,π
+=
+m
+27
+�
+3(m − 1)(2τ1 − 1) + 2i
+�
+3(1 − τ1)τ1
+�
+(2τ1 − 1),
+(38)
+f (n1,2,0)
+(m,m,m)(T (2))
+θ=0,π
+=
+−m
+8 [(3m ��� 1)τ2 − 2m] (3τ2 − 2)τ2,
+(39)
+Here, we note that the suppression functions of Eqs. (36)(38) have a common constant root τ1 = 1/2, and those of
+Eqs. (37),(39) the common constant root τ2 = 2/3. These suppression laws are plotted in Fig. 3(b),(d) in the main
+text as the points and circles on the dashed line. Here, only the suppression laws obtained from Eqs. (38),(39) are
+related to the symmetry principle of Refs. [2, 3].
+SUPPRESSION LAWS FROM PERMUTATION SYMMETRY
+In Refs. [2, 3] were developed suppression laws for interferometers related to some input symmetries. Now we will
+show that only a part of the suppression laws we found are related to these symmetries. First of all, denoting SM as
+the group of permutations of M elements and σ their elements, we define the action of the permutation operator Pσ
+in a M-dimensional vector as follows
+Pσ
+�
+�
+�
+�
+�
+�
+�
+�
+x1
+...
+xM
+�
+�
+�
+�
+�
+�
+�
+�
+=
+�
+�
+�
+�
+�
+�
+�
+�
+xσ−1(1)
+...
+xσ−1(M)
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(40)
+Let an input configuration which is symmetric under the operation σ(m) = m and an interferometer U that satisfies:
+PσU = ZUΛ,
+(41)
+where Z is a diagonal unitary matrix related to external phases and Λ a diagonal matrix that contains the eigenvectors
+of Pσ. Then, according to [2], the outputs n satisfying λn1
+1 ...λnM
+M
+̸= 1 are suppressed and considering our choice of
+input/outputs, these laws are shown in Table I (a). Similarly, if we have outputs symmetrical under the operation
+σ(n) = n and an interferometer satisfying
+UP †
+σ = Λ∗UZ∗,
+(42)
+we have suppression for inputs m such that λm1
+1 ...λmM
+M
+̸= 1. These suppression laws are shown in Table I (b) for our
+choice of inputs/outputs.
+For the interference in a beamsplitter, we need to consider the group S2 = {I, (12)}. From our results, only the
+suppression laws for the amplitudes b⟨n1, 1|m, m⟩a are related to the symmetry principle, since they are zero for
+τ = 1/2, which corresponds to the beamsplitter symmetrical under the permutation (12).
+For the interference in a tritter, we need to consider the permutation group S3 = {I, (12), (13), (23), (123), (132)}.
+From our method, the suppression laws obtained for the amplitudes b⟨n1, 2, 0|m, m, m⟩a are related to the symmetry
+principle, since they are zero for the tritter Ts, which is symmetric under the permutations (123) and (321). Our
+tritters also can recover the suppression laws due to the permutations (12) and (23), however, these results are the
+trivial cases, where some τj = 0 or τj = 1. Now, denoting our tritters by T (k) = T (k)(τk, θ), these last suppression
+laws are shown in Table I.
+
+7
+TABLE I: Suppression laws for tritter from permutation symmetry
+a) Output suppression configurations for symmetric inputs Pσ(m) = m
+σ
+U
+Z
+Λ
+Suppressions from Eq.(41)
+Suppressions from f n
+m(U)
+(12)
+T (2)(1, θ)
+diag(−1, −1, 1)
+diag(−1, 1, 1)
+⟨n1, 1, 1|m, m, m⟩ and
+⟨n1, 2, 0|m, m, m⟩ for odd n1
+⟨1, 1, 1|1, 1, 1⟩ and
+⟨1, 2, 0|1, 1, 1⟩
+(12)
+T (2)(0, 0)
+diag(i, −i, 1)
+diag(1, −1, 1)
+⟨n1, 1, 1|m, m, m⟩ for any n1
+Same
+(12)
+T (2)(0, π)
+diag(i, −i, 1)
+diag(1, 1, −1)
+⟨n1, 1, 1|m, m, m⟩ for any n1
+Same
+(123)
+Ts
+I
+diag(1, ei2π/3, ei4π/3)
+⟨n1, 2, 0|m, m, m⟩ for any n1
+Same
+(321)
+Ts
+I
+diag(1, ei4π/3, ei2π/3)
+⟨n1, 2, 0|m, m, m⟩ for any n1
+Same
+b) Input suppression configurations for symmetric outputs Pσ(n) = n
+σ
+U
+Z
+Λ
+Suppressions from Eq.(42)
+Suppressions from f n
+m(U)
+(23)
+T (1)(1, θ)
+I
+diag(1, −1, 1)
+⟨n1, 1, 1|n1, 1, 1⟩ and
+⟨n1, 1, 1|m, m, m⟩ for any n1
+Same
+(23)
+T (1)(0, θ)
+I
+diag(−1, 1, 1)
+⟨n1, 1, 1|n1, 1, 1⟩ and
+⟨n1, 1, 1|m, m, m⟩ for odd n1
+⟨1, 1, 1|1, 1, 1⟩
+(23)
+T (2)(1, θ)
+diag(1, −1, −1)
+diag(1, 1, −1)
+⟨n1, 1, 1|n1, 1, 1⟩ and
+⟨n1, 1, 1|m, m, m⟩ for any n1
+Same
+SUPPRESSION LAWS AND PARTIAL DISTINGUISHABILITY
+Photons are partially distinguishable due to degrees of freedom not acted upon by the interferometer, which are
+called the internal states. In Ref. [4] it has been conjectured that the zero probability in the output of multi-photon
+interference with partially distinguishable photons is invariably the result of an exact cancellation of the quantum
+amplitudes of only the completely indistinguishable photons. This conjecture generalizes the well-known HOM effect
+[5] to more than two photons and arbitrary interferometer (and also to non-ideal detectors) and the observations made
+in Ref. [6]. It has been confirmed by all suppression laws in Refs. [2, 3]. Thus, by the conjecture, any suppression
+law which is not broken by partial the distinguishability of photons needs other suppression laws for smaller total
+numbers of photons.
+Now, this effect will be illustrated for a simple case. Let an experimental setup where N photons are prepared
+from independent sources in either N pure internal states |φi⟩, i = 1, . . . , N. If, for instance, an input has one mode
+occupied by one photon and this photon is partially distinguishable from the rest of N − 1 photons, we can use just
+two internal states |1⟩ and |2⟩, with |φk⟩ = |1⟩ for 1 ≤ k ≤ N − 1 and |φN⟩ = cosα|1⟩ + sinα|2⟩. Note that, the last
+photon becomes indistinguishable from the others when α = 0 and distinguishable when α = π/2. Therefore, we have
+the following state at the input:
+ˆρm = 1
+m!
+N−1
+�
+i=1
+ˆa†
+ki,1ˆa†
+kN,φN |0⟩⟨0|
+N−1
+�
+i=1
+ˆaki,1ˆakN,φN , m! = m1!m2!...mM!,
+(43)
+where the first index of the creation/annihilation operators is related to the spatial mode and the second index to the
+internal state. The creation operator of the N-th photon is then given by:
+ˆa†
+kN,φN = cosα ˆa†
+kN,1 + sinα ˆa†
+kN,2,
+(44)
+We define a set of POVMs ˆΠn related to the detection of the photons in the configurations n at the output:
+ˆΠn = 1
+n!
+�
+j
+N
+�
+i=1
+ˆb†
+li,ji|0⟩⟨0|
+N
+�
+i=1
+ˆbli,ji , n! = n1!n2!...nM!,
+(45)
+
+8
+where the sum in j is over the internal states ji = 1, 2. Then, after some calculations we can get the following
+expression for the probability:
+P(n|m, α) =
+�
+j
+Tr
+�
+ˆρm ˆΠn
+�
+=
+1
+m!n!
+�
+j
+�����⟨0|
+N
+�
+i=1
+ˆbli,ji
+N−1
+�
+i=1
+ˆa†
+ki,1
+�
+cosα ˆa†
+kN,1 + sinα ˆa†
+kN,2
+�
+|0⟩
+�����
+2
+= cos2α
+m!n!
+���⟨0
+�����
+N
+�
+i=1
+ˆbli,1
+N
+�
+i=1
+ˆa†
+ki,1|0⟩
+�����
+2
++ sin2α
+m!n!
+�
+j
+�����⟨0|
+N
+�
+i=1
+ˆbli,ji
+N−1
+�
+i=1
+ˆa†
+ki,1ˆakN,2|0⟩
+�����
+2
+= cos2α |b⟨n|m⟩a|2 + sin2α
+m!n!
+�
+j
+�����⟨0|
+N
+�
+i=1
+ˆbli,ji
+N−1
+�
+i=1
+ˆa†
+ki,1
+� M
+�
+l=1
+Uklˆb†
+l,2
+�
+|0⟩
+�����
+2
+= cos2α |b⟨n|m⟩a|2 + sin2α
+M
+�
+l=1
+|Ukl|2 |b⟨n − 1l|m − 1k⟩a|2 .
+(46)
+In the previous equation, we have developed suppression laws for the amplitudes b⟨n|m⟩a in the main text. However,
+in principle, the other terms b⟨n − 1l|m − 1k⟩a are non zero and then we need to use another sequence of recurrence
+relations to eliminate the photons at n − 1l.
+Let us focus on the distinguishable projection of the previous equation. The sum over l has M non-zero terms, each
+one being a product of two probabilities: a probability of the transition of one distinguishable photon to one output
+mode l (such that nl > 0 in n) multiplied by the probability of detecting the remaining N −1 indistinguishable photons
+to the reduced output n − 1l. Except the trivial case of the single-photon probability being zero, all probabilities of
+detecting N − 1 photons in the outputs n − 1l should be zero for zero output probability of such N photons.
+To illustrate this effect in our results, let us consider the simple example, where have m1 + m2 photons interfer-
+ing in a beamsplitter and we want to calculate the probability P(n1, 1|m1, m2, α). Considering that the partially
+distinguishable photon is injected at the input mode k = 1, we arrive at the following probability:
+P(n1, 1|m1, m2, α) = cos2α|b⟨n|m⟩a|2 +
++ sin2α
+�
+|U11|2|b⟨n1 − 1, 1|m1 − 1, m2⟩a|2 + |U12|2|b⟨n1, 0|m1 − 1, m2⟩a|2�
+,
+(47)
+where the first term is zero for τ = m1/(m1 +m2), according to the main text. However, ignoring the trivial solutions
+τ = 0, 1, the second term is zero when τ = (m1 − 1)/(m1 + m2 − 1) and the last is zero only for trivial solutions.
+Therefore the suppression law is broken, as the probability P(n1, 1|m1, m2, α) is no longer zero, because the three
+terms cannot be simultaneously zero for τ ̸= 0, 1.
+Now, let us consider the interference in the tritter T (1), with phase θ = π/2, and the probability P(n1, 1, 1|n1, 1, 1, α).
+If the partially distinguishable photon is injected at k = 1, we have
+P(n1, 1, 1|n1, 1, 1, α) = cos2α|b⟨n|m⟩a|2 + sin2α
+�
+|U11|2|b⟨n1 − 1, 1, 1|n1, 1, 0⟩a|2 +
++|U12|2|b⟨n1, 0, 1|n1, 1, 0⟩a|2 + |U13|2|b⟨n1, 1, 0|n1, 1, 0⟩a|2�
+,
+(48)
+where the first term is zero for τ1 = 3n1/(4n1 + 1), according to the Table I in the main text. The other three need
+to satisfy respectively the following equations
+(n1 + 1)
+�
+τ1(1 − τ1) +
+√
+3(n1 + 1)τ1 −
+√
+3n1 = 0,
+(n1 + 1)
+�
+τ1(1 − τ1) −
+√
+3(n1 + 1)τ1 +
+√
+3n1 = 0,
+4(n1 + 1)τ1
+√
+1 − τ1 − 3
+√
+1 − τ1 = 0.
+(49)
+where the last lead to τ1 = 1 or τ1 = 3/4(n1 + 1), which are not solutions of the first two equations. Therefore, the
+probability P(n1, 1, 1|n1, 1, 1, α) cannot be zero.
+[1] R. A. Campos, Three-photon Hong-Ou-Mandel interference at a multiport mixer, Phys. Rev. A 62, 013809 (2000)
+
+9
+[2] C. Dittel, G. Dufour, M. Walschaers, Totally destructive many-particle interference, Phys. Rev. Lett. 120, 240404 (2018).
+[3] C. Dittel, G. Dufour, M. Walschaers, G. Weihs, A. Buchleitner, R. Keil, Totally destructive interference for permutation-
+symmetric many-particle states, Phys. Rev. A 97, 062116 (2018).
+[4] V. S. Shchesnovich, Partial indistinguishability theory for multiphoton experiments in multiport devices, Phys. Rev. A 91,
+013844 (2015).
+[5] C. K. Hong, Z. Y. Ou, and L. Mandel, Measurement of subpicosecond time intervals between two photons by interference,
+Phys. Rev. Lett. 59, 2044 (1987).
+[6] M. C. Tichy, Sampling of partially distinguishable bosons and the relation to the multidimensional permanent, Phys. Rev.
+A 91, 022316 (2015).
+

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@@ -0,0 +1,1622 @@
+arXiv:2301.04897v1  [gr-qc]  12 Jan 2023
+GEOMETRICAL PROPERTIES OF A POINT-LIKE GLOBAL MONOPOLE
+SPACETIME
+ABSOS ALI SHAIKH1, FAIZUDDIN AHMED2 AND BISWA RANJAN DATTA3
+Abstract. The objective of the paper is to study the geometric properties of the point-
+like global monopole (briefly, PGM) spacetime, which is a static and spherically symmetric
+solution of Einstein field equation. It is shown that PGM spacetime admits various types of
+pseudosymmetric structures, such as, pseudosymmetry due to Weyl conformal curvature tensor,
+pseudosymmetry due to concircular curvature tensor, pseudosymmetry due to conharmonic
+curvature tensor, Ricci generalized conformal pseudosymmetric due to projective curvature
+tensor, Ricci generalized projective pseudosymmetric etc. Moreover, it is proved that PGM
+spacetime is 2-quasi Einstein, generalized quasi-Einstein, Einstein manifold of degree 2 and
+its Weyl conformal curvature 2-forms are recurrent. It is also shown that the stress energy
+momentum tensor of the PGM spacetime realizes several types of pseudosymmetry, and its Ricci
+tensor is compatible for Riemann curvature, Weyl conformal curvature, projective curvature,
+conharmonic curvature and concircular curvature. Further, it is shown that PGM spacetime
+admits motion, curvature collineation and Ricci collineation. Also, the notion of curvature
+inheritance (resp., curvature collineation) for the (1,3)-type curvature tensor is not equivalent
+to the notion of curvature inheritance (resp., curvature collineation) for the (0,4)-type curvature
+tensor as it is shown that such distinctive properties are possessed by PGM spacetime. Hence
+the notions of curvature inheritance defined by Duggal [1] and Shaikh and Datta [2] are not
+equivalent.
+1. Introduction
+Let M be a smooth and connected manifold of dimension n (≥ 3) equipped with a semi-
+Riemannian metric g of signature (δ, n−δ), 0 ≤ δ ≤ n. If δ = 1 or n−1 (resp., δ = 0 or n), then
+M is known as a Lorentzian (resp., Riemannian) manifold, and spacetimes are the mathemati-
+cal models of 4-dimensional connected Lorentzian manifolds. Throughout the paper ∇, R, S, κ
+respectively denote the Levi-Civita connection, Riemann curvature tensor of type (0, 4), Ricci
+tensor of type (0, 2) and the scalar curvature of M.
+Date: January 13, 2023.
+2020 Mathematics Subject Classification. 53B20, 53B25, 53B30, 53B50, 53C15, 53C25, 53C35, 83C15.
+Key words and phrases. Point-like global monopole metric, Einstein field equation, semisymmetric type
+tensor, Weyl conformal curvature tensor, pseudosymmetric type curvature condition, 2-quasi-Einstein manifold.
+1
+
+2
+A. A. SHAIKH, F. AHMED & B. R. DATTA
+To understand the symmetry of a semi-Riemannian manifold, curvature rigorously perform
+a crucial role as in 1926 Cartan introduced the notion of locally symmetry [3] by the relation
+∇R = 0 and the notion of semisymmetry [4] by the relation R · R = 0 in 1946 (later classified
+by Szab´o [5–7]). During last eight decades, several differential geometers and physicists debil-
+itated such curvature conditions to generalize the concept of symmetry in various directions,
+which infers different generalized notions of symmetry, such as, pseudosymmetric manifolds
+by Chaki [8, 9], pseudosymmetric manifolds by Adam´ow and Deszcz [10], weakly symmetric
+manifolds by Tam´assy and Binh [11, 12], recurrent manifolds by Ruse [13–16], curvature 2-
+forms of recurrent manifolds [17,18], several kinds of generalized recurrent manifolds by Shaikh
+et al. [19–24] etc. We note that Som-Raychaudhuri spacetime [25], Robertson-Walker space-
+time [26,27], G¨odel spacetime [28], Siklos spacetime [29], Robinson-Trautman spacetime [30] and
+Reissner-Nordstr¨om spacetime [31] admit different pseudosymmetric type geometric structures.
+There are two major aspects of geometric structures of a certain spacetime, one is for geom-
+etry and another is its physical nature due to the Einstein field equation (briefly, EFE). The
+main moto of this paper is to explore the geometric structures of PGM spacetime in terms of
+curvatures appearing by means of first order as well as higher order covariant derivatives.
+Again, to constitute gravitational potentials satisfying EFE, imposing of symmetry is a vital
+tool, which implies that the geometrical symmetries play a crucial role in the theory of general
+relativity. A geometric quantity is preserved along a vector field if the Lie derivative of certain
+tensor vanishes with respect to that vector field, and the vanishing Lie derivative illustrates ge-
+ometrical symmetries. Motion, curvature collineation, Ricci collineation etc. are the notions of
+such symmetries. Katzin et al. [32,33] rigorously investigated the role of curvature collineation
+in general relativity. In 1992, Duggal [1] introduced the notion of curvature inheritance general-
+izing the concept of curvature collineation for the (1,3)-type curvature tensor. Recently, Shaikh
+and Datta [2] introduced the concept of generalized curvature inheritance, which is a general-
+ization of curvature collineation as well as curvature inheritance for the (0,4)-type curvature
+tensor. During last three decades, a plenty of papers (see, [2,34–45]) appeared in the literature
+regarding the investigations of such kinds of symmetries. In this papar, it is also found that
+the PGM spacetime admits several symmetries, such as, motion, curvature collineation, Ricci
+collineation and curvature inheritance. Also, it is shown that the notions of curvature inheri-
+tance as well as curvature collineation for the (1,3)-type curvature tensor by Duggal [1] and for
+the (0,4)-type curvature tensor by Shaikh and Datta [2] are not equivalent as PGM spacetime
+
+GEOMETRICAL PROPERTIES OF A POINT-LIKE GLOBAL MONOPOLE SPACETIME
+3
+realizes such distinctive properties.
+Some Grand Unified Theories suggested that topological defects may have been produced
+during the phase transition in the early universe through a spontaneous symmetry breaking
+mechanism [46, 47]. There are different kinds of topological defects, and among them, cosmic
+strings [48, 49] and global monopoles [50–53] have been widely studied in the literature. A
+global monopole is a heavy object characterized by spherically symmetry and divergent mass.
+The gravitational field of a static global monopole was found by Barriola et al. [50] and is
+expected to be stable against spherical as well as polar perturbation.
+The line element of PGM spacetime, which is a static and spherically symmetric metric in
+(t, r, , θ, φ) coordinates, is described by [50]
+ds2 = −dt2 + dr2
+α2 + r2 (dθ2 + sin2 θ dφ2),
+(1.1)
+where α2 =
+�
+1 − 8πη2
+0
+�
+< 1 depends on the energy scale η0. The parameter η0 represents the
+dimensionless volumetric mass density of the PGM defect. Here, the different coordinates are
+in the ranges −∞ < t < +∞,
+0 ≤ r < ∞,
+0 ≤ θ ≤ π
+2, and 0 ≤ φ < 2 π. The PGM spacetime
+reveals some interesting features delineated as follows:
+(i) it is not globally flat, and possesses a naked curvature singularity on the axis given by
+the Ricci scalar κ = 2 (α2−1)
+r2
+,
+(ii) the area of a sphere of radius r in this manifold is not 4 π r2 but rather it is equal to
+4 π α2 r2,
+(iii) the surface θ = π
+2 presents the geometry of a cone with the deficit angle ∇ φ = 8 π2 η2
+0
+and
+(iv) there is no Newtonian-like gravitational potential: gtt = −1.
+Furthermore, in this topological defect space-time geometry the solid angle of a sphere of
+radius r is 4 π2 r2 α2 which is smaller than 4 π2 r2, and hence, there is a solid angle deficit
+∇ Ω = 32 π2 η2
+0. Other interesting features of this PGM spacetime have been given in details
+in [54]. If α → 1, one can obtain the spherically symmetric Minkowski flat space line element.
+The effects of global monopole in quantum mechanical systems have been studied in the liter-
+ature (see, [54–80]).
+The present paper exhibits several curvature properties of PGM spacetime accomplished by
+the metric (1.1), such as, pseudosymmetry due to Weyl conformal curvature, pseudosymme-
+try due to concircular curvature, pseudosymmetry due to conharmonic curvature etc. Also,
+
+4
+A. A. SHAIKH, F. AHMED & B. R. DATTA
+it is neither Einstein nor quasi-Einstein but it is an Einstein manifold of level 2, generalized
+quasi Einstein, 2-quasi-Einstein manifold and conformal curvature 2-forms are recurrent, etc.
+Moreover, it is shown that this metric is Ricci generalized conformal pseudosymmetric due to
+projective curvature tensor, Ricci generalized projective pseudosymmetric etc. Ricci tensor is
+neither Codazzi type nor cyclic parallel. Additionally, it is shown that the energy momentum
+tensor also admits several types of pseudosymmetric structures and Ricci tensor is compatible
+for Riemann, conformal, projective, concircular and conharmonic curvature tensors. Finally, it
+is shown that with respect to certain vector fields the PGM spacetime reveals motion, curvature
+collineation, Ricci collineation and curvature inheritance. Also, by exhibiting some distinctive
+properties of PGM spacetime, it is shown that the notions of curvature inheritance for (1,3)-
+type curvature tensor and the (0,4)-type curvature tensor are not equivalent.
+The paper is organized in the following way: in Section 2 we have discussed various rudi-
+mentary facts regarding various curvature tensors and their derivatives, which are essential
+throughout the paper to investigate the geometric properties of PGM spacetime. Section 3 is
+devoted to the study of several geometric structures of PGM spacetime and several interesting
+findings are obtained. In Section 4, we determine some geometric structures due to energy
+momentum tensor of the spacetime. In Section 5 it is shown that the PGM spacetime admits
+some symmetries, such as, motion, curvature collineation, Ricci collineation and curvature in-
+heritance. Also, the distinctness of the notions of curvature inheritance for (1,3)-type curvature
+tensor and for (0,4)-type curvature tensor, has been exhibited for a PGM spacetime. Finally,
+the last section consists of the conclusion of the paper briefly.
+2. Preliminaries
+The aim of this section is to explain different kinds of geometric structures originated by
+appointing restrictions on the curvatures and their covariant derivatives, which are effective to
+elaborate the symmetry of the PGM spacetime having certain geometric meanings. Also, the
+notions of motion, curvature inheritance, Ricci inheritance are illustrated in this section.
+For two symmetric second order covariant tensors ν1 and ν2, the Kulkarni-Nomizu product
+ν1 ∧ ν2 is defined by (see, [81–83])
+(ν1 ∧ ν2)(η1, η2, ι1, ι2)
+=
+ν1(η1, ι2)ν2(η2, ι1) + ν1(η2, ι1)ν2(η1, ι2)
+−
+ν1(η1, ι1)ν2(η2, ι2) − ν1(η2, ι2)ν2(η1, ι1),
+
+GEOMETRICAL PROPERTIES OF A POINT-LIKE GLOBAL MONOPOLE SPACETIME
+5
+where η1, η2, ι1, ι2 ∈ χ(M), the Lie algebra of all smooth vector fields on M. Now, We define
+some endomorphisms given as follows: (see, [28,31,81,82,84,85])
+(η1 ∧ν η2)ι
+=
+ν(η2, ι)η1 − ν(η1, ι)η2,
+IR(η1, η2)
+=
+[∇η1, ∇η2] − ∇[η1,η2],
+IC(η1, η2)
+=
+IR(η1, η2) −
+1
+n − 2
+×
+�
+E η1 ∧g η2 + η1 ∧g E η2 −
+κ
+n − 1η1 ∧g η2
+�
+,
+IK(η1, η2)
+=
+IR(η1, η2) −
+1
+n − 2 (E η1 ∧g η2 + η1 ∧g E η2) ,
+IW(η1, η2)
+=
+IR(η1, η2) −
+κ
+n(n − 1) η1 ∧g η2,
+IP(η1, η2)
+=
+IR(η1, η2) −
+1
+n − 1 η1 ∧S η2,
+where ν is a (0,2) type symmetric tensor and E is the Ricci operator defined by S(η1, η2) =
+g(η1, E (η2)).
+Throughout this study we suppose that the smooth vector fields η, η1, η2 · · · ,
+ι, ι1, ι2 · · · ∈ χ(M). Now, corresponding to an endomorphism I(ι1, ι2), a (0, 4)-tensor I can be
+defined as
+I(ι1, ι2, ι3, ι4) = g(I(ι1, ι2)ι3, ι4).
+If the corresponding endomorphism I is replaced by IR (resp., IC, IK, IW and IP), the (0, 4)-
+tensor I turns into the Riemann curvature tensor R (resp., conformal curvature C, conharmonic
+curvature K, concircular curvature W and projective curvature P).
+On a (0, r)-tensor ζ, r ≥ 1, we simulate an endomorphism I(ι1, ι2) to define (0, r + 2)-type
+tensor I · ζ given as follows ( [25,86–89]):
+(I · ζ)(η1, η2, · · · , ηr, ι1, ι2)
+=
+(I(ι1, ι2)ζ)(η1, η2, · · · , ηr)
+=
+−ζ(I(ι1, ι2)η1, η2, · · · , ηr) − · · · − ζ(η1, η2, · · · , I(ι1, ι2)ηr).
+If we take I(ι1, ι2) = ι1 ∧ν ι2, then the (0, r + 2)-type tensor Q(ν, ζ) is known as Tachibana
+tensor defined as ( [89–92])
+Q(ν, ζ)(η1, η2, · · · , ηr, ι1, ι2) = ((ι1 ∧ν ι2)ζ)(η1, η2, · · · , ηr)
+= ν(ι1, ι1)ζ(ι2, η2, · · · , ηr) + · · · + ν(ι1, ιr)ζ(η1, η2, · · · , ι2)
+−ν(ι2, η1)ζ(ι1, η2, · · · , ηr) − · · · − ν(ι2, ηr)ζ(η1, η2, · · · , ι1).
+
+6
+A. A. SHAIKH, F. AHMED & B. R. DATTA
+In terms of the local coordinates, the tensor I · ζ and the Tachibana tensor Q(ν, ζ) can be
+rewritten as
+(I · ζ)b1b2...brαβ
+=
+−guv[Iαβb1vζub2...br + · · · + Iαβblvζb1b2...u],
+Q(ν, ζ)b1b2...brαβ
+=
+νb1βζαb2...br + · · · + νbrβζb1b2...α
+−
+νb1αζβb2...br − · · · − νbrαζb1b2...β.
+Definition 2.1. [10, 91, 93–99] If the condition I · ζ = fζQ(g, ζ) holds for a smooth scalar
+function fζ on M, i.e., the tensors I · ζ and Q(g, ζ) are linearly dependent on M, then M is
+called a ζ-pseudosymmetric manifold due to the tensor I. Also, if the tensors I · ζ and Q(S, ζ)
+are linearly dependent by the relation I · ζ = �fζQ(S, ζ) with a smooth scalar function �fζ on
+M, then M is called a Ricci generalized ζ-pseudosymmetric manifold due to the tensor I. In
+particular, a ζ-semisymmetric manifold due to the tensor I is defined by the relation I · ζ = 0.
+In the relation I · ζ = fζQ(g, ζ) if I = ζ = R, then M is simply called a pseudosymmetric
+manifold and for I = R and ζ = K (resp., S, P, W and C), it is called conharmonic (resp.,
+Ricci, projective, concircular and conformal) pseudosymmetric manifold.
+Similarly, various
+types of Ricci generalized pseudosymmetric and semisymmetric manifolds can be obtained by
+considering I and ζ as others curvature tensors.
+Again, if the Ricci tensor S is proportional to the metric tensor g on M, i.e., S = κ
+ng, then M is
+said to be an Einstein manifold [17], and the manifold M is called m-quasi-Einstein [85,100–102]
+if the rank of (S−αg) is m for some scalar α, and in this case Ricci tensor locally takes the form
+S = αg +βΓ⊗Γ with some scalars α, β and 1-form Γ. Also if α = 0, then the m-quasi-Einstein
+manifold turns into a Ricci simple manifold.
+We note that Morris-Thorne spacetime [103]
+and G¨odel spacetime [28] are Ricci simple manifolds, Robertson-Walker spacetime [26] and
+Siklos spacetime [29] are quasi-Einstein manifolds, Kantowski-Sachs spacetime [84] and Som-
+Raychaudhuri spacetime [25] are 2-quasi Einstein manifolds and Kaigorodov spacetime [29] is
+an Einstein manifold. For curvature properties of Robinson-Trautman metric, Melvin magnetic
+metric and generalized pp-wave metric, etc., we refer the reader to see [30,104–106].
+Definition 2.2. [107] The manifold M is said to be generalized quasi-Einstein if
+S = αg + βΘ ⊗ Θ + γ(Θ ⊗ Σ + Σ ⊗ Θ)
+holds for some smooth scalar functions α, β, γ and mutually orthogonal 1-forms Θ and Σ.
+
+GEOMETRICAL PROPERTIES OF A POINT-LIKE GLOBAL MONOPOLE SPACETIME
+7
+In the literature, there are other notions of generalized quasi-Einstein manifolds (see, Shaikh
+[100]). But throughout the paper we will consider the generalized quasi-Einstein manifold by
+Chaki [107] as given in Definition 2.2.
+Definition 2.3. ( [17, 85, 91, 108]) If S2, S3, S4, defined by Sλ+1(ι1, ι2) = Sλ(ι1, E ι2) with
+λ = 1, 2, 3, are linearly dependent by the relation
+ℵ1g + ℵ2S + ℵ3S2 + ℵ4S3 + S4 = 0
+(resp.,
+ℵ5g + ℵ6S + ℵ7S2 + S3 = 0 and ℵ8g + ℵ9S + S2 = 0)
+on M for some scalar functions ℵi (1 ≤ i ≤ 9), then M is called an Ein(4) (resp., Ein(3) and
+Ein(2)) manifold.
+It is noteworthy to mention that Melvin magnetic spacetime [109] and Siklos spacetime [29]
+are Ein(2) manifolds, while Lifshitz spacetime [110] and Som-Raychaudhuri spacetime [25] are
+Ein(3) manifolds.
+Definition 2.4. [25, 85, 89, 111–114] If the Riemann-Christoffel curvature tensor R can be
+expressed as a linear combination of the tensors g ∧g, g ∧S, S ∧S, g ∧S2, S ∧S2 and S2 ∧S2,
+given by
+R = (B1S2 + B2S + B3g) ∧ S2 + (B4S + B5g) ∧ S + B6(g ∧ g)
+(resp.,
+R = (B7S + B8g) ∧ S + B9g ∧ g)
+for some scalars Bi, 1 ≤ i ≤ 9, then M is called a generalized Roter type (resp., Roter
+type [86,114–117]) manifold.
+We mention that Vaidya-Bonner spacetime [106] and Lifshitz spacetime [110] are generalized
+Roter type manifold, and Nariai spacetime [105] and Melvin magnetic spacetime [109] are Roter
+type manifold.
+Definition 2.5. [11,12] A manifold M is called a weakly symmetric in the sense of Tam´assy
+and Binh if the covariant derivative of Riemann curvature tensor R can be expressed in the
+form
+(∇XR)(η1, η2, η3, η4)
+=
+Π(X) ⊗ R(η1, η2, η3, η4) + Φ(η4) ⊗ R(η1, η2, η3, X)
++
+Φ(η3) ⊗ R(η1, η2, X, η4) + Ψ(η2) ⊗ R(η1, X, η3, η4)
++
+Ψ(η1) ⊗ R(X, η2, η3, η4),
+
+8
+A. A. SHAIKH, F. AHMED & B. R. DATTA
+where Π, Φ and Ψ are associated 1-forms on M. In particular, if Π = 2Φ = 2Ψ, it is a Chaki
+pseudosymmetric manifold [8,9].
