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+Time-frequency metrology with two single-photon states: phase space picture and the
+Hong-Ou-Mandel interferometer
+´Eloi Descamps1,2, Arne Keller2,3, P´erola Milman2
+1D´epartement de Physique de l’´Ecole Normale Sup´erieure - PSL, 45 rue d’Ulm, 75005 Paris, France
+2Universit´ee Paris Cit´e, CNRS, Laboratoire Mat´eriaux et Ph´enom`enes Quantiques, 75013 Paris, France and
+3D´epartement de Physique, Universit´e Paris-Saclay, 91405 Orsay Cedex, France
+(Dated: January 30, 2023)
+We use time-frequency continuous variables as the standard framework to describe states of light
+in the subspace of individual photons occupying distinguishable auxiliary modes. We adapt to this
+setting the interplay between metrological properties and the phase space picture already extensively
+studied for quadrature variables. We also discuss in details the Hong-Ou-Mandel interferometer,
+which was previously shown to saturate precision limits, and provide a general formula for the co-
+incidence probability of a generalized version of this experiment. From the obtained expression, we
+systematically analyze the optimality of this measurement setting for arbitrary unitary transforma-
+tions applied to each one of the input photons. As concrete examples, we discuss transformations
+which can be represented as translations and rotations in time-frequency phase space for some
+specific states.
+PACS numbers:
+I.
+INTRODUCTION
+Much has been discovered since the first proposals to
+use quantum systems in metrology.
+From the role of
+entanglement [1–4] to the one of modes, for pure and
+noisy systems and measurements, several main results
+have been established, and the most important one is
+the fact that quantum mechanical protocols can provide
+a better scaling in precision with the number of probes
+than classical ones. Nevertheless, much still remains to
+be done, in particular concerning the application and the
+adaptation of such results to specific physical configura-
+tions. Of practical importance, for instance, is the issue
+of finding measurement strategies that lead to the opti-
+mal calculated limits, and this is far from being obvious
+for general states.
+Another relevant problem concerns
+adapting the general principles to physical constraints,
+as energy or temperature limits and thresholds [5, 6].
+Those are the main issues of this paper: in one hand, we
+deeply study the conditions for optimality of a specific
+measurement set-up and on the other hand, we consider a
+specific physical system, consisting of individual photons,
+for measuring time and frequency related parameters.
+In order to measure a given parameter κ one performs
+an experiment producing different outcomes x with asso-
+ciated probabilities Pκ(x) and build an unbiased estima-
+tor K such that κ = ⟨K⟩κ is recovered. Here the index κ
+means that we take the average for the probability distri-
+bution Pκ. The Cram´er-Rao bound (CRB) [7] imposes a
+limit on the precision of parameter estimation:
+δκ ≥
+1
+√
+NF
+,
+(1)
+where, δκ is the standard deviation in the estimation of
+κ: δκ =
+�
+Varκ(K), N is the number of independent
+measurements which were performed to estimate κ and
+F is the quantity known as the Fisher information (FI),
+defined by : F =
+�
+dx
+1
+Pκ(x)
+�
+∂Pκ(x)
+∂κ
+�2
+.
+In a quantum setting, one can use as a probe a quan-
+tum state |ψ⟩ which can evolve under the action of
+an operator ˆU(κ) = e−iκ ˆ
+H generated by an Hamilto-
+nian ˆH.
+By optimizing the precision over all possible
+quantum measurements of a parameter κ, one obtains a
+bound, called the quantum Cram´er-Rao bound (QCRB)
+[8] which reads:
+δκ ≥
+1
+√NQ,
+(2)
+where Q is a quantity known as the quantum Fisher
+information (QFI) which for pure states and uni-
+tary
+evolutions
+(as
+the
+ones
+considered
+in
+the
+present
+paper),
+is
+equal
+to
+Q
+=
+4(∆ ˆH)2,
+with
+(∆ ˆH)2 = ⟨ψ(κ)| ˆH2 |ψ(κ)⟩ − ⟨ψ(κ)| ˆH |ψ(κ)⟩2.
+The FI indicates the precision of a given measurement,
+whereas the QFI is the maximum precision obtainable
+with any measurement. For a given setting, we can thus
+compute both quantities (FI and QFI) to have an idea if
+the measurement is optimal (QFI=FI) or not (QFI>FI).
+Determining the QFI is a mathematical task much
+easier than finding a physical experimental set-up that
+reaches it. In quantum optical systems, several propos-
+als and implementations exist where the QFI is indeed
+achieved [4, 9–11], and one example where this is possi-
+ble is the Hong-Ou-Mandel (HOM) experiment [12–15].
+In this experiment, one focus on simple physical systems
+composed of two photons occupying distinguishable spa-
+tial modes with a given spectral distributions. This state
+is a particular example of a state defined in the single
+photon subspace (where each mode is populated by at
+most one photon), in which a general pure state that can
+arXiv:2301.11755v1  [quant-ph]  27 Jan 2023
+
+2
+be expanded as:
+|ψ⟩ =
+�
+dω1 · · · dωnF(ω1, · · · , ωn) |ω1, · · · , ωn⟩ .
+(3)
+In this formula, the indexes 1,2, ..n, label different aux-
+iliary degrees of freedom (as for instance polarization or
+the propagation direction). The state |ω1, · · · , ωn⟩ is a
+pure state where each photon propagating in the mode
+α is exactly at the frequency ωα. The spectral function
+F also known as the joint spectral amplitude (JSA) is
+normalized to one:
+�
+|F(ω1, ..., ωn)|2dω1...dωn = 1.
+In this setting one can introduce time and frequency
+operators for each mode α: ˆωα and ˆtα. They correspond
+respectively to the generators of time and frequency
+shifts of the photon in the mode labeled by α.
+An
+important property of these operators is that, in the
+considered single photon subspace they satisfy the
+commutation relation [ˆωα, ˆtβ] = iδα,β analogous to the
+one observed for the quadrature operators ˆXα and ˆPα.
+Notice that we are using throughout this paper dimen-
+sionless operators, which are relative to particular time
+and frequency scales of the associated implementation.
+For a more complete description of the time frequency
+continuous variables one can refer to Appendix A and
+to [16].
+Previous works on quantum metrology using the
+electromagnetic field quadratures or particles’ posi-
+tion and momentum have shown how the phase space
+(x1, · · · , xn, p1, · · · , pn) can provide not only insight but
+also an elegant geometrical picture of the measurement
+precision [16–18]. Indeed the QFI can also be defined in
+terms of the Bures distance [19] s(|ψ(κ)⟩ , |ψ(κ + dκ)⟩):
+Q = 4( s(|ψ(κ)⟩,|ψ(κ+dκ)⟩)
+dκ
+)2. In the case of pure states,
+this distance is simply expressed in terms of the overlap
+s(|ψ⟩ , |φ⟩) =
+�
+2(1 − |⟨φ|ψ⟩|). Since the overlap of two
+states can be computed as the overlap of their respective
+Wigner function, one can interpret the QFI as a measure
+of how much the Wigner function must be shifted so as it
+becomes orthogonal to the initial one. A consequence of
+this is that the maximum precision of a measurement can
+be seen geometrically on the Wigner function, by looking
+at their typical size of variation in the direction of an evo-
+lution [17]. Since in the case of single photon states one
+can also define a time-frequency phase space associated
+to the variables (τ1, · · · , τn, ϕ1, · · · , ϕn), it is natural to
+investigate wether the same type of interpretation makes
+sense in this context.
+The present paper purposes are thus twofold: in the
+first place, we provide general conditions for the HOM to
+saturate precision limits using time-frequency (TF) vari-
+ables. For such, we consider arbitrary evolution operators
+acting on TF variables of single photons. In second place,
+we provide a phase-space picture and interpretation of
+the QFI for this type of system. Indeed, as shown in [20],
+there is an analogy between the quadrature phase space
+and the TF phase space from which metrological proper-
+ties of time and frequency states can be inferred. Never-
+theless, in the present case, photons have both spectral
+classical wave-like properties and quantum particle-like
+ones. Interpreting from a quantum perspective both the
+role of the spectral distribution and of collective quantum
+properties as entanglement in the single photon subspace
+has shown to demand taking a different perspective on
+the TF phase space [21]. Having this in mind, we in-
+vestigate how relevant examples of evolution operators,
+taken from the universal set of continuous variables quan-
+tum gates, can be implemented and represented in phase
+space, as well as the precision reached when one measures
+them using the HOM experiment. We’ll concentrate on
+single-mode Gaussian operations, analogously to what
+was done in [5], even though we provide a general for-
+mula for any transformation.
+This paper is organized as follows:
+In Section II
+we provide a description of the TF phase space and
+introduce the states we’ll discuss in details as well as
+their representation. In Section III we discuss the HOM
+experiment and the conditions for it to reach optimal
+precision limits. Finally, in Sections IV and V we discuss
+two different Gaussian operations in phase space as well
+as their implementation and the associated precision
+reached in the HOM experiment.
+II.
+TIME FREQUENCY PHASE SPACE
+We consider pure two-photon states which can be writ-
+ten in the form: |ψ⟩ =
+�
+dω1dω2F(ω1, ω2) |ω1, ω2⟩. The
+Wigner function in variables (τ1, τ2, ϕ1, ϕ2) of such states
+can be defined as
+W|ψ⟩(τ1, τ2, ϕ1, ϕ2) =
+�
+dω1dω2e2i(ω1τ1+ω2τ2)F(ϕ1 + ω1, ϕ2 + ω2)F ∗(ϕ1 − ω1, ϕ2 − ω2).
+(4)
+Evolutions generated by ˆωα and ˆtα (α = 1, 2) correspond
+to translations in phase space:
+We−iˆ
+ω1κ|ψ⟩(τ1, τ2, ϕ1, ϕ2) = W|ψ⟩(τ1 − κ, τ2, ϕ1, ϕ2),
+(5a)
+We−iˆt1κ|ψ⟩(τ1, τ2, ϕ1, ϕ2) = W|ψ⟩(τ1, τ2, ϕ1 − κ, ϕ2),
+(5b)
+and analogously for ˆω2 and ˆt2.
+
+3
+Using the QFI formulation based on the Bures
+distance, we can safely state that the precision of
+a measurement device is related to its capability of
+distinguishing between an initial state |ψ(κ)⟩ and a state
+|ψ(κ + dκ)⟩ that has evolved according to a parameter
+κ. This precision is then directly related to how small
+the parameter dκ should be such that these two states
+can be distinguished i.e. the overlap |⟨ψ(κ)|ψ(κ + dκ)⟩|
+gets close to zero. This can be also elegantly interpreted
+using the overlap of the two states’s respective Wigner
+functions, that describe trajectories in the phase space
+that are governed by the interaction Hamiltonian and
+the parameter dκ.
+To gain some familiarity with the studied problem
+we start with the case of a single-photon state |ψ⟩ =
+�
+dωS(ω) |ω⟩. Although using this type of state is not
+current in metrology, this simpler case can be seen as a
+building block and will help understanding the role of the
+spectrum in the present configuration.
+For a single photon, the Wigner function is defined
+as: W(τ, ϕ) =
+�
+dωe2iωτS(ϕ + ω)S∗(ϕ − ω). In the case
+of a Gaussian state |ψG⟩ with spectral wave function
+SG(ω) =
+e
+− ω2
+4σ2
+(2πσ2)1/4 its Wigner function is also Gaussian:
+WG(τ, ϕ) = exp
+�
+−2σ2τ 2 − ϕ2
+2σ2
+�
+. It is characterized by
+its width in the orthogonal directions τ and ϕ: 1/2σ and
+σ respectively.
+An evolution generated by ˆω corresponds to a trans-
+lation in the direction τ in phase space. The associated
+measurement precision is given by the smallest value of
+dκ such that the initial Wigner function is almost orthog-
+onal to the translated one in the corresponding direction.
+Since the width of the Wigner function in the direction
+of evolution is proportional to 1/σ, we have dκ ∼ 1/σ
+leading to a QFI of the order of Q ∼ σ2. Alternatively if
+one considers the generator ˆt, the associated width of the
+state will be σ leading to a QFI of the order of Q ∼ 1/σ2.
+We thus remark that the estimated QFI depends on the
+width of the state in phase space in the direction of evo-
+lution. We notice as well the similarities and differences
+with the quadrature phase space case: even though the
+relation between the phase space geometrical properties
+and metrological interest are common to both variables,
+in the case of quadrature they are related to some abso-
+lute quantum resource dependent quantity, the number
+of photons of the state. In the present case, the single
+photon spectrum is a classical resource and its width can
+only set a relative size scale in phase space.
+It is interesting to notice that this type of interpreta-
+tion is also possible for classical fields, as studied in [22–
+24]. In this classical context, the electromagnetic field
+amplitude replaces the function F and one can also re-
+late spectral metrological properties to the phase space
+structures. Nevertheless, as discussed in [21], this picture
+is merely associated to classical metrological properties
+of single mode fields (their spectrum) and no interesting
+scaling can be observed in this context. As a matter of
+fact, the classical single mode field and the single photon
+phase space can be mapped into one another.
+In the present paper, the multi-modal character of the
+quantum field is an essential ingredient for the discussion
+of the quantum metrological advantage, since it is a con-
+sequence of the multi-photon state. We will see in par-
+ticular how these two features (spectral and particle-like)
+of the considered single photon subspace are combined in
+the QFI.
+The situation is different and richer for bi-photon
+states, since the phase space is of dimension 4.
+One
+can thus imagine different directions of translation as
+for instance the ones generated by operators ˆω1, ˆω2,
+ˆω1 − ˆω2, . . . Then, optimizing the measurement precision
+involves, for a given spectral distribution, choosing a di-
+rection of evolution for which the Wigner function of the
+state has the smallest scale structures. This direction, as
+we’ll see, will depend on the number of photons, and can
+display a non-classical scaling.
+III.
+THE HOM AS A MEASUREMENT DEVICE
+A.
+The setup
+In the setup proposed by Hong, Ou and Mandel [25]
+two photons impinge into a balanced beam splitter (BS),
+each one of them from a different port, as represented on
+figure 1. By measuring the output of the beam-splitter
+using single-photon detectors we can compute the prob-
+ability of obtaining coincidences (when the two photons
+exit the BS by different paths) or anti-coincidences (when
+they bunch and exit the BS at the same path).