+Definition 2.6. The Ricci tensor of a manifold M is cyclic parallel (see, [118–121]) if
+(∇η1S)(η2, η3) + (∇η2S)(η3, η1) + (∇η3S)(η1, η2) = 0
+holds and Codazzi type if the Ricci tensor realizes the relation (see, [122,123])
+(∇η1S)(η2, η3) = (∇η2S)(η1, η3).
+We note that the Ricci tensor of G¨odel spacetime [28] is cyclic parallel and the Ricci tensor
+of (t − z)-type plane wave metric [104] is of Codazzi type.
+Definition 2.7. ( [87,90,111,124–127])
+Let ζ be a (0, 4)-type tensor on M. Then a symmetric (0,2)-type tensor ν corresponding to
+the endomorphism Iν is said to be ζ-compatible if
+ζ(Iνη1, ι, η2, η3) + ζ(Iνη2, ι, η3, η1) + ζ(Iνη3, ι, η1, η2) = 0,
+holds on M. Again, if ϕ ⊗ ϕ is ζ-compatible for an 1-form ϕ, then ϕ is called a ζ-compatible.
+Replacing ζ by the curvature tensor R (resp., C, W, P and K), the Riemann (resp., confor-
+mal, concircular, projective and conharmonic) compatibility of ν can be obtained.
+Definition 2.8. For a tensor I of type (0, 4), the curvature 2-forms Ωm
+(I)l [128] are called
+recurrent [18,129–131] if
+S
+η1,η2,η3(∇η1I)(η2, η3, ι, η) =
+S
+η1,η2,η3 σ(η1)I(η2, η3, ι, η)
+holds on M, where S is the cyclic sum over η1, η2, η3 and for a (0, 2) tensor field ν, the 1-forms
+∧(ν)l ( [128]) are called recurrent if (∇η1ν)(η2, ι) − (∇η2ν)(η1, ι) = σ(η1)ν(η2, ι) − σ(η2)ν(η1, ι),
+for some 1-form σ.
+Definition 2.9. ( [25,132,133]) For a (0, 4)-type tensor I if M admits the relation
+S
+η1,η2,η3 σ(η1) ⊗ I(η2, η3, ι, η) = 0,
+where S is the cyclic sum over η1, η2, η3 and L(M) is the vector space of all 1-forms with
+dimension ≥ 1, then M is called I-space by Venzi.
+Now, we give some definitions of geometrical symmetries, such as, motion, curvature collineation,
+curvature inheritance, Ricci collineation and Ricci inheritance, which are originated from the
+Lie derivatives of several tensors.
+
+GEOMETRICAL PROPERTIES OF A POINT-LIKE GLOBAL MONOPOLE SPACETIME
+9
+Definition 2.10. A manifold M admits motion with respect to some vector field η if £ηg = 0,
+where £η represents the Lie derivative with respect to η.
+In 1969, Katzin et al. [32, 33] defined the notion of curvature collineation by vanishing Lie
+derivative of the Riemann curvature tensor with respect to some vector field. Again, in 1992,
+Duggal [1] generalizes the concept of curvature collineation by introducing the notion of curva-
+ture inheritance.
+Definition 2.11. ( [1]) A manifold M possesses curvature inheritance if there is a vector field
+η which satisfies
+£η �R = λ �R,
+where λ is a scalar function and the (1,3)-type curvature tensor �R is associated with the (0,4)-
+type curvature tensor R by the relation R(ι1, ι2, ι3, ι4) = g( �R(ι1, ι2)ι3, ι4). In particular, if λ = 0,
+i.e., £η �R = 0, then it turns into curvature collineation [32,33].
+Definition 2.12. ( [1]) A manifold M admits Ricci inheritance if it realizes the relation
+£ηS = λS
+for some vector field η and scalar function λ. Further, if λ = 0, it turns into Ricci collineation
+(i.e., £ηS = 0).
+Again, recently Shaikh and Datta [2] introduced the notion of generalized curvature inheri-
+tance, which is defined as follows:
+Definition 2.13. ( [2]) A manifold M admits generalized curvature inheritance if there is a
+vector field η which possesses
+£ηR = λR + λ1g ∧ g + λ2g ∧ S + λ3S ∧ S,
+where λ, λ1, λ2, λ3 are the scalar functions. In particular, if λi = 0 for i = 1, 2, 3, then it M
+admits curvature inheritance. Further, if λ = 0 = λi for i = 1, 2, 3, then it turns into curvature
+collineation.
+In this paper, it is shown that the notions of curvature inheritance in Definition 2.11 and
+Definition 2.13 are not equivalent as shown by PGM spacetime.
+
+10
+A. A. SHAIKH, F. AHMED & B. R. DATTA
+3. Global monopole spacetime admitting geometric structures
+In static and spherical coordinates (t, r, θ, φ) the line element of PGM spacetime is given by
+(c = ℏ = G)
+ds2 = −dt2 + dr2
+α2 + r2 (dθ2 + sin2 θ dφ2),
+(3.1)
+which can be written as ds2 = gµν dxµ dxν, where g11 = −1, g22 =
+1
+α2, g33 = r2, g44 = r2 sin2 θ
+and gij = 0, i ̸= j for i, j = 1, 2, 3, 4.
+The non-vanishing components Γγ
+µν of the Christoffel symbols of the second kind are given by
+Γ3
+23 = 1
+r = Γ4
+24,
+Γ2
+33 = −rα2,
+Γ4
+34 = cot θ,
+Γ2
+44 = −rα2 sin2 θ,
+Γ3
+44 = − cos θ sin θ.
+The non-vanishing components (upto symmetry) of Riemann curvature tensor Rµνσλ and the
+Ricci tensor of Sµν are obtained as follows:
+R3434 = −r2(−1 + α2) sin2 θ;
+(3.2)
+S33 = −1 + α2, S44 = (−1 + α2) sin2 θ.
+(3.3)
+The scalar curvature κ is given by κ = 2(−1+α2)
+r2
+.
+This leads to the following:
+Proposition 3.1. The PGM spacetime (3.1) is neither Einstein nor quasi-Einstein manifold
+but
+(i) it is an Einstein manifold of degree 2, i.e., it fulfills the condition S2 = (−1+α2)
+r2
+S,
+(ii) it is 2-quasi Einstein and generalized quasi-Einstein manifold,
+(iii) its Riemann curvature can be decomposed by R =
+r2
+2(−1+α2)S ∧ S,
+(iv) its Ricci tensor is Riemann compatible, conharmonic compatible, concircular compatible,
+projective compatible and conformal compatible.
+Let V1 = ∇R and V2 = ∇S. Then the non-vanishing components (upto symmetry) of the
+covariant derivatives of the Riemann curvature tensor R and the Ricci tensor S are given by
+V1
+2334,4 = −r(−1 + α2) sin2 θ = −V1
+2434,3, V1
+3434,2 = 2r(−1 + α2) sin2 θ,
+V2
+23,3 = 1−α2
+r , V2
+24,4 = −(−1+α2) sin2 θ
+r
+, V2
+33,2 = 2−2α2
+r
+, V2
+44,2 = −2(−1+α2) sin2 θ
+r
+.
+
+GEOMETRICAL PROPERTIES OF A POINT-LIKE GLOBAL MONOPOLE SPACETIME
+11
+The components (upto symmetry) other than zero of the conformal curvature tensor C are
+given below:
+C1212 = −1+α2
+3r2α2 , C1313 = 1−α2
+6 , C1414 = −(−1+α2) sin2 θ
+6
+,
+C2323 = 1
+6 −
+1
+6α2, C2424 = (−1+α2) sin2 θ
+6α2
+, C3434 = −r2(−1+α2) sin2 θ
+3
+.
+The components (upto symmetry) other than zero of the projective curvature tensor P are
+shown as follows:
+P1331 = −1+α2
+3
+, P1441 = 1
+3(−1 + α2) sin2 θ, P2442 = −(−1+α2) sin2 θ
+3α2
+,
+P2332 = 1
+3(−1 +
+1
+α2), P3434 = −2
+3r2(−1 + α2) sin2 θ = −P3443.
+If V3 = ∇C, then the components other than zero of the covariant derivative of conformal
+curvature tensor C are given by
+V3
+1212,2 = −2(−1+α2)
+3r3α2
+, V3
+1213,3 = −1+α2
+2r
+, V3
+1214,4 = (−1+α2) sin2 θ
+2r
+,
+V3
+1313,2 = −1+α2
+3r
+, V3
+1414,2 = (−1+α2) sin2 θ
+3r
+, V3
+2323,2 = −−1+α2
+3rα2 , V3
+2424,2 = −(−1+α2) sin2 θ
+3rα2
+,
+V3
+2334,4 = −r(−1+α2) sin2 θ
+2
+= −V3
+2434,3, V3
+3434,2 = 2r(−1+α2) sin2 θ
+3
+.
+From the above tensor components, we can state the following:
+Proposition 3.2. The PGM spacetime (3.1) realizes the following:
+(i) its Ricci 1-forms are recurrent, i.e., ∇η1S(η2, η3) − ∇η2S(η1, η3) = ϑ(η1) ⊗ S(η2, η3) −
+ϑ(η2) ⊗ S(η1, η3) for ϑ =
+�
+0, −1
+r, 0, 0
+�
+,
+(ii) its conformal curvature C is recurrent for the 1-form
+�
+0, 1
+r, 0, 0
+�
+,
+(iii) it is a R-space by Venzi for {0, 0, 1, 1},
+(iv) it is Chaki pseudosymmetric for the 1-form Π =
+�
+0, −1
+r, 0, 0
+�
+,
+(v) it is semisymmetric as R · R = 0. Therefore, it is Ricci semisymmetric, conharmonic
+semisymmetric, projective semisymmetric, concircular semisymmetric and conformal
+semisymmetric, and hence it is also pseudosymmetric, Ricci pseudosymmetric, confor-
+mal pseudosymmetic in the sense of Deszcz.
+Let Z1 = C · R, Z2 = C · C, Z3 = P · C, H1 = Q(g, R), H2 = Q(g, C) and H3 = Q(S, C).
+Then the components other than zero of Z1, Z2, Z3, H1, H2 and H3 are computed as follows:
+Z1
+1434,13 = −(−1+α2)2 sin2 θ
+6
+= −Z1
+1334,14, Z1
+2434,23 = (−1+α2)2 sin2 θ
+6α2
+= −Z1
+2334,24;
+Z2
+1223,13 = −(−1+α2)2
+12r2α2
+= −Z2
+1213,23, Z2
+1434,13 = −(−1+α2)2 sin2 θ
+12
+= −Z2
+1334,14,
+Z2
+1224,14 = −(−1+α2)2 sin2 θ
+12r2α2
+= −Z2
+1214,24, Z2
+2434,23 = (−1+α2)2 sin2 θ
+12α2
+= −Z2
+2334,24;
+
+12
+A. A. SHAIKH, F. AHMED & B. R. DATTA
+Z3
+1223,13 = −(−1+α2)2
+9r2α2
+= −Z3
+1213,23, Z3
+1434,13 = −(−1+α2)2 sin2 θ
+18
+= −Z3
+1334,14,
+Z3
+1224,14 = −(−1+α2)2 sin2 θ
+9r2α2
+= −Z3
+1214,24, Z3
+2434,23 = (−1+α2)2 sin2 θ
+18α2
+= −Z3
+2334,24,
+Z3
+1223,31 = (−1+α2)2
+9r2α2
+= −Z3
+1213,32, Z3
+1434,31 = (−1+α2)2 sin2 θ
+18
+= −α2Z3
+2434,32,
+Z3
+1224,41 = (−1+α2)2 sin2 θ
+9r2α2
+= −Z3
+1214,42, Z3
+1334,41 = −(−1+α2)2 sin2 θ
+18
+= −α2Z3
+2334,42;
+H1
+1434,13 = r2(−1 + α2) sin2 θ = −H1
+1334,14, H1
+2434,23 = −r2(−1+α2) sin2 θ
+α2
+= −H1
+2334,24;
+H2
+1223,13 = 1
+2 −
+1
+2α2 = −H2
+1213,23, H2
+1434,13 = r2(−1+α2) sin2 θ
+2
+= −H2
+1334,14,
+H2
+1224,14 = (−1+α2) sin2 θ
+2α2
+= −H2
+1214,24, H2
+2434,23 = −r2(−1+α2) sin2 θ
+2α2
+= −H2
+2334,24;
+H3
+1223,13 = (−1+α2)2
+3r2α2
+= −H3
+1213,23, H3
+1434,13 = (−1+α2)2 sin2 θ
+6
+= −H3
+1334,14,
+H3
+1224,14 = (−1+α2)2 sin2 θ
+3r2α2
+= −H3
+1214,24, H3
+2434,23 = −(−1+α2)2 sin2 θ
+6α2
+= −H3
+2334,24.
+The above calculation of tensors leads to the following:
+Proposition 3.3. The PGM spacetime (3.1) satifies the pseudosymmetric type curvature con-
+ditions
+C · R = −(−1 + α2)
+6r2
+Q(g, R),
+C · C = −(−1 + α2)
+6r2
+Q(g, C) and P · C = −1
+3Q(S, C),
+i.e., it is pseudosymmetric due to conformal curvature tensor, pseudosymmetric Weyl curvature
+tensor and also Ricci generalized conformal peudosymmetric due to projective curvature tensor.
+The components other than zero of the concircular curvature tensor W of PGM spacetime
+are given by
+W1212 = −(−1+α2)
+6r2α2 , W1313 = −(−1+α2)
+6
+, W1414 = −(−1+α2) sin2 θ
+6
+,
+W2323 = 1
+6 −
+1
+6α2, W2424 = (−1+α2) sin2 θ
+6α2
+, W3434 = −5r2(−1+α2) sin2 θ
+6
+.
+If V4 = ∇W, then the components other than zero of the covariant derivative of concircular
+curvature tensor W are given by
+V4
+1212,2 = −1+α2
+3r3α2 , V4
+1313,2 = −1+α2
+3r
+, V4
+1414,2 = (−1+α2) sin2 θ
+3r
+,
+V4
+2323,2 = −(−1+α2)
+3rα2 , V4
+2334,4 = −r(−1 + α2) sin2 θ = −V4
+2434,3, V4
+2424,2 = −(−1+α2) sin2 θ
+3rα2
+,
+V4
+3434,2 = 5(−1+α2)r sin2 θ
+3
+.
+Let Z4 = W · R, Z5 = P · W, H4 = Q(S, W). Then the components other than zero of the
+tensor Z4, Z5, H4 are given as follows:
+Z4
+1434,13 = −(−1+α2)2 sin2 θ
+6
+= −Z4
+1334,14, Z4
+2434,23 = (−1+α2)2 sin2 θ
+6α2
+= −Z4
+2334,24;
+
+GEOMETRICAL PROPERTIES OF A POINT-LIKE GLOBAL MONOPOLE SPACETIME
+13
+Z5
+1223,13 = (−1+α2)2
+18r2α2
+= Z5
+1213,32, Z5
+1434,13 = −(−1+α2)2 sin2 θ
+18
+= Z5
+1334,41,
+Z5
+1224,14 = (−1+α2)2 sin2 θ
+18r2α2
+= −Z5
+1214,42, Z5
+1334,14 = (−1+α2)2 sin2 θ
+18
+= Z5
+1434,31,
+Z5
+1213,23 = −(−1+α2)2
+18r2α2
+= Z5
+1223,31, Z5
+2434,23 = (−1+α2)2 sin2 θ
+18α2
+= −Z5
+2334,42,
+Z5
+1214,24 = −(−1+α2)2 sin2 θ
+18r2α2
+= Z5
+1224,41, Z5
+2334,23 = −(−1+α2)2 sin2 θ
+18α2
+= Z5
+2434,32,
+H4
+1223,13 = −(−1+α2)2
+6r2α2
+= −H4
+1213,23, H4
+1434,13 = (−1+α2)2 sin2 θ
+6
+= −H4
+1334,14,
+H4
+1224,14 = −(−1+α2)2 sin2 θ
+6r2α2
+= −H4
+1214,24, H4
+2434,23 = −(−1+α2)2 sin2 θ
+6α2
+= −H4
+2334,24.
+From the above calculation of tensors we can infer the following:
+Proposition 3.4. The PGM spacetime fulfills the curvature conditions
+W · R = −(−1 + α2)
+6r2
+Q(g, R)
+and
+P · W = −1
+3Q(S, W)
+i.e., the spacetime is pseudosymmetric due to concircular curvature tensor and also Ricci gen-
+eralized concircular peudosymmetric due to projective curvature tensor.
+The components other than zero of the conharmonic curvature tensor K of PGM spacetime
+are given below:
+K1313 = 1−α2
+2 , K1414 = −(−1+α2) sin2 θ
+2
+,
+K2323 = 1
+2 −
+1
+2α2, K2424 = (−1+α2) sin2 θ
+2α2
+.
+If V5 = ∇K, then the components other than zero of the covariant derivative of conharmonic
+curvature tensor K are given by
+V5
+1213,3 = −1+α2
+2r
+, V5
+1214,4 = (−1+α2) sin2 θ
+2r
+= 1
+2V5
+1414,2, V5
+1313,2 = −1+α2
+r
+,
+V5
+2323,2 = −
+1+ 1
+α2
+r
+, V5
+2334,4 = −1
+2r(−1 + α2) sin2 θ = −V5
+2434,3, V5
+2424,2 = −(−1+α2) sin2 θ
+rα2
+.
+Let Z6 = K · R, Z7 = P · K, Z8 = P · P, H5 = Q(S, K) and H6 = Q(S, P). Then the
+components other than zero of the tensors Z6, Z7, Z8, H5 and H6 are computed as follows:
+Z6
+1434,13 = −(−1+α2)2 sin2 θ
+2
+= −Z6
+1334,14, Z6
+2434,23 = (−1+α2)2 sin2 θ
+2α2
+= −Z6
+2334,24;
+Z7
+1434,13 = −(−1+α2)2 sin2 θ
+6
+= −Z7
+1334,14, Z7
+2434,23 = (−1+α2)2 sin2 θ
+6α2
+= −Z7
+2334,24,
+Z7
+1434,31 = (−1+α2)2 sin2 θ
+6
+= −Z7
+1334,41, Z7
+2434,32 = −(−1+α2)2 sin2 θ
+6α2
+= −Z7
+2334,42;
+Z8
+1333,13 = (−1+α2)2
+9
+= −Z8
+1333,31, Z8
+1443,13 = (−1+α2)2 sin2 θ
+9
+= Z8
+3441,13 = Z8
+1334,14 = Z8
+3431,41,
+Z8
+1444,14 = (−1+α2)2 sin4 θ
+9
+= −Z8
+1444,41, Z8
+3431,14 = −(−1+α2)2 sin2 θ
+9
+= Z8
+1443,31 = Z8
+3441,31 = Z8
+1334,41,
+Z8
+2333,23 = −(−1+α2)2
+9α2
+= −Z8
+2333,32, Z8
+2443,23 = −(−1+α2)2 sin2 θ
+9α2
+= Z8
+3442,23 = Z8
+2334,24 = Z8
+3432,24,
+Z8
+2444,24 = −(−1+α2)2 sin4 θ
+9α2
+= −Z8
+2444,42, Z8
+3432,24 = (−1+α2)2 sin2 θ
+9α2
+= Z8
+2443,32 = Z8
+3442,32 = Z8
+2334,42;
+H5
+1434,13 = (−1+α2)2 sin2 θ
+2
+= −H5
+1334,14, H5
+2434,23 = −(−1+α2)2 sin2 θ
+2α2
+= −H5
+2334,24;
+
+14
+A. A. SHAIKH, F. AHMED & B. R. DATTA
+H6
+1333,13 = −(−1+α2)2
+3
+, H6
+1443,13 = −(−1+α2)2 sin2 θ
+3
+= H6
+3441,13 = H6
+1334,14 = −H6
+3431,14,
+H6
+1444,14 = −(−1+α2)2 sin4 θ
+3
+, H6
+2333,23 = (−1+α2)2
+3α2
+,
+H6
+2443,23 = (−1+α2)2 sin2 θ
+3α2
+= H6
+3442,23 = H6
+2334,24 = −H6
+3432,24,
+H6
+2444,24 = (−1+α2)2 sin4 θ
+3α2
+;
+The above computation of tensors leads to the following:
+Proposition 3.5. The PGM spacetime (3.1) fulfills the following pseudosymmetric type cur-
+vature conditions:
+K · R = −(−1 + α2)
+2r2
+Q(g, R),
+P · K = −1
+3Q(S, K) and
+P · P = −1
+3Q(S, P),
+i.e., the spacetime is pseudosymmetric due to conharmonic curvature tensor, Ricci generalized
+conharmonic peudosymmetric due to projective curvature tensor and Ricci generalized projective
+pseudosymmetric.
+From the above propositions, we can state that the PGM spacetime (3.1) admits the following
+curvature restricted geometric properties:
+Theorem 3.1. The PGM spacetime (3.1) reveals the following curvature properties:
+(i) it is pseudosymmetric due to conformal curvature tensor as C · R = −(−1+α2)
+6r2
+Q(g, R).
+Hence C · S = −(−1+α2)
+6r2
+Q(g, S), C · C = −(−1+α2)
+6r2
+Q(g, C) (i.e., pseudosymmetric Weyl
+conformal curvature tensor), C · W = −(−1+α2)
+6r2
+Q(g, W), C · P = −(−1+α2)
+6r2
+Q(g, P) and
+C · K = −(−1+α2)
+6r2
+Q(g, K),
+(ii) it realizes pseudosymmetry due to concircular curvature tensor as W·R = −(−1+α2)
+6r2
+Q(g, R).
+Hence W · S = −(−1+α2)
+6r2
+Q(g, S), W · C = −(−1+α2)
+6r2
+Q(g, C), W · W = −(−1+α2)
+6r2
+Q(g, W),
+W · P = −(−1+α2)
+6r2
+Q(g, P) and W · K = −(−1+α2)
+6r2
+Q(g, K),
+(iii) it admits pseudosymmetry due to conharmonic curvature tensor as K·R = −(−1+α2)
+2r2
+Q(g, R).
+Hence K · S = −(−1+α2)
+2r2
+Q(g, S), K · C = −(−1+α2)
+2r2
+Q(g, C), K · W = −(−1+α2)
+2r2
+Q(g, W),
+K · P = −(−1+α2)
+2r2
+Q(g, P) and K · K = −(−1+α2)
+2r2
+Q(g, K),
+(iv) it is Ricci generalized conformal pseudosymmetric due to projective curvature tensor as
+P · C = −1
+3Q(S, C). Hence P · P = −1
+3Q(S, P), P · W = −1
+3Q(S, W) and P · K =
+−1
+3Q(S, K),
+(v) it is a Venzi space for {0, 0, 1, 1}, hence its curvature 2-forms are recurrent,
+(vi) its conformal curvature 2-forms are recurrent for the 1-form
+�
+0, 1
+r, 0, 0
+�
+,
+(vii) its Ricci 1-forms are recurrent for the 1-form
+�
+0, −1
+r, 0, 0
+�
+,
+(viii) it is Chaki pseudosymmetric for the 1-form
+�
+0, −1
+r, 0, 0
+�
+,
+(ix) it is Chaki pseudo Ricci symmetric for the 1-form
+�
+0, −1
+r, 0, 0
+�
+,
+
+GEOMETRICAL PROPERTIES OF A POINT-LIKE GLOBAL MONOPOLE SPACETIME
+15
+(x) its Riemann curvature can be decomposed as R =
+r2
+2(−1+α2)S ∧ S. Hence, it is an Ein(2)
+spacetime with S2 = (−1+α2)
+r2
+S,
+(xi) it is a generalized quasi-Einstein manifold for α = 1
+2(r2+
+√
+4 + r4), β = 1
+2(r2−
+√
+4 + r4),
+γ = 1, Θ=
+�
+−
+√
+(2+r4+r2√
+4+r4)
+√
+2
+, 1, 0, 0
+�
+and Σ=
+�
+(r2−
+√
+4+r4)√
+(2+r4+r2√
+4+r4)
+2
+√
+2
+, 0, 0, 0
+�
+and
+(xii) its Ricci tensor is compatible for the curvature R, C, K, W and P.
+Corollary 3.1. The PGM spacetime is Chaki pseudosymmetric and hence it is weakly symmet-
+ric in the sense of Tam´assy and Binh for the associated 1-forms Π=
+�
+0, −2
+r, 0, 0
+�
+, Ψ=
+�
+0, −1
+r, 0, 0
+�
+and Φ=
+�
+0, −1
+r, 0, 0
+�
+.
+Remark 3.1. From the calculation with various tensors, it can be mentioned that the PGM
+spacetime (3.1) does not admit certain geometric structures, which are described as follows:
+(i) it is neither recurrent nor recurrent for C, P, W, K,
+(ii) its Ricci tensor is neither of Codazzi type nor cyclic parallel,
+(iii) it is not a Venzi space for C, P, W, K,
+(iv) it is not Ricci generalized pseudosymmetric (i.e., R · R and Q(S, R) are linearly inde-
+pendent),
+(v) it is neither Einstein nor quasi-Einstein and
+(vi) its curvature 2-forms are not recurrent for K, P and W.
+4. Energy momentum tensor of PGM spacetime
+From the EFE the stress energy momentum tensor T of a spacetime is given by
+T = 1
+τ
+�
+S −
+���
+2 − Λ
+�
+g
+�
+,
+where τ = 8πG
+c4 , c is the velocity of light in vacuum, Λ is the cosmological constant and G is
+the gravitational constant. The only non-vanishing components (upto symmetry) of the energy
+momentum tensor T are given by
+T11 = −1+α2
+8r2 , T22 = −(−1+α2)
+8r2α2 .
+Hence the non-vanishing components of covariant derivative of the energy momentum tensor
+T are given by
+T11,2 = −(−1+α2)
+4r3
+, T22,2 = (−1+α2)
+4r3α2 ,
+T23,3 = −(−1+α2)
+8r
+, T24,4 = −(−1+α2) sin2 θ
+8r
+.
+Let C · T = C1, W · T = W1, K · T = K1 and Q(g, T) = Q1. Then the components other
+than zero of the tensor C1, W1, K1 and Q1 are calculated as follows:
+C1
+1313 = (−1+α2)2
+48r2
+=
+1
+sin2 θC1
+1414,
+C1
+2323 = −(−1+α2)2
+48r2α2
+=
+1
+sin2 θC1
+2424;
+
+16
+A. A. SHAIKH, F. AHMED & B. R. DATTA
+W1
+1313 = (−1+α2)2
+48r2
+=
+1
+sin2 θW1
+1414,
+W1
+2323 = −(−1+α2)2
+48r2α2
+=
+1
+sin2 θW1
+2424;
+K1
+1313 = (−1+α2)2
+16r2
+=
+1
+sin2 θK1
+1414,
+K1
+2323 = −(−1+α2)2
+16r2α2
+=
+1
+sin2 θK1
+2424;
+Q1
+1313 = −(−1+α2)
+8
+=
+1
+sin2 θQ1
+1414,
+Q1
+2323 = (−1+α2)
+8α2
+=
+1
+sin2 θQ1
+2424.
+From the above calculations we can state the following:
+Theorem 4.1. The PGM spacetime (3.1) admits certain pseudosymmetric type curvature con-
+ditions for the energy momentum tensor T given as follows:
+(i) C ·T = −(−1+α2)
+6r2
+Q(g, T), i.e., the nature of the energy momentum tensor is conformally
+pseudosymmetric,
+(ii) W·T = −(−1+α2)
+6r2
+Q(g, T), i.e., the nature of the energy momentum tensor is concircularly
+pseudosymmetric,
+(iii) K · T = −(−1+α2)
+2r2
+Q(g, T), i.e., the nature of the energy momentum tensor is conhar-
+monically pseudosymmetric,
+(iv) the energy momentum tensor T is compatible for Riemann, projective, conharmonic,
+conformal and concircular curvature tensors.
+5. Curvature inheritance realized by PGM spacetime
+Let χ(M) be the Lie algebra of all smooth vector fields on an n-dimensional smooth semi-
+Riemannian manifold M. Then the Lie subalgebra K(M) of all Killing vector fields contains at
+most n(n+1)/2 linearly independent Killing vector fields. If M is of constant scalar curvature,
+K(M) consists of exactly n(n+1)/2 linearly independent vector fields. In Section 3 it has been
+shown that the PGM spacetime possesses non-constant scalar curvature κ given by 2(α2−1)/r2.
+In this section some Killing vector fields on PGM spacetime are exhibited and it is shown that
+the PGM spacetime admits curvature collineation, Ricci collineation and curvature inheritance
+for some non-Killing vector fields.
+Proposition 5.1. The PGM spacetime admits motion for the vector fields
+∂
+∂t and
+∂
+∂φ, i.e., the
+vector fields
+∂
+∂t and
+∂
+∂φ on PGM spacetime are Killing (£ ∂
+∂tg = 0 and £ ∂
+∂φ g = 0).
+Corollary 5.1. As
+∂
+∂t and
+∂
+∂φ are Killing vector fields, the vector field λ ∂
+∂t + µ ∂
+∂φ is also Killing
+for any constants λ and µ, i.e., £λ ∂
+∂t +µ ∂
+∂φg = 0 for all real numbers λ and µ.
+In this section we have considered the non-Killing vector fields
+∂
+∂r,
+∂
+∂θ, λ ∂
+∂r + µ ∂
+∂θ (λ and µ
+are constants), in the direction of which the Lie derivative of various tensors are computed.
+The non-zero components of the (1,3)-type curvature tensor �R are given as follows:
+�R3
+434 = (1 − α2) sin2 θ,
+�R3
+334 = −(1 − α2)
+(5.1)
+
+GEOMETRICAL PROPERTIES OF A POINT-LIKE GLOBAL MONOPOLE SPACETIME
+17
+and the non-vanishing components of the (0,4)-type curvature tensor R are given in (3.2). From
+the components of �R provided in (5.1), we have £ ∂
+∂r �R = 0, which leads to the following:
+Proposition 5.2. The PGM spacetime admits curvature collineation for the non-Killing vector
+field ξ =
+∂
+∂r as it possesses £ξ �R = 0.
+Again, Duggal (Theorem 3, [1]) proved that if a manifold admits curvature inheritance, it
+also realizes Ricci inheritance, and hence the above proposition implies the following:
+Corollary 5.2. The PGM spacetime realizes Ricci collineation for the non-Killing vector field
+ξ =
+∂
+∂r, i.e., £ξS = 0.
+For the non-Killing vector field η =
+∂
+∂θ, the non-vanishing components of £η �R and £ηS, are
+computed as follows:
+(£η �R)3
+434 = (1 − α2) sin 2θ = −(£η �R)3
+443,
+(5.2)
+(£ηS)44 = −(1 − α2) sin 2θ.
+(5.3)
+From the tensor components in (5.2) and (5.3), we note the following remarks:
+Remark 5.1. For the non-Killing vector field η =
+∂
+∂θ, there exists no scalar function λ such
+that the PGM spacetime possesses the relation £η �R = λ �R, i.e., with respect to the non-
+Killing vector field
+∂
+∂θ the PGM spacetime admits neither curvature collineation nor curvature
+inheritance (in sense of Definition 2.11) for the (1,3)-type curvature tensor �R.
+Remark 5.2. For the non-Killing vector field η =
+∂
+∂θ, there exists no scalar function λ such
+that the PGM spacetime realizes £ηS = ��S, i.e., with respect to the non-Killing vector field
+∂
+∂θ the PGM spacetime possesses neither Ricci collineation nor Ricci inheritance.
+Now, for the non-Killing vector fields ξ =
+∂
+∂r and η =
+∂
+∂θ, the non-vanishing components of
+£ξR and £ηR, are calculated as follows:
+(£ξR)3434 = (£ξR)4343 = −2(α2 − 1)r sin2 θ = −(£ξR)3443 = −(£ξR)4334,
+(£ηR)3434 = (£ηR)4343 = −(α2 − 1)r2 sin 2θ = −(£ηR)3443 = −(£ηR)4334,
+This leads to the following:
+Proposition 5.3. The PGM spacetime admits curvature inheritance (in the sense of Definition
+2.13) for the vector fields ξ =
+∂
+∂r and η =
+∂
+∂θ as it realizes the relations
+£ξR = 2
+rR
+and
+£ηR = 2 cot θ R.
+
+18
+A. A. SHAIKH, F. AHMED & B. R. DATTA
+If λ and µ are any non-zero constants, the non-zero components of £ξR for the vector field
+V = λ ∂
+∂r + µ ∂
+∂θ are given as follows:
+(£V R)3434 = (£V R)4343 = −2(α2 − 1)r sin θ(µr cos θ + λ sin θ) = −(£V R)3443 = −(£V R)4334.
+The above components of £λ ∂
+∂r +µ ∂
+∂θ R lead to the following:
+Proposition 5.4. For the vector field ξ = λ ∂
+∂r + µ ∂
+∂θ, the PGM spacetime possesses curvature
+inheritance (Definition 2.13) in the sense of Shaikh and Datta [2] as it satisfy the relation
+£ξR = 2(λ + µr cot θ)
+r
+R.
+Incorporating the above propositions and their consequences, we can state the following:
+Theorem 5.1. The PGM spacetime reveals the following symmetry properties:
+(i) it admits motion for the vector fields
+∂
+∂t and
+∂
+∂φ,
+(ii) if λ, µ are any non-zero constants, it possesses motion for the vector field λ ∂
+∂t + µ ∂
+∂φ,
+(iii) it admits curvature collineation (in the sense of Definition 2.11) and hence Ricci collineation
+with respect to the non-Killing vector field
+∂
+∂r, in fact, £ ∂
+∂r �R = 0 and £ ∂
+∂r S = 0,
+(iv) it admits curvature inheritance (in the sense of Definition 2.13) for the non-Killing
+vector fields
+∂
+∂r and
+∂
+∂θ, in fact,
+£ ∂
+∂r R = 2
+rR
+and
+£ ∂
+∂θ R = 2 cot θ R,
+(v) for any non-zero constants λ, µ it realizes curvature inheritance (in the sense of Defini-
+tion 2.13) for the non-Killing vector field λ ∂
+∂r + µ ∂
+∂θ, in fact,
+£λ ∂
+∂r +µ ∂
+∂θ R = 2(λ + µr cot θ)
+r
+R.
+Remark 5.3. It is interesting to note that the PGM spacetime with respect to the non-Killing
+vector field
+∂
+∂r, admits curvature collineation for the (1,3)-type curvature tensor �R. But it does
+not realize curvature collineation for the (0,4)-type curvature tensor R, whereas it possesses
+curvature inheritance for the (0,4)-type curvature tensor R. Also, we mention that with respect
+to the non-Killing vector field ∂
+∂θ, the PGM spacetime admits curvature inheritance for the (0,4)-
+type curvature tensor R, but it realizes neither curvature collineation nor curvature inheritance
+for the (1,3)-type curvature tensor �R. Hence it follows that the notion of curvature inheritance
+(resp., curvature collineation) for (1,3)-type curvature tensor (Definition 2.11) in the sense of
+Duggal [1] and the notion of curvature inheritance (resp., curvature collineation) for (0,4)-type
+curvature tensor (Definition 2.13) in the sense of Shaikh and Datta [2] are not equivalent.
+
+GEOMETRICAL PROPERTIES OF A POINT-LIKE GLOBAL MONOPOLE SPACETIME
+19
+6. Conclusions
+In this paper, we have investigated various curvature restricted geometric properties of PGM
+spacetime. It is proved that the spacetime is not Ricci generalized pseudosymmetric, but it
+admits various type of pseudosymmetric type curvature conditions, such as, pseudosymmetry
+due to Weyl conformal curvature tensor, pseudosymmetry due to conharmonic curvature tensor,
+Ricci generalized conformal pseudosymmetry due to projective curvature tensor. Also, it is
+proved that the spacetime is Einstein manifold of degree 2, generalized quasi-Einstein and
+2-quasi Einstein manifold (see, Theorem 3.1).
+Moreover, the energy momentum tensor of
+PGM spacetime satisfies several pseudosymmetric type curvature conditions, and both of its
+Ricci tensor and energy momentum tensor are compatible for Riemann, conformal, projective,
+conharmonic and concircular curvature (see, Theorem 4.1). Finally, it is shown that the PGM
+spacetime admits curvature collineation, Ricci collineation for the (1,3)-curvature tensor and
+curvature inheritance for the (0,4) curvature tensor with respect to certain non-Killing vector
+fields (see, Theorem 5.1). Also, some non-Killing vector fields are exhibited (see, Remark 5.3),
+with respect to which it is shown that the notions of curvature inheritance (also, of curvature
+collineation) for the (1,3)-type curvature tensor by Duggal [1] and for the (0,4)-type curvature
+tensor by Shaikh and Datta [2] are distinct (see, Remark 5.1 and Remark 5.3).
+Acknowledgment
+B. R. Datta is grateful to the Council of Scientific and Industrial Research (CSIR File No.:
+09/025(0253)/2018-EMR-I), Govt. of India, for the award of SRF (Senior Research Fellow-
+ship). All the algebraic computations of Section 3 to 5 are performed by a program in Wolfram
+Mathematica.
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+
+GEOMETRICAL PROPERTIES OF A POINT-LIKE GLOBAL MONOPOLE SPACETIME
+25
+1,3 Department of Mathematics,
+University of Burdwan, Golapbag,
+Burdwan-713104, West Bengal, India
+Email address: [email protected] ; [email protected]
+Email address: [email protected]
+2 Department of Physics,
+University of Science & Technology Meghalaya,
+Ri-Bhoi, Meghalaya-793101, India,
+Email address: [email protected] ; [email protected]
+

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+Ground state of superheavy elements with 120 ≤ Z ≤ 170: systematic study of the
+electron-correlation, Breit, and QED effects
+I. M. Savelyev,1 M. Y. Kaygorodov,1 Y. S. Kozhedub,1 A. V. Malyshev,1 I. I. Tupitsyn,1 and V. M. Shabaev1, 2
+1Department of Physics, St.
+Petersburg State University,
+7/9 Universitetskaya nab., 199034 St.
+Petersburg, Russia
+2National Research Centre “Kurchatov Institute” B.P. Konstantinov Petersburg
+Nuclear Physics Institute, Gatchina, Leningrad district 188300, Russia
+(Dated: January 5, 2023)
+For superheavy elements with atomic numbers 120 ≤ Z ≤ 170, the concept of the ground-state
+configuration is being reexamined. To this end, relativistic calculations of the electronic structure
+of the low-lying levels are carried out by means of the Dirac-Fock and configuration-interaction
+methods. The magnetic and retardation parts of the Breit interaction as well as the QED effects are
+taken into account. The influence of the relativistic, QED, and electron-electron correlation effects
+on the determination of the ground-state is analyzed.
+I.
+INTRODUCTION
+Mendeleev’s Periodic Table is an empirically sup-
+ported scheme which allows one to categorize the phys-
+ical and chemical properties of the elements by linking
+them with the rule of the successive occupation of the
+atomic orbitals. With increasing the atomic number Z,
+relativistic effects grow substantially. They can signifi-
+cantly alter various properties of the elements as com-
+pared with their lighter homologues. A classical example
+of the relativistic effects is the yellow color of gold [1–3].
+In the region of the superheavy elements (SHEs) belong-
+ing to the 7th period of the Table, the interplay between
+the relativistic and electron-electron correlation effects
+gives rise to the trends of deviation from the periodic
+law [4–8]. Some of these deviations, such as a different
+ground-state configuration of Lr (Z = 103) relative to
+its lighter homologue Lu (Z = 71), are confirmed exper-
+imentally [9], the others, like a positive electron affinity
+in Og (Z = 118), are predicted only theoretically [10–
+14]. Whether the 8th-period SHEs (with Z > 118) fit
+the Periodic Table and obey the periodic law is an open
+intriguing question. For instance, this period brings into
+play the previously-never-met 5g shell, and the corre-
+sponding electronic-structure feature no doubt must be
+presented in possible extensions of the Periodic Table. In
+addition, the influence of the quantum-electrodynamics
+(QED) effects on the SHE ground states has not been
+investigated systematically so far.
+A review of the current status of the problem and
+an extension of the Periodic Table up to Z = 172 based
+on the Dirac-Fock (DF) calculations, also known as the
+relativistic Hartree-Fock ones, are presented in Ref. [15],
+see also a very recent review [16]. This upper bound is
+determined by the fact that at higher values of Z the
+lowest (1s) Dirac level “dives” into the negative-energy
+continuum, provided a reasonable model for the nuclear
+charge distribution is employed [17–28].
+The key point for the description of the SHE prop-
+erties is determination of the ground-state configuration.
+The first attempts to advance the study of the SHEs be-
+yond the 7th period and to conjecture on their chemical
+and physical properties were undertaken in the 1970s [29–
+34]. Throughout the years, the issue was addressed by
+using various one-configuration methods [35, 36]. It soon
+became clear that in many cases the total energies of var-
+ious configurations differ very little from each other, and
+more sophisticated configuration-interaction calculations
+are necessary. Taking into account the correlation effects
+may lead to a change in the ground-state configuration.
+Some SHEs from the 8th period were studied within
+the many-configuration approaches [37–46]. The only pa-
+per that went beyond the one-configuration approxima-
+tion for all the 8th period elements is Ref. [47].
+The
+multiconfiguration Dirac–Fock method was used there to
+account for the interaction between energetically close
+configurations in the SHEs with Z ≤ 164. As a result, in
+about 50% of cases the ground-state configurations found
+in Ref. [47] differ from the ones obtained in the previous
+Dirac-Fock-Slater calculations [34], where no electron-
+electron correlation effects were considered. In Ref. [47],
+only the ground-state configurations were reported with-
+out any information on the level structure. The stability
+of the obtained results with respect to the accuracy of
+the correlation treatment was not discussed in that work
+as well.
+There are also some other issues that need to be clar-
+ified when discussing the SHE ground states. Does the
+one-configuration description remain valid for so com-
+plex systems possessing quite a large number of valence
+shells with, in particular, the 5g one among them? In
+other words, it seems reasonable that the ground-state
+arXiv:2301.01740v1  [physics.atom-ph]  4 Jan 2023
+
+2
+level of the SHEs generally can not be found without
+taking into account the electron-correlation effects, but
+is it possible, in principle, to describe with a sufficient
+accuracy this state using a single configuration?
+Can
+previously unaccounted QED effects change the ground
+state of the SHEs? The present paper aims to investigate
+these points in the framework of the relativistic Dirac-
+Fock method and the configuration-interaction method in
+the basis of the Dirac-Fock-Sturm orbitals [48–50]. In our
+calculations, in order to investigate a possible influence
+of the QED effects on the electronic structure and the
+ground-state configuration, both the methods are paired
+with the model-QED-operator approach [51, 52] which
+has been recently extended to the region 120 ≤ Z ≤ 170
+in Ref. [53].
+The paper is organized as follows.
+In Sec. II, an
+overview of the methods and their main implementation
+features are presented.
+The numerical details and the
+particular aspects of the calculation procedure are given
+in Sec. III. We discuss the obtained results and compare
+them with the previous calculations in Sec. IV. Sec. V
+concludes the paper with a brief summary.
+The atomic units are used throughout the paper.
+II.
+THEORETICAL APPROACHES AND
+METHODS
+In the present work, to explore the SHE ground
+states we use the Dirac-Fock (DF) and configuration-
+interaction (CI) methods. The issue is studied from the
+different perspectives using the one- as well as the many-
+configuration approaches.
+We consider a relativistic configuration K defined
+by the occupation numbers {qa}Ns
+a=1, where the index a
+enumerates the relativistic shells. For the list of the rela-
+tivistic shells (n1l1j1)q1 . . . (nNslNsjNs)qNs (n is the prin-
+cipal quantum number, l and j are the orbital and total
+angular momenta, respectively), we first determine the
+DF energy obtained within the relativistic-configuration-
+average (RAV) approximation, also known in the litera-
+ture as the jj-average one [54]. The corresponding total
+DF energy can be formally written as
+EDF
+RAV(K) = 1
+Nd
+�
+α∈K
+⟨α| ˆHDC|α⟩ ,
+(1)
+where α ≡ detα{ϕDF
+i
+} are the Slater determinant con-
+structed from the one-electron DF orbitals ϕDF
+i
+belong-
+ing to the configuration K, Nd is the number of these de-
+terminants, and ˆHDC ≡ ˆHD + ˆV C is the many-electron
+Dirac-Coulomb Hamiltonian. In the DF-RAV approxi-
+mation, the summation in the functional (1), which can
+be performed analytically, is equivalent to the summa-
+tion over the relativistic terms J of the configuration K
+taking into account their multiplicities [55, 56]. The DF
+equations can be derived by varying Eq. (1) with the ev-
+ident constraints due to the orthonormality conditions.
+The DF orbitals ϕDF
+i
+and the energy EDF
+RAV(K) are then
+obtained by solving the corresponding DF equations. We
+note that only the Coulomb-interaction operator ˆVC is in-
+cluded self-consistently into the DF equations.
+The Breit-interaction correction to the average en-
+ergy EDF
+RAV(K) of the configuration K is evaluated per-
+turbatively as
+∆EB
+RAV(K) = 1
+Nd
+�
+α∈K
+⟨α| ˆV B|α⟩ ,
+(2)
+where the determinants α are constructed from ϕDF
+i
+and
+ˆV B is the Breit-interaction operator.
+The QED cor-
+rections are treated using the model-QED-operator ap-
+proach developed in Refs. [51–53].
+We mention also
+some alternative methods to approximately account for
+the QED effects in many-electron systems, see, e.g.,
+Refs. [40, 44, 57–62]. For very recent applications and de-
+velopments of the model-QED-operator methods, which
+include the calculations of the QED effects in molecules,
+we refer to Refs. [63–67]. Like the Breit-interaction con-
+tribution, the QED correction in the RAV approximation
+is calculated as the relativistic-configuration-average ex-
+pectation value of the model-QED operator ˆV Q,
+∆EQ
+RAV(K) = 1
+Nd
+�
+α∈K
+⟨α| ˆV Q|α⟩ .
+(3)
+Finally, for the configuration K, we consider the average
+energy including the Breit correction and QED effects,
+EDCBQ
+RAV (K) = EDF
+RAV(K) + ∆EB
+RAV(K) + ∆EQ
+RAV(K) .
+(4)
+Hereinafter, this scheme is referred to as the DCBQ-RAV
+one.
+To resolve the level structure for the configura-
+tion K, one can try to find a single-configuration DF wave
+function and corresponding energy for the jj-coupling
+term using the energy functional constructed for a given
+value of J. However, if this approach is employed for an
+open-shell system possessing a complex level structure, it
+often proves impossible to adequately select a proper lin-
+ear combination of many-electron wave functions solely
+from symmetry considerations. It turns out that most
+of the SHEs have several open shells, and, therefore, this
+straightforward scheme may result in a wrong level struc-
+ture. In the present work, the level structure of the con-
+figuration K is resolved by means of the CI approach
+for the Dirac-Coulomb-Breit (DCB) Hamiltonian sup-
+plemented with the model-QED operator. The DCBQ
+
+3
+Hamiltonian reads as
+ˆHDCBQ = Λ+ �
+ˆHD + ˆV C + ˆV B + ˆV Q�
+Λ+ ,
+(5)
+where Λ+ is the product of one-electron projectors on
+the positive-energy solutions of the DF-RAV equations.
+The eigenvalue problem induced by the Hamiltonian (5)
+in the space of all the determinants α arising from the
+single relativistic configuration (SRC) K describes the
+splitting of the levels,
+ˆHDCBQΨSRC(K, JM) = EDCBQ
+SRC
+ΨSRC(K, JM) ,
+(6)
+where M means the projection of J.
+However, in case of energetically close configura-
+tions, their strong interaction and mixing may result in
+changes of the level structure. To account for the corre-
+lation effects, we consider a larger CI problem,
+ˆHDCBQΨCI(JM) = EDCBQ
+CI
+(J)ΨCI(JM) ,
+(7)
+in the space spanned by the Slater determinants gener-
+ated not only from the configuration K but also from a
+given list of relativistic configurations (see details in the
+next section). In the present calculations, the CI method
+in the basis of the Dirac-Fock-Sturm orbitals is used (CI-
+DFS) [48–50]. At the CI level, the Breit and QED cor-
+rections are taken into account, according to Eqs. (5) –
+(7), by including the corresponding terms into the Dirac-
+Coulomb Hamiltonian.
+Finally, we emphasize that the main goal of the
+present study is not to obtain the most accurate the-
+oretical predictions for the SHE energy-level structure,
+since in the cases of complex configurations this can be
+a separate extremely complicated task. Instead, we aim
+at a reliable determination of the ground-state levels and
+the configurations they belong to within a series of the
+adequate relativistic calculations.
+Having discussed the methods, let us proceed to de-
+tails of their application in the scope of the present work.
+III.
+DETAILS OF THE CALCULATIONS
+In the present work, all the calculations of the en-
+ergy levels of SHE are performed employing the Fermi
+model for the nuclear-charge distribution.
+The root-
+mean-square radius of the nucleus (in fm) is given by
+R =
+�
+3
+5Rsphere,
+Rsphere = 1.2A1/3,
+(8)
+where for the nucleon number A we use the approximate
+formula from Ref. [20],
+A = 0.00733Z2 + 1.30Z + 63.6.
+(9)
+The value of A obtained from Eq. (9) is rounded to the
+nearest integer. This choice of the nuclear size is consis-
+tent with the one made in Ref. [53].
+The SHE ground-state configuration is a priori un-
+known. As described in the previous section, we use three
+schemes to define the ground-state configuration. The
+first scheme is based on the DF-RAV method. Probing
+various configurations K, we determine the ground-state
+one as the configuration K∗ with the lowest average en-
+ergy EDCBQ
+RAV (K∗).
+For each Z, the list of all possible
+configuration-candidates for the role of the ground one
+is constructed by distributing Ne = Z − Ncore valence
+electrons over the valence relativistic shells. The number
+of the core-shell electrons Ncore and the list of the va-
+lence shells are presented in Table I. We consider as the
+valence shells those ones which, according to our prelim-
+inary calculations, are most likely to be occupied in the
+ground state.
+TABLE I. The list of the relativistic shells used to generate
+the relativistic configurations for which the DF-RAV equa-
+tions are solved.
+The absence of the lower index j in the
+column “Core shells” means that all relativistic orbitals cor-
+responding to the nonrelativistic one are fully occupied. The
+probe configurations are generated according to the following
+rule: the core shells are fully occupied, the Z − Ncore valence
+electrons are distributed over the valence shells. The nota-
+tions [Rn] and [Og] represent the closed-shell configurations
+of radon and oganesson atoms, respectively.
+Z
+Core shells
+Ncore
+Valence shells
+120 – 121
+[Rn]5f 6d 7s 7p1/2
+114
+7p3/2 8s 8p1/2 7d3/2
+122 – 123
+[Og]
+118
+8s 8p1/2 7d3/2 6f5/2
+124 – 133
+[Og]8s
+120
+8p1/2 6f5/2 7d3/2 5g7/2
+134 – 144
+[Og]8s 5g7/2
+128
+8p1/2 6f5/2 7d3/2 5g9/2
+145 – 146
+[Og]8s 8p1/2 5g7/2
+130
+6f5/2 7d3/2 5g9/2 9s
+147 – 155
+[Og]8s 8p1/2 5g
+140
+6f5/2 7d3/2 6f7/2 9s
+156 – 160
+[Og]8s 8p1/2 5g 6f5/2
+146
+6f7/2 7d3/2 9s 7d5/2
+161 – 165
+[Og]8s 8p1/2 5g 6f
+154
+7d3/2 7d5/2 9s 8p3/2
+166 – 168
+[Og]8s 8p1/2 5g 6f 7d3/2
+158
+7d5/2 9s 8p3/2 9p1/2
+169 – 170
+[Og]8s 8p1/2 5g 6f 7d
+164
+9s 8p3/2 9p1/2 7f5/2
+As an example, in Table II for the SHEs with Z =
+125 and Z = 140, seven configurations K with the lowest
+relativistic-configuration-average energies EDF
+RAV(K) are
+presented.
+For each configuration K given relative to
+the closed-shell one, the total DF-RAV energy and the
+energies obtained by successively adding the Breit and
+QED corrections are placed in the fourth, fifth, and sixth
+columns of Table II, respectively. The configurations are
+sorted in ascending order of the energy EDF
+RAV(K), i.e.,
+the first entry corresponds to the configuration with the
+lowest energy. The relative changes in the order of the
+configurations after addition of the corrections are indi-
+cated by the symbols ▽ (down) and △ (up). The absence
+
+4
+of these symbols corresponds to the case when the con-
+figuration position in the sorted list does not change.
+TABLE II. The relativistic-configuration-average energies for the SHEs with Z = 125 and Z = 140 evaluated for the configura-
+tions K using the DF method, EDF
+RAV, with the addition of the Breit-interaction correction, +∆EB
+RAV, and with the additional
+accounting for the QED correction, +∆EQ
+RAV, (a.u.). The configurations are presented relative to the closed-shell configuration
+and sorted in the ascending order of the energy EDF
+RAV. In the last two columns, the symbols ▽ (down) and △ (up) indicate the
+change in the order of the configurations relative to the order in the preceding column. The absence of these symbols means
+that there is no change in the position of the configuration. In particular, the QED effects do not affect the order.
+Z
+Closed Shells
+K
+EDF
+RAV
++∆EB
+RAV
++∆EQ
+RAV
+125
+[Og]8s2
+1/2
+8p1
+1/26f 4
+5/2
+−64 846.3116
+−64 718.7639 ▽3
+−64 627.5323
+8p1
+1/26f 3
+5/25g1
+7/2
+−64 846.3061
+−64 718.7781 △1
+−64 627.5496
+8p1
+1/27d1
+3/26f 2
+5/25g1
+7/2
+−64 846.3033
+−64 718.7718 △1
+−64 627.5421
+8p2
+1/26f 2
+5/25g1
+7/2
+−64 846.3007
+−64 718.7680 △1
+−64 627.5377
+7d1
+3/26f 4
+5/2
+−64 846.2769
+−64 718.7300 ▽1
+−64 627.4988
+8p1
+1/27d1
+3/26f 3
+5/2
+−64 846.2697
+−64 718.7182 ▽1
+−64 627.4854
+7d1
+3/26f 3
+5/25g1
+7/2
+−64 846.2680
+−64 718.7408 △2
+−64 627.5127
+140
+[Og]8s2
+1/28p2
+1/25g8
+7/2
+7d2
+3/26f 2
+5/25g6
+9/2
+−93 548.8504
+−93 320.7516 ▽1
+−93 179.0799
+7d1
+3/26f 3
+5/25g6
+9/2
+−93 548.8479
+−93 320.7539 △1
+−93 179.0837
+7d1
+3/26f 4
+5/25g5
+9/2
+−93 548.7928
+−93 320.6719 ▽1
+−93 178.9986
+7d3
+3/26f 1
+5/25g6
+9/2
+−93 548.7738
+−93 320.6697 ▽1
+−93 178.9965
+6f 4
+5/25g6
+9/2
+−93 548.7689
+−93 320.6791 △2
+−93 179.0103
+6f 5
+5/25g5
+9/2
+−93 548.7559
+−93 320.6398
+−93 178.9679
+7d2
+3/26f 3
+5/25g5
+9/2
+−93 548.7488
+−93 320.6229
+−93 178.9480
+As one can see from the examples given in Ta-
+ble II, the Breit-interaction correction can change the
+ground-state configuration within the RAV approxima-
+tion.
+For Z = 125, Table II demonstrates that when
+only the Coulomb interaction is taken into account the
+ground-state configuration is 8p1
+1/26f 4
+5/2. However, when
+we add the Breit-interaction correction evaluated accord-
+ing to Eq. (2), the configuration 8p1
+1/26f 3
+5/25g1
+7/2 turns
+out to be the lowest-energy one. All the other six con-
+figurations change their order as well.
+A similar con-
+figuration reordering occurs for the SHE with Z = 140
+as well.
+However, in the latter case some configu-
+rations retain their positions.
+Without the Breit in-
+teraction, the ground-state configuration for Z = 140
+is 7d2
+3/26f 2
+5/25g6
+9/2, but with this correction taken into ac-
+count the lowest-energy configuration is 7d1
+3/26f 3
+5/25g6
+9/2.
+For both demonstrated cases, the QED correction does
+not change the ground-state configuration and the sorted
+configuration list as a whole.
+After the configurations with the lowest average en-
+ergies EDCBQ
+RAV (K) are found, we explore their level struc-
+ture using the DCBQ-SRC approach. To determine the
+ground-state level, we choose seven configurations with
+the lowest DCBQ-RAV energies and for each level belong-
+ing to these configurations solve the DCBQ-SRC prob-
+lem (6). The configuration with the lowest DCBQ-SRC
+energy is considered to be the ground-state configura-
+tion in the DCBQ-SRC approach, and the corresponding
+level gives the ground-state level. Since for most SHEs, a
+few selected DCBQ-RAV energies are close to each other,
+we bear in mind that the DCBQ-SRC consideration can
+change the ground-state configuration.
+-64627.65
+-64627.60
+-64627.55
+-64627.50
+-64627.45
+-64627.40
+8p1
+1/26f3
+5/25g1
+7/2
+RAV
+8p1
+1/27d1
+3/26f2
+5/25g1
+7/2
+RAV
+J = 13/2
+J = 17/2
+EDCBQ (a.u.)
+FIG. 1.
+Relativistic-configuration-average energies EDCBQ
+RAV
+calculated
+for
+the
+configurations
+8p1
+1/26f 3
+5/25g1
+7/2
+(left)
+and 8p1
+1/27d1
+3/26f 2
+5/25g1
+7/2 (right) of the SHE with Z = 125
+and all the possible levels which contribute to these average
+energies. For the lowest DCBQ-SRC levels the total angular
+momentum quantum numbers J are shown.
+
+5
+The last statement is illustrated in Fig. 1, where the
+average energies EDCBQ
+RAV (K) of the SHE with Z = 125
+are presented for two configurations having the lowest
+average energies: K = 8p1
+1/26f 3
+5/25g1
+7/2 (left) and ˜K =
+8p1
+1/27d1
+3/26f 2
+5/25g1
+7/2 (right). For each configuration, we
+show all the levels EDCBQ
+SRC
+(K, J) contributing to the
+relativistic-configuration-average energy.
+For the low-
+est DCBQ-SRC levels, the total angular momenta J
+are presented.
+One can see, that the average energy
+of the configuration K is lower than the energy of the
+configuration ˜K by about EDCBQ
+RAV ( ˜K) − EDCBQ
+RAV (K) =
+0.0065 a.u. (see also Table II). However, the lowest
+level J = 17/2 of the configuration ˜K lies lower than
+the lowest level J = 13/2 of the configuration K by
+about EDCBQ
+SRC
+(K, 13/2) − EDCBQ
+SRC
+( ˜K, 17/2) = 0.0245 a.u.
+This kind of behavior is not a specific feature of the SHE
+with Z = 125, but rather an example of some general
+trend observed for many other SHEs as well.
+The DCBQ-RAV and DCBQ-SRC schemes are one-
+configuration approaches and therefore they do not take
+into account the electron-correlation effects i.e. mixing of
+the different configurations. Below, we discuss the influ-
+ence of the electron-electron correlations on the order of
+the SHE lowest levels. To this end, for each Z we perform
+the independent CI-DFS calculations for seven reference
+configurations with the lowest DCBQ-RAV energies se-
+lected at the previous stage. For each configuration, we
+evaluate the three lowest levels and determine their to-
+tal angular momenta J. The level with the lowest en-
+ergy EDCBQ
+CI
+(J) is the ground-state level. Within this ap-
+proach, the ground-state level depends on how accurately
+we solve the CI problem. We investigate the dependence
+of the level order on the number of virtual orbitals by
+performing two different CI-DFS calculations referred to
+as “DCBQ-CI1” and “DCBQ-CI2”. In both schemes, the
+single (S) and double (D) excitations from the reference
+configuration determined at the DCBQ-RAV stage are
+considered. The DCBQ-CI1 scheme can be thought of
+as a CI problem with a relatively small number of the
+virtual orbitals, whereas the DCBQ-CI2 scheme includes
+more virtual orbitals. The list of the active and virtual
+orbitals used in both CI calculations is presented in Ta-
+ble III. The occupied orbitals, which are not mentioned
+in the table, are treated as the frozen core. The active
+orbitals as well as the virtual orbitals, which are involved
+in the most important configurations, are taken as the
+solutions of the DF equations, whereas the other virtual
+orbitals are obtained from the solutions of the DFS equa-
+tions.
+TABLE III. The list of the active and virtual relativistic shells employed in the “DCBQ-CI1” and “DCBQ-CI2” calculations.
+The absence of the lower index indicates that for a given l both the relativistic orbitals with l − 1/2 and l + 1/2 are included
+in the CI problem.
+Z
+Active valence shells
+Virtual shells
+DCBQ-CI1
+DCBQ-CI2
+120 – 121
+7p3/28s1/28p1/27d3/2
+8p3/27d5/2
+9s1/29p8d6f5g
+122 – 123
+8s1/28p1/27d3/26f5/2
+8p3/27d5/26f7/2
+9s1/29p8d7f5g
+124 – 133
+8p1/26f5/27d3/25g7/2
+8p3/27d5/26f7/25g9/2
+9s1/29p8d7f6g
+134 – 144
+8p1/26f5/27d3/25g9/2
+8p3/26f7/27d5/2
+9s1/29p1/28d3/27f5/26g7/2
+145 – 146
+6f5/27d3/25g9/29s1/2
+8p3/26f7/27d5/2
+10s1/29p8d7f6g
+147 – 155
+6f5/27d3/26f7/29s1/2
+8p3/27d5/2
+10s1/29p8d7f6g
+156 – 160
+6f7/27d3/29s1/27d5/2
+8p3/2
+10s1/29p8d7f6g
+161 – 165
+7d3/27d5/29s1/28p3/2
+10s1/29p8d7f6g
+166 – 168
+7d3/27d5/29s1/28p3/29p1/2
+9p3/2
+10s1/210p8d7f6g
+169 – 170
+7d5/29s1/28p3/29p1/27f5/2
+7f7/29p3/2
+10s1/210p8d8f6g
+The correlation effects can result in exotic scenarios
+for the level structure. An example is presented in Ta-
+ble IV, where the lowest levels with J = 0, 1, 2, 3 and
+the related configurations of the SHE with Z = 168
+are shown.
+One can see that within the DCBQ-SRC
+approximation, when the electron-electron correlations
+are neglected, the level with J = 1, EDCBQ
+SRC
+(J = 1) =
+−202904.8013 a.u., lies below the levels with J = 2,
+EDCBQ
+SRC
+(J = 2)
+=
+−202904.7871 a.u. and J
+=
+0,
+EDCBQ
+SRC
+(J = 0) = −202904.7789 a.u.
+If we account
+for the electronic correlations by means of the DCBQ-
+CI1 scheme, the level with J = 1 rises above the level
+with J = 2 and exceeds it by about 0.025 a.u. When we
+improve the description of the electron-electron correla-
+tions using the DCBQ-CI2 scheme, the level with J = 1
+becomes the lowest one again and the level with J = 0
+falls below the level with J = 2. In this case, the dif-
+ference between the levels with J = 0 and J = 1 consti-
+tutes about 0.01 a.u. This demonstrates that with the
+
+6
+improvement of the correlation treatment the changes in
+the ground-state levels may occur. More importantly, the
+dominant configurations of levels with J = 0, 1, 2 do not
+coincide with each other. Therefore, one can expect that
+not only levels which belong to the same configuration
+can interchange, but also the rearrangements involving
+the levels of other configurations are possible.
+TABLE IV. The lowest-level energies EDCBQ(J) for levels J = 0, 1, 2, 3 calculated by means of the DCBQ-SRC, DCBQ-CI1,
+and DCBQ-CI2 schemes for the SHE with Z = 168 (a.u.). For the DCBQ-SRC values, the configurations of these levels are
+presented. For the DCBQ-CI1 and DCBQ-CI2 results, the configurations contributing with the weights of at least 0.05 are
+given.