+FIG. 1: Schematic representation of HOM experiment.
+Since its original proposal and implementation, many
+modifications and adaptations were made to the HOM
+set-up, which was shown to be very versatile to reveal
+different aspects of quantum optics using two-photon in-
+terference [26]: it can be used to witness particle [27] and
+
+A
+BS
+B4
+spectral [28] entanglement, to saturate precision bounds
+on time delay measurements[12, 13] or to directly mea-
+sure the Wigner function of the incoming state [29, 30].
+We’re interested in quantum metrological tasks, so
+we’ll start by discussing the results obtained in [12],
+where the authors provided experimental evidence that
+the HOM device can saturate precision limits on time
+measurements. To achieve this result, the authors con-
+sidered the initial state:
+|ψU⟩ =
+1
+√
+2
+�
+dΩf(Ω)
+� ��ω0
+1 + Ω, ω0
+2 − Ω
+�
+−
+��ω0
+2 + Ω, ω0
+1 − Ω
+� �
+,
+(6)
+where ω0
+1 and ω0
+2 are the central frequencies of the pho-
+tons. Due to the energy conservation and to the phase-
+matching conditions, the support of the JSA associated
+to (6) is the line ω1 + ω2 = 0 in the plane (ω1, ω2). It
+is anti-diagonal in the plane (ω1, ω2) and infinitely thin
+along the diagonal direction ω− = ω1 − ω2. Adding a
+delay in the arm 1 of the HOM interferometer corre-
+sponds to an evolution generated by the operator ˆω1,
+corresponding to a translation κ in the τ1 direction. The
+QFI is simply calculated as: Q = 4∆(ˆω1)2. After the
+beam-splitter, the measurement can lead to two out-
+comes: coincidence or anti-coincidence, with probability
+Pc and Pa, respectively.
+The FI is thus expressed as:
+F =
+1
+Pc
+� ∂Pc
+∂κ
+�2 +
+1
+Pa
+� ∂Pa
+∂κ
+�2. The authors of [12] thus
+showed that using the input state (6) in the HOM inter-
+ferometer, the two quantities F and Q are the same.
+In [13] the HOM interferometer was also used and
+shown to lead to the QFI in a two-parameter estimation
+experiment. Finally, in [14] biphoton states were classi-
+fied as metrological resources according to their spectral
+width, still in the situation where the HOM experiment
+is used as a measurement apparatus.
+B.
+Generalization: the HOM as an optimal
+measurement device for quantum metrology with
+biphotons
+We now make a general description of the HOM ex-
+periment as a parameter estimation device and try to
+understand and determine when it corresponds to an op-
+timal measurement strategy. In [13], the authors tackle a
+part of this problem by studying the HOM as a measure-
+ment apparatus for two parameter estimation by estab-
+lishing conditions on frequency correlation states. In this
+reference, the authors restrict themselves to time delay
+evolutions.
+In the present paper, we are interested in studying any
+evolution that can be described by a two photon unitary
+|ψ(κ)⟩ = ˆU(κ) |ψ⟩ = e−i ˆ
+Hκ |ψ⟩ (see figure 2). We will see
+that under a symmetry assumption on the JSA of the
+state, it is possible to obtain an explicit formula for the
+FI, and this formula can be used to compute at a glance
+if the measurement setup considered is optimal or not.
+FIG. 2: HOM setup where we apply a general gate ˆU
+before the BS.
+For any input state |ψ⟩, the QFI will then be expressed
+as:
+Q = 4∆( ˆH)2.
+(7)
+On the other hand, one can show that the coincidence
+probability is:
+Pc = 1
+2(1 − ⟨ψ| ˆU † ˆS ˆU |ψ⟩).
+(8)
+(see Appendix B) where we introduced the hermitian
+swap operators ˆS whose action on the states is given
+by ˆS |ω1, ω2⟩ = |ω2, ω1⟩. Furthermore we can compute
+the associated FI. If the state |ψ⟩ is symmetric or anti-
+symmetric (i.e. ˆS |ψ⟩ = ± |ψ⟩) the FI at κ = 0 it is given
+by:
+F = ∆( ˆH − ˆS ˆH ˆS)2.
+(9)
+(see Appendix B). This means that under the symme-
+try assumption on the JSA, comparing the QFI and
+the FI is done simply by comparing the variance of
+two different operators, mainly:
+2 ˆH and
+ˆH − ˆS ˆH ˆS.
+Equation (9) implies that if [ ˆH, ˆS] = 0, then F = 0
+and no information can be obtained about κ from the
+measurements.
+However, if { ˆH, ˆS} = 0 then F = Q
+since ˆS ˆH ˆS = − ˆS2 ˆH = − ˆH.
+In this last case, the
+measurement strategy is optimal. In [31], general con-
+ditions for reaching the QFI were also obtained in the
+context of amplitude correlation measurements. These
+conditions are based on a quantum state’s symmetry
+under (unphysical) path exchange.
+The previous calculations form a simple tool that can
+be applied to different evolution Hamiltonians ˆH. We’ll
+now discuss examples taken from the universal set of
+quantum gates in continuous variables: translations (gen-
+erated by operator ˆωα’s) and rotations (generated by
+ˆH = (ˆω2 +ˆt2)/2). These gates have already been studied
+in [5] in the case of quadrature or position and momen-
+tum. In the present physical configuration, they corre-
+spond to the free evolution of single photons in free space
+
+A
+U
+2
+BS
+B5
+(translations) or in a dispersive medium, as for instance
+an optical fiber combined to time lenses (rotation).
+IV.
+TIME-FREQUENCY PHASE-SPACE
+TRANSLATIONS
+A.
+Different types of translations
+Since we’re considering two-photon states, translations
+can be represented by any linear combination of the cor-
+responding operators, that is : ˆH = αˆω1+βˆω2+��ˆt1+δˆt2.
+To illustrate our results we choose to focus on the four
+operators ˆω1, ˆω2 and ˆω± = ˆω1 ± ˆω2, since they are the
+most easily implemented in HOM experiment.
+Notice
+that ˆω± are collective operators acting in both input
+photons while ˆω1,2 act in a single photon only.
+If we consider a state which is (anti-)symmetric and
+separable in the variables ω± = ω1 ± ω2, we can write:
+|ψ⟩ =
+1
+√
+2
+�
+dω+dω−f(ω+)g(ω−)
+����
+ω+ + ω−
+2
+, ω+ − ω−
+2
+�
+,
+(10)
+with g satisfying g(−ω) = ±g(ω) and the functions g
+and f being normalized to one.
+The specific form of
+each function is related to the phase-matching conditions
+and the energy conservation of the two-photon generation
+process and this type of state can be experimentally pro-
+duced in many set-ups [32, 33]. Using equations (7) and
+(9) we can compute the QFI and FI associated to each
+type of evolution:
+• For ˆH = ˆω1, we get Q = ∆(2ˆω1)2 = ∆(ˆω++ˆω−)2 =
+∆(ˆω−)2 + ∆(ˆω+)2, while F = ∆(ˆω−)2. Thus this
+situation is optimal only if ∆(ˆω+)2 = 0, which was
+the case for the state |ψU⟩ of Eq. (6) used in [12]).
+We obtain the same type of result for ˆω2.