+J
+DCBQ-SRC
+DCBQ-CI1
+DCBQ-CI2
+K
+EDCBQ
+SRC
+(K, J)
+K
+EDCBQ
+CI
+(J)
+K
+EDCBQ
+CI
+(J)
+0
+9s2
+1/29p2
+1/2
+−202904.7789
+0.87 · 9s2
+1/29p2
+1/2+
+0.07 · 9s2
+1/28p2
+3/2+
+0.05 · 8p2
+3/29p2
+1/2
+−202904.8206
+0.91 · 9s2
+1/29p2
+1/2
+−202904.9561
+1
+9s2
+1/28p1
+3/29p1
+1/2 −202904.8013
+0.92 · 9s2
+1/28p1
+3/29p1
+1/2+
+0.07 · 8p3
+3/29p1
+1/2
+−202904.8247
+0.94 · 9s2
+1/28p1
+3/29p1
+1/2
+−202904.9652
+2
+9s1
+1/28p2
+3/29p1
+1/2 −202904.7871
+0.55 · 9s1
+1/28p2
+3/29p1
+1/2+
+0.24 · 9s1
+1/28p1
+3/29p2
+1/2+
+0.20 · 9s1
+1/28p3
+3/2
+−202904.8491
+0.81 · 9s1
+1/28p2
+3/29p1
+1/2+
+0.09 · 9s1
+1/28p1
+3/29p2
+1/2+
+0.07 · 9s1
+1/28p3
+3/2
+−202904.9542
+3
+9s1
+1/28p2
+3/29p1
+1/2 −202904.7319
+0.99 · 9s1
+1/28p2
+3/29p1
+1/2
+−202904.7385
+0.96 · 9s1
+1/28p2
+3/29p1
+1/2
+−202904.8835
+IV.
+RESULTS
+To begin with, we calculate the average energies of
+the configurations including the Breit interaction and
+QED effects within the DCBQ-RAV method.
+For all
+the SHEs in the range 120 ≤ Z ≤ 170, the configura-
+tions with the lowest average energy are presented in Ta-
+ble V. The ground-state configurations are shown rela-
+tive to the closed-shell configurations given in the sec-
+ond column.
+Our results (the column “DCBQ-RAV”)
+are compared with the results of Ref. [34], where the
+calculations were performed using the Dirac-Fock-Slater
+method for Z up to 173, the results of Ref. [29], where
+the DF method was employed for Z up to 131, and, fi-
+nally, the results of Ref. [35], where the calculations were
+based on the relativistic density functional theory and
+121 ≤ Z ≤ 131 were considered. We note that in Ref. [29]
+the Gaunt-interaction correction was taken into account
+perturbatively, while the Coulomb electron-electron in-
+teraction was treated self-consistently within the density
+functional theory.
+Moreover, since the non-relativistic
+notations were used in Refs. [29, 35], we retain them in
+Table V without any changes.
+Within the DCBQ-RAV approximation, the general
+trend of the occupation rule with the growth of Z is as
+follows: after the 8s1/2 shell is filled for Z = 120, the elec-
+trons begin to occupy the 8p1/2, 7d3/2, and 6f5/2 shells.
+According to our calculations, the first electron in the
+5g7/2 shell appears for Z = 125.
+Starting from this
+atomic number, the 5g series begins.
+For Z = 126,
+the 8p1/2 shell becomes the closed one.
+With a fur-
+ther increase for Z, the 5g7/2 and 5g9/2 shells are subse-
+quently occupied with the electrons. The partially occu-
+pied 7d3/2 and 6f5/2 shells remain the valence ones and
+their occupation numbers exhibit little changes. This 5g-
+occupation process is completed at Z = 144 which has
+the fully occupied 5g7/2 and 5g9/2 shells and partially
+occupied 7d3/2 and 6f5/2 shells.
+After both the relativistic 5g shells are filled, the
+6f shells begin to be occupied quite systematically: 6f5/2
+and 6f7/2 become fully occupied for Z
+= 148 and
+Z = 155, respectively.
+The 7d3/2 shell is unoccupied
+for Z = 155 and it remains partially occupied for the
+other Z in this region becoming closed only for Z = 158.
+Finally, the SHE with Z
+= 164 has the configura-
+tion [Og]8s2
+1/28p2
+1/25g8
+7/25g10
+9/26f 6
+5/26f 8
+7/27d4
+3/27d6
+5/2 with
+all the relativistic shells being occupied.
+Notably, the 8p3/2 shell is not filled along the se-
+quence Z = 120 − 166 that is due to a large spin-orbital
+splitting of the 8p shell.
+The DF-RAV calculations
+
+7
+TABLE V. The ground-state configurations of superheavy elements with atomic numbers 120 ≤ Z ≤ 170 evaluated within the
+DCBQ-RAV method, taking into account the Breit interaction and QED effects. The configurations are shown relative to the
+closed-shell ones, which are presented in the column “Closed shells”. [Og] corresponds to the configuration of the oganesson
+atom. The succeeding records in this column show the relativistic orbitals which have to be added to the previous ones to
+obtain the closed-shell configurations for heavier atoms. The results of the present work, “DCBQ-RAV”, are compared with the
+results of Refs. [29, 34, 35].
+Z
+Closed shells
+DCBQ-RAV
+Fricke and Soff [34]
+Mann and Waber [29]
+Umemoto and Saito [35]
+120
+[Og]
+8s2
+1/2
+8s2
+1/2
+8s2
+121
++8s2
+1/2
+8p1
+1/2
+8p1
+1/2
+8p1
+8p1
+122
+8p1
+1/2 7d1
+3/2
+8p1
+1/2 7d1
+3/2
+8p17d1
+8p2
+123
+8p1
+1/2 7d1
+3/2 6f 1
+5/2
+8p1
+1/2 7d1
+3/2 6f 1
+5/2
+8p17d16f 1
+8p17d16f 1
+124
+8p1
+1/2 6f 3
+5/2
+8p1
+1/2 6f 3
+5/2
+8p16f 3
+8p26f 2
+125
+8p1
+1/2 6f 3
+5/2 5g1
+7/2
+8p1
+1/2 6f 3
+5/2 5g1
+7/2
+8p16f 35g1
+8p16f 4
+126
+8p2
+1/2 6f 2
+5/2 5g2
+7/2
+8p1
+1/2 7d1
+3/2 6f 2
+5/2 5g2
+7/2
+8p26f 25g2
+8p16f 45g1
+127
++8p2
+1/2
+6f 2
+5/2 5g3
+7/2
+6f 2
+5/2 5g3
+7/2
+8p26f 25g3
+8p26f 35g2
+128
+6f 2
+5/2 5g4
+7/2
+6f 2
+5/2 5g4
+7/2
+8p26f 25g4
+8p26f 35g3
+129
+6f 2
+5/2 5g5
+7/2
+6f 2
+5/2 5g5
+7/2
+8p26f 25g5
+8p26f 35g4
+130
+6f 2
+5/2 5g6
+7/2
+6f 2
+5/2 5g6
+7/2
+8p26f 25g6
+8p26f 35g5
+131
+6f 2
+5/2 5g7
+7/2
+6f 2
+5/2 5g7
+7/2
+8p26f 25g7
+8p26f 35g6
+132
+7d1
+3/2 6f 1
+5/2 5g8
+7/2
+6f 2
+5/2 5g8
+7/2
+133
++5g8
+7/2
+6f 3
+5/2
+6f 3
+5/2
+134
+6f 4
+5/2
+6f 4
+5/2
+135
+6f 4
+5/2 5g1
+9/2
+6f 4
+5/2 5g1
+9/2
+136
+6f 4
+5/2 5g2
+9/2
+6f 4
+5/2 5g2
+9/2
+137
+7d1
+3/2 6f 3
+5/2 5g3
+9/2
+7d1
+3/2 6f 3
+5/2 5g3
+9/2
+138
+7d1
+3/2 6f 3
+5/2 5g4
+9/2
+7d1
+3/2 6f 3
+5/2 5g4
+9/2
+139
+7d1
+3/2 6f 3
+5/2 5g5
+9/2
+7d2
+3/2 6f 2
+5/2 5g5
+9/2
+140
+7d1
+3/2 6f 3
+5/2 5g6
+9/2
+7d1
+3/2 6f 3
+5/2 5g6
+9/2
+141
+7d2
+3/2 6f 2
+5/2 5g7
+9/2
+7d2
+3/2 6f 2
+5/2 5g7
+9/2
+142
+7d2
+3/2 6f 2
+5/2 5g8
+9/2
+7d2
+3/2 6f 2
+5/2 5g8
+9/2
+143
+7d2
+3/2 6f 2
+5/2 5g9
+9/2
+7d2
+3/2 6f 2
+5/2 5g9
+9/2
+144
+7d3
+3/2 6f 1
+5/2 5g10
+9/2
+7d3
+3/2 6f 1
+5/2 5g10
+9/2
+145
++5g10
+9/2
+7d2
+3/2 6f 3
+5/2
+7d2
+3/2 6f 3
+5/2
+146
+7d2
+3/2 6f 4
+5/2
+7d2
+3/2 6f 4
+5/2
+147
+7d2
+3/2 6f 5
+5/2
+7d2
+3/2 6f 5
+5/2
+148
+7d2
+3/2 6f 6
+5/2
+7d2
+3/2 6f 6
+5/2
+149
++6f 6
+5/2
+7d3
+3/2
+7d3
+3/2
+150
+7d4
+3/2
+7d4
+3/2
+151
+7d3
+3/2 6f 2
+7/2
+7d3
+3/2 6f 2
+7/2
+152
+7d3
+3/2 6f 3
+7/2
+7d3
+3/2 6f 3
+7/2
+153
+7d2
+3/2 6f 5
+7/2
+7d2
+3/2 6f 5
+7/2
+154
+7d2
+3/2 6f 6
+7/2
+7d2
+3/2 6f 6
+7/2
+155
+9s1
+1/2 6f 8
+7/2
+7d2
+3/2 6f 7
+7/2
+156
++6f 8
+7/2
+7d2
+3/2
+7d2
+3/2
+157
+7d3
+3/2
+7d3
+3/2
+158
+7d4
+3/2
+7d4
+3/2
+159
++7d4
+3/2
+9s1
+1/2
+9s1
+1/2
+160
+7d1
+5/2 9s1
+1/2
+7d1
+5/2 9s1
+1/2
+161
+7d2
+5/2 9s1
+1/2
+7d2
+5/2 9s1
+1/2
+162
+7d4
+5/2
+7d4
+5/2
+163
+7d5
+5/2
+7d5
+5/2
+164
+7d6
+5/2
+7d6
+5/2
+165
++7d6
+5/2
+9s1
+1/2
+9s1
+1/2
+166
+9s2
+1/2
+9s2
+1/2
+167
++9s2
+1/2
+8p1
+3/2
+9p1
+1/2
+168
+8p1
+3/2 9p1
+1/2
+9p2
+1/2
+169
+8p1
+3/2 9p2
+1/2
+8p1
+3/2 9p2
+1/2
+170
+8p2
+3/2 9p2
+1/2
+8p2
+3/2 9p2
+1/2
+
+8
+for Z = 166 shows that the spin-orbital splitting of the
+8p shell is about 80 eV. The first 8p3/2 electron appears in
+the SHE with Z = 167, after the 9s1/2 shell becomes the
+closed one. However, for Z = 168 the 9p1/2 shell turns
+out to be more energetically advantageous than the 8p3/2
+one. The 8p3/2 shell is not fully occupied even for the last
+considered SHE with Z = 170. Another remarkable ob-
+servation found in our DCBQ-RAV calculations is that
+for Z = 155 the configuration with the valence 9s1/2 elec-
+tron turn out to be more energetically beneficial than
+the configuration with 7d3/2 electrons, whereas its neigh-
+bors — Z = 154 and Z = 156 — have two electrons in
+the 7d3/2 shell. Next time the 9s1/2 electron appears in
+the series Z = 159 – 161, and, finally, the 9s1/2 shell es-
+tablishes on the regular basis starting from the element
+with Z = 165.
+Throughout the calculations we found that the
+ground-state configuration may change due to the Breit-
+interaction corrections, see the discussion in Sec. III. This
+kind of changes is observed for Z = 125 and Z = 140 and
+never occurs for the other values of Z. Concerning the
+QED corrections, we deduce that within the DCBQ-RAV
+approach they never change the ground-state configura-
+tion for the SHEs under consideration. In general, our
+DCBQ-RAV ground-state configurations coincide with
+the DF-RAV ones, obtained without the Breit and QED
+corrections, in all cases except for Z = 125 and Z = 140.
+Our DCBQ-RAV results for Z = 120 – 131 are in
+full agreement with the results of Ref. [29]. The obtained
+ground-state configurations agree with the related results
+of Ref. [35] for the SHEs with 121 ≤ Z ≤ 123, but differ
+for the other available values of Z. Our DF-RAV results
+differ from the results of Ref. [34] obtained without the
+Breit and QED corrections for eight of the considered
+SHEs, namely, for Z = 125, 126, 132, 139, 140, 155, 167,
+and 168.
+For Z = 155, our results predict that 9s1/2
+electron unexpectedly jumps in the 6f7/2-occupation se-
+quence.
+Perhaps the configuration with the valence
+9s1/2 electron was not considered in Ref. [34].
+As
+for the other discrepancies, they seem to have a non-
+systematical nature and might be due to the Slater
+exchange-interaction approximation used in Ref. [34].
+Nevertheless, the real reasons for these deviations remain
+unclear to us.
+Proceeding with the analysis, we are aimed at find-
+ing the configuration of the lowest-energy level.
+We
+employ the CI-DFS method using the one-configuration
+scheme DCBQ-SRC as well as the more elaborated
+schemes DCBQ-CI1 and DCBQ-CI2. As in the DCBQ-
+RAV approach, in these schemes the Breit and QED cor-
+rections are included, however, in the non-perturbative
+manner. The thorough description of the CI calculations
+is presented in Sec. III.
+In Table VI, we give the levels with the lowest DCBQ
+energies for the SHEs with 120 ≤ Z ≤ 170 obtained in
+three considered CI schemes. Additionally, the quantum
+numbers J of these levels are listed. For the DCBQ-SRC
+results, the configurations which the ground-state levels
+belong to are given. The DCBQ-CI1 and DCBQ-CI2 re-
+sults include the electron-electron correlation effects. For
+these data, we list the configurations contributing to the
+ground levels with the weight of at least 0.05. Following
+the structure of the previous table, the configurations are
+given relative to the closed-shell ones. The obtained re-
+sults are compared with the results of the previous mul-
+ticonfiguration Dirac-Fock calculations [47].
+The non-
+relativistic notations of Ref. [47] are retained.
+A comparison of Tables V and VI shows that the
+configurations of the ground levels obtained within the
+SRC approach differ from the RAV ground-state config-
+urations in almost half of the cases (for convenience, the
+corresponding values of Z are typed in a bold font). This
+result indicates the complex level structure of the SHEs,
+which is discussed in details for Z = 125 in Sec. III.
+The subsequent discussion consists of two parts. At
+first, we identify general trends for the results of the
+many-configuration calculations and compare them with
+the single-configuration ones. We note, that the DCBQ-
+CI1 and DCBQ-CI2 results are, in general, not much
+different. Therefore, in this part we often drop the in-
+dices "1" or "2" and use the generalized designation
+"DCBQ-CI" for the many-configuration calculations. In
+the second part, we compare the results obtained by the
+configuration-interaction method CI1 and CI2 with each
+other.
+Exactly as the DCBQ-RAV scheme predicts, our
+many-configuration DCBQ-CI results detect the first ap-
+pearance of the 5g electron in the ground state for the
+SHE with Z = 125. However, in contrast to the DCBQ-
+RAV results, the DCBQ-CI schemes predict that the 5g
+shell becomes closed for Z = 145 instead of Z = 144. In
+the range Z = 125 – 132, the many-configuration calcula-
+tions reveal that the dominant configurations of the ob-
+tained ground-state levels in all eight cases differ from the
+ones obtained within the DCBQ-RAV approach. More-
+over, a configuration mixing in the ground states takes
+place for some SHEs in range Z = 125 – 145 as the 5g
+shells are gradually occupied.
+In most of the consid-
+ered cases, the weights of the dominant configurations
+lie in range 0.80 – 0.90. The configurations with the dif-
+ferent occupation numbers for the 8p1/2, 7d3/2, 6f5/2,
+and 5g7/2,9/2 shells are admixed. The DCBQ-CI schemes
+show that starting from Z = 130 the dominant con-
+figuration of the ground-state level has the 8p1/2 shell
+fully occupied. However, the configurations with the par-
+tially occupied 8p1/2 shell contribute (with the weights
+
+9
+about 0.05 or higher) to these levels up approximately
+Z ≈ 135 − 137.
+A mixture of the configurations with the partially oc-
+cupied 7d3/2 and 6f5/2 shells occurs also in the range Z =
+147 – 151. The situation with the ground states becomes
+more clear starting from the SHE with Z = 152, when the
+6f5/2 shell turns out to be fully occupied. Up to Z = 165,
+the weights of the dominant configurations are larger
+than 0.90, and in most of the cases the dominant con-
+figurations of the ground-state levels coincide with the
+ground-state configurations obtained within the DCBQ-
+RAV approach.
+In particular, the fact that the SHE
+with Z = 164 possesses the ground-state configuration
+with all the relativistic shells closed is confirmed by the
+more elaborated methods.
+The SHEs with Z = 168
+and Z = 169 demonstrate within the DCBQ-CI1 scheme
+poorly resolved dominant configurations of the ground-
+state levels. For instance, the DCBQ-CI1 weight of the
+dominant configuration for Z = 168 is only 0.55, which
+was not the case even for the SHEs with the open 5g7/2
+and 5g9/2 shells. However, increasing the number of the
+active orbitals remedies the situation, and for Z = 168
+the DCBQ-CI2 scheme yields the dominant-configuration
+weight equal to 0.92. This is due to the fact that the levels
+interchange, see the corresponding discussion in Sec. III.
+The
+overall
+trends
+obtained
+in
+our
+many-
+configuration calculations are the following.
+First,
+the configurations which have the lowest levels within
+the DCBQ-SRC approach are the dominant ones con-
+tributing to the ground-state levels within the DCBQ-CI
+approach in about 80% of the considered cases. Second,
+the ground-state levels obtained without the electronic
+correlations using the DCBQ-SRC scheme in about 75%
+of the cases coincide with the ones obtained by means
+of the DCBQ-CI approach. The deviations are mainly
+concentrated in the range Z = 131 − 138, where the
+5g7/2 and 5g9/2 shells are partially occupied and strong
+interaction between several configurations takes place.
+The simultaneous change of the dominant configuration
+and the ground-state level when passing from the DCBQ-
+SRC to the DCBQ-CI method occurs for, e.g., Z = 131.
+In this case the first scheme yields JSRC
+=
+25/2
+of the configuration KSRC
+=
+8p1
+1/27d1
+3/26f 3
+5/25g6
+7/2,
+whereas the second scheme predicts the lowest level
+to be JCI
+= 21/2 with the dominant configuration
+being KCI
+=
+8p2
+1/26f 3
+5/25g6
+7/2
+with the weight of
+about 0.82 − 0.85.
+Now we proceed to contrast of the two DCBQ-CI
+schemes results. Compared to the DCBQ-CI1 data, the
+more accurate treatment of the electron-electron corre-
+lations by means of the DCBQ-CI2 approach results in
+the changes of the ground-state level in 4 of 51 cases.
+In 3 of these 4 cases, the configuration which gives the
+maximum contribution to the ground-state level changes
+as well. These SHEs, which need particular attention,
+are the ones with Z = 130, 137, 143, and 168. For in-
+stance, for Z = 130, the level J = 14 of the dom-
+inant configuration K = 8p1
+1/27d1
+3/26f 3
+5/25g5
+7/2 is pre-
+dicted to be the ground-state one in both DCBQ-SRC
+and DCBQ-CI1 schemes: KSRC = KCI1 = K.
+How-
+ever, the electronic correlations evaluated by means of the
+DCBQ-CI2 scheme change the ground-state level, and it
+becomes J = 12 with the dominant configuration be-
+ing KCI2 = 8p2
+1/26f 3
+5/25g5
+7/2 ̸= K.
+We compare our DCBQ-CI2 results with the only
+available systematic many-configuration calculations of
+Ref. [47] where the Breit interaction was taken into ac-
+count as well. Since the quantum numbers J which char-
+acterize the ground-state levels are not presented in that
+paper, we are able to compare only the configurations.
+We found a disagreement in the configurations contribut-
+ing to the ground states for the SHEs with Z = 123 – 128,
+Z = 130, Z = 136 – 137, Z = 143 – 144, Z = 152 – 156,
+and Z = 163. It is difficult to reveal a possible reason
+of the discrepancy due to the lack of the computational
+details given in Ref. [47].
+The changes of the ground-state levels in transition
+from the DCBQ-CI1 to the DCBQ-CI2 calculations and
+the deviations from the previous results raise the fol-
+lowing question: can hypothetical larger CI calculations
+change the obtained ground states as the DCBQ-CI2
+scheme changes the ground states in comparison with
+the DCBQ-CI1 one?
+A comprehensive answer can be
+given only within the scope of the corresponding large-
+scale CI calculations.
+However, to get an idea of the
+cases for which the correlation effects may change the
+dominant configuration of the ground-state level, we in-
+vestigate the behavior of the energy difference between
+the ground-state level and the closest level belonging to a
+different dominant configuration for both our DCBQ-CI
+calculations. This study allows us to determine whether
+the ground-state level is in some sense isolated from levels
+of other configurations and whether the electronic corre-
+lations break down this isolation. The absolute values of
+the corresponding differences are presented in Table VII.
+The SHE with Z = 120 is omitted in Table VII, since it
+possesses the ground-state configuration K∗ = [Og]8s2
+1/2
+that causes no doubt.
+As it is seen from Table VII, some SHEs have
+a clear separation of the ground-state level from lev-
+els of other configurations which almost does not de-
+pend on the correlation-treatment scheme. For instance,
+for Z = 121, the separations of the levels in the DCBQ-
+
+10
+TABLE VII. The absolute values of the energy difference between the ground-state level of the dominant configuration K∗
+and the closest excited level belonging to the dominant configuration which is different from K∗ for the SHEs in the range
+121 ≤ Z ≤ 170 (a.u.). The results are presented for the DCBQ-CI1 and DCBQ-CI2 schemes.
+Z
+CI1
+CI2
+Z
+CI1
+CI2
+Z
+CI1
+CI2
+Z
+CI1
+CI2
+Z
+CI1
+CI2
+121
+0.0399
+0.0378
+131
+0.0047
+0.0069
+141
+0.0195
+0.0106
+151
+0.0176
+0.0114
+161
+0.0103
+0.0059
+122
+0.0127
+0.0107
+132
+0.0083
+0.0118
+142
+0.0143
+0.0046
+152
+0.0059
+0.0122
+162
+0.0094
+0.0016
+123
+0.0299
+0.0351
+133
+0.0254
+0.0204
+143
+0.0015
+0.0116
+153
+0.0062
+0.0129
+163
+0.0111
+0.0176
+124
+0.0050
+0.0046
+134
+0.0313
+0.0270
+144
+0.0013
+0.0129
+154
+0.0107
+0.0008
+164
+0.0492
+0.0542
+125
+0.0062
+0.0057
+135
+0.0165
+0.0125
+145
+0.0539
+0.0376
+155
+0.0108
+0.0257
+165
+0.0424
+0.0443
+126
+0.0106
+0.0126
+136
+0.0062
+0.0095
+146
+0.0352
+0.0273
+156
+0.0136
+0.0110
+166
+0.0153
+0.0305
+127
+0.0070
+0.0089
+137
+0.0011
+0.0043
+147
+0.0398
+0.0349
+157
+0.0208
+0.0212
+167
+0.0022
+0.0096
+128
+0.0116
+0.0123
+138
+0.0248
+0.0303
+148
+0.0715
+0.0760
+158
+0.0529
+0.0555
+168
+0.0244
+0.0091
+129
+0.0071
+0.0034
+139
+0.0370
+0.0322
+149
+0.0583
+0.0568
+159
+0.0568
+0.0581
+169
+0.0066
+0.0087
+130
+0.0022
+0.0009
+140
+0.0237
+0.0088
+150
+0.0295
+0.0300
+160
+0.0308
+0.0285
+170
+0.0287
+0.0324
+CI1 and DCBQ-CI2 schemes constitute 0.0399 a.u.
+and 0.0378 a.u., respectively. In other cases, the ground-
+state level becomes more isolated from the levels of other
+configurations as the correlation treatment is improved.
+So, for Z = 155, the separation increases from 0.0108 a.u.
+in the DCBQ-CI1 scheme to 0.0257 a.u. in the DCBQ-
+CI2 one. In spite of this, it is difficult to formulate for
+all the systems under consideration a reliable criteria to
+determine if the dominant configuration contributing to
+the ground-state level does change with increase of the
+configuration-space. From this point of view, the most
+suspicious SHEs are the ones which have a small (less
+than a few thousandths of a.u.) separation between the
+considered levels within the DCBQ-CI2 scheme. In addi-
+tion, we also include in this category the cases where the
+separation between the levels significantly decreases in
+the DCBQ-CI2 scheme compared to the DCBQ-CI1 re-
+sults. Analyzing the data in Table VII, we consider the
+SHEs with Z = 129, 130, 137, 140, 142, 154, 161, 162, 168,
+and 169 as those that can possibly have a different domi-
+nant configuration of the ground-state level than the one
+obtained within the DCBQ-CI2 scheme. These elements
+have to be studied within the more elaborated electron-
+correlation calculations.
+V.
+CONCLUSION
+In the scope of the present paper, we have performed
+the extensive relativistic study of the ground states of
+the superheavy elements in the range 120 ≤ Z ≤ 170.
+The Breit interaction is rigorously taken into in the cal-
+culations, and the QED effects are considered within
+the model-QED-operator approach [51–53]. The ground-
+state configurations are first determined by means of
+the Dirac-Fock method in the relativistic-configuration-
+average approximation. It is deduced that the QED ef-
+fects can not change the ground-state configuration in
+contrast to the Breit interaction.
+To resolve the level structure of the configurations,
+the ground-state levels are found using the configuration-
+interaction method in the basis of the Dirac-Fock-Sturm
+orbitals. We study the general trends in the order of oc-
+cupation of orbitals in the SHE. We obtain that in spite of
+the complex electronic structure of the considered SHEs,
+the ground-state levels have distinct dominant configura-
+tions with the weights exceeding 0.85. Finally, we demon-
+strate that the electron-correlation effects can change the
+dominant configuration of the ground-state level.
+For
+some SHEs, the large-scale calculations are needed in or-
+der to more reliably determine the ground states and
+the structure of low-lying energy levels.
+Nevertheless,
+the ground-state configurations of the SHEs obtained in
+the present work within the many-configuration approach
+can be used as a solid basis for accurate calculations of
+various atomic properties of these elements as well as
+to examine the role of the electron-electron correlations,
+QED, and relativistic effects on the Periodic Law.
+VI.
+ACKNOWLEDGEMENTS
+We thank Yu. Ts. Oganessian for stimulating dis-
+cussions. Valuable conversations with E. Eliav, V. Per-
+shina, and A. V. Titov are also gratefully acknowl-
+edged. The work was supported by the Ministry of Sci-
+ence and Higher Education of the Russian Federation
+within Grant No. 075-10-2020-117.
+
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+
+13
+TABLE VI: The levels with lowest total energies, the main configurations contributing to them, and the total angular momenta
+J evaluated by means of the DCBQ-SRC, DCBQ-CI1, and DCBQ-CI2 schemes for the SHEs with the atomic numbers 120 ≤
+Z ≤ 170. For the DCBQ-CI1 and DCBQ-CI2 results, the configurations with weights of at least 0.05 are presented. The
+configurations obtained are given relative to the closed-shell configurations listed in the column “Closed shells”. In the first
+column, the values of Z typed in the bold font indicate the cases when the ground-state configurations obtained within the
+DCBQ-RAV and DCBQ-SRC methods differ. In addition, the following notations around Z are adopted to assist the reader
+in navigation through the table. The underline “
+” means that the ground-state JSRC level evaluated using the DCBQ-SRC
+approach differs from the JCI1 one calculated by means of the DCBQ-CI1 approach, JSRC ̸= JCI1. The left vertical line “|
+”
+signalizes that the configuration KSRC which the ground-state SRC level belongs to differs from the dominant configuration KCI1
+of the ground-state CI1 level, KSRC ̸= KCI1. The overline “
+” represents the fact that JCI1 ̸= JCI2. Finally, the right vertical
+line “
+|” stands for the case KCI1 ̸= KCI2. The obtained ground-state levels are compared with the results of Ref. [47]. The
+non-relativistic notation of Ref. [47] are retained.
+Z
+Closed
+DCBQ-SRC
+JSRC
+DCBQ-CI1
+JCI1
+DCBQ-CI2
+JCI2
+Ref. [47]
+Shells
+120
+[Og]
+8s2
+1/2
+0
+0.94 · 8s2
+1/2
+0
+0.93 · 8s2
+1/2
+0
+8s2
+121
+8s2
+1/28p1
+1/2
+1/2
+0.92 · 8s2
+1/28p1
+1/2
+1/2
+0.91 · 8s2
+1/28p1
+1/2
+1/2
+8s28p1
+122
+8s2
+1/28p1
+1/27d1
+3/2
+2
+0.85 · 8s2
+1/28p1
+1/27d1
+3/2+
+0.09 · 8s1
+1/28p1
+1/27d2
+3/2
+2
+0.84 · 8s2
+1/28p1
+1/27d1
+3/2+
+0.07 · 8s1
+1/28p1
+1/27d2
+3/2
+2
+7d18p1
+123
+8s2
+1/28p1
+1/27d1
+3/26f1
+5/2
+9/2
+0.83 · 8s2
+1/28p1
+1/27d1
+3/26f1
+5/2+
+0.14 · 8s1
+1/28p1
+1/27d2
+3/26f1
+5/2
+9/2
+0.82 · 8s2
+1/28p1
+1/27d1
+3/26f1
+5/2+
+0.09 · 8s1
+1/28p1
+1/27d2
+3/26f1
+5/2
+9/2
+6f28p1
+|124
++8s2
+8p1
+1/27d1
+3/26f2
+5/2
+6
+0.83 · 8p1
+1/26f3
+5/2+
+0.10 · 8p1
+1/26f2
+5/26f1
+7/2
+5
+0.85 · 8p1
+1/26f3
+5/2+
+0.07 · 8p1
+1/26f2
+5/26f1
+7/2
+5
+6f28p2
+125
+8p1
+1/27d1
+3/26f2
+5/25g1
+7/2 17/2
+0.96 · 8p1
+1/27d1
+3/26f2
+5/25g1
+7/2
+17/2
+0.94 · 8p1
+1/27d1
+3/26f2
+5/25g1
+7/2
+17/2
+0.81 · 5g16f28p2+
+0.17 · 5g16f17d28p1+
+0.02 · 6f37d18p1
+126
+8p1
+1/27d1
+3/26f2
+5/25g2
+7/2
+10
+0.93 · 8p1
+1/27d1
+3/26f2
+5/25g2
+7/2
+10
+0.92 · 8p1
+1/27d1
+3/26f2
+5/25g2
+7/2
+10
+0.998 · 5g26f38p1+
+0.002 · 5g26f28p2
+127
+8p1
+1/27d1
+3/26f2
+5/25g3
+7/2 27/2
+0.95 · 8p1
+1/27d1
+3/26f2
+5/25g3
+7/2
+27/2
+0.94 · 8p1
+1/27d1
+3/26f2
+5/25g3
+7/2
+27/2
+0.88 · 5g36f28p2+
+0.12 · 5g36f17d28p1
+128
+8p1
+1/27d1
+3/26f2
+5/25g4
+7/2
+14
+0.80 · 8p1
+1/27d1
+3/26f2
+5/25g4
+7/2+
+0.14 · 8p1
+1/27d1
+3/26f2
+5/25g3
+7/25g1
+9/2
+14
+0.81 · 8p1
+1/27d1
+3/26f2
+5/25g4
+7/2+
+0.13 · 8p1
+1/27d1
+3/26f2
+5/25g3
+7/25g1
+9/2
+14
+0.88 · 5g46f28p2+
+0.12 · 5g46f17d28p1
+129
+8p1
+1/27d1
+3/26f3
+5/25g4
+7/2 29/2
+0.93 · 8p1
+1/27d1
+3/26f3
+5/25g4
+7/2+
+0.06 · 8p1
+1/27d1
+3/26f2
+5/25g4
+7/26f1
+7/2
+29/2
+0.94 · 8p1
+1/27d1
+3/26f3
+5/25g4
+7/2
+29/2
+5g46f37d18p1
+130|
+8p1
+1/27d1
+3/26f3
+5/25g5
+7/2
+14
+0.92 · 8p1
+1/27d1
+3/26f3
+5/25g5
+7/2+
+0.06 · 8p1
+1/27d1
+3/26f2
+5/25g5
+7/26f1
+7/2
+14
+0.85 · 8p2
+1/26f3
+5/25g5
+7/2+
+0.06 · 7d2
+3/26f3
+5/25g5
+7/2
+12
+5g56f37d18p1
+|131
+8p1
+1/27d1
+3/26f3
+5/25g6
+7/2 25/2
+0.82 · 8p2
+1/26f3
+5/25g6
+7/2+
+0.06 · 7d2
+3/26f3
+5/25g6
+7/2+
+0.06 · 8p1
+1/27d2
+3/26f2
+5/25g6
+7/2+
+0.05 · 8p2
+1/26f2
+5/25g6
+7/26f1
+7/2
+21/2
+0.85 · 8p2
+1/26f3
+5/25g6
+7/2+
+0.05 · 7d2
+3/26f3
+5/25g6
+7/2
+21/2
+0.86 · 5g66f38p2+
+0.14 · 5g66f27d28p1
+|132
+8p1
+1/27d1
+3/26f3
+5/25g7
+7/2
+10
+0.84 · 8p2
+1/26f3
+5/25g7
+7/2+
+0.08 · 8p1
+1/27d2
+3/26f2
+5/25g7
+7/2+
+0.06 · 7d2
+3/26f3
+5/25g7
+7/2
+6
+0.87 · 8p2
+1/26f3
+5/25g7
+7/2+
+0.05 · 7d2
+3/26f3
+5/25g7
+7/2
+6
+5g76f38p2
+
+14
+TABLE VI. (Continuation.)
+Z
+Closed
+DCBQ-SRC
+JSRC
+DCBQ-CI1
+JCI1
+DCBQ-CI2
+JCI2
+Ref. [47]
+Shells
+|133
+8p1
+1/27d1
+3/26f3
+5/25g8
+7/2 13/2
+0.83 · 8p2
+1/26f3
+5/25g8
+7/2+
+0.09 · 8p1
+1/27d2
+3/26f2
+5/25g8
+7/2
+9/2
+0.87 · 8p2
+1/26f3
+5/25g8
+7/2
+9/2
+5g86f38p2
+|134 +5g8
+7/2
+8p1
+1/27d1
+3/26f4
+5/2
+6
+0.82 · 8p2
+1/26f4
+5/2+
+0.07 · 8p1
+1/27d2
+3/26f3
+5/2+
+0.06 · 7d2
+3/26f4
+5/2
+4
+0.84 · 8p2
+1/26f4
+5/2+
+0.05 · 8p1
+1/27d2
+3/26f3
+5/2+
+0.05 · 7d2
+3/26f4
+5/2
+4
+5g86f48p2
+|135
+8p2
+1/27d1
+3/26f3
+5/25g1
+9/2 13/2
+0.82 · 8p2
+1/26f4
+5/25g1
+9/2+
+0.08 · 8p1
+1/27d2
+3/26f3
+5/25g1
+9/2+
+0.05 · 7d2
+3/26f4
+5/25g1
+9/2
+5/2
+0.85 · 8p2
+1/26f4
+5/25g1
+9/2+
+0.06 · 8p1
+1/27d2
+3/26f3
+5/25g1
+9/2
+5/2
+5g96f48p2
+136
+8p2
+1/27d1
+3/26f3
+5/25g2
+9/2
+3
+0.91 · 8p2
+1/27d1
+3/26f3
+5/25g2
+9/2+
+0.05 · 7d3
+3/26f3
+5/25g2
+9/2
+4
+0.89 · 8p2
+1/27d1
+3/26f3
+5/25g2
+9/2
+4
+5g106f48p2
+|137|
+8p2
+1/27d1
+3/26f3
+5/25g3
+9/2 19/2
+0.80 · 8p2
+1/26f4
+5/25g3
+9/2+
+0.11 · 8p1
+1/27d2
+3/26f3
+5/25g3
+9/2
+13/2
+0.89 · 8p2
+1/27d1
+3/26f3
+5/25g3
+9/2
+17/2
+5g116f48p2
+138
++8p2
+1/2
+7d1
+3/26f3
+5/25g4
+9/2
+6
+0.91 · 7d1
+3/26f3
+5/25g4
+9/2
+7
+0.89 · 7d1
+3/26f3
+5/25g4
+9/2
+7
+5g126f37d18p2
+139
+7d1
+3/26f3
+5/25g5
+9/2
+13/2
+0.92 · 7d1
+3/26f3
+5/25g5
+9/2
+13/2
+0.91 · 7d1
+3/26f3
+5/25g5
+9/2
+13/2
+5g136f37d18p2
+|140
+7d2
+3/26f2
+5/25g6
+9/2
+6
+0.90 · 7d1
+3/26f3
+5/25g6
+9/2+
+0.05 · 7d1
+3/26f2
+5/25g6
+9/26f1
+7/2
+6
+0.90 · 7d1
+3/26f3
+5/25g6
+9/2
+6
+5g146f37d18p2
+141
+7d2
+3/26f2
+5/25g7
+9/2
+9/2
+0.93 · 7d2
+3/26f2
+5/25g7
+9/2
+9/2
+0.91 · 7d2
+3/26f2
+5/25g7
+9/2
+9/2
+5g156f27d28p2
+142
+7d2
+3/26f2
+5/25g8
+9/2
+2
+0.91 · 7d2
+3/26f2
+5/25g8
+9/2
+2
+0.89 · 7d2
+3/26f2
+5/25g8
+9/2
+2
+5g166f27d28p2
+143
+7d2
+3/26f3
+5/25g8
+9/2
+5/2
+0.93 · 7d2
+3/26f3
+5/25g8
+9/2
+3/2
+0.92 · 7d2
+3/26f3
+5/25g8
+9/2
+1/2
+5g176f27d28p2
+144
+7d2
+3/26f3
+5/25g9
+9/2
+7
+0.96 · 7d2
+3/26f3
+5/25g9
+9/2
+7
+0.94 · 7d2
+3/26f3
+5/25g9
+9/2
+7
+0.95 · 5g176f27d38p2+
+0.05 · 5g176f47d18p2
+145
+7d2
+3/26f3
+5/25g10
+9/2
+13/2
+0.96 · 7d2
+3/26f3
+5/25g10
+9/2
+13/2
+0.93 · 7d2
+3/26f3
+5/25g10
+9/2
+13/2
+5g186f37d28p2
+146
++5g10
+9/2
+7d2
+3/26f4
+5/2
+6
+0.95 · 7d2
+3/26f4
+5/2
+6
+0.91 · 7d2
+3/26f4
+5/2
+6
+6f47d28p2
+147
+7d2
+3/26f5
+5/2
+7/2
+0.89 · 7d2
+3/26f5
+5/2+
+0.07 · 7d2
+3/26f4
+5/26f1
+7/2
+9/2
+0.88 · 7d2
+3/26f5
+5/2+
+0.05 · 7d2
+3/26f4
+5/26f1
+7/2
+9/2
+6f57d28p2
+148
+7d2
+3/26f6
+5/2
+2
+0.94 · 7d2
+3/26f6
+5/2
+2
+0.93 · 7d2
+3/26f6
+5/2
+2
+6f67d28p2
+149
+7d3
+3/26f6
+5/2
+3/2
+0.96 · 7d3
+3/26f6
+5/2
+3/2
+0.93 · 7d3
+3/26f6
+5/2
+3/2
+6f67d38p2
+150
+7d3
+3/26f6
+5/26f1
+7/2
+2
+0.85 · 7d3
+3/26f6
+5/26f1
+7/2+
+0.11 · 7d3
+3/26f5
+5/26f2
+7/2
+2
+0.89 · 7d3
+3/26f6
+5/26f1
+7/2+
+0.06 · 7d3
+3/26f5
+5/26f2
+7/2
+2
+6f77d38p2
+151
+7d3
+3/26f6
+5/26f2
+7/2
+9/2
+0.89 · 7d3
+3/26f6
+5/26f2
+7/2+
+0.09 · 7d3
+3/26f5
+5/26f3
+7/2
+9/2
+0.89 · 7d3
+3/26f6
+5/26f2
+7/2+
+0.06 · 7d3
+3/26f5
+5/26f3
+7/2
+9/2
+6f87d38p2
+152 +6f6
+5/2
+7d2
+3/26f4
+7/2
+6
+0.95 · 7d2
+3/26f4
+7/2
+6
+0.92 · 7d2
+3/26f4
+7/2
+6
+6f97d38p2
+153
+7d2
+3/26f5
+7/2
+11/2
+0.96 · 7d2
+3/26f5
+7/2
+11/2
+0.93 · 7d2
+3/26f5
+7/2
+11/2
+6f107d38p2
+
+15
+TABLE VI. (Continuation.)
+Z
+Closed
+DCBQ-SRC
+JSRC
+DCBQ-CI1
+JCI1
+DCBQ-CI2
+JCI2
+Ref. [47]
+Shells
+154
+7d2
+3/26f6
+7/2
+6
+0.98 · 7d2
+3/26f6
+7/2
+6
+0.95 · 7d2
+3/26f6
+7/2
+6
+6f117d38p2
+|155
+7d2
+3/26f7
+7/2
+7/2
+0.99 · 9s1
+1/26f8
+7/2
+1/2
+0.94 · 9s1
+1/26f8
+7/2
+1/2
+6f127d38p2
+156
+7d2
+3/26f8
+7/2
+2
+0.97 · 7d2
+3/26f8
+7/2
+2
+0.97 · 7d2
+3/26f8
+7/2
+2
+6f137d38p2
+157
++6f8
+7/2
+7d3
+3/2
+3/2
+0.96 · 7d3
+3/2
+3/2
+0.96 · 7d3
+3/2
+3/2
+6f147d38p2
+158
+7d4
+3/2
+0
+0.98 · 7d4
+3/2
+0
+0.96 · 7d4
+3/2
+0
+6f147d48p2
+159 +7d4
+3/2
+9s1
+1/2
+1/2
+0.98 · 9s1
+1/2
+1/2
+0.96 · 9s1
+1/2
+1/2
+6f147d48p29s1
+160
+7d1
+5/29s1
+1/2
+3
+0.96 · 7d1
+5/29s1
+1/2
+3
+0.95 · 7d1
+5/29s1
+1/2
+3
+6f147d58p29s1
+161
+7d2
+5/29s1
+1/2
+9/2
+0.97 · 7d2
+5/29s1
+1/2
+9/2
+0.92 · 7d2
+5/29s1
+1/2
+9/2
+6f147d68p29s1
+162
+7d3
+5/29s1
+1/2
+5
+0.98 · 7d3
+5/29s1
+1/2
+5
+0.96 · 7d3
+5/29s1
+1/2
+5
+6f147d78p29s1
+163
+7d5
+5/2
+5/2
+0.96 · 7d5
+5/2
+5/2
+0.95 · 7d5
+5/2
+5/2
+6f147d88p29s1
+164
+7d6
+5/2
+0
+0.98 · 7d6
+5/2
+0
+0.96 · 7d6
+5/2
+0
+6f147d108p2
+165 +7d6
+5/2
+9s1
+1/2
+1/2
+0.98 · 9s1
+1/2
+1/2
+0.96 · 9s1
+1/2
+1/2
+166
+9s2
+1/2
+0
+0.84 · 9s2
+1/2 + 0.10 · 8p2
+3/2
+0
+0.90 · 9s2
+1/2
+0
+|167
+9s1
+1/28p1
+3/29p1
+1/2
+3/2
+0.88 · 9s2
+1/28p1
+3/2 + 0.06 · 8p3
+3/2
+3/2
+0.91 · 9s2
+1/28p1
+3/2
+3/2
+|168|
+9s2
+1/28p1
+3/29p1
+1/2
+1
+0.55 · 9s1
+1/28p2
+3/29p1
+1/2+
+0.24 · 9s1
+1/28p1
+3/29p2
+1/2+
+0.20 · 9s1
+1/28p3
+3/2
+2
+0.94 · 9s2
+1/28p1
+3/29p1
+1/2
+1
+169
+9s2
+1/28p2
+3/29p1
+1/2
+3/2
+0.76 · 9s2
+1/28p2
+3/29p1
+1/2+
+0.16 · 9s2
+1/28p1
+3/29p2
+1/2+
+0.06 · 9s2
+1/28p3
+3/2
+3/2
+0.83 · 9s2
+1/28p2
+3/29p1
+1/2+
+0.07 · 9s2
+1/28p1
+3/29p2
+1/2
+3/2
+170
+9s2
+1/28p2
+3/29p2
+1/2
+2
+0.96 · 9s2
+1/28p2
+3/29p2
+1/2
+2
+0.93 · 9s2
+1/28p2
+3/29p2
+1/2
+2
+

INE2T4oBgHgl3EQf_Qm3/content/tmp_files/2301.04247v1.pdf.txt

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+arXiv:2301.04247v1  [math.OA]  10 Jan 2023
+Projective Hilbert modules and sequential approximations
+Lawrence G. Brown and Huaxin Lin
+January 12, 2023
+Abstract
+We show that, when A is a separable C∗-algebra, every countably generated Hilbert
+A-module is projective (with bounded module maps as morphisms).
+We also study the
+approximate extensions of bounded module maps. In the case that A is a σ-unital simple
+C∗-algebra with strict comparison and every strictly positive lower semicontinuous affine
+function on quasitraces can be realized as the rank of an element in Cuntz semigroup, we show
+that the Cuntz semigroup is the same as unitarily equivalent class of countably generated
+Hilbert A-modules if and only if A has stable rank one.
+1
+Introduction
+We study Hilbert module over a C∗-algebra A. A Hilbert A -module H is said to be projective
+(with bounded module maps as morphisms) if, for any Hilbert A-modules H1 and H2, any
+bounded module map ϕ : H → H1 and any surjective bounded module map s : H2 → H1, there
+is always a bounded module map ψ : H → H2 such that ϕ = s ◦ ψ. It is easy to see that, when
+A is a unital C∗-algebra, a direct sum A(n) of n copies of A is a projective Hilbert A-module.
+However, it is not known that this holds for non-unital C∗-algebras. One observes that, for a
+non-unital C∗-algebra A, it itself is not algebraically finitely generated as an A-module. The
+characterization of projective Hilbert modules over a C∗-algebra A seems to remain elusive for
+decades. Evidence, on the other hand, suggests that most countably generated Hilbert modules
+are projective.
+In this paper, we will confirm that, for any separable C∗-algebra A, every
+countably generated Hilbert A-module is projective.
+It is attempting to classify countably generated Hilbert A-modules. It was shown in [11] that,
+when A has stable rank one, unitary equivalence classes of Hilbert A-modules are determined by
+the Cuntz semigroup of A. One might prefer a more quantified description of Hilbert A-modules
+using some version of dimension functions. Indeed, in the case that A is a separable simple
+C∗-algebra such that its purely non-compact elements in the Cuntz semigroup are determined
+by strictly positive lower semicontinuous affine functions on its quasitraces, then one wishes to
+use the same functions to described those Hilbert A-modules which are not algebraically finitely
+generated. We show that in this case, Cuntz equivalence classes are the same as unitary equiva-
+lence classes of countably generated Hilbert A-modules if and only if A has tracial approximate
+oscillation zero, or equivalently, in this case, A has stable rank one. This is a partial converse
+of the theorem in [11] mentioned above.
+Injective Hilbert A-modules were studied in [33]. If A is not an AW ∗-algebra, then A itself
+is not an injective Hilbert A-module (with bounded module maps as morphisms) (see Theorem
+3.14 of [33]). Even if we only consider bounded module maps with adjoints, there are only very
+few injective Hilbert modules. It was shown in [33] that a countably generated Hilbert A-module
+is *-injective if only if it is orthogonal complementary. Let H be a countably generated Hilbert
+A-module. Suppose that H0 ⊂ H1 are Hilbert A-modules and ϕ : H0 → H is a bounded module
+1
+
+map.
+As we mentioned ϕ may not be extended to a bounded module map from H1 to H.
+However, we show that one may find a sequence of bounded module maps ϕn : H1 → H with
+∥ϕn∥ ≤ ∥ϕ∥ such that limn→∞ ∥ϕn(x) − ϕ(x)∥ = 0 for all x ∈ H0 (note that H0 is not assumed
+to be separable). This result may be stated as every countably generated Hilbert module over
+a σ-unital C∗-algebra A is “sequentially approximately injective”. With the same spirit, we
+show that every countably generated Hilbert module over a σ-unital C∗-algebra is “sequentially
+approximately projective”.
+The paper is organized as follows: Section 2 collects some easy facts about projective Hilbert
+modules and algebraically finitely generated Hilbert modules over a C∗-algebra. Section 3 dis-
+cusses some basic results about countably generated Hilbert A-modules. In Section 4, we show
+that, under the assumption that A is a σ-unital simple C∗-algebra with strict comparison and
+the canonical (dimension function) map Γ from Cuntz semigroup to lower semi-continuous affine
+functions on quasitraces �
+QT(A) is surjective, countably generated (but not algebraically gener-
+ated) Hilbert A-modules can be classified by these lower semi-continuous affine functions if and
+only if A has stable rank one. This is a partial converse of a theorem in [11]. We also show that,
+assume that A is a σ-unital simple C∗-algebra with finite radius of comparison, then a countably
+generated Hilbert A-module with infinite quasitrace is unitarily equivalent to l2(A). In section
+5, we show that every countably generated Hilbert module over a separable C∗-algebra A is
+always projective. Section 6 shows that every Hilbert A-module is “approximately injective,”
+and every countably generated Hilbert module over a σ-unital C∗-algebra A is “approximately
+projective”.
+Acknowledgement: This work is based on a preprint [35] of 2010. A draft of the current
+paper was formed in 2014. The second named author was partially supported by a NSF grant
+(DMS-1954600). Both authors would like to acknowledge the support during their visits to the
+Research Center of Operator Algebras at East China Normal University which is partially sup-
+ported by Shanghai Key Laboratory of PMMP, Science and Technology Commission of Shanghai
+Municipality (#13dz2260400 and #22DZ2229014).
+2
+Easy facts about projective Hilbert modules
+Definition 2.1. Let A be a C∗-algebra. For an integer n ≥ 1, denote by A(n) the (right) Hilbert
+A-module of orthogonal direct sum of n copies of A. If x = (a1, a2, ..., an), y = (b1, b2, ..., bn),
+then
+⟨x, y⟩ =
+n
+�
+i=1
+a∗
+nbn.
+Denote by HA, or l2(A), the standard countably generated Hilbert A-module
+HA = {{an} :
+k
+�
+n=1
+a∗
+nan converges in norm as k → ∞},
+where the inner product is defined by
+⟨{an}, {bn}⟩ =
+∞
+�
+n=1
+a∗
+nbn.
+Let H be a Hilbert A-module. denote by H♯ the set of all bounded A-module maps from H
+to A. For each y ∈ H, define ˆy(x) = ⟨y, x⟩ for all x ∈ H. Then ˆy ∈ H♯. We say H is self-dual, if
+every f ∈ H♯ has the form ˆy for some y ∈ H.
+2
+
+If H1, H2 are Hilbert A-modules, denote by B(H1, H2) the space of all bounded module maps
+from H1 to H2. If T ∈ B(H1, H2), denote by T ∗ : H2 → H♯
+1 the bounded module maps defined
+by
+T ∗(y)(x) = ⟨y, Tx⟩ for all x ∈ H1 and y ∈ H2.
+If T ∗ ∈ B(H2, H1), one says that T has an adjoint T ∗. Denote by L(H1, H2) the set of all
+bounded A-module maps in B(H1, H2) with adjoints. Let H be a Hilbert A-module. In what
+follows, denote B(H) = B(H, H) and L(H) = L(H, H). B(H) is a Banach algebra and L(H) is
+a C∗-algebra.
+For x, y ∈ H, define θx,y ∈ L(H) by θx,y(z) = x⟨y, z⟩ for all z ∈ H. Denote by F(H) the linear
+span of those module maps with the form θx,y, where x, y ∈ H. Denote by K(H) the closure of
+F(H). K(H) is a C∗-algebra. It follows from a result of Kasparov ([26]) that L(H) = M(K(H)),
+the multiplier algebra of K(H), and, by [31], B(H) = LM(K(H)), the left multiplier algebra of
+K(H).
+Suppose that H1 and H2 are Hilbert A-submodules of a Hilbert A-module H. Then K(Hi)
+is a hereditary C∗-subalgebra of K(H), i = 1, 2, (see Lemma 2.13 [33]). Denote by K(H1, H2)
+the subspace of K(H) consists of module maps of the form STL, where S ∈ K(H2), T ∈ B(H)
+and L ∈ K(H1), where K(H2) and K(H1) are viewed as hereditary C∗-subalgebras of K(H).
+Note that K(H1, H2) is the closure of the linear span of those module maps with the form θx,y,
+where x ∈ H2 and y ∈ H1.
+It is convenient to have an example that H ̸= H♯. Let A be a C∗-algebra with a sequence
+{dn} of mutually orthogonal positive elements with ∥dn∥ = 1. Define f({an}) = �∞
+n=1 dnan for
+all {an} ∈ HA. Then f is a bounded module map but f ̸∈ H.
+Two Hilbert A-modules are said to be unitarily equivalent, or isomorphic, if there is an
+invertible map U ∈ B(H1, H2) such that
+⟨U(x1), U(x2)⟩ = ⟨x1, x2⟩ for all x1, x2 ∈ H1.
+(see [41] for further basic information).
+Definition 2.2. Let A be a C∗-algebra and H a Hilbert A-module and H1 the A∗∗-Hilbert
+module extension of H constructed in Section 4 of [41], Denote by H∼ the self-dual Hilbert A∗∗-
+module H♯
+1 (see Section 4 of [41]). Every bounded module map in B(H, H♯) can be uniquely
+extended to a bounded module map in B(H∼). (This easily follows from the construction of H∼
+and 3.6 of [41]. See also 1.3 of [31].) If H is self-dual, then B(H) = L(E) (see 3.5 of [41]). Thus
+M(K(H)) = LM(K(H)) = QM(K(H)). If in addition, A is a W ∗-algebra, B(H) is also a W ∗
+-algebra. In particular, B(H∼) is a W ∗-algebra. Since all maps in B(H, H♯) can be uniquely
+extended to a maps in B(H∼), we conclude that B(H∼) is a W ∗ -algebra containing K(H),
+M(K(H)), LM(K(H) and QM(K(H)).
+Definition 2.3. Let H be a Hilbert A-module and F ⊂ H be a subset. We say that H is
+generated by F (as a Hilbert A-module), if the linear combination of elements of the form
+{za : z ∈ F, a ∈ A} is dense in H.
+We say H is algebraically generated by F, if every element x ∈ H is a linear combination of
+elements of the form {za : z ∈ F, a ∈ ˜A}. Note that we write a ∈ ˜A instead a ∈ A to include the
+element z. H is algebraically finitely generated if H is algebraically generated by a finite subset
+F. See Corollary 2.13 for some clarification.
+Now we turn to the projectivity of Hilbert modules.
+Definition 2.4. Let A be a C∗-algebra and H be a Hilbert A-module.
+We say that H is
+∗-projective, if, for any Hilbert A-modules H1 and H2, any module map ϕ ∈ L(H, H2) and any
+3
+
+surjective module map s ∈ L(H1, H2), there exists a module map ψ ∈ L(H, H1) such that
+s ◦ ψ = ϕ.
+(e 2.1)
+Note here we assume that s, ϕ and ψ have adjoint module maps.
+Theorem 2.5. Let A be a C∗-algebra.
+(1) Suppose that H and H1 are Hilbert A-modules and s ∈ L(H1, H) is surjective. Then
+there is ψ1 ∈ L(H, H1) such that
+s ◦ ψ1 = idH.
+(e 2.2)
+Moreover, ψ1(H) is an orthogonal summand of H1 and ψ1(H) is unitarily equivalent to H.
+(2) Every Hilbert A-module is ∗-projective.
+Proof. For (1), let s be as described. We note that s has a closed range. Define T : H1 ⊕ H →
+H1⊕H by T(h1 ⊕h) = 0⊕s(h1) for h1 ∈ H1 and h ∈ H. Then T ∈ L(H1⊕H) = M(K(H1⊕H))
+(see Theorem 1.5 of [31]). It follows from Lemma 2.4 of [33] that
+H1 ⊕ H = kerT ⊕ |T|(H1 ⊕ H).
+(e 2.3)
+Let T = V |T| be the polar decomposition in (K(H1 ⊕ H))∗∗. Note that the proof of Lemma
+2.4 of [33] shows that 0 is an isolated point of |T| or |T| is invertible. So the same holds for
+(T ∗T), and hence the same holds for (TT ∗). Let S = (TT ∗)−1, where the inverse is taken in the
+hereditary C∗-subalgebra L(H) ⊂ L(H1 ⊕ H). Since s is surjective,
+|TT ∗|(H) = H.
+(e 2.4)
+Note that T ∗ = |T|V ∗, V |T|V ∗ = |TT ∗|1/2 and V ∗|TT ∗|1/2 = T ∗. Hence
+L1 := V ∗(TT ∗)−1/2 = V ∗(TT ∗)1/2S = T ∗S ∈ L(H1 ⊕ H).
+(e 2.5)
+This also implies that L1(H) = V ∗((TT ∗)−1/2(H)) = V ∗(H) is closed. Moreover, V ∗ gives a
+unitary equivalence of H and L1(H). One then checks that
+TL1 = V |T|V ∗(TT ∗)−1/2 = P,
+(e 2.6)
+where P is the range projection of (TT ∗)1/2 which gives the identity map on H. Since both T
+and L1 are in L(H1 ⊕ H), so is P. One then defines ψ1 = L1P|H : H → H1. Thus s ◦ ψ1 = idH.
+Next we note that H ⊂ kerT and |T|(H1⊕H) = |T|(H1) ⊂ H1. It follows that kerT = H0⊕H.
+By (e 2.3) again, H1 ⊕ H = H0 ⊕ H ⊕ |T|(H1). It follows that |T|(H) is an orthogonal summand
+of H1.
+On the other hand, since ψ1(H) = L1(H) = V ∗(H), ψ1(H) is closed. But we also have
+ψ1(H) = T ∗S(H) = T ∗T(H1) = |T|(H1). As we have shown that L1(H) is unitarily equivalent
+to H, this proves the “Moreover” part of (1).
+To show that every Hilbert A-module H is ∗-projective, let H1 and H2 be Hilbert A-modules,
+ϕ : H → H1 and s : H2 → H1 be bounded module maps with adjoints, where s is surjective.
+We now apply the statement of (1).
+Note that (1) holds for any Hilbert A-module, in particular, it holds for H1 (in place of H).
+Since s is surjective, we obtain ψ1 ∈ L(H1, H2) such that
+s ◦ ψ1 = idH1.
+(e 2.7)
+Define ψ = ψ1 ◦ ϕ : H → H2. Then s ◦ ψ = s ◦ ψ1 ◦ ϕ = ϕ.
+4
+
+Remark 2.6. (i) It is probably worth noting that, in part (1) of Theorem 2.5, ∥ψ∥ = ∥|ss∗|−1/2∥
+(the inverse is taken in L(H)), and, for the second part, ∥ψ∥ = ∥|ss∗|−1/2 ◦ ϕ∥. This, probably,
+may not be improved.
+The next three propositions are easy facts, but perhaps, not entirely trivial. We include
+here for the clarity of our further discussion. After an earlier version ([35]) of this note was first
+posted (in 2010), Leonel Robert informed us that, using Proposition 2.5 above, he has a proof
+that the converse of the following also holds, i.e., if H is algebraically finitely generated, then
+K(H) has an identity (see Proposition 2.11).
+Proposition 2.7. Let A be a C∗-algebra and H be a Hilbert A-module. Suppose that 1H ∈
+K(H). Then H is algebraically finitely generated.
+Proof. Let F(H) be the linear span of rank one module maps of the form θξ,ζ for ξ, ζ ∈ H. Since
+F(H) is dense in K(H), there is T ∈ F(H) such that
+∥1H − T∥ < 1/4.
+(e 2.8)
+It follows that T is invertible. There are ξ1, ξ2, ..., ξn, ζ1, ζ2, ..., ζn ∈ H such that
+T(ξ) =
+n
+�
+j=1
+ξj < ζj, ξ > for all ξ ∈ H.
+(e 2.9)
+But TH = H. This implies that �n
+j=1 ξjA = H.
+Definition 2.8. Let A be a C∗-algebra and H be a Hilbert A-module. We say H is projective
+(with bounded module maps as morphisms), if for any Hilbert modules H1 and H2 and any
+bounded surjective module map s : H2 → H1 and any bounded module map ϕ : H → H1,
+there is a bounded module map ψ : H → H2 such that s ◦ ψ = ϕ. So we have the following
+commutative diagram:
+H
+H1
+H2
+ϕ
+ψ
+s
+(e 2.10)
+Proposition 2.9. Let A be a C∗-algebra and H be a Hilbert A-module for which K(H) has an
+identity. Then H is a projective Hilbert A-module.
+Proof. Denote by ˜A the minimum unitization of A. Note that H is a Hilbert ˜A-module. We
+may write ˜A(N) = e1 ˜A ⊕ e2 ˜A ⊕ · · · ⊕ eN ˜A, where ⟨ei, ei⟩ = 1 ˜
+A. Let us first show that ˜A(N) is a
+projective Hilbert ˜A-module.
+Suppose that H1 and H2 are Hilbert ˜A-modules, s : H2 → H1 is a surjective bounded
+module map and ϕ : ˜A(N) → H1 is a bounded module map. Let xi = ϕ(ei), 1 ≤ i ≤ N. Since s
+is surjective, one chooses yi ∈ H2 such that
+s(yi) = xi, 1 ≤ i ≤ N.
+(e 2.11)
+Define a module map ψ : ˜A(N) → H1
+ψ(h) =
+N
+�
+i=1
+yi⟨ei, h⟩ for all h = ⊕N
+i=1ai,
+5
+
+where ai ∈ A, 1 ≤ i ≤ N. Then ϕ ∈ K(H, H2). Moreover
+s ◦ ψ(ei) = xi⟨ei, ei⟩ = xi = ϕ(ei), 1 ≤ i ≤ N.
+(e 2.12)
+It follows that s ◦ ψ = ϕ.
+Now we consider the general case. From 2.7, H is algebraically finitely generated. Therefore,
+a theorem of Kasparov ([26]) shows that H = PH ˜
+A for some projection P ∈ L(H ˜
+A). The fact
+that 1H ∈ K(H) implies that P ∈ K(H ˜
+A). Therefore there is an integer N ≥ 1 and a projection
+P1 ∈ MN( ˜A) such that PH is unitarily equivalent to P1H ˜
+A. In other words, one may assume
+that H is a direct summand of ˜A(N).
+Write ˜A(N) = H ⊕ H′ for some Hilbert ˜A-module H′. Let H0 and H1 be Hilbert A-modules,
+s : H0 → H1 be a surjective bounded module map and ϕ : H → H1 be another bounded module
+map. We view them as ˜A-modules and ˜A-module maps. Let ι : H → H ⊕ H′ be the embedding
+and j : H ⊕ H′ → H be the projection. Then j ◦ ι = idH. We then have the following diagram:
+H
+←j
+H ⊕ H′
+←ι H
+↓ϕ
+H1
+ևs
+H0
+Recall that these are Hilbert ˜A-modules and bounded ˜A-module maps. Since we have shown
+H ⊕ H′ = ˜A(N) is projective, we obtain a bounded module map ψ1 : H ⊕ H′ → H1 such that
+the following commutes:
+H
+←j
+H ⊕ H′
+←ι H
+↓ϕ
+↓ψ1
+H1
+ևs
+H0
+So s ◦ ψ1 = ϕ ◦ j. Define ψ = ψ1 ◦ ι. So s ◦ ψ = s ◦ ψ1 ◦ ι = ϕ ◦ j ◦ ι = ϕ.
+Remark 2.10. (1) One may notice that we do not provide an estimate of ∥ψ∥ in Proposition
+2.9. Note that s induces a one-to one and surjective Banach module map ˜s : H2/kers → H1.
+Therefore its inverse ˜s−1 is bounded. In the proof, for any ǫ > 0, one may choose yi ∈ H2 such
+that ∥yi∥ ≤ ∥˜s−1 ◦ ϕ∥ + ǫ/
+√
+N. Then one may be able to estimate that ∥ψ∥ ≤
+√
+N∥˜s−1 ◦ ϕ∥ + ǫ.
+It is not clear one can do better than that in these lines of proof.
+(2) The fact that ⟨ei, ei⟩ = 1 ˜
+A is crucial in the proof. It should be noted that, when A is
+not unital, the argument in the proof of Proposition 2.9 does not imply that A(n) is projective
+(with bounded module maps as morphisms). However, we do not assume that A is unital in
+Proposition 2.9 (but K(H) is unital). On the other hand, one should be warned that, if A is not
+unital and H = A(n), then K(H) = Mn(A) which is never unital. There are projective Hilbert
+modules for which K(H) is not unital. In fact, in Theorem 5.3 below, we show that A(n) is
+always projective when A is separable.