+• For ˆH = ˆω+, Q = 4∆(ˆω+)2, while F = ∆(ˆω+ −
+ˆω+)2 = 0.
+In this situation the precision of the
+measurement is zero, and the reason for that is that
+variables ω+ cannot be measured using the HOM
+experiment (we notice that [ˆω+, ˆS] = 0).
+• For ˆH = ˆω−, we get Q = 4∆(ˆω−)2, while F =
+∆(ˆω− + ˆω−)2 = 4∆(ˆω−)2. This time we have F =
+Q, which means that the measurement is optimal.
+In this case, we have that {ˆω−, ˆS} = 0.
+We now illustrate these general expressions and inter-
+pret them using different quantum states and their phase
+space representations.
+B.
+Example: Gaussian and Schr¨odinger cat-like
+state
+To illustrate our point we discuss as an example two
+states |ψG⟩ and |ψC⟩ that can be expressed in the form
+of equation (10). For |ψG⟩, f and g are Gaussians:
+fG(ω+) = e
+−
+(ω+−ωp)2
+4σ2
++
+(2πσ2
++)1/4
+gG(ω−) =
+e
+−
+ω2
+−
+4σ2
+−
+(2πσ2
+−)1/4 ,
+(11)
+where σ± is the width of the corresponding function and
+ωp is a constant, which is also the photon’s central fre-
+quency. As for state |ψC⟩, it can be seen as the general-
+ization of (6). We consider f to be Gaussian and g to be
+the sum of two Gaussians:
+fC(ω+) = fG(ω+)
+gC(ω−) =
+1
+√
+2
+�
+gG(ω− + ∆/2) − gG(ω− − ∆/2)
+�
+,
+(12)
+where ∆ is the distance between the two Gaussian peaks
+of gC. We assume that the two peaks are well separated:
+∆ ≫ σ−.
+Consequently, gC is approximately normal-
+ized to one. We can verify that with these definitions
+the function gG is even while gC is odd by exchange of
+variables ω1 and ω2. We first compute the variances for
+both states (table I) and then apply the formula (7) and
+(9).
+State
+|ψG⟩
+|ψC⟩
+∆(ˆω1)2 or ∆(ˆω2)2
+1
+4σ2
++ + 1
+4σ2
+−
+1
+16∆2 + 1
+4σ2
++ + 1
+4σ2
+−
+(∆ˆω+)2
+σ2
++
+σ2
++
+(∆ˆω−)2
+σ2
+−
+1
+4∆2 + σ2
+−
+TABLE I: Variance of various time translation operators
+for states |ψG⟩ and |ψC⟩. See Appendix C for details.
+So for the case of an evolution generated by ˆω1, for
+|ψG⟩ we obtain:
+Q = σ2
++ + σ2
+−
+F = σ2
+−,
+(13)
+while for |ψC⟩ we have:
+Q = 1
+4∆2 + σ2
++ + σ2
+−
+F = 1
+4∆2 + σ2
+−.
+(14)
+We thus see that time precision using the HOM measure-
+ment and the quantum state evolution generated by ˆω1 is
+optimal only if the parameter σ+ is negligible compared
+to ∆ or σ−. This is exactly the case for the state (6)
+where σ+ = 0.
+In addition, we see that there is a difference between
+the QFI associated to |ψC⟩ and |ψG⟩ involving the param-
+eter ∆. This difference can be interpreted, as discussed
+in [14], as a spectral effect. In this reference, the spectral
+width is considered as a resource, and for a same spectral
+width state |ψC⟩ has a larger variance than state |ψG⟩.
+Nevertheless, as discussed in [21], this effect has a clas-
+sical spectral engineering origin and choosing to use one
+rather than the other depends on the experimentalists
+constraints.
+
+6
+(a) Projection on the plane τ−,
+ω−
+(b) Projection on the plane τ1,
+ω1
+FIG. 3: Wigner function of the cat-like state |ψC⟩
+projected in different variables.
+C.
+Interpretation of translations in the
+time-frequency phase space
+We now discuss the dependency of precision on the
+direction of translation. For such, we can consider the
+Wigner function associated to a JSA which is separable
+in the ω± variables.
+Its Wigner function will also be
+separable on these variables:
+W(τ1, τ2, ϕ1, ϕ2) = W+(τ+, ϕ+)W−(τ−, ϕ−),
+(15)
+where the phase space variables τ± and ϕ± are defined
+as: ϕ± =
+ϕ1±ϕ2
+2
+and τ± = τ1 ± τ2. Even though the
+Wigner function W+ (resp. W−) can be associated to
+the one of a single variable (ω+ (ω−)) and spectral wave
+function f (resp. g), it displays some differences with the
+single photon one. This fact is well illustrated in Fig. 3.
+For state |ψC⟩, according to (15) the projection of the
+Wigner function W− in the plane τ−, φ− of the phase
+space can be represented as show in Figure 3 (a). We see
+that it is composed of two basic shapes: two Gaussian
+peaks and an oscillation pattern in between. Figure 3 (b)
+represents another way to project this very same Wigner
+function onto the plane τ1, φ1 of the phase space. One can
+observe that in this case the distance between the peaks
+is larger than in the previous representation by a factor
+of 2. As precision is directly related to the size of the
+Wigner function structures in phase space, we observe
+that the interference fringes are closer apart in the phase
+space associated to the minus variable than in the one
+associated to mode 1. Thus, the precision in parameter
+estimation will be better using ˆω− as the generator of the
+evolution than when using ˆω1. This phase space based
+observations explain well the result of the computation
+of the QFI:
+4∆(ˆω1)2 = ∆(ˆω−)2.
+(16)
+with the assumption that σ+ ≪ ∆, σ−.
+The reason for the appearance of a factor 2 difference
+in fringe spacing for the Wigner function associated to
+variable ω− is the fact that it is a collective variable,
+and translations in the phase space associated to these
+variables are associated to collective operators, acting on
+both input photons (instead of a single one, as is the case
+of translations generated by operator ˆω1, for instance).
+Thus, one can observe, depending on the biphoton quan-
+tum state (i.e., for some types of frequency entangled
+states), a scaling depending on the number of particles (in
+this case, two). As analyzed in [21] for general single pho-
+ton states composed of n individual photons, we have for
+frequency separable states a scaling corresponding to the
+shot-noise one (i.e., proportional to √n). A Heisenberg-
+like scaling (proportional to n) can be achieved for non-
+physical maximally frequency correlated states, and con-
+sidering a physical non-singular spectrum leads to a non-
+classical scaling in between the shot-noise and the Heisen-
+berg limit.
+Experimentally, such collective translation can be im-
+plemented by adding a delay of τ in arm 1 and of −τ in
+arm 2. Notice that this situation is different from cre-
+ating a delay of 2τ in only one arm, even though both
+situations lead to the same experimental results in the
+particular context of the HOM experiment.
+V.
+TIME-FREQUENCY PHASE SPACE
+ROTATIONS
+We now move to the discussion of the phase space ro-
+tations. For this, we’ll start by providing some intuition
+by discussing in first place the single photon (or single
+mode) situation. In this case, time-frequency phase space
+rotations are generated by the operators ˆR = 1
+2(ˆω2 + ˆt2).
+As previously mentioned, we consider here dimension-
+less observables. Physically, time-frequency phase space
+rotations correspond to performing a fractional Fourier
+transform of the JSA. While for transverse variable of
+single photons the free propagation or a combination of
+lenses can be used for implementing this type of oper-
+ation [34, 35], in the case of time and frequency this
+transformation corresponds to the free propagation in a
+dispersive medium [36–40] combined to temporal lenses
+[41–43].
+A.
+Single mode rotations
+In this Section, we compute the QFI associated to
+a rotation
+ˆR for a single photon, single mode state
+using the variance of this operator for different states
+|ψ⟩ =
+�
+dωS(ω) |ω⟩. As for the translation, this simpler
+configuration is used as a tool to better understand the
+two photon case.
+
+1.5
+1.0
+0.5
+0.0
+T
+-0.5
+-1.0
+1.5
+-10
+-5
+0
+5
+101.5
+1.0
+0.5
+0.0
+T
+-0.5
+-1.0
+1.5
+-5
+-10
+0
+5
+107
+1.
+Gaussian state:
+We start by discussing a single-photon Gaussian state
+at central frequency ω0 and spectral width σ:
+|ψG(ω0)⟩ =
+1
+(2πσ2)1/4
+�
+dωe− (ω−ω0)2
+4σ2
+|ω⟩ .
+(17)
+For this state, we have that:
+∆( ˆR)2 = σ2ω2
+0 + 1
+8
+� 1
+4σ4 + 4σ4 − 2
+�
+.
+(18)
+Eq. (18) has two types of contributions that we can in-
+terpret:
+• The first term σ2ω2
+0 corresponds to the distance in
+phase space (ω0) of the center of the distribution, to
+the origin of the phase space (ω = 0, τ = 0), times
+the width of the state σ in the direction of rotation
+(see Figure 4 (a)). This term is quite intuitive. The
+Wigner function of a state which is rotated by an
+angle θ = 1/2σω0 has an overlap with the Wigner
+function of the initial one which is close to zero.
+• The term
+1
+4σ4 + 4σ4 − 2 reaches 0 as a minimum
+when σ =
+1
+√
+2. For this value the Wigner function
+is perfectly rotationally symmetric.
+Its meaning
+can be intuitively understood if we consider that
+ω0 = 0, so that this term becomes the only con-
+tribution to the variance(see Figure 4 (b)). In this
+case, we are implementing a rotation around the
+center of the state. If the state is fully symmetric
+then this rotation has no effect, and the variance
+is 0. Only in the case where the distributions ro-
+tational symmetry is broken we obtain a non zero
+contribution.
+2.
+Schr¨odinger cat-like state centered at the origin (ω = 0):
+We now consider the superposition of two Gaussian
+states:
+��ψ0
+C
+�
+=
+1
+√
+2(|ψG(∆/2)⟩ − |ψG(−∆/2)⟩).
+(19)
+This state is of course non physical as a single-photon
+state, since it contains negative frequencies.
+However,
+since it can be be well defined using collective variables
+(as for instance ω−) for a two or more photons state,
+we still discuss it. Assuming that the two peaks are well
+separated (∆ ≫ σ), we can ignore the terms proportional
+to e− ∆2
+8σ2 , and this leads to:
+∆( ˆR)2 = 1
+8
+� 1
+4σ4 + 4σ4 − 2
+�
++ 1
+4∆2σ2.
+(20)
+We see that there is no clear metrological advantage
+when using this state compared to the Gaussian state:
+the quantity ∆/2 plays the same role as ω0. This can
+be understood geometrically once again, with the help
+of the Wigner function.
+We see in Figure 4 (c) how
+the considered state evolves under a rotation.
+In this
+situation the interference fringes are rotated around
+their center so even though they display a small scale
+structure, they are moved only by a small amount,
+resulting in a non significant precision improvement.
+3.
+Schr¨odinger cat-like state centered at any frequency:
+We can now discuss the state formed by the superpo-
+sition of two Gaussian states whose peaks are at frequen-
+cies ω0 − ∆/2 and ω0 + ∆/2, and with the same spectral
+width σ as previously considered:
+|ψC⟩ =
+1
+√
+2
+�
+|ψG(ω0 + ∆/2)⟩ − |ψG(ω0 − ∆/2)⟩
+�
+. (21)
+Still under the assumption of a large separation between
+the two central frequencies (∆ ≫ σ), we obtain:
+∆( ˆR)2 = 1
+8
+� 1
+4σ4 + 4σ4 − 2
+�
++ 1
+4∆2(σ2 + ω2
+0) + σ2ω2
+0.
+(22)
+We can notice that by setting ω0 = 0 we recover the
+variance corresponding to the same state rotated around
+its center. Nevertheless, in the present case ω0 ̸= 0, and
+we have two additional terms: σ2ω2
+0 and ∆2ω2
+0/4. Both
+terms can be interpreted as a product of the state’s dis-
+tance to the origin and its structure in phase space. How-
+ever, while the first one is simply the one corresponding
+to the Gaussian state, the second one is a product of the
+states’ distance to the origin and its small structures in
+phase space, created by the interference between the two
+Gaussian states (see Figure 4 (d)). The interference pat-
+tern is thus rotated by an angle θ corresponding to an arc
+of length ω0θ, and since the distance between the fringes
+is of order ∆, if θ ∼ 1/ω0∆ (corresponding to the term
+∆2ω2
+0/4 in the expression of the variance) the rotated
+state is close to orthogonal to the initial one.
+In all this section, we have considered rotations about
+the time and frequency origin of the phase space. Never-
+theless, it is of course possible to displace this origin and
+consider instead rotations about different points of the
+TF phase space. In this case, for a rotation around an
+arbitrary point τ0 and ϕ0, the generator would be given
+by (ˆω − ϕ0)2/2 + (ˆt − τ0)2/2.
+B.
+Different types of rotations
+We now move to the case of two single photons
+(biphoton states). As for the case of translations, there
+are many possible variables and can consider rotations
+in different planes of the phase space:
+ˆR1,
+ˆR2,
+ˆR±,
+
+8
+(a) Gaussian state centered at
+ω0. For θω0 ∼ 1/2σ the initial
+state and the rotate one are
+distinguishable.
+(b) Gaussian state centered at
+the origin. The rotated state will
+be distinguishable from the
+initial one only in the absence of
+rotational symmetry.
+(c) Superposition of two Gaussian
+states (cat-like state) centered the
+origin. The small structures of
+the fringes do not play a relevant
+role since they are only moved by
+a small distance under rotation.
+(d) Superposition of two
+Gaussian states (cat-like state)
+centered at ω0. The fringes play
+an important role, since with
+θω0 ∼ 1/∆, the two states are
+nearly orthogonal.
+FIG. 4: Schematic representation of the Wigner
+function of various states under rotation. The ellipses
+represent the typical width of Gaussians. The doted
+lines represent the rotated states.
+ˆR1 ± ˆR2 . . . where ˆR1 =
+1
+2(ˆω2
+1 + ˆt2
+1) (and similarly for
+ˆR2) and ˆR± = 1
+4(ˆω2
+± + ˆt2
+±) (recall that ˆω± = ˆω1 ± ˆω2 and
+ˆt± = ˆt1 ± ˆt2). For all these operators we can as before
+apply the general formula for the QFI and of the FI to
+the corresponding HOM measurement. The results are
+displayed in table II.