+(3) Let H1 = xA and H2 = yA. Define a module map ψ : xA → yA by ϕ(xa) = ya for all
+a ∈ A. In general such a module map may not be bounded. Consider A = C([0, 1]) and x(t) = t
+and y = 1A. Let ψ : xA → yA be defined by ψ(xf) = f for all f ∈ C([0, 1]). This module map
+is not bounded. We do not know that A(n) is projective in general when A is not unital and not
+separable (see Lemma 6.7).
+(4) One may also mention that if H1, H2, ..., Hn are projective Hilbert A-modules then
+�n
+i=1 Hi is also projective.
+Proposition 2.11 (L. Robert). Let A be a C∗-algebra and H be an algebraically finitely gen-
+erated Hilbert A-module. Then K(H) is unital and H is self-dual.
+6
+
+Proof. Write H = �n
+i=1 ξi ˜A (where ξi ∈ H). Put H1 = ˜A(n). We view both H and H1 as Hilbert
+˜A-modules. Write H1 = �n
+i=1 ei ˜A, where ei = 1 ˜
+A, 1 ≤ i ≤ n. Define a module map s : H1 → H
+by s(ei) = ξi, 1 ≤ i ≤ n. Then
+s(h) =
+n
+�
+i=1
+ξi⟨ei, h⟩ for all h ∈ H1.
+(e 2.13)
+In other words s = �n
+i=1 θξi,ei ∈ K(H1, H). In particular, s ∈ L(H1, H). Since H is algebraically
+generated by {ξ1, ξ2, .., ξn}, s is surjective. We then proceed the same proof of Theorem 2.5
+to obtain the bounded module map L1 = T ∗S. Note, in this case, T ∈ K(H1 ⊕ H) and hence
+L1 ∈ K(H1 ⊕ H). It follows that s ◦ ψ1 ∈ K(H) (with notation in the proof of 2.5). Since
+s ◦ ψ1 = idH, we conclude that K(H) is unital.
+By Theorem 2.5, ψ1(H) is an orthogonal summand of H1 = ˜A(n). Let U : H → ψ1(H) be a
+bounded module map which implements the unitary equivalence. Note that U ∈ L(H, ψ1(H)).
+Let Q : H1 → ψ1(H) be the projection of H1 onto the orthogonal summand ψ1(H).
+Suppose that f ∈ H♯, i.e., f : H → A is a bounded module map. Define ˜f : H1 → A
+by ˜f(h) = f ◦ U −1 ◦ Q(h)) for all h ∈ H1. Note that f(x) = ˜f(U(x)) for all x ∈ H. Since
+H1 = ˜A(n) is self-dual, there is g ∈ H1 such that ˜f(h) = ⟨g, h⟩ for all h ∈ H1. If x ∈ H, then
+f(x) = ˜f(U(x)) = ⟨g, Q(U(x))⟩. Hence f(x) = ⟨(QU)∗(g), x⟩ for all x ∈ H. Since (QU)∗(g) ∈ H,
+this shows that H = H♯ and H is self-dual.
+Lemma 2.12 (cf. Proposition 1.4.5 of [42]). Let A be a C∗-algebra and H a Hilbert A-module
+and x ∈ H. Then, for any 0 < α < 1/2, there exists y ∈ xA ⊂ H with ⟨y, y⟩1/2 = ⟨x, x⟩1/2−α
+such that x = y · ⟨x, x⟩α.
+Proof. We first consider Hilbert ˜A module x ˜A. Put a = ⟨x, x⟩, an = (1/n + a)−α and xn =
+x · (1/n + a)−α, n ∈ N. We will show that {xn} is a Cauchy sequence.
+Put βn,m = (1/n + a)−α − (1/m + a)−α, n, m ∈ N. Then
+∥xβn,m∥ = ∥(βn,m⟨x, x⟩βn,m)1/2∥ = ∥a1/2βn,m∥.
+Since a1/2(1/n+a)−α converges to a1/2−α in norm. we conclude that ∥xβn,m∥ → 0 as n, m → ∞.
+It follows that xn → yα for some yα ∈ H. Hence xn⟨x, x⟩α → yα⟨x, x⟩α. Moreover, ⟨yα, yα⟩1/2 =
+limn→∞ a1/2an = a1/2−α (converge in norm).
+Similarly, limn→∞ ∥x − xanaα∥ = limn→∞ ∥a1/2(1 − anaα)∥ = 0. Since xnaα = xanaα,
+we obtain that limn→∞ xnaα = x. It follows that x = yα⟨x, x⟩α. Choose α < α′ < 1/2. Then
+(1/n+a)−α′aα′−α ∈ A. Since limn→∞ x(1/n+a)−α′aα′−α = limn→∞ x(1/n+a)−α′(1/n+a)α′−α =
+limn→∞ x(1/n + a)−α = yα, yα ∈ xA.
+Let us end this section with the following clarification on Definition 2.3.
+Corollary 2.13. Let H be a Hilbert A-module and F ⊂ H be a subset.
+(1) Then F ⊂ {z · a : z ∈ F, a ∈ A}.
+(2) If H is algebraically finitely generated, then there are x1, x2, ..., xn ∈ H such that H =
+{�n
+i=1 xiai : ai ∈ A} (see Definition 2.3).
+Proof. For (1), let z ∈ F. By Lemma 2.12, we may write z = y⟨z, z⟩α for some 0 < α < 1/2 and
+y ∈ H. Let {eλ} ⊂ A be an approximate identity. Then zeλ = y⟨z, z⟩αeλ → y⟨z, z⟩ = z. Hence
+z ∈ {z · a : z ∈ F, a ∈ A} (So in Definition 2.3, F is in the closure of {za : z ∈ F and a ∈ A}.)
+For (2), suppose that H = �n
+i=1 xi ˜A. By Lemma 2.12, we may write xi = yi⟨xi, xi⟩α for
+some 0 < α < 1/2 and yi ∈ H, 1 ≤ i ≤ n. Let H0 = �n
+i=1 yiA ⊂ H. Since xi = yi⟨xi, xi⟩α ∈ H0,
+H ⊂ H0. It follows that H = �n
+i=1 yiA.
+7
+
+3
+Countably generated Hilbert modules
+Let A be a C∗-algebra and H1 and H2 be a pair of Hilbert A-modules. In this section we discuss
+the question when H1 is unitarily equivalent to a Hilbert A-submodule of H2. The main result
+of this section is Theorem 3.6 which provide some answer to the question. For the connection
+to Cuntz semigroup, see Corollary 3.9.
+Lemma 3.1. Let A be a C∗-algebra and H a Hilbert A-module. Let {Eλ} be an approximate
+identity for K(H). Then, for any ǫ > 0 and any finite subset F ⊂ H, there is λ0 such that, for
+all λ ≥ λ0,
+∥Eλ(x) − x∥ < ǫ.
+(e 3.14)
+Proof. Let F = {ξ1, ξ2, ..., ξk} and ǫ > 0. Without loss of generality, we may assume that
+0 ̸= ∥ξi∥ ≤ 1, 1 ≤ i ≤ k. There is 1/3 < α < 1/2 such that
+∥⟨ξi, ξi⟩1−α − ⟨ξi, ξi⟩α∥ = ∥⟨ξi, ξi⟩α(⟨ξi, ξi⟩1−2α − 1)∥ < ǫ/3, 1 ≤ i ≤ k.
+(e 3.15)
+By Lemma 2.12, write ξi = ζi⟨ξi, ξi⟩α with ⟨ζi, ζi⟩ = ⟨ξi, ξi⟩1−2α, 1 ≤ i ≤ k. Put Si = θζi,ζi,
+1 ≤ i ≤ k. Then, for 1 ≤ i ≤ k,
+∥Si(ξi) − ξi∥ = ∥ζi⟨ζi, ξi⟩ − ξi∥ = ∥ζi⟨ξi, ξi⟩1−α − ζi⟨ξi, ξi⟩α∥ < ǫ/3.
+There exists λ0 such that, for all λ ≥ λ0,
+∥EλSi − Si∥ < ǫ/3, 1 ≤ i ≤ k.
+(e 3.16)
+It follows that, for all λ ≥ λ0,
+∥Eλ(ξi) − ξi∥
+≤
+∥Eλ(ξi) − EλSi(ξi)∥ + ∥EλSi(ξi) − Si(ξi)∥ + ∥Si(ξi) − ξi∥
+<
+∥Eλ∥∥ξi − Si(ξi)∥ + ǫ/3 + ǫ/3 < ǫ,
+1 ≤ i ≤ k.
+At least part of the following is known, in fact, (1) ⇔ (2) is known as Corollary 1.5 of [39].
+We also think that the rest of them are known. Note that even H is countably generated it may
+not be separable when A is not separable.
+Proposition 3.2 (cf. Corollary 1.5 of [39]). Let A be a C∗-algebra and H be a Hilbert A-module.
+Then the following are equivalent.
+(1) H is countably generated,
+(2) K(H) is σ-unital and
+(3) There exists an increasing sequence of module maps En ∈ K(H)+ with ∥En∥ ≤ 1 such
+that
+lim
+n→∞ ∥Enx − x∥ = 0 for all x ∈ H.
+(e 3.17)
+Proof. As mentioned above, by Corollary 1.5 of [39], (1) ⇔ (2).
+(1) ⇒ (3): Let {xn} be a sequence of elements in the unit ball of H such that H is generated
+by {xn} as Hilbert A-module. Let Fn be the set of those elements with the form xia, 1 ≤ i ≤ n
+and a ∈ A with ∥a∥ ≤ n. Note that ∪∞
+n=1span(Fn) is dense in H. Warning: Fn is not a finite
+subset.
+8
+
+For each n ∈ N, there exists 0 < 1/3 < αn < 1/2 such that, for any 0 ≤ b ≤ 1 in A,
+∥b1/2−αn(1 − b1−2αn)bαn∥ ≤ 1/n2.
+(e 3.18)
+By Lemma 2.12, choose ξi,n ∈ xiA such that xi = ξi,n⟨xi, xi⟩αn with ⟨ξi,n, ξi,n⟩ = ⟨xi, xi⟩1−2αn.
+Put bi = ⟨xi, xi⟩, n ∈ N. Then
+θξi,n,ξi,n(xia) = ξi,n⟨ξi,n, ξi,nbαn
+i ⟩a = ξi,n⟨ξi,n, ξi,n⟩bαn
+i a = ξi,nb1−αn
+i
+a.
+Then, for any ∥a∥ ≤ n and i, n ∈ N, by (e 3.18),
+∥θξi,n,ξi,n(xia) − xia∥2
+=
+∥ξi,n(b1−αn
+i
+− bαn
+i )a∥2 = ∥ξi,n(b1−2αn
+i
+− 1)bαn
+i a∥2
+(e 3.19)
+=
+∥a∗bαn
+i (b1−2αn
+i
+− 1)⟨ξi,n, ξi,n⟩(b1−2αn
+i
+− 1)bαn
+i a∥
+(e 3.20)
+=
+∥b1/2−αn
+i
+(1 − b1−2αn
+i
+)bαn
+i a∥2 ≤ (1/n)4∥a∥2 < 1/n2. (e 3.21)
+By using an approximate identity of K(H), we obtain an increasing sequence of {En} ∈ A+
+with ∥En∥ ≤ 1 such that
+∥Enθξi,nξi,n − θξi,n,ξi,n∥ < 1/n2, 1 ≤ i ≤ n.
+(e 3.22)
+It follows that, for any m ≥ n and a ∈ A with ∥a∥ ≤ n, by (e 3.21) and (e 3.22),
+∥Em(xia) − xia∥
+≤
+∥Em(xia − θξi,m,ξi,m(xia))∥ + ∥Em ◦ θξi,m,ξi,m(xia) − θξi,m,ξi,m(xia)∥
++∥θξi,mξi,m(xia) − xia∥
+<
+1/m + ∥xa∥/m2 + 1/m2 < 2/m + 1/m2.
+(e 3.23)
+Since ∪∞
+i=1span(Fi) is dense in H, this proves (3).
+(3) ⇒ (2): Let {En} be an increasing sequence in K(H)+ with ∥En∥ ≤ 1 such that
+lim
+n→∞ ∥En(x) − x∥ = 0 for all x ∈ H.
+(e 3.24)
+Fix x1, x2, ..., xN, y1, y2, ..., yN ∈ H. Let S = �N
+j=1 θxj,yj and
+M = N · max ∥{∥yj∥; 1 ≤ j ≤ N}.
+For any ǫ > 0, there exists n0 ∈ N such that
+∥Em(xj) − xj∥ <
+ǫ
+(M + 1), 1 ≤ j ≤ N.
+(e 3.25)
+whenever m ≥ n0. Then, for any m ≥ n0 and, for any z ∈ H,
+∥EmS(z) − S(z)∥ = ∥
+N
+�
+j=1
+(Emxj − xj)⟨yj, z⟩∥ ≤ (
+ǫ
+(M + 1))
+m
+�
+j=1
+∥yj∥∥z∥ < ∥z∥ǫ.
+(e 3.26)
+It follows that limn→∞ ∥EmS − S∥ = 0. Since F(H) is dense in K(H), this implies that K(H)
+is σ-unital.
+Definition 3.3. Let H be a Hilbert A-module and G ⊂ H be a subset. We say G is A-weakly
+dense in H, if, for any finite subset F ⊂ H, ǫ > 0, and any x ∈ H, there is g ∈ G such that
+∥⟨(x − g), z⟩∥ < ǫ for all z ∈ F.
+(e 3.27)
+(see Remark 6.2).
+9
+
+One may compare the next proposition with Lemma 1.3 of [39].
+Proposition 3.4. Let A be a σ-unital C∗-algebra and H be a Hilbert A-module.
+(1) Let s ∈ K(H)+. Then s is a strictly positive element of K(H) if s(H) is A-weakly dense
+in H, and, if s is a strictly positive element of K(H), then s(H) is dense in H.
+(2) Let H1 and H2 be Hilbert A-modules and T ∈ K(H1, H2) such that T(H1) is A-weakly
+dense in H2, then TT ∗ ∈ K(H2) is a strictly positive element of K(H2).
+Proof. For (1), we note that, by Lemma 1.3 of [39], s is a strictly positive element if and only
+s(H) is dense in H.
+We now suppose that s(H) is A-weakly dense in H. Without loss of generality, we may
+assume that ∥s∥ ≤ 1. Let p be the open projection associated with s, i.e., s1/n ր p. Put
+ηn = ∥s1/ns − s∥, n ∈ N. Then limn→∞ ηn = 0.
+Fix ǫ > 0. Let x, y ∈ H such that ∥x∥, ∥y∥ ≤ 1. Choose ξ1, ξ2 ∈ H such that ∥ξ1∥, ∥ξ2∥ ≤ 1.
+Then, there is ξ ∈ H such that
+∥⟨s(ξ) − ξ1, y⟩∥ < ǫ/2.
+(e 3.28)
+Then
+∥⟨θξ1,ξ2(x) − θs(ξ),ξ2(x), y⟩∥ = ∥⟨ξ1⟨ξ2, x⟩, y⟩ − ⟨s(ξ)⟨ξ2, x⟩, y⟩∥
+(e 3.29)
+= ∥⟨ξ2, x⟩∗⟨ξ1 − s(ξ), y⟩∥ < ǫ/2.
+(e 3.30)
+We estimate that
+∥⟨s1/nθξ1,ξ2(x) − θξ1,ξ2(x), y⟩∥
+=
+∥⟨s1/n(θξ1,ξ2 − θs(ξ),ξ2)(x), y⟩∥
++∥⟨(θs1/ns(ξ),ξ2(x) − θs(ξ1),ξ2(x)), y⟩∥ + ∥⟨(θs(ξ),ξ2(x) − θξ1,ξ2(x)), y⟩∥
+<
+∥s1/n∥∥(θξ1,ξ2 − θs(ξ),ξ2)(x), y⟩∥ + ηn + ǫ/2
+<
+ǫ + ηn.
+(e 3.31)
+Fix n ∈ N, let ǫ → 0. We obtain that
+∥⟨s1/nθξ1,ξ2(x) − θξ1,ξ2(x), y⟩∥ ≤ ηn.
+This holds for all y ∈ H with ∥y∥ ≤ 1. It follows that
+∥s1/nθξ1,ξ2(x) − θξ1,ξ2(x)∥ < ηn, n ∈ N.
+(e 3.32)
+Since (e 3.32) holds for every x ∈ H with ∥x∥ ≤ 1, we obtain that
+∥s1/nθξ1,ξ2 − θξ1,ξ2∥ < ηn, n ∈ N.
+(e 3.33)
+Since linear combinations of module maps of the form θξ1,ξ2 is dense in K(H), we obtain, for
+any k ∈ K(H),
+lim
+n→∞ ∥s1/nk − k∥ = 0 = lim
+n→∞ ∥k1/2(1 − s1/n)k1/2∥.
+(e 3.34)
+It follows that ∥k1/2(1 − p)k1/2∥ = 0, or pk = k for all k ∈ K(H). Thus p is the open projection
+associated with the hereditary C∗-subalgebra K(H) (see Lemma 2.13 of [33]). It then follows
+that s is a strictly positive element of K(H).
+For (2), let H = H1 ⊕ H2. Define ˜T ∈ K(H) by ˜T(h1 ⊕ h2) = 0 ⊕ T(h1) for h1 ∈ H1 and
+h2 ∈ H2. Let ˜T = V ( ˜T ∗ ˜T)1/2 be the polar decomposition of ˜T in K(H)∗∗. Then we may write
+T = ˜T|H1 = V (T ∗T)1/2. Since T ∈ K(H1, H2), T ∗ ∈ K(H2, H1).
+10
+
+Recall that T ∗ = |T|V ∗, V |T|V ∗ = |TT ∗|1/2 and V ∗|TT ∗|1/2 = T ∗. Hence T = (TT ∗)1/2V.
+We then write T = (TT ∗)1/4(TT ∗)1/4V. Note that (TT ∗)1/4V ∈ K(H) and (TT ∗)1/4V (H) ⊂ H2.
+It follows that (TT ∗)1/4(H2) is A -weakly dense in H2, as T(H) = (TT ∗)1/4((TT ∗)1/4V (H)) is
+A-weakly dense in H2. By (1), this implies that (TT ∗)1/4 is a strictly positive element of K(H2).
+Hence TT ∗ is a strictly positive element of K(H2).
+3.5. Let A be a C∗-algebra, H1, H2 be Hilbert A-modules and T ∈ B(H1, H2). Note T ∗ ∈
+B(H2, H♯
+1) and, for any y ∈ H1, T ∗(y)(x) = ⟨y, Tx⟩ for all x ∈ H1. We may view H1 ⊂ H♯
+1. We
+say T ∗(H2) is dense in H1, if T ∗(H2) ⊃ H1 (even though T ∗(H2) may not be in H1). Recall that
+TK∗
+0 ∈ K(H1, H2), for any K0 ∈ K(H1) which has an adjoint (TK∗
+0)∗ = K0T ∗ ∈ K(H2, H1). If
+we write T ∈ LM(K(H1, H2)), then, in general, T ∗ ∈ RM(K(H2, H1)). In particular, K0T ∗ ∈
+K(H2, H1) for all K0 ∈ K(H1). These will be used in the next theorem.
+The equivalence of (a) and (d) in part (2) of next theorem is known (see Theorem 4.1 of
+[19]).
+Theorem 3.6. Let A be a C∗-algebra and H1 and H2 be Hilbert A-modules. Suppose that H2
+is countably generated.
+(1) Then H2 is unitarily equivalent to a Hilbert A-submodule of H1 if and only if there is
+bounded module map T : H1 → H2 such that T(H1) is A-weakly dense in H2.
+(2) Suppose that H1 is also countably generated. Then the following are equivalent:
+(a) H1 and H2 are unitarily equivalent as Hilbert A-module;
+(b) there exists a bounded module map T ∈ B(H1, H2) such that T(H1) is A-weakly dense in
+H2 and (K0T ∗)(H2) is A-weakly dense in H1 for any strictly positive element K0 ∈ K(H1);
+(c) there exists a bounded module map T ∈ B(H1, H2) such that T(H1) is dense in H2 and
+T ∗(H2) is dense in H1 (see 3.5);
+(d) There is an invertible bounded module map T ∈ B(H1, H2).
+Proof. For (1), the “only if” part is clear. Let us assumes that there is T ∈ B(H1, H2) whose
+range is A-weakly dense in H2.
+Suppose that H2 is generated by {x1, x2, ..., xn, ...}. For each n ∈ N, there exists a sequence
+{yn,k} in H2 such that,
+∥⟨T(yn,k) − xn, xi⟩∥ < (
+1
+k + n) max{∥xi∥ : 1 ≤ i ≤ n + k}, 1 ≤ i ≤ n + k.
+Hence, for any fixed n and, any y ∈ H, limk→∞ ∥⟨T(yn,k) − xn, y⟩∥ = 0. It follows that the set
+of linear combinations of elements in {T(yn,k)a : n, k ∈ N, a ∈ A} is A-weakly dense in H2. Let
+H3 ⊂ H1 be the closure of A-submodule generated by {yn,k : k, n ∈ N}. Then H3 is a countably
+generated Hilbert A-submodule of H1 such that T(H3) is A-weakly dense in H2.
+Put H = H3 ⊕ H2. In what follows we may identify T with the map ˜T(h3 ⊕ h2) = 0 ⊕ T(h3)
+for h3 ∈ H3 and h2 ∈ H2, whenever it is convenient. We may also view K(H2) and K(H3) as
+hereditary C∗-subalgebras of K(H) (see Lemma 2.13 of [33]).
+Applying Corollary 1.5 of [39] (see Lemma 3.2 above for convenience), we obtain strictly
+positive elements K0 ∈ K(H3) ⊂ K(H) and L ∈ K(H2) ⊂ K(H), respectively. If follows from
+(1) of Proposition 3.4 that K0(H3) is dense in H3. Thus TK0(H3) is A-weakly dense in H2.
+Applying part (2) of Proposition 3.4, we conclude that S = (TK0)(TK0)∗ is a strictly positive
+element of K(H2).
+Therefore the range projection Ps of S in K(H3)∗∗ is the same as the range projection
+of L. Write (TK0)∗ = US1/2 in the polar decomposition of (TK0)∗ in K(H)∗∗. Note that
+(TK0)∗ = K1/2
+0
+(TK1/2
+0
+)∗ = US1/2. It follows that U(S1/2(H)) ⊂ H3. Since S1/2 is a strictly
+11
+
+positive element of K(H2), H2 = S1/2H2. Hence U(H2) ⊂ H3. Note that UY ∈ K0 · K(H) for
+any Y ∈ K(H). In particular, US′U ∗ ∈ K0 · K(H) · K0 = K(H3) for all S′ ∈ K(H2). We also
+have UH2 ⊂ H3. Moreover
+US1/nU ∗ ր UPsU ∗ = Q,
+(e 3.35)
+where Ps is the range projection of S (in K(H)∗∗) and Q is an open projection of K(H)∗∗, and
+QUPs = U. It follows that U ∗U = Ps and UH2 ⊂ H3 is unitarily equivalent to H2. This proves
+(1)
+For (2). Let us first show that (b) ⇒ (a). We keep the notation used in the proof of part (1).
+But since H1 is now assumed to be countably generated, we choose H3 = H1. In particular, K0
+is a strictly positive element of K(H1). With notation in (1), it suffices to show that U(H2) is
+dense in H1. Since K0T ∗(H) is A-weakly dense in H1, by Proposition 3.4, K0T ∗TK0 is a strictly
+positive element in K(H1). By (1) of Proposition 3.4, K0T ∗TK0(H1) is dense in H1. Because
+TK0(H1) ⊂ H2, this implies that K0T ∗(H2) is actually dense in H2.
+Since K0T ∗ = US1/2, and K0T ∗(H2) is dense in H1, U(S1/2(H2)) is dense in H1. As in the
+proof of (1) above, S1/2(H2) ⊂ H2. Hence, indeed, U(H2) is dense in H1.
+Now consider (c) ⇒ (b). It suffices to show that K0T ∗(H2) is dense in H1.
+Fix z ∈ H1. Since K0 is a strictly positive element, by Proposition 3.4, K0(H1) is dense in
+H1. There exists a sequence zn ∈ H1 such that
+lim
+n→∞ ∥K0(zn) − z∥ = 0
+(e 3.36)
+By the assumption of (c), for each n ∈ N, there exists a sequence of elements yn,k ∈ H2 such
+that (we do not assume that T ∗(yn,k) ∈ H1)
+lim
+k→∞ ∥T ∗(yn,k) − zn∥ = 0.
+(e 3.37)
+Thus, one obtain a subsequence {ym(n)} ⊂ {yn,k : k, n ∈ N} such that
+lim
+n→∞ ∥K0T ∗(ym(n)) − z∥ = 0.
+(e 3.38)
+This shows that K0T ∗(H2) is dense in H1, as desired.
+It is clear that (a) ⇒ (b), (a) ⇒ (c) and (a) ⇒ (d).
+It remains to show that (d) ⇒ (b). We will apply the proof of (1) first. But since H1 is
+countably generated, choose H3 = H1. Let K0 be a strictly positive element of K(H1). Since T
+is invertible, T(H1) is dense in H2. Therefore, it suffices to show that K0T ∗(H2) is A-weakly
+dense in H1. Let H = H1 ⊕ H2. We will view K(H1) and K(H2) as hereditary C∗-subalgebras
+of K(H). So T is identified with the bounded module map (h1 ⊕ h2) �→ 0 ⊕ T(h1) and T −1
+is identified with the bounded module map (h1 ⊕ h2) �→ T −1(h2) ⊕ 0. In what follows, we
+use the fact that (T −1)∗ = (T ∗)−1 and we will also work in H∼, if necessary (see Definition
+2.2). In this way, we may write T ∗, (T −1)∗ ∈ RM(K(H)). Hence, if E ∈ K(H2) ⊂ K(H),
+E(T ∗)−1 = E(T −1)∗ ∈ K(H).
+Let {En} be an approximate identity for K(H2). For any finite subset F ⊂ H2 in the unit
+ball of H2, and ǫ > 0, by Lemma 3.1, there is n0 ∈ N such that, for all n ≥ n0,
+∥T −1En(x) − T −1(x)∥ < ǫ/(∥T ∗∥ + 1)(∥K0∥ + 1) for all x ∈ F.
+(e 3.39)
+It follows that, for any y ∈ H1 with ∥y∥ ≤ 1,
+∥⟨(En(T ∗)−1 − (T ∗)−1)(y), x⟩∥
+=
+∥⟨y, (T −1En − T −1)(x)⟩∥
+(e 3.40)
+≤
+∥y∥∥∥T −1En(x) − T −1(x)∥
+(e 3.41)
+<
+ǫ/(∥T ∗∥ + 1)(∥K0∥ + 1)
+(e 3.42)
+12
+
+for all x ∈ F. Thus
+∥⟨T ∗En(T ∗)−1(y) − y, x⟩∥
+=
+∥⟨T ∗En(T ∗)−1(y) − T ∗(T ∗)−1(y), x⟩∥
+(e 3.43)
+<
+ǫ/(∥K0∥ + 1).
+(e 3.44)
+Hence
+∥⟨K0T ∗En(T ∗)−1(y) − K0(y), x⟩∥ < ǫ.
+(e 3.45)
+Since En(T ∗)−1 = En(T −1)∗ ∈ K(H), we have En(T ∗)−1(H1) ⊂ H2. Also, by (1) of Lemma 3.4,
+K0(H1) is dense in H1. It follows from (e 3.45) that K0T ∗(H2) is A-weakly dense in H1.
+Remark 3.7. Theorem 3.6 may provide a correction to 2.2 of [33] (which was not used there).
+As the reader may expect that Theorem 3.6 will lead some discussion about the Cuntz
+semigroups. Indeed we will discuss this in next section. But let us end this section with the
+following corollary.
+Definition 3.8. Let A be a C∗-algebra and b ∈ A+. Denote by Her(b) = bAb the hereditary
+C∗-subalgebra of A. Suppose also a ∈ A+. Let us write a
+∼< b if there is x ∈ A such that x∗x = a
+and xx∗ ∈ Her(b) (see [15] and this notation in [32]).
+Corollary 3.9. Let A be a C∗-algebra and let a, b ∈ A+. Suppose that H1 = aA and H2 = bA.
+Then a
+∼< b, if there is T ∈ B(H2, H1) whose range is A-weakly dense, and if a
+∼< b, then there
+is T ∈ B(H2, H1) which has dense range.
+Proof. Suppose that a
+∼< b, i.e., there is x ∈ A such that
+x∗x = a and xx∗ ∈ Her(b) = bAb.
+(e 3.46)
+Then, for any c ∈ A, (x∗bc)(x∗bc)∗ = x∗bcc∗bx ∈ aAa. It follows that x∗bc ∈ aA = H2 for all
+c ∈ A. Define T : H2 → H1 by T(bc) = x∗bc for all c ∈ A. Then T ∈ B(H2, H1). Since x∗x = a,
+by Dini’s theorem, x∗b1/nx converges to a in norm. Since T is a bounded module map, we
+conclude that T(H2) is dense in H1.
+Now we assume that there is T ∈ B(H2, H1) whose range is A-weakly dense in H1. Note
+that a is a strictly positive element of K(H1) and b is a strictly positive element of K(H2),
+respectively. By Proposition 3.4, b(H2) is dense in H2. It follows that Tb(H2) is A-weakly dense
+in H1. By (2) of Proposition 3.4, c = (Tb)(Tb)∗ is a strictly positive element of Her(a). Write
+(Tb)∗ = uc1/2 as polar decomposition in A∗∗. Then uy ∈ A for all y ∈ aA. Put x = ua1/2. Then
+x∗x = a and xx∗ ∈ K(H2) = Her(b).
+(e 3.47)
+4
+Equivalence classes of Hilbert modules and stable rank one
+Now let us discuss the possibility to use quasitraces to measure Hilbert A-modules. We will
+consider the question stated in 4.5. Moreover, we will discuss the case that unitary equivalence
+classes of Hilbert A-submodules can be determined by the Cuntz semigroup.
+Definition 4.1. Let A be a σ-unital C∗-algebra. Then one may identify A ⊗ K with K(HA).
+Denote by CH(A) the unitary equivalence classes [H] of countably generated Hilbert A-modules
+(where H is a countably generated Hilbert A-module).
+13
+
+One may define [H1] + [H2] to be the unitarily equivalent class [H1 ⊕ H2]. Then CH(A)
+becomes a semigroup.
+Let a, b ∈ (A ⊗ K)+. We write a≈∼b, if there exists x ∈ A ⊗ K such that x∗x = a and xx∗ is
+a strictly positive element of Her(b). In the terminology of [40], a ≈∼ b is the same as a ∼ b′ ∼= b
+in [40].
+Next proposition states that “≈∼” is an equivalence relation.
+Let a ∈ (A ⊗ K)+. Define Ha = a(HA). Let H be a countably generated Hilbert A-module.
+Then by [39] (see Proposition 3.2, for convenience), K(H) is σ-unital. We may view H as a
+Hilbert A-submodule of HA = l2(A). Let b ∈ K(H) be a strictly positive element of K(H).
+Then, by Proposition 3.4, Hb := bHA = bH = H. If c ∈ K(H) is another strictly positive
+element, then Hc = Hb.
+Let pa, pb be the open projections corresponding to a and b (in (A ⊗ K)∗∗), respectively.
+Define pa ≈cu pb if there is v ∈ (A ⊗ K)∗∗ such that v∗v = pa and vv∗ = pb and, for any
+c ∈ Her(a) and d ∈ Her(b), vc, v∗d ∈ A ⊗ K. See also [15] and [16].
+We would like to remind the reader of the following statement.
+Proposition 4.2 (Proposition 4.3 and 4.2 of [40]). Let A be a σ-unital C∗-algebra and a, b ∈
+(A ⊗ K)+. Then the following are equivalent.
+(1) a ≈∼ b;
+(2) pa ≈cu pb and
+(3) [Ha] = [Hb] in CH(A).
+Moreover, “ ≈∼ ” is an equivalence relation.
+4.3. By subsection 4 of [9], there are examples of stably finite and separable C∗-algebras A
+which contain positive elements a ∼ b in Cu(A) but a ̸≈∼ b.
+4.4. (1) Let A be a σ-unital C∗-algebra of stable rank one. By Theorem 3 of [11], a ∼ b in
+Cu(A) if and only if Ha and Hb are unitarily equivalent. One may ask whether the converse
+holds. Theorem 4.9 provides a partial answer to the question.
+(2) Let A be a σ-unital simple C∗-algebra and Ped(A) be its Pedersen ideal.
+Let a ∈
+Ped(A) ∩ A+. Then Her(a) ⊗ K ∼= A ⊗ K, by Brown’s stable isomorphism theorem ([4]). One
+notes that Her(a) is algebraically simple. To study the Cuntz semigroup of A, or the semigroup
+CH(A), one may consider Her(a) instead of A.