+Operator
+QFI
+FI
+ˆR1
+4∆( ˆR1)2
+∆( ˆR1 − ˆR2)2
+ˆR±
+4∆( ˆR±)2
+0
+ˆR1 + ˆR2
+4∆( ˆR1 + ˆR2)2
+0
+ˆR1 − ˆR2
+4∆( ˆR1 − ˆR2)2
+4∆( ˆR1 − ˆR2)2
+TABLE II: QFI and FI of various rotation operators.
+We see that the only two situations where the HOM
+can indeed be useful as a measurement device for metro-
+logical applications are ˆR1 and ˆR1 − ˆR2. The reason for
+that is the symmetry of ˆR± and ˆR1+ ˆR2, which commute
+with the swap operator ˆS. As for ˆR1, it corresponds to
+the rotation of only one of the photons and may not be
+the optimal strategy. Finally, ˆR1 − ˆR2 corresponds to
+the simultaneous rotation in opposite directions of both
+photons sent into the two different input spatial modes.
+As ˆR1 − ˆR2 anti-commutes with ˆS then we can affirm
+that the HOM measurement is optimal for this type of
+evolution.
+C.
+QFI and FI computation with Gaussian and
+cat-like state
+We now compute the QFI and FI using the variance
+of ˆR1 and ˆR1 − ˆR2 calculated for states |ψG⟩ and |ψC⟩.
+For |ψG⟩:
+We have:
+∆( ˆR1)2 = 1
+32
+�� 1
+σ2
++
++ 1
+σ2
+−
+�2
++ (σ2
++ + σ2
+−)2 − 8
+�
++ 1
+16ω2
+p(σ2
++ + σ2
+−)
+∆( ˆR1 − ˆR2) = 1
+4
+�
+1
+σ2
++σ2
+−
++ σ2
++σ2
+− − 2
+�
++ 1
+4σ2
+−ω2
+p. (23)
+For |ψC⟩:
+We have:
+∆( ˆR1)2 = 1
+32
+�� 1
+σ2
++
++ 1
+σ2
+−
+�2
++ (σ2
++ + σ2
+−)2 − 8
+�
++ 1
+64(4ω2
+p + ∆2)(σ2
++ + σ2
+−)
++ 1
+64∆2ω2
+p + ∆2
+128
+� 1
+σ2
+−
++ σ2
+−
+�
+∆( ˆR1 − ˆR2) = 1
+4
+�
+1
+σ2
++σ2
+−
++ σ2
++σ2
+− − 2
+�
++ 1
+4σ2
+−ω2
+p. (24)
+We notice that for both states 4∆( ˆR1)2 ≥ ∆( ˆR1− ˆR2)2,
+meaning that the measurement of a rotation imple-
+mented in only one mode using the HOM is not an
+optimal measurement.
+Experimentally realizing an evolution generated by ˆR1
+is easier than implementing the one associated to ˆR1− ˆR2.
+Furthermore we see that for the Gaussian state |ψG⟩ a
+dominant term is ω2
+pσ2
+− which appears with the same
+factor in 4∆( ˆR1)2 and ∆( ˆR1 − ˆR2)2, meaning that one
+could perform a measurement which although not opti-
+mal would be pretty efficient. The same applies to the
+Schr¨odinger cat-like state |ψC⟩ where one dominant term
+is ∆2ω2
+p.
+D.
+Phase space interpretation
+We now provide a geometrical interpretation of the
+previous results.
+If we consider that σ− ≫ σ+ in
+
+T
+0
+6
+1
+2g
+03T
+个
+6
+1
+2g7
+0T
+039
+the case of a Gaussian state or ∆ ≫ σ+ in the case
+of a Schr¨odinger cat-like state, the projection of the
+Wigner function on the plane corresponding to collective
+minus variables (τ−, φ−) is the one presenting a relevant
+phase space structure.
+Thus, it would be interesting
+to consider, as in the case of translations, that these
+states are manipulated using operators acting on modes
+associated to this collective variable.
+A na¨ıve guess
+would then trying to apply the rotation operator ˆR−.
+However it comes with many difficulties.
+Indeed it
+first poses an experimental problem, since this rotation
+corresponds to a non-local action which would be very
+hard to implement. In addition, the HOM is not able to
+measure such evolution. Finally, it turns out that this is
+not the operator with the greatest QFI. This fact can be
+understood by taking a more careful look at the Wigner
+function of the considered states. The Wigner function
+for separable states can be factorized as the product
+of two Wigner functions defined in variables plus and
+minus, and we have that W+ is the Wigner function
+of a Gaussian state centered at ωp (corresponding to
+the situation (a) in Figure (4). As for W−, it is either
+the Wigner function of a Gaussian state or the one
+associated to a superposition of two Gaussian states
+centered around zero (corresponding to the situation
+(b) and (c) in Figure 4).
+The QFI increases with the
+distance of the states to the rotation point.
+For this
+reason, states |ψG⟩ and |ψC⟩ under a rotation using ˆR−,
+do not lead to a high QFI.
+A higher QFI is obtained using rotations around a
+point which is far away from the center of the state. In
+this case, the QFI displays a term which is proportional
+to the distance from the center of rotation squared di-
+vided by the width of the state squared.
+Both terms
+ω2
+pσ2
+− and ∆2ω2
+p which were dominant in the expression
+of the variance of ˆR1 and ˆR1 − ˆR2 can be interpreted
+as such.
+This means that the rotation ˆR1, whose ac-
+tion is not easily seen in the variables plus and minus,
+can be interpreted as a rotation which moves W− around
+the distance ωp from the origin of the TF phase space
+(ω = 0).
+For both states then, the main numerical contribution
+to the QFI comes from a classical effect, related to the
+intrinsic resolution associated to the central (high) fre-
+quency of the field. In general, in phase space rotations,
+both in the quadrature and in the TF configuration, the
+distance from the phase space origin plays an important
+role. While in the quadrature configuration this distance
+has a physical meaning that can be associated both to the
+phase space structure and to the number of probes. In
+the case of TF phase space, the distance from the origin
+and the phase space scaling are independent. In partic-
+ular, the distance from the origin can be considered as a
+classical resource that plays no role on the scaling with
+the number of probes.
+E.
+A discussion on scaling properties of rotations
+The different types of FT phase space rotations have
+different types of interpretation in terms of scaling. The
+combined rotations of the type ˆR1 ± ˆR2, for instance,
+can be generalized to an n photon set-up through oper-
+ators as ˆR = �n
+i αi ˆRi, with αi = ±1. In this situation,
+we have that rotation operators are applied individually
+and independently to each one of the the n photons. In
+this case, we can expect, in first place, a collective (clas-
+sical) effect, coming simply from the fact that we have n
+probes (each photon). In addition, it is possible to show
+that a Heisenberg-like scaling can be obtained by con-
+sidering states which are maximally mode entangled in a
+mode basis corresponding to the eigenfunctions of oper-
+ators ˆRi. Indeed, for each photon (the i-th one), we can
+define a mode basis such that ˆRi |φk⟩i = (k + 1/2) |φk⟩i,
+with |φk⟩i =
+1
+√
+2kk!
+1
+π1/4
+�
+dωe− ω2
+2 Hk(ω) |ω⟩i with Hk(ω)
+being the k-th Hermite polynomial associated to the i-
+th photon. For a maximally entangled state in this mode
+basis, i.e. , a state of the type |φ⟩ = �∞
+k=0 Ak
+�n
+i=1 |φk⟩i,
+(where we recall that the subscript i refers to each pho-
+ton and k to the rotation eigenvalues) the ˆR eigenvalues
+behave as random classical variables and we can show
+that the QFI scales as n2.
+As for rotations of the type ˆR±, they cannot be de-
+composed as independently acting on each photon, but
+consist of entangling operators that can be treated ex-
+actly as ˆR1 and ˆR2 but using variables ω± = ω1 ± ω2
+instead of ω1 and ω2.
+We can also compute the scal-
+ing of operators as ˆJ = �
+Ωβ ˆRΩβ where Ωβ = �n
+i αiωi,
+αi = ±1 and β is one of the 2n−1 ways to define a collec-
+tive variable using the coefficients αi. For such, we can
+use the same techniques as in the previous paragraph but
+for the collective variables Ωβ. Nevertheless, the exper-
+imental complexity of producing this type of evolution
+and the entangled states reaching the Heisenberg limit is
+such that we’ll omit this discussion here.
+VI.
+CONCLUSION
+We have extensively analyzed a quantum optical set-
+up, the HOM interferometer, in terms of its quantum
+metrological properties. We provided a general formula
+for the coincidence probability of this experiment which
+led to a general formula for the associated FI. We used
+this formula to analyze different types of evolution and
+showed when it is possible to reach the QFI in this set-
+up. In particular, we made a clear difference between col-
+lective quantum effects that contribute to a better than
+classical precision scaling and classical only effects, asso-
+ciated to single mode spectral properties. We then briefly
+discussed the general scaling properties of the QFI asso-
+ciated to the studied operators.
+Our results provide a complete recipe to optimize the
+HOM experiment with metrological purposes. They rely
+
+10
+on the symmetry properties of quantum states that are
+revealed by the HOM interferometer. An interesting per-
+spective is to generalize this type of reasoning for differ-
+ent set-ups where different symmetries play a role on the
+measurement outputs.
+Acknowledgements
+The French gouvernement through the action France
+2030 from Agence Nationale de la Recherche, reference
+“ANR-22-PETQ-0006” provided financial support to this
+work. We thanks Nicolas Fabre for fruitful discussions
+and comments on the manuscript.
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+Appendix A: Time frequency formalism
+In quantum mechanics, light is described with the help of modes [44], representing the various physical properties
+a photon can have: frequency, position, spectral shape, wave vector, polarization... Mathematically we associate
+to each mode α a creation and annihilation operators ˆa†
+α and ˆaα which satisfy the familiar bosonic commutation
+relation [ˆaα, ˆa†
+β] = δα,β. The quantum states are then obtained by acting with the creation operators on the vacuum
+|vac⟩, which can be interpreted as adding a photon in the corresponding mode.
+In time frequency continuous variables we look at modes parameterized by the frequency [16]. We will thus adapt
+the terminology: for us a mode will correspond to all physical parameter needed to describe a photon excluding
+the frequency (position, wave vector, polarization...). In the following we will look at interferometers, and thus the
+parameter α will describe in which arm the photon is propagating. We will thus describe single photon states in
+a given mode α with frequency ω with the help of a creation operator acting on the vacuum state: ˆa†
+α(ω). In this
+situation the commutation relation is written as
+[ˆaα(ω), ˆa†
+β(ω′)] = δ(ω − ω′)δα,β,
+(A1)
+the other commutation relations (between two creation or two annihilation operators) vanishing. It’s useful to intro-
+duce the conjugated temporal variable t, by the use of the Fourier transform:
+ˆaα(t) =
+1
+√
+2π
+�
+dωˆaα(ω)e−iωt.
+(A2)
+We can verify that the creation and annihilation operators in the temporal domain verify the same commutation
+relation as the one in the spectral domain: [ˆaα(t), ˆa†
+β(t′)] = δ(t − t′)δα,β.
+a.
+States in time-frequency variables
+The creation operators allow to define general single photon states on a single mode via:
+|ψ⟩ =
+�
+dωS(ω)ˆa†(ω) |vac⟩ =
+�
+dωS(ω) |ω⟩ .
+(A3)
+The spectrum S(ω) is the Fourier transform of the time of arrival distribution and it can be recovered from the
+state S(ω) = ⟨ω|ψ⟩. If we are interested in a collection of n single photons states in n different modes, we can work
+with the state:
+|ψ⟩ =
+�
+dω1 · · · dωnF(ω1, · · · , ωn)ˆa†
+1(ω1) · · · ˆa†
+n(ωn) |vac⟩ =
+�
+dω1 · · · dωnF(ω1, · · · , ωn) |ω1, · · · , ωn⟩ ,
+(A4)
+where the spectral function F is normalised to one:
+�
+|F(ω1, ω2|2dω1dω2 = 1.
+
+13
+b.
+Time-frequency operators
+We can introduce two very useful operators as follows:
+ˆtα =
+�
+dt tˆa†
+α(t)ˆaα(t)
+ˆωα =
+�
+dω ωˆa†
+α(ω)ˆaα(ω).
+(A5)
+The fundamental property of these operators is the fact that they verify the familiar commutation relation on the
+subspace of single photons:
+[ˆωα, ˆtα] = i.
+(A6)
+More precisely, we have the general result:
+[ˆωα, ˆtα] = i
+∞
+�
+−∞
+dωˆa†
+α(ω)ˆaα(ω) = i ˆNα,
+(A7)
+where the operator ˆNα count the number of photon operator in the mode α.
+The action of the both operators ˆω and ˆt can be computed on the JSA and we have:
+ˆω : S(ω) �→ ωS(ω)
+ˆt : S(ω) �→ −i∂ωS(ω).
+(A8)
+Appendix B: Appendix: Derivation of equations (8) and (9)
+a.
+Equation (8)
+To show equation (8) we start with the state before the BS:
+ˆU |ψ⟩ =
+�
+dω1dω2F(ω1, ω2) |ω1, ω2⟩ .
+(B1)
+The usual balanced BS relation reads:
+|ω1⟩1 |ω2⟩2 �→ 1
+2
+�
+|ω1⟩1 |ω2⟩1 − |ω1⟩1 |ω2⟩2 + |ω1⟩2 |ω2⟩1 − |ω1⟩2 |ω2⟩2
+�
+.
+(B2)
+To be able to use it, we introduce two mode changing operators ˆT1 and ˆT2 defined by:
+ˆT1 |ω1⟩1 |ω2⟩2 = |ω1⟩1 |ω2⟩1
+ˆT2 |ω1⟩1 |ω2⟩2 = |ω1⟩2 |ω2⟩2 .
+(B3)
+With these definition the BS splitter relation is equivalent to applying the operator:
+1
+2( ˆT1 − ˆ1 + ˆS − ˆT2),
+(B4)
+where ˆS is the swap operator, defined as ˆS |ω1, ω2⟩ = |ω2, ω1⟩ So the state coming out of the BS is:
+|ψout⟩ = 1
+2
+�
+dω1dω2F(ω1, ω2)
+�
+ˆT1 ˆU − ˆU + ˆS ˆU − ˆT2 ˆU
+�
+|ω1, ω2⟩ .