+(3) Let us now assume that A is algebraically simple. Let �
+QT(A) be the set of all 2-quasi-
+traces defined on the Pedersen ideal of A⊗K. Let QT(A) be the set of 2-quasi-traces τ ∈ �
+QT(A)
+such that ∥τ|A∥ = 1. Let us assume that QT(A) ̸= ∅. It follows from Proposition 2.9 of [20] that
+0 ̸∈ QT(A)
+w. Recall that QT(A)
+w is a compact set (see Proposition 2.9 of [20]).
+Denote by Aff+(QT(A)
+w) the set of all continuous affine functions f on the compact con-
+vex set QT(A)
+w such that f = 0, or f(τ) > 0 for all τ ∈ QT(A)
+w. Let LAff+(QT(A)
+w)
+be the set of those lowe-semi-continuous affine functions f on QT(A)
+w such that there are
+fn ∈ Aff+(QT(A)
+w) such that fn ր f (point-wisely). We allow f has value ∞.
+(4) In what follows, for any 0 < δ < 1, denote by fδ a function in C([0, ∞)) with 0 ≤ fδ(t) ≤ 1
+for all t ∈ [0, ∞), fδ(t) = 0 if t ∈ [0, δ/2], fδ(t) = 1 if t ∈ [δ, ∞) and fδ(t) is linear on (δ/2, δ).
+Let a, b ∈ (A ⊗ K)+. Define
+dτ(a) = lim
+n→∞ τ(f1/n(a)) for all τ ∈ �
+QT(A).
+Note f1/n(a) ∈ Ped(A)+ for each n ∈ N.
+14
+
+(5) Let A be an algebraically simple C∗-algebra with QT(A) ̸= ∅. We say A has strict
+comparison (of Blackdar, see [3]), if, for any a, b ∈ (A ⊗ K)+, dτ(a) < dτ(b) for all τ ∈ QT(A)
+w
+implies that a ≲ b in Cu(A).
+(6) A is said to have finite radius of comparison, if there is 0 < r < ∞ such that, for any
+a, b ∈ (A ⊗ K)+, a ≲ b, whenever dτ(a) + r < dτ(b) for all τ ∈ QT(A)
+w (see [50]).
+If A is σ-unital simple which is not algebraically simple, we may pick a nonzero element
+e ∈ Ped(A)+ and consider Her(a) ⊗ K ∼= A ⊗ K. Then we say that A has strict comparison (or
+has finite radius comparison), if Her(e) does. It should be noted that this definition does not
+depend on the choice of e. Note that, in both cases above, we always assume that �
+QT(A) ̸= {0}.
+4.5. In this section, we will consider the following question: Suppose that dτ(b) is much larger
+than dτ(a) for all τ ∈ QT(A)
+w. Does it follow that a
+∼< b?, or equivalently, that Ha is unitarily
+equivalent to a Hilbert A-submodule of Hb?
+4.6. (7) Define a map Γ : Cu(A) → LAff+(QT(A)
+w) by Γ([a])(τ) = �
+[a](τ) = dτ(a) for all
+τ ∈ QT(A)
+w.
+(8) It is proved by M. Rordam [46] that, for a unital finite simple C∗-algebra A, if A
+is Z-stable then A has stable rank one. Robert [44] showed that any σ-unital simple stably
+projectionless Z-stable C∗-algebra A has almost stable rank one. Recently, it is shown that,
+any σ-unital finite simple Z-stable C∗-algebra has stable rank one (see [21]).
+A part of the Toms-Winter conjecture states that, for a separable amenable simple C∗-
+algebra A, A is Z-stable if and only if A has strict comparison. It follows from [46] and [18]
+that if A is Z-stable, then A has strict comparison and Γ is surjective. The remaining open
+question is whether a separable amenable simple C∗-algebra A with strict comparison is always
+Z-stable. There are steady progresses to resolve the remaining problem ([28], [47], [52], [55],
+[48], [14], [36] and [37], for example).
+Since strict comparison is a property for Cu(A), one may ask the question whether a separable
+amenable simple C∗-algebra A whose Cu(A) behaves like a separable simple Z-stable C∗-algebra
+is in fact Z-stable. To be more precise, let us assume that A is a separable simple C∗-algebra
+such that Cu(A) = V (A) ⊔ (LAff+(�
+QT(A)) \ {0}) = CH(A), where V (A) is the sub-semigroup
+of Cu(A) whose elements are represented by projections, i.e., A has strict comparison, Γ is
+surjective and Cu(A) = CH(A). The question is whether such A is Z-stable, if we also assume A
+is amenable. Theorem 4.9 shows that such C∗-algebras always have stable rank one, by showing
+that these C∗-algebras have tracial approximate oscillation zero (see 4.7). It should be also
+mentioned that if A is a separable unital simple C∗-algebra with stable rank one, then Γ is
+surjective (see [48] and [2]).
+To state the next lemma, let us recall the definition of tracial approximate oscillation zero.
+Definition 4.7. Let A be a C∗-algebra with �
+QT(A) \ {0} ̸= ∅. Let S ⊂ �
+QT(A) be a compact
+subset. Define, for each a ∈ (A ⊗ K)+,
+ω(a)|S
+=
+lim
+n→∞ sup{dτ(a) − τ(f1/n(a)) : τ ∈ S}
+(e 4.48)
+(see A1 of [17] and Definition 4.1 of [20]). We will assume that A is algebraically simple and only
+consider the case that S = QT(A)
+w, and in this case, we will write ω(a) instead of ω(a)|QT(A)
+w,
+in this paper. It should be mentioned that ω(a) = 0 if and only if dτ(a) is continuous (and
+finite) on QT(A)
+w.
+In this case, (when ∥a∥
+2,QT (A)w < ∞, for example, a ∈ Ped(A ⊗ K)), we write ΩT(a) =
+ΩT (a)|S = 0, if there is a sequence bn ∈ Ped(A ⊗ K) ∩ Her(a)+ such that ∥bn∥ ≤ ∥a∥,
+lim
+n→∞ ω(bn) = 0 and
+lim
+n→∞ ∥a − bn∥
+2,QT (A)w = 0,
+(e 4.49)
+15
+
+where ∥x∥
+2,QT (A)w = sup{τ(x∗x)1/2 : τ ∈ QT(A)
+w} (see Proposition 4.8 of [20]). If ω(a) = 0,
+then ΩT (a) = 0 (see the paragraph after Definition 4.7 of [20]).
+Even if A is not algebraically simple (but σ-unital and simple), we may fix a nonzero element
+e ∈ Ped(A) ∩ A+ (so that Her(e) ⊗ K ∼= A ⊗ K) and choose S = QT(Her(e))
+w. Then, for any
+a ∈ (A ⊗ K)+ with ∥a∥2,S < ∞, we write ΩT (a) = 0 if ΩT (a)|S = 0 (this does not depend on the
+choice of e, see Proposition 4.9 of [20]).
+A σ-unital simple C∗-algebra A is said to have tracial approximate oscillation zero, if ΩT (a) =
+0 for every positive element a ∈ Ped(A ⊗ K).
+Lemma 4.8. Let A be a σ-untal algebraically simple C∗-algebra with QT(A) ̸= ∅, strict com-
+parison and surjective Γ. For any a ∈ (A ⊗ K)+, there is b ∈ (A ⊗ K)+, such that Γ([a]) = Γ([b])
+in Cu(A) such that ΩT(b) = 0.
+Proof. Recall that we assume that QT(A) ̸= ∅. Let g ∈ LAff+(QT(A)
+w) be such that �
+[a] = g.
+There is a sequence of increasing gn ∈ Aff+(QTA) such that gn ր g (pointwisely).
+Put f1 = g1 − (1/4)g1 ∈ Aff+(QT(A)
+w) and α1 = 1/4. Then g2(τ) > f1(τ) for all τ ∈
+QT(A)
+w. Choose 0 < α2 < 1/22+1 such that (1 − α2)g2(τ) > f1(τ) for all τ ∈ QT(A)
+w.
+Put f2 = (1 − α2)g2. Suppose that 0 < αi < 1/22+i is chosen, i = 1, 2, ..., n, such that (1 −
+αi+1)gi+1(τ) > (1−αi)gi(τ) for all τ ∈ QT(A)
+w, i = 1, 2, ..., n−1. Then gn+1(τ) > (1−αn)gn(τ)
+for all τ ∈ QT(A)
+w. Choose 0 < αn+1 < 1/22+n such that (1 − αn+1)gn+1(τ) > (1 − αn)gn(τ)
+for all τ ∈ QT(A)
+w. Define fn+1 = (1 − αn+1)gn+1 ∈ Aff+(QT(A)
+w). By induction, we obtain a
+strictly increasing sequence fn ∈ Aff+(QT(A)
+w) such that fn ր g.
+Define h1 = f1 and hn = fn − fn−1 for n ≥ 2. Then hn ∈ Aff+(QT(A)
+w) and g = �∞
+n=1 hn
+(converges point-wisely). Since Γ is surjective, there are an ∈ (A ⊗ K)+ such that �
+[an](τ) =
+dτ(an) = hn, n ∈ N. We may assume that aiaj = 0 if i ̸= j. Define b ∈ (A ⊗ K)+ by b =
+�∞
+n=1 an/n.
+Then �[b] = g. In other words, Γ([a]) = Γ([b]).
+For each m ∈ N, since �
+[an] is continuous on QT(A)
+w, (by Lemma 4.5 of [20]), choose en,m ∈
+Her(an) with 0 ≤ en,m ≤ 1 such that {en,m}m∈N forms an approximate identity (for Her(an)),
+dτ(an) − τ(en,m) < 1/2n+m for all τ ∈ QT(A)
+w, and ω(en,m) < 1/2n+m.
+(e 4.50)
+Define en = �n
+j=1 ej,n. Since aiaj = 0, if i ̸= j, we have that 0 ≤ en ≤ 1, and {en} forms an
+approximate identity for Her(b). By (2) of Proposition 4.4 of [20], we compute that
+ω(en) ≤
+n
+�
+j=1
+ω(ej,n) <
+n
+�
+j=1
+1/2j+n = 1/2n.
+(e 4.51)
+Hence limn→∞ ω(en) = 0 and, by Proposition 5.7 of [20], ΩT (b) = 0.
+Theorem 4.9. Let A be a σ-unital simple C∗-algebra with strict comparison and surjective Γ
+(which is not purely infinite). Then the following are equivalent:
+(1) for any a, b ∈ (A ⊗ K)+, [a] = [b] in Cu(A) if and only if a ≈∼ b;
+(2) A has stable rank one;
+(3) Cu(A) = CH(A);
+(4) for any a, b ∈ (A ⊗ K)+, [a] = [b] in Cu(A) if and only if Hilbert A-module Ha and Hb
+are unitarily equivalent.
+Proof. (2) ⇒ (3) follows from Theorem 3 of [11].
+(3) ⇔ (4) follows from the definition.
+16
+
+(1) ⇔ (4) follows from Proposition 4.2.
+(3) ⇒ (2): We will show that A has tracial approximate oscillation zero (see Definition 5.1
+of [20]).
+Let a ∈ (Ped(A ⊗ K)) with 0 ≤ a ≤ 1. Suppose that Γ([a]) is continuous. Then ω([a]) = 0.
+Hence ΩT (a) = 0. Now suppose that Γ([a]) is not continuous.
+In particular, [a] cannot be
+represented by a projection. By Lemma 4.8, there exists b ∈ (A ⊗ K)+ such that ΩT(b) = 0 and
+Γ([a]) = Γ([b]). Since we now assume that Γ([a]) is not continuous neither is Γ([b]). Hence both
+[a] and [b] are not represented by projections. As we assume that A has strict comparison, this
+implies that [a] = [b].
+Since (3) holds, a(HA) is unitarily equivalent to b(HA). We have shown (3) ⇔ (4) and (1) ⇔
+(4). By Proposition 4.2, there is a partial isometry v ∈ (A⊗K)∗∗ such that v∗cv ∈ a(A ⊗ K)a for
+all c ∈ b(A ⊗ K)b and v∗bv is a strict positive element of a(A ⊗ K)a. It follows that ΩT(a) = 0.
+Since this holds for every a ∈ Ped(A ⊗ K), A has tracial approximate oscillation zero.
+By
+Theorem 9.4 of [20], A has stable rank one, i.e., (2) holds.
+Let us end this section with the following partial answer to the question in 4.5. Note that
+dτ(b) is as large as one can possibly have.
+Theorem 4.10. Let A be a σ-unital simple C∗-algebra with finite radius of comparison. Suppose
+that a, b ∈ (A ⊗ K)+ such that dτ(b) = ∞ for all τ ∈ �
+QT(A) \ {0}. Then
+(1) a
+∼< b and Ha is unitarily equivalent to an orthogonal summand of Hb,
+(2) Hb ∼= HA as Hilbert A-module, and
+(3) Her(b) ∼= A ⊗ K.
+Proof. Put B = b(A ⊗ K)b. Let us show that B is stable.
+We will use the characterization
+of stable C∗-algebras of [25].
+Moreover, we use the following fact: If A has finite radius of
+comparison and c, b ∈ (A ⊗ K)+ such that dτ(d) < ∞ and dτ(b) = ∞ for all τ ∈ �
+QT(A) \ {0},
+then d ≲ b.
+Let c ∈ B+ such that there exists e ∈ B+ such that ec = c. Working in the commutative
+C∗-subalgebra generated by c and e, we conclude that, if 0 < δ < 1/2, fδ(e)c = c.
+Let b1 := (1 − f1/8(e))1/2b(1 − f1/8(e))1/2. Then
+b1c = cb1 = 0.
+(e 4.52)
+Note that f1/16(e) ∈ Ped(A ⊗ K)+. Therefore
+dτ(f1/8(e)) ≤ τ(f1/16(e)) < ∞ for all τ ∈ �
+QT(A).
+We also have
+b ≲ b1/2(1 − f1/8(e))b1/2 ⊕ b1/2f1/8(e)b1/2.
+(e 4.53)
+Since b1/2f1/8(e)b1/2 ≲ f1/8(e), then dτ(b1/2f1/8(e)b1/2) < ∞ for all τ ∈ �
+QT(A). Hence
+dτ(b1) = dτ(b1/2(1 − f1/8(e))b1/2) = ∞ for all τ ∈ �
+QT(A) \ {0}.
+(e 4.54)
+Since A has finite radius of comparison, as mentioned above, we have that
+f1/8(e) ≲ b1.
+(e 4.55)
+There is, by Lemma 2.2 of [45], x ∈ A ⊗ K such that
+x∗x = f1/16(f1/8(e)) and xx∗ ∈ Her(b1).
+(e 4.56)
+17
+
+Let x = v|x| be the polar decomposition of x in A∗∗. Then ϕ : Her(f1/6(f1/8(e))) → Her(b1) de-
+fined by ϕ(d) = vdv∗ for all d ∈ Her(f1/6(f1/8(e))) is homomorphism. Note that f1/6(f1/8(e)) ≤
+f1/4(e) and f1/4(e)c = c. Put y = vc1/2. Then y ∈ A ⊗ K and y∗y = c and yy∗ ∈ Her(b1). Hence
+c = yy∗ and c ⊥ yy∗.
+(e 4.57)
+Since c is chosen arbitrarily in B+ with the property that there is e ∈ B+ such that ec = c, it
+follows from Theorem 2.1 of [25] that B is stable. By Brown’s stable isomorphism theorem ([4]),
+B ∼= A ⊗ K. This proves (3).
+Since B is stable, there is a sequence of mutually orthogonal nonzero elements b0,n ∈ B+
+(n ∈ N) such that b0,1
+≈∼ b0,n for all n ∈ N and b0 = �∞
+n=1 b0,n/n ∈ B is a strictly positive
+element. We have Hb0,n ∼= Hb0,m for all n, m ∈ N. It follows that Hb = Hb0 ∼= l2(Hb0) as Hilbert
+A-modules. By Proposition 7.4 of [29], l2(Hb0) ∼= l2(A) = HA as Hilbert A-module. This proves
+(2).
+For (1), since we have shown that Hb ∼= HA, we may apply Kasparov’s absorbing theorem
+([26]).
+5
+Projective Hilbert Modules
+The main result of this section is Theorem 5.6 which states that, for separable C∗-algebra A,
+every countably generated Hilbert A-module is projective.
+Note that in the following statement, we use the fact that B(H) = LM(H) (see Theorem
+1.5 of [33]).
+Lemma 5.1. Let A be a C∗-algebra and H be Hilbert A-modules. Suppose that T ∈ B(H) is
+a bounded module map such that T(H) is a Hilbert A-submodule of H and L ∈ LM(K(H)) the
+corresponding left multiplier. Then LK(H) = K(T(H)).
+Proof. Let H1 = T(H) and F(H) be the linear span of bounded module maps of the form θx,y
+for x, y ∈ H. Note that Lb(x) = Tb(x) for all b ∈ K(H) and x ∈ H.
+Let ξ, ζ ∈ H1 ⊂ H. Since T : H → H1 is surjective, there is x ∈ H such that T(x) = ξ. Then
+T ◦ θx,ζ = θξ,ζ. This implies that LF(H) = F(H1). Let H0 = kerT be as a Hilbert submodule of
+H. Then K(H0) ⊂ K(H) is a hereditary C∗-subalgebra of K(H) (see Lemma 2.13 of [33]). Let
+p be the open projection in K(H)∗∗ corresponding to K(H0). Then (working in H∼ if necessary)
+Lp = 0 and Lb = L(1 − p)b for all b ∈ K(H). We identify H/H0 with (1 − p)H (⊂ H∼). Let
+˜T : (1 − p)H → H1 be the bounded module map induced by T which has a bounded inverse
+˜T −1 as T is surjective. Note that ˜T −1(T(z)) = (1 − p)z for all z ∈ H.
+To show that T(K(H)) = K(H1), fix S ∈ K(H1). Let Sn ∈ F(H1) be such that
+limn→∞ ∥Sn − S∥ = 0. Since LF(H) = F(H1), we may choose Fn ∈ F(H) such that LFn = Sn,
+n ∈ N. Then, for any x ∈ H, ˜T −1(LFn(x)) = (1 − p)Fn(x).
+We claim that {(1 − p)Fn} converges in norm to an element of the form (1 − p)b for some
+b ∈ K(H). Note that, for any n, m ∈ N, and any x ∈ H,
+∥(1 − p)Fn(x) − (1 − p)Fm(x)∥ = ∥ ˜T −1(LFn(x) − LFm(x))∥ ≤ ∥ ˜T −1∥∥Sn − Sm∥∥x∥.
+This implies that {(1−p)Fn} is Cauchy in norm and it must converges to an element of the form
+(1 − p)b for some b ∈ K(H), as (1 − p)K(H) is closed. It then follows that, for any b ∈ K(H),
+Lb = L(1 − p)b = lim
+n→∞ L(1 − p)Fn = lim
+n→∞ Sn = S.
+This shows that LK(H) = K(H1).
+18
+
+Definition 5.2. Let H and H1 be Hilbert A-modules and T : H1 → H be a surjective bounded
+module map. Then T induces an invertible bounded map ˜T : H1/kerT → H from Banach A-
+modules H1/kerT onto H. Denote by ˜T −1 : H → H1/kerT the inverse (which is also bounded).
+Theorem 5.3. Let A be a σ-unital C∗-algebra.
+For any countably generated Hilbert A-modules H1, H2, H3, any bounded module maps T1 :
+H1 → H3 and T2 : H2 → H3. Suppose that T1 is surjective, then there is a bounded module map
+T3 : H2 → H1 such that
+T1 ◦ T3 = T2.
+Moreover, ∥T3∥ = ∥ ˜T −1
+1
+◦ T2∥.
+Proof. Without loss of generality, we may assume that ∥T2∥ = 1. Let H4 = H1 ⊕ HA, H5 =
+H2 ⊕ HA and H6 = H3 ⊕ HA. Let T4 = T1 ⊕ idHA : H4 → H6 and T5 = T2 ⊕ idHA : H5 → H6.
+Suppose T1 is surjective. Then T4 is surjective. Suppose that there is a bounded module map
+T ′ : H5 → H4 such that
+T4 ◦ T ′ = T5 and ∥T ′∥ = ∥ ˜T −1
+4
+◦ T5∥
+One computes that
+∥ ˜T −1
+4
+◦ T5∥ = ∥ ˜T −1
+1
+◦ T2∥.
+One also has that
+T4 ◦ T ′|H2 = T5|H2 = T2.
+Since range(T2) ⊂ H3, T4(T ′(H2)) = T5(H2) ⊂ H3. Let P1 : H4 → H1, PHA : H4 → HA
+and P3 : H6 → H3 be the orthogonal projections.
+Then P3 ◦ T4 ◦ T ′|H2 = T2. Write T ′ =
+P1T ′ + (1H4 − P1)T ′. Note that 1H4 − P1 = PHA and P3T4(1H4 − P1) = 0. Then
+P3 ◦ T4 ◦ T ′|H2
+=
+P3 ◦ T4 ◦ P1 ◦ T ′|H2 + P3 ◦ T4 ◦ (1H4 − P1) ◦ T ′|H2
+(e 5.58)
+=
+P3 ◦ T4 ◦ P1 ◦ T ′|H2 + 0
+(e 5.59)
+=
+T1 ◦ P1 ◦ T ′|H2.
+(e 5.60)
+Define T = P1 ◦ T ′|H2. Then, by (e 5.60),
+T1 ◦ T = T2 and ∥T∥ = ∥P1 ◦ T ′|H2∥ = ∥ ˜T −1
+1
+◦ T2∥.
+(e 5.61)
+Therefore, without loss of generality, we may assume that H1 ∼= H2 ∼= H3 = HA. Put B =
+A⊗K. B is a σ-unital C∗-algebra. We view T1 as a bounded module map from HA onto HA. Let
+H0 = ker T1. Then H0 is a Hilbert A-submodule of HA and K(H0) ⊂ K(HA) = B is a hereditary
+C∗-subalgebra (see Lemma 2.13 of [33]). Let p ∈ B∗∗ be the open projection corresponding to
+the hereditary C∗-subalgebra K(H0). Let L1 ∈ LM(B) be given by the bounded module map
+T1 (see Theorem 1.5 of [31]). Note that L1b(z) = T1(b(z)) for all z ∈ HA and b ∈ B. For any
+a ∈ K(H0) ⊂ B, L1a = 0. It follows that L1p = 0 and L1(1 − p) = L1.
+Let J = K(H0)B be the right ideal of K(HA) = B. Consider the quotient Banach B-
+module B/J. Note that the module map ¯b �→ (1 − p)b is an isometry.
+So we may identify
+B/J with (1 − p)B. Define ˜L1 : B/J → B by ˜L1((1 − p)b) = L1b for b ∈ B. Viewing B
+as a Hilbert B-module, one has K(B) = B. By Lemma 5.1, viewing B as Hilbert B-module,
+L1B = L1K(B) = K(B) = B. It follows that ˜L1 is also a surjective map and hence has bounded
+inverse as a bounded linear map.
+Denote its inverse as ˜L−1
+1 . Note that ˜L−1
+1 L1b = (1 − p)b
+for b ∈ B (recall that L1 is surjective).
+In particular, ˜L−1
+1
+is a bounded module map from
+K(HA) = B to (1 − p)B.
+We may also use the identification HA/H0 = (1 − p)HA. Now T1 induces a bounded module
+map ˜T1 from (1 − p)HA onto HA which has a bounded inverse ˜T −1
+1 . Let L2 ∈ LM(B) be given
+19
+
+by the bounded module map T2. Then the map L2 : B → B given by L2(b) = L2b for all b ∈ B
+is a bounded B-module map from B into B = K(HA).
+Consider the bounded B-module map ˜L−1
+1
+◦ L2 : B → B/J. Then, by 3.11 of [6], there exists
+a bounded module map L : B → B such that
+(1 − p)L = ˜L−1
+1
+◦ L2 and ∥L∥ = ∥˜L−1
+1
+◦ L2∥.
+(e 5.62)
+It follows that ˜L1(1 − p)L = L2. Recall that ˜L1(1 − p) = L1. Therefore
+L1 ◦ L = L2.
+(e 5.63)
+We identify L with a left multiplier in LM(B) (see Theorem 1.5 of [31]). Let T : H1 = HA →
+HA = H3 be the bounded module map given by L (see again Theorem 1.5 of [31]). Then we
+obtain that
+T1 ◦ T = T2 and ∥T∥ = ∥ ˜T −1
+1
+◦ T2∥.
+(e 5.64)
+The commutative diagrams on B level may be illustrated as follows (with π : B → B/J the
+quotient map given by (1 − p))
+B
+↓L2
+ցL
+B
+ևL1
+B
+�L−1
+1
+↕�L1
+ւπ
+B/J
+and
+B
+−→L
+B
+�L−1
+1
+L2 ↓
+ւπ
+B/J
+.
+Corollary 5.4. Let A be a σ-unital C∗-algebra and let
+0 → H1 →ι H2 →s H3 → 0
+be a short exact sequence of countably generated Hilbert A-modules. Then it splits. Moreover,
+the splitting map j : H3 → H2 has ∥j∥ = ∥˜s−1∥.
+Proof. Consider the diagram:
+H2
+↓s
+H3
+idH3
+։
+H3
+By Theorem 5.3, there is a bounded module map T : H3 → H2 such that
+s ◦ T = idH3 and ∥T∥ = ∥˜s−1∥.
+We need the following lemma which the authors could not locate a reference.
+Lemma 5.5. Let X be a Banach space and let H be a separable Banach space. Suppose that
+T : X → H is a surjective bounded linear map. Then there is a separable subspace Y ⊂ X such
+that TX = H.
+20
+
+Proof. Note that the Open Mapping Theorem applies here. From the open mapping theorem
+(or a proof of it), there is δ > 0 for which T(B(0, a)) is dense in O(0, aδ) for any a > 0, where
+B(0, a) = {x ∈ X : ∥x∥ ≤ a} and O(0, b) = {h ∈ H : ∥h∥ < b}. For each rational number r > 0,
+since H is separable, one may find a countable set Er ⊂ B(0, r) such that T(Er) is dense in
+O(0, rδ). Let Y be the closed subspace generated by ∪r∈Q+Er.
+Let d = δ/2 and let y0 ∈ O(0, d). Then T(Y ∩ B(0, 1/2)) is dense in O(0, d). Choose ξ1 ∈
+Y ∩ B(0, 1/2) such that
+∥y0 − Tξ1∥ < δ/22.
+(e 5.65)
+In particular,
+y1 = y0 − Tξ1 ∈ O(0, δ/22).
+(e 5.66)
+Since T(Y ∩ B(0, 1/22)) is dense in O(0, δ/22), one obtains ξ2 ∈ Y ∩ B(0, 1/22) such that
+∥y1 − Tξ2∥ < δ/23.
+(e 5.67)
+In other words,
+y2 = y1 − Tξ2 = y0 − (Tξ1 + Tξ2) ∈ O(0, δ/23).
+(e 5.68)
+Continuing this process, one obtains a sequence of elements {ξn} ⊂ Y for which ξn ∈ B(0, 1/2n)
+and
+∥y0 − (Tξ1 + Tξ2 + · · · + Tξn)∥ < δ/2n+1, n = 1, 2, ....
+(e 5.69)
+Define ξ0 = �∞
+n=1 ξn. Note that the sum converges in norm and therefore ξ0 ∈ Y. By the
+continuity of T,
+Tξ0 = y0.
+(e 5.70)
+This implies that T(Y ) ⊃ O(0, d). It follows that T(Y ) = H.
+Theorem 5.6. Let A be a separable C∗-algebra. Then every countably generated Hilbert A-
+module is projective in the following sense:
+For any Hilbert A-modules H1, H2, H3, any bounded module maps T1 : H1 → H3 and T2 :
+H2 → H3. Suppose that T1 is surjective and H2 is countably generated, then there is a bounded
+module map T3 : H2 → H1 such that
+T1 ◦ T3 = T2.
+Moreover, ∥T3∥ = ∥ ˜T −1
+1
+◦ T2∥.
+The statement above may be illustrated as follows:
+H2
+H3
+H1
+T2
+T3
+T1
+(e 5.71)
+21
+
+Proof. Let H′
+3 be the Hilbert A-module generated by T2(H2). Since H2 is countably generated,
+so is H′
+3. Since A is separable, H′
+3 is separable. Let H′
+1 = T −1(H′
+3), the pre-image of H′
+3 under
+T1. So T1|H′
+1 : H′
+1 → H′
+3 is surjective. Then, by Lemma 5.5, there is a separable Banach subspace
+of S ⊂ H′
+1 such that T1(S) = H′
+3. Let {xn} be a dense subset of S. Define H′′
+1 to be the closure of
+span{xna : a ∈ A, n = 1, 2, ....}. Then H′′
+1 is a countably generated Hilbert A-module. Moreover,
+S ⊂ H′′
+1 ⊂ H′
+1 ⊂ H1. But T1(S) = H′
+3. Therefore T1(H′′
+1 ) = H′
+3. We have the following diagram:
+H2
+H′′
+3
+H′′
+1
+T2
+T1
+(e 5.72)
+Now H′
+1, H2 and H′′
+3 are all countably generated. By 5.3, there exists T3 : H2 → H′′
+1 ⊂ H1 such
+that
+T1 ◦ T3 = T2 and ∥T3∥ = ∥ ˜T −1
+1
+◦ T2∥.
+(e 5.73)
+Corollary 5.7. Let A be a separable C∗-algebra and let
+0 → H1 →ι H2 →s H3 → 0
+be a short exact sequence of Hilbert A-modules. Suppose that H3 is countably generated. Then
+the short exact sequence splits. Moreover, the splitting map j : H3 → H2 has ∥j∥ = ∥˜s−1∥.
+Corollary 5.8. Let A be a separable C∗-algebra.
+(1) Suppose that H is a countably generated Hilbert A-module. Then, for any short exact
+sequence of Hilbert A-modules
+0 → H0
+j→ H1
+s→ H2 → 0,
+one has the following short exact sequence:
+0 → B(H, H0)
+j∗
+→ B(H, H1) s∗
+→ B(H, H2) → 0,
+where j∗(ϕ) = j ◦ ϕ for ϕ ∈ B(H, H0) and s∗(ψ) = s ◦ ψ for ψ ∈ B(H, H1).
+(2) Suppose that H2 is countably generated. Then, for any Hilbert A-module H, and any
+short exact sequence of Hilbert A-modules:
+0 → H0
+j→ H1
+s→ H2 → 0,
+One has the splitting short exact sequence
+0 → B(H, H0)
+j∗
+→ B(H, H1) s∗
+→ B(H, H2) → 0,
+(e 5.74)
+Proof. For (1), let us only show that s∗ is surjective. Let ψ ∈ B(H, H2). Since s : H1 → H2 is
+surjective, by Theorem 5.6, H is projective. There is ˜ψ ∈ B(H, H1) such that s ◦ ˜ψ = ψ. This
+implies that s∗ is surjective.
+For (2), by Corollary 5.7, there exists ψ : H2 → H1 such that s ◦ ψ = idH2. For each
+T ∈ B(H, H2), define ψ ◦ T : H → H1. Then s ◦ (ψ ◦ T) = T. Hence s∗ is surjective. This
+implies that (e 5.74) is a short exact. Moreover, map ψ above also gives a splitting map ψ∗ :
+B(H, H2) → B(H, H1).
+22
+
+Definition 5.9. A finitely generated free Hilbert A-module is a Hilbert A-module with the form
+H = e1A⊕e2A⊕· · · ⊕enA, where ⟨ei, ei⟩ = pi is a projection in A, i = 1, 2, ..., n. Such a module
+is always self-dual (see, for example, Proposition 2.11).
+There is a notion of torsion free modules. Every Hilbert A-module is torsion free in the
+following sense. Let x ∈ H be a non-zero element and xa = 0 for some a ∈ A. Then a must be
+a left zero divisor. In fact, if xa = 0, then a⟨x, x⟩a = 0, or ⟨x, x⟩1/2a = 0. Then a must be a left
+zero divisor.
+One may define a Hilbert A-module H to be flat, if, for any bounded module map T : F → H,
+where F is a finitely generated free Hilbert A-module, there exists a free Hilbert A-module G
+and a bounded module map ϕ : F → G and a bounded module map ψ : G → H such that
+ϕ(kerT) = 0 and ψ ◦ ϕ = T as described as the following commutative diagram:
+G
+րϕ
+ցψ
+kerT ֒→
+F
+T
+−→
+H
+ϕ(kerT) = 0
+This is equivalent to say the following: for any x1, x2, ..., xm ∈ H, if there are ai ∈ A, 1 ≤ i ≤ m,
+such that �m
+i=1 xiai = 0, there must be some integer n, yj ∈ H, 1 ≤ j ≤ m and bi,j ∈ A,
+1 ≤ j ≤ m, 1 ≤ i ≤ n, such that �m
+i=1 aibi,j = 0 and xi = �n
+j=1 ai,jyj, 1 ≤ i ≤ m.
+Proposition 5.10. Let A be a σ-unital C∗-algebra and H be any Hilbert A-module. Suppose
+that F is a self-dual Hilbert A-module and T ∈ B(F, H). Then there are bounded module maps
+ϕ : F → F and ψ : F → H such that ψ ◦ ϕ = T and ϕ|kerT = 0. In particular, every Hilbert
+A-module is flat.
+Proof. Since F is self-dual, Therefore T ∗ maps H into F (instead into F ♯), or T ∗ ∈ B(H, F). In
+other words, T ∈ L(F, H). Put H1 = F ⊕ H. Define S ∈ B(H1) by
+S(f ⊕ h) = 0 ⊕ T(f) for all ∈ F and h ∈ H.
+Then S∗ ∈ B(H1). Let S = V |S| be the polar decomposition of S in L(H)∗∗. Then V |S|1/2, |S|1/2 ∈
+L(H1). In other words, |T|1/2 ∈ L(F) and V |T|1/2 ∈ L(F, H).
+Now define ϕ : F → F by ϕ = |T|1/2 and ψ : F → H by ψ = V |T|1/2. Then ϕ|kerT = 0 and
+ψ ◦ ϕ = V |T| = T.
+If F is a finitely generated free Hilbert A-module, then F is self-dual. So the above applies.
+Consequently H is flat.
+6
+Sequential approximation
+Let A be a C∗-algebra and H0 ⊂ H be Hilbert A-modules. Suppose that ϕ : H0 → A is a
+bounded module map. Consider the Hahn-Banach type of extension question: whether there
+is a bounded module map ψ : H → A such that ψ|H0 = ϕ (and ∥ψ∥ = ∥ϕ∥). If H0 ⊂ H is
+an arbitrary pair of Hilbert A-modules. This is to ask whether A is an injective Hilbert A-
+module with bounded module maps as morphisms. By Theorem 3.8 of [33], if A is a monotone
+complete C∗-algebra, then A is an injective Hilbert A-module. Conversely, if A is an injective
+Hilbert A-module, then A must be an AW ∗-algebra (see Theorem 3.14 of [33]). In fact, injective
+Hilbert A-modules are rare. It may never happen when A is a separable but infinite dimensional
+C∗-algebra (see [33] for some further discussion).
+In what follows, when x, y ∈ H and ǫ > 0, we may write x ≈ǫ y if ∥x − y∥ < ǫ.
+23
+
+Let H be a Hilbert A-module and H♯ = B(H, A), the Banach A-module of all bounded
+A-module maps from H to A. Recall that H♯ ̸= H in general (see the end of 2.1). However, let
+us include the following approximation result.
+Theorem 6.1. Let H be a Hilbert A-module and ϕ : H → A be a bounded module map. Then,
+for any finite subset F ⊂ H, and any ǫ > 0, there exists x ∈ H such that
+∥ϕ(ξ) − ⟨x, ξ⟩∥ < ǫ for all ξ ∈ F and ∥x∥ ≤ ∥ϕ∥.
+(e 6.75)
+Proof. We may assume that ∥ϕ∥ ̸= 0. Let H1 = H ⊕ A and P1, PA ∈ L(H1) be the projections
+from H1 onto H and onto A, respectively. Define T ∈ B(H1) by T = ϕ ◦ P1. Note that, for
+x ∈ H, T(x) = ϕ(x) and ∥T∥ = ∥ϕ∥.
+Let F = {ξ1, ξ2, ..., ξk} ⊂ H and ǫ > 0. Without loss of generality, we may assume that
+0 ̸= ∥ξi∥ ≤ 1, 1 ≤ i ≤ k. Let {Eλ} be an approximate identity for K(H). By Lemma 3.1, there
+exists λ0 such that, for all λ ≥ λ0,
+∥Eλ(x) − x∥ < ǫ/4(∥ϕ∥ + 1) for all x ∈ F
+(e 6.76)
+Fix λ ≥ λ0. Applying Theorem 1.5 of [31], B(H1) = LM(K(H1)). It follows that TEλP1 ∈
+K(H1). Note also that ∥TEλP1∥ ̸= 0. Therefore, there are y′
+1, y′
+2, ..., y′
+m, z1, z2, ..., zm ∈ H1 such
+that
+∥TEλP1 −
+m
+�
+j=1
+θy′
+j,zj∥ < ǫ/8, 1 ≤ j ≤ m.
+Then
+∥
+m
+�
+j=1
+θy′
+j,zj∥ ≤ ∥TEλP1∥ + ǫ/8.
+(e 6.77)
+Put β =
+∥TEλP1∥
+∥TEλP1∥+ǫ/8 and yj = βy′
+j, 1 ≤ j ≤ m. Then
+m
+�
+j=1
+θyj,zj = β
+m
+�
+j=1
+θy′
+j,zj and ∥
+m
+�
+j=1
+θyj,zj∥ ≤ ∥TEλP1∥ ≤ ∥ϕ∥.
+(e 6.78)
+Moreover,
+∥TEλP1 −
+m
+�
+j=1
+θyj,zj∥ ≤ ∥TEλP1 −
+m
+�
+j=1
+θy′
+j,z′
+j∥ + (1 − β)∥
+m
+�
+j=1
+θy′
+j,z′
+j∥
+(e 6.79)
+< ǫ/8 + (1 − β)(∥TEλP1∥ + ǫ/8) = ǫ/8 + ǫ/8 = ǫ/4.
+(e 6.80)
+Denote L = �m
+j=1 θyj,zj. Then, by (e 6.78), ∥L∥ ≤ ∥ϕ∥. Note that P1 ∈ L(H1). Therefore, for
+all z ∈ H, θyj,zjP1(z) = yj⟨zj, P1z⟩ = yj⟨P1zj, z⟩ = θyj,P1zj(z), 1 ≤ j ≤ m. By replacing zj by
+P1zj, we may assume that zj ∈ H. Recall PATEλP1 = TEλP1. Hence PAyj ∈ A, 1 ≤ j ≤ m.
+Put aj = (P1yj)∗, 1 ≤ j ≤ m. Then, by (e 6.76), (e 6.80),
+ϕ(ξi) = T(ξi) ≈ǫ/4 TEλP1(ξi) ≈ǫ/4 L(ξi) =
+m
+�
+j=1
+a∗
+j⟨zj, ξi⟩.
+(e 6.81)
+Choose x = �m
+j=1 zjaj. Then L(ξi) = ⟨x, ξi⟩, i ∈ N. Hence
+∥ϕ(ξi) − ⟨x, ξi⟩∥ < ǫ,
+i = 1, 2, ..., k.
+(e 6.82)
+24
+
+It remains to show that ∥x∥ ≤ ∥ϕ∥. However, we have
+∥x∥ = ∥⟨x, x/∥x∥⟩∥ = ∥L(x/∥x∥)∥ ≤ ∥L∥ ≤ ∥ϕ∥.
+(e 6.83)
+Remark 6.2. Theorem 6.1 also shows that H may not be A-weakly closed, in the case that
+H ̸= H♯ (see 3.3).
+In the following statement, note that, H may not be a separable Banach space when A is
+not separable.
+Theorem 6.3. Let H be a countably generated Hilbert A-module and ϕ : H → A be a bounded
+module map. Then, there exists xn ∈ H, n ∈ N, such that ∥xn∥ ≤ ∥ϕ∥ and
+lim
+n→∞ ∥ϕ(ξ) − ⟨xn, ξ⟩∥ = 0 for all ξ ∈ H.
+(e 6.84)
+Proof. We need to modify the proof of Theorem 6.1. Let {En} be an approximate identity
+for K(H) (by Theorem 3.2). Let H1 = H ⊕ A, P1, PA and T be as in the proof of Theorem
+6.1. Consider TEnP1 ∈ K(H). As in the proof of Theorem 6.1, we obtain Ln ∈ K(H) with
+∥Ln∥ ≤ ∥TEnP1∥ ≤ ∥ϕ∥ such that
+∥TEnP1 − Ln∥ < 1/2n,
+(e 6.85)
+where Ln = �m(n)
+j=1 θxn,j,yn,j. Put PA(xn,j) = a∗
+n,j, 1 ≤ j ≤ m(n), n ∈ N. For any ǫ > 0 and any
+z ∈ H, we may choose n(x) such that, for all n ≥ n(x),
+∥EnP1(z) − z∥ < ǫ/2(∥ϕ∥ + 1).
+(e 6.86)
+Then, if n ≥ n(x),
+ϕ(z) ≈ǫ/2 TEnP1(y) ≈∥z∥/2n Ln(z) =
+m(n)
+�
+j=1
+a∗
+n,j⟨yn,j, z⟩.
+(e 6.87)
+Choose xn = �m(n)
+j=1 yn,jan,j, n ∈ N. The same proof as the end of the proof of Theorem 6.1
+shows that, for all n ≥ n(x),
+ϕ(z) ≈ǫ/2+∥z∥/2n L(z) = ⟨xn, z⟩ and ∥xn∥ ≤ ∥ϕ∥.
+(e 6.88)
+The theorem then follows.
+As mentioned at the beginning of this section, injective Hilbert A-modules are rare. However,
+one may have some approximate extensions of bounded module maps. Let B0, B1 and C be C∗-
+algebras and ϕ : B0 → C be a contractive completely positive linear map. If C is amenable (or
+B0 is), then, for any ǫ > 0 and any finite subset F ⊂ B0, there exists a contractive completely
+positive linear map ψ : B → C such that
+∥ϕ(b) − ψ(b)∥ < ǫ for all b ∈ F.
+It is very useful feature of amenable C∗-algebras.
+With the same spirit, let us present the
+following approximate extension result:
+25
+
+Theorem 6.4. Let A be a C∗-algebra, H0, H1, H be Hilbert A-modules such that H0 ⊂ H1 and
+ϕ : H0 → H be a bounded module map.
+Then, for any ǫ > 0 and any finite subset F ⊂ H0, there exists a bounded module map
+ψ : H1 → H such that ∥ψ∥ ≤ ∥ϕ∥ and
+∥ψ(x) − ϕ(x)∥ < ǫ for all x ∈ F.
+(e 6.89)
+The above may be illustrated as follows:
+H1
+H0
+H
+ψ
+⟲ε
+ϕ
+on F.
+(e 6.90)
+If, in addition, H0 is countably generated, then there exists ψn : H1 → H, n ∈ N, such that
+∥ψn∥ ≤ ∥ϕ∥ and
+lim
+n→∞ ∥ψn(x) − ϕ(x)∥ = 0 for all x ∈ H0.
+(e 6.91)
+Proof. We may assume that ∥ϕ∥ ̸= 0. Let H2 = H0 ⊕ H and H3 = H1 ⊕ H. We view H2
+as a Hilbert A-submodule of H3. Let P0 ∈ L(H2) be the projection from H2 onto H0, and let
+P1, PH ∈ L(H3) be the projections from H3 onto H1 and onto H, respectively. Define T ∈ B(H2)
+by T = ϕ ◦ P0. Let F ⊂ H0 be a finite subset and ǫ > 0.
+Let {Eλ} be an approximate identity for K(H0). By Lemma 3.1, there exists λ0 such that,
+for all λ ≥ λ0,
+∥Eλ(x) − x∥ < ǫ/(∥ϕ∥ + 1) for all x ∈ F.
+(e 6.92)
+Applying Theorem 1.5 of [31], we have that B(H2) = LM(K(H2)). It follows that (for a fixed
+λ ≥ λ0) TEλP0 ∈ K(H2). Applying Lemma 2.13 of [33], we obtain Tλ ∈ K(H3) such that
+∥Tλ∥ = ∥TEλP0∥ and Tλ|H2 = TEλP0.
+(e 6.93)
+Put ψ = PHTλ. Note that (recall that T(H2) ⊂ H)
+∥ψ∥ = ∥TEλP0∥ and ψ|H2 = TEλP0.
+(e 6.94)
+We may view ψ as a bounded module map from H1 to H with ∥ψ∥ ≤ ∥T∥ = ∥ϕ∥. We estimate
+that, for x ∈ F, by (e 6.92) and (e 6.94),
+∥ψ(x) − ϕ(x)∥
+=
+∥ψ(x) − T(x)∥ = ∥ψ(x) − TEλP0(x)∥ + ∥TEλP0(x) − T(x)∥
+<
+0 + ∥T∥∥Eλ(x) − x∥ < ǫ.
+(e 6.95)
+This proves the first part of the theorem.
+For the second part of the theorem, by Theorem 3.2, let {En} be an approximate identity
+for K(H0) with En+1En = En, n ∈ N. Replacing Eλ by En, we obtain Tn ∈ K(H3) such that
+∥Tn∥ = ∥TEnP0∥ and Tn|H2 = TEnP0.
+(e 6.96)
+Put ψn = PHTn, n ∈ N. Then
+∥ψn∥ ≤ ∥ϕ∥ and ψn|H2 = TEnP0, n ∈ N.
+(e 6.97)
+We then use the fact that limn→∞ ∥En(x) − x∥ = 0 for all x ∈ H0 (see again Theorem 3.2).
+26
+
+Corollary 6.5. Let A be a C∗-algebra and H0 ⊂ H be Hilbert A-modules. Suppose that there
+is a bounded module map ϕ : H0 → A. Then, for any ǫ > 0 and any finite subset F ⊂ H0, there
+exists a bounded module map ψ : H → A such that
+∥ϕ(x) − ψ(x)∥ < ǫ for all x ∈ F and ∥ψ∥ ≤ ∥ϕ∥.
+(e 6.98)
+If, in addition, H0 is countably generated Hilbert A-module, then there exists a sequence of
+bounded module maps ψn : H → A such that
+lim
+n→∞ ∥ψn(x) − ϕ(x)∥ = 0 for all x ∈ H and ∥ψn∥ ≤ ∥ϕ∥ for all n ∈ N.
+(e 6.99)
+Remark 6.6. In the light of Theorem 6.4, one may think that every Hilbert A-module is
+“approximately injective”. Next, let us turn to “approximate projectivity”.
+Lemma 6.7. Let A be a σ-unital C∗-algebra and H = A(k) (for some k ∈ N). Suppose that
+H1, H2 are Hilbert A-modules, ϕ : H → H1 is a bounded module map and s : H2 → H1 is a
+surjective bounded module map. Then there exists a sequence of bounded module maps ψn : H →
+H2 and an increasing sequence of Hilbert A-submodules Xn ⊂ H such that ∪∞
+n=1Xn = H,
+s ◦ ψm|Hn = ϕ|Hn
+for all m ≥ n.
+(e 6.100)
+Moreover there exists a sequence of module maps Tn : H → Xn such that ∥Tn∥ ≤ 1 and
+lim
+n→∞ ∥Tn(h) − h∥ = 0, T2n+j|Hn = idHn, n, j ∈ N,
+(e 6.101)
+and s ◦ ϕn(h) = ϕ(Tn(h)) for all h ∈ H.
+(e 6.102)
+In particular,
+lim
+n→∞ ∥s ◦ ψn(x) − ϕ(x)∥ = 0 for all x ∈ H.
+(e 6.103)
+Proof. Fix a strictly positive element e ∈ A. Write A(n) = e1A⊕e2A⊕· · ·⊕enA, where ei = e1/2
+(1 ≤ i ≤ n). Let f1/n ∈ C([0, ∞)) be defined in (4) of 4.4, n ∈ N.
+For each n ∈ N, define Xn = �k
+i=1 f1/n(ei)A. Note that Xn ⊂ Xn+1, n ∈ N, and ∪∞
+n=1Xn =
+A(k) = H.
+Let xn,i = ϕ(f1/2n(ei)), 1 ≤ i ≤ k, and n ∈ N. Choose ym,i ∈ H2 such that s(ym.i) = xm,i,
+1 ≤ i ≤ m and m ∈ N. Define ψn : H → H2 by
+ψn(h) =
+k
+�
+i=1
+yn,i⟨f1/2n(ei), h⟩ for all h ∈ H.
+(e 6.104)
+It follows that ψn ∈ K(H, H2). If h = �k
+i=1 f1/n(e)ai, where ai ∈ A, then
+ψn(h) =
+k
+�
+i=1
+yn,if1/n(e)ai.
+(e 6.105)
+27
+
+Hence, for h = �k
+i=1 f1/n(ei)ai, where ai ∈ A (1 ≤ i ≤ k), and for any j ≥ 0,
+s(ψn+j(
+k
+�
+i=1
+f1/n(ei)ai))
+=
+s(
+k
+�
+i=1
+yn+j,if1/n(e)ai)
+(e 6.106)
+=
+k
+�
+i=1
+ϕ(f1/2(n+j)(ei))f1/n(e)ai
+(e 6.107)
+=
+k
+�
+i=1
+ϕ(f1/2(n+j)(ei)f1/n(e))ai
+(e 6.108)
+=
+k
+�
+i=1
+ϕ(f1/n(ei))ai = ϕ(
+k
+�
+i=1
+f1/n(ei)ai).
+(e 6.109)
+It follows that, for any m ≥ n,
+s ◦ ψm|Hn = ϕ|Hn.
+(e 6.110)
+Next consider the module map Tn : H → Xn defined by Tn(h) = �k
+i=1 f1/n(ei)⟨f1/n(ei), h⟩
+for all h ∈ H. In other words, Tn ∈ K(H, H2) and Tn(�k
+i=1 ai) = �k
+i=1 f1/n(e)2ai for any
+�k
+i=1 ai ∈ A(k). Hence ∥Tn∥ = 1. Moreover (for any ai ∈ A),
+T2n+j(
+k
+�
+i=1
+f1/n(e)ai)
+=
+k
+�
+i=1
+f1/(2n+j)(e)2f1/n(e)ai
+(e 6.111)
+=
+k
+�
+i=1
+f1/n(e)ai.
+(e 6.112)
+Hence
+T2n+j|Xn = idXn, n, j ∈ N.
+(e 6.113)
+It follows, for any h = �k
+i=1 ai, also by (e 6.104), that
+s ◦ ψn+j(h)
+=
+k
+�
+i=1
+xn+j,i⟨f1/2(n+j)(ei), ai⟩
+(e 6.114)
+=
+k
+�
+i=1
+ϕ(f1/2(n+j)(ej))f1/2(n+j)(e)ai) = ϕ(Tn+j(h)).
+(e 6.115)
+To see the last part of the lemma, fix h = �k
+i=1 ai ∈ A(k). Let ǫ > 0. Choose n ∈ N such
+that
+∥f1/n(e)2ai − ai∥ < ǫ/k(∥ϕ∥ + 1), 1 ≤ i ≤ k.
+(e 6.116)
+Hence, for any j ∈ N,
+∥ϕ(Tn+j(h)) − ϕ(h)∥ < ǫ
+(e 6.117)
+It follows that, for this h and for any j ∈ N,
+∥s ◦ ψn+j(h) − ϕ(h)∥
+≤
+∥s ◦ ψn+j(h) − ϕ(Tn+j(h))∥ + ∥ϕ(Tn+j(h)) − ϕ(h)∥
+=
+0 + ǫ.
+28
+
+From (e 6.116), we also have
+lim
+n→∞ ∥Tn(h) − h∥ = 0 for all h ∈ H.
+(e 6.118)
+Theorem 6.8. Let A be a σ-unital C∗-algebra and H a countably generated Hilbert A-module.
+Suppose that H1 and H2 are Hilbert A-modules, and ϕ : H → H1 is a bounded module map
+and s : H2 → H1 is a surjective bounded module map. Then there exists a sequence of bounded
+module maps ψn : H → H2 such that
+lim
+n→∞ ∥s ◦ ψn(x) − ϕ(x)∥ = 0 for all x ∈ H.
+(e 6.119)
+The following approximately commutative diagram may illustrate the statement of Theorem
+6.8 (with ǫn → 0):
+H
+H1
+H2 .
+ψn
+ϕ
+⟲εn
+s
+(e 6.120)
+Proof. We first prove the theorem for H = HA = l2(A).
+So now we assume that H = l2(A). Put Yn = A(n), n ∈ N. We view Yn ⊂ Yn+1 as an
+increasing sequence of Hilbert A-submodules of H = l2(A) and the closure of the union is H.
+By applying Lemma 6.7, we obtain, for each n ∈ N, an increasing sequence of Hilbert A-
+submodules Xn,j ⊂ Xn,j+1 such that ∪∞
+j=1Xn,j = Yn, and there exists a sequence of bounded
+module maps ψn,j : Yn → H2 and module maps Tn,j : Yn → Xn,j with ∥Tn,j∥ ≤ 1 such that
+s ◦ ψn,j+i|Xn,j = ϕ|Xn,j, j, i ∈ N,
+(e 6.121)
+Tn,2j+i|Xn,j = id|Xn,j, j, i ∈ N, and
+(e 6.122)
+lim
+j→∞ ∥Tn,j(h) − h∥ = 0 for all h ∈ Yn and
+(e 6.123)
+s ◦ ψn,j(h) = ϕ(Tn,j(h)) for all h ∈ Yn and j ∈ N.
+(e 6.124)
+Note that Yn = A(n) ⊂ A(n+1) = Yn+1, n ∈ N, therefore we may arrange so that
+Xn,j ⊂ Xn+1,j for all j ∈ N and n ∈ N.
+(e 6.125)
+Define Pn : H → Yn by Pn(�∞
+n=1 an) = �n
+i=1 ai, where �∞
+n=1 a∗
+nan converges, i.e, �∞
+n=1 an ∈
+l2(A) = H. Then Pn is a projection from H = HA onto A(n).
+Now define
+ψn = ψn,2n ◦ Pn : H → H2, n ∈ N.
+(e 6.126)
+We will verify that ψn meets the requirements.
+Fix h = �∞
+n=1 an ∈ l2(A) = H. There is n0 ∈ N such that, for all n ≥ m ≥ n0,
+∥Pm(h) − h∥ < ǫ/6(∥ϕ∥ + 1) and ∥Pn(h) − Pm(h)∥ < ǫ/6(∥ϕ∥ + 1).
+(e 6.127)
+For Pn0(h) ∈ A(n0), there is n1 ∈ N such that, for all n ≥ n1
+∥Tn0,n(Pn0(h)) − Pn0(h)∥ < ǫ/6(∥ϕ∥ + 1).
+(e 6.128)
+Since Tn0,n(Pn0(h)) ∈ Xn0,n ⊂ Xn,n (see (e 6.125)), by (e 6.122),
+Tn,2n(Tn0,n(Pn0(h))) = Tn0,n(Pn0(h)).
+(e 6.129)
+29
+
+Hence, if n ≥ n1 + n0, by (e 6.129) and by (e 6.128),
+∥Tn,2n(Pn0(h)) − Pn0(h)∥ = ∥Tn,2n(Pn0(h)) − Tn,2n(Tn0,n(Pn0(h)))∥
+(e 6.130)
++∥Tn,2n((Tn0,n(Pn0(h)))) − Pn0(h)∥
+(e 6.131)
+≤
+∥Tn,2n∥∥Pn0(h) − Tn0,n(Pn0(h))∥ + ∥Tn0,n(Pn0(h)) − Pn0(h)∥
+(e 6.132)
+<
+ǫ/6(∥ϕ∥ + 1) + ǫ/6(∥ϕ∥ + 1)
+(e 6.133)
+=
+ǫ/3(∥ϕ∥ + 1).
+(e 6.134)
+We also estimate that, if n ≥ n1 +n0, by both parts of (e 6.127) and the inequalities right above,
+∥Tn,2n(Pn(h)) − h∥
+≤
+∥Tn,2n(Pn(h)) − Pn0(h)∥ + ∥Pn0(h) − h∥
+<
+∥Tn,2n(Pn(h) − Pn0(h))∥ + ∥Tn,2n(Pn0(h)) − Pn0(h)∥ + ǫ/6(∥ϕ∥ + 1)
+<
+∥Tn,2n∥(ǫ/3(∥ϕ∥ + 1)) + ǫ/3(∥ϕ∥ + 1)) + ǫ/3(∥ϕ∥ + 1)
+=
+ǫ/(∥ϕ∥ + 1).
+(e 6.135)
+We then have, by (e 6.126), (e 6.124) and (e 6.135),
+∥s ◦ ψn(h) − ϕ(h)∥
+≤
+∥s ◦ ψn,2n(P 2
+n(h)) − ϕ(Tn,2n(Pn(h)))∥ + ∥ϕ(Tn,2n(Pn(h))) − ϕ(h)∥
+< 0 + ǫ.
+Next we consider the general case that H is an arbitrary countably generated Hilbert A-
+module. By Kasparov’s absorbing theorem ([26]), there is a Hilbert module isomorphism U :
+H⊕l2(A) → l2(A). Define j : H → H⊕l2(A) to be the obvious embedding and p : H⊕l2(A) → H
+the projection. Thus p◦j = idH. Define Φ = ϕ◦p◦U −1 : l2(A) → H1. Then we have the following
+commutative diagram:
+l2(A)
+U ↕U−1
+ցΦ
+H ⊕ l2(A)
+ϕ◦p
+−→
+H1.
+In particular, Φ ◦ U(x) = ϕ(x) for all x ∈ H. By what have been proved above, there exists a
+sequence of module maps Ψn : l2(A) → H2 such that
+lim
+n→∞ ∥s ◦ Ψn(z) − Φ(z)∥ = 0 for all z ∈ l2(A).
+(e 6.136)
+Define ψn : H → H2 by ψn(x) = Ψn ◦ U(x) for all x ∈ H. It follows that
+, lim
+n→∞ ∥s ◦ ψn(x) − ϕ(x)∥ = lim
+n→∞ ∥s ◦ Ψn(U(x)) − Φ(U(x))∥ = 0.
+(e 6.137)
+Remark 6.9. Note that, in Lemma 6.7, for any ǫ > 0, we may choose ∥yn∥ ≤ ∥˜s−1 ◦ ϕ∥ + ǫ. So
+we may estimate that ∥ψn∥ ≤
+√
+k(∥˜s−1 ◦ ϕ∥ + ǫ), which unfortunately depends on k. Therefore
+we do not have an estimate on the norm of ψn in Theorem 6.8. This is also the reason that the
+proof seems a little more involved than expected (as we need a complicated statement in Lemma
+6.7). If {∥ψn∥} were, at least, uniformly bounded, then the proof would be much shorter.
+However, under the assumption that A has real rank zero, in the next statement, we then
+have a better estimate on the norm on ψn.
+30
+
+Theorem 6.10. Let A be a σ-unital C∗-algebra of real rank zero and H a countably generated
+Hilbert A-module. Suppose that ϕ : H → H1 is a bounded module map and s : H2 → H1 is a
+surjective bounded module map. Then there exists a sequence of module maps ψn : H → H2
+such that
+∥ψn∥ ≤ ∥˜s−1 ◦ ϕ∥ and
+lim
+n→∞ ∥s ◦ ψn(x) − ϕ(x)∥ = 0 for all x ∈ H.
+(e 6.138)
+Proof. Since H is countably generated, by Corollary 1.5 of [39] (see (Lemma 3.2, for convenience),
+K(H) is a σ-unital hereditary C∗-subalgebra of A ⊗ K (see also Lemma 2.13 of [33]). Let {Pm}
+be an approximate identity for K(H) consisting of projections.
+Put H0,m = Pm(H). Then
+K(H0,m) = PmK(H)Pm is unital. It follows from Proposition 2.7 that H0,m is algebraically
+finitely generated, say, by {ξm,i : 1 ≤ i ≤ d(m)}, where ∥ξm,i∥ = 1. It is important to note that
+H0,m ⊂ H0,m′ if m < m′.
+Fix m ∈ N. Let H1,m be the Hilbert A-submodule of H1 generated by {xm,i = ϕ(ξm,i) : 1 ≤
+i ≤ d(m)}. Since H0,m ⊂ H0,m′, H1,m ⊂ H1,m′, if m < m′.
+Then K(H1,m) is a σ-unital hereditary C∗-subalgebra of A ⊗ K (see Lemma 3.2), m ∈ N.
+Since A has real rank zero, so is K(H1,m), m ∈ N. Let {pm,k}∞
+k=1 be an approximate identity
+for K(H1,m). Let H1,m,k = pm,k(H1,m), k, m ∈ N. Since H1,m ⊂ H1,m′, if m ≤ m′, K(H1,m) ⊂
+K(H1,m′) (see Lemma 2.13 of [33]). So we may assume that
+∥pm+1,kpm,k − pm,k∥ <
+1
+2k+m+1(∥ϕ∥ + 1), k, m ∈ N.
+(e 6.139)
+Note that K(H1,m,k) = pm,kK(H1,m)pm,k which is unital. It follows from Proposition 3.2
+that H1,m,k is algebraically finitely generated, say, by zm,k,1, zm,k,2, ..., zm,k,r(m,k) as A-module.
+Choose ym,k,j ∈ H2 such that s(ym,k,j) = zm,k,j, 1 ≤ j ≤ r(m, k), k, m ∈ N. Let H2,m,k be the
+Hilbert A-submodule of H2 generated by {ym,k,j : 1 ≤ k = j ≤ r(m, k)} as Hilbert A-module.
+Then, since H1,m,k is algebraically generated by zm,k,1, zm,k,2, ..., zm,k,r(i,k) as A-module,
+s(H2,m,k) = H1,m,k. In other words, s|H2,m,k is surjective (onto H1,m,k).
+Fix an integer m ∈ N. Define ϕm,k : H → H1,m,k by
+ϕm,k(x) = pm,k ◦ ϕ(Pm(x))
+(e 6.140)
+for all x ∈ H. Note that ∥ϕm,k∥ ≤ ∥ϕ∥, m, k ∈ N.
+Now H2,m,k is countably generated, it follows from Theorem 5.3 that there is a module map
+ψm,k : H → H2,m,k such that
+s ◦ ψm,k = ϕm,k for all k, m ∈ N, and
+(e 6.141)
+∥ψm,k∥ ≤ ∥˜s−1 ◦ ϕm,k∥ ≤ ∥˜s−1 ◦ ϕ∥.
+(e 6.142)
+Define ψn = ψn,n, n ∈ N. Then ∥ψn∥ ≤ ∥˜s−1 ◦ ϕ∥ for all n ∈ N. We will check that
+lim
+n→∞ ∥s ◦ ψn(x) − ϕ(x)∥ = 0 for all x ∈ H.
+(e 6.143)
+To check this, let ǫ > 0 and x ∈ H with ∥x∥ ≤ 1. Let M = (∥s∥ + 1)(∥˜s−1 ◦ ϕ∥ + 1). By Lemma
+3.1 , there is n0 ∈ N such that
+∥x − Pm(x)∥ < ǫ/8M and ∥Pm(x) − Pm′(x)∥ < ǫ/8M for all m′, m ≥ n0.
+(e 6.144)
+There is also k0 ∈ N with 1/k0 < ǫ/4 such that, for any k ≥ k0,
+∥pn0,k(ϕ(Pn0(x))) − ϕ(Pn0(x))∥ < ǫ/4.
+(e 6.145)
+31
+
+If n ≥ n0 and for all k ∈ N, we have, if k ≥ k0 + n0, by (e 6.145) and (e 6.139),
+pk,k(ϕ(Pn0(x)))
+≈ǫ/4
+pk,kpn0,k0(ϕ(Pn0(x)))
+(e 6.146)
+≈1/2k0
+pn0,k0(Pn0(x)) ≈ǫ/4 Pn0(x).
+(e 6.147)
+Then (recall that H0,n0 ⊂ H0,n, if n0 < n), by (e 6.144), (e 6.140), (e 6.141) and (e 6.146), when
+n ≥ n0 + k0 + (4/ǫ),
+∥s ◦ ψn(x) − ϕ(x)∥
+≤
+∥s ◦ ψn(x) − s ◦ ψn,n(Pn0(x))∥ + ∥s ◦ ψn,n(Pn0(x)) − ϕ(x)∥
+<
+∥s ◦ ψn∥∥x − Pn0(x)∥ + ∥s ◦ ψn,n(Pn0(x)) − ϕ(Pn0(x))∥ +
++∥ϕ(Pn0(x)) − ϕ(x)∥
+(e 6.148)
+<
+ǫ/8 + ∥s ◦ ψn,n(Pn0(x)) − ϕn,n(Pn0(x))∥
++∥pn,n(ϕ(Pn0(x))) − ϕ(Pn0(x))∥ + ǫ/4
+(e 6.149)
+<
+ǫ/8 + 0 + (1/2k0 + ǫ/2) + ǫ/8 < ǫ.
+Remark 6.11. This paper was based on a preprint [35] of 2010. However, some of the original
+part of [35] have been out of dated. A draft of the current version was made in 2014 including
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+

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