+(B5)
+If we do selection on coincidence, we only keep the part of the state with one photon in each mode. We get the state:
+|ψfin⟩ = −1
+2
+�
+dω1dω2F(ω1, ω2)
+�
+ˆU − ˆS ˆU
+�
+|ω1, ω2⟩
+(B6a)
+= 1
+2
+�
+ˆS ˆU − ˆU
+�
+|ψ⟩ .
+(B6b)
+
+14
+We can finally compute the coincidence probability by taking the norm square of |ψfin⟩:
+Pc = ⟨ψfin|ψfin⟩
+(B7a)
+= 1
+4 ⟨ψ|
+�
+ˆU † − ˆU † ˆS
+��
+ˆU − ˆS ˆU
+�
+|ψ⟩
+(B7b)
+= 1
+4 ⟨ψ|
+�
+ˆU † ˆU
+����
+=1
+−2 ˆU † ˆS ˆU + ˆU † ˆS ˆS ˆU
+� �� �
+= ˆU † ˆU=1
+�
+|ψ⟩
+(B7c)
+= 1
+2
+�
+1 − ⟨ψ| ˆU † ˆS ˆU |ψ⟩
+�
+.
+(B7d)
+b.
+Equation (9)
+The expression for Q is a direct consequence of the expression of the QFI for pure state.
+The proof of the expression of F is a little bit more involved. We have to compute:
+FI(κ) = 1
+Pc
+�∂Pc
+∂κ
+�2
++ 1
+Pa
+�∂Pa
+∂κ
+�2
+.
+(B8)
+We have seen the expression of the (anti)-coincidence probability Pc and Pa that depends on ⟨ψ| ˆU † ˆS ˆU |ψ⟩. If we
+make the assumption that the state |ψ⟩ is either symmetric or anti-symmetric we known that we have: ⟨ψ| ˆU † ˆS ˆU |ψ⟩ =
+± ⟨ψ| ˆU † ˆS ˆU ˆS |ψ⟩ = ⟨ψ| ˆV (κ) |ψ⟩ where we denote ˆV (κ) = ˆU † ˆS ˆU ˆS = eiκ ˆ
+He−iκ ˆS ˆ
+H ˆS. We first start by expanding this
+scalar product up to the second order in κ, using the short hand notation ⟨·⟩ = ⟨ψ| · |ψ⟩.
+⟨ψ| ˆV (κ) |ψ⟩ =
+�
+eiκ ˆ
+He−iκ ˆS ˆ
+H ˆS�
+(B9a)
+≃
+��
+1 + iκ ˆH − κ2
+2
+ˆH2��
+1 − iκ ˆS ˆH ˆS − κ2
+2 ( ˆS ˆH ˆS)2��
+(B9b)
+=
+�
+1 + iκ ˆH − iκ ˆS ˆH ˆS − κ2
+2
+ˆH2 − κ2
+2 ( ˆS ˆH ˆS)2 + κ ˆH ˆS ˆH ˆS
+�
+(B9c)
+Since the state |ψ⟩ is (anti)-symmetric, for any operators ˆG, we have
+�
+ˆS ˆG
+�
+= ±
+�
+ˆG
+�
+=
+�
+ˆG ˆS
+�
+, which allows some
+simplifications.
+= 1 − κ2
+2
+� �
+ˆH2�
++
+�
+( ˆS ˆH ˆS)2�
+−
+�
+ˆH ˆS ˆH ˆS
+�
+−
+�
+ˆS ˆH ˆS ˆH
+� �
+(B9d)
+= 1 − κ2
+2
+�
+( ˆH − ˆS ˆH ˆS)2�
+(B9e)
+= 1 − κ2
+2 ∆( ˆH − ˆS ˆH ˆS)2
+(B9f)
+Since thanks to the symmetry of |ψ⟩,
+�
+ˆH − ˆS ˆH ˆS
+�
+=
+�
+ˆH − ˆH ˆS2�
+= 0
+
+15
+By defining ˆG = ˆH − ˆS ˆH ˆS it remains to compute the FI:
+FI(κ = 0) = 1
+Pc
+�∂Pc
+∂κ
+�2
++ 1
+Pa
+�∂Pa
+∂κ
+�2
+(B10a)
+=
+1
+4Pc
+�
+κ∆( ˆG)2�2
++
+1
+4Pa
+�
+κ∆( ˆG)2�2
+(B10b)
+= κ2∆( ˆG)4
+4
+� 1
+Pc
++ 1
+Pa
+�
+(B10c)
+= κ2∆( ˆG)4
+4
+Pa + Pc
+PcPa
+(B10d)
+= κ2∆( ˆG)4
+4
+4
+�
+1 + ⟨ψ| ˆV (κ) |ψ⟩
+� �
+1 − ⟨ψ| ˆV (κ) |ψ⟩
+�
+(B10e)
+= κ2∆( ˆG)4
+1
+1 − ⟨ψ| ˆV (κ) |ψ⟩2
+(B10f)
+= κ2∆( ˆG)4
+1
+κ2∆( ˆG)2
+(B10g)
+= ∆( ˆG)2
+(B10h)
+It is interesting to note that the computation of the Fisher information is singular. Indeed for the HOM interfer-
+ometer around κ = 0 the derivative of the probabilities vanishes ∂κPc,a = 0, while one of the two probability (Pc
+if the state is symmetric or Pa if its anti-symmetric) is also equal to zero. We thus obtain here the FI at zero by
+computing it at κ ̸= 0 and taking the limit. As a result we see that the FI is proportional to the second derivative
+of the coincidence probability. This means that for such a measurement what is important is the curvature of the
+probability peak/dip.
+Appendix C: Appendix: Details on the computation of the various variances
+To compute explicitly the various variances of this paper on the two states |ψG⟩ and |ψG⟩ one can note that
+since these states are separable in the variables ω±, if we consider two operators ˆH+ and ˆH− which are respectively
+functions of ˆω+ and ˆt+ or ˆω− and ˆt− we have:
+�
+ˆH+ ˆH−
+�
+=
+�
+ˆH+
+� �
+ˆH−
+�
+. Where for a fixed state |ψ⟩,
+�
+ˆH
+�
+= ⟨ψ| ˆH |ψ⟩.
+In order to compute any variance, one only has to compute some expectation values. By expanding and using
+the independence property from above, one only need to compute as building block expectation value of the form:
+�
+ˆωk
+±ˆtl
+±
+�
+. Indeed we can use the commutation relation to reorder any product such that the frequency operators are
+on the left of the time operators. One has to pay attention that due to the choice of normalisation in the definition of
+ˆω± = ˆω1 ± ˆω2 and ˆt± = ˆt1 ± ˆt2 we have [ˆω±, ˆt±] = 2i. Such expectation values can be obtained systematically using
+a software (here we used Mathematica), we have the following values:
+
+16
+Operator
+Variable +
+Variable − for |ψG⟩
+Variable − for |ψC⟩
+ˆω
+ωp
+0
+0
+ˆω2
+ω2
+p + σ2
++
+σ2
+−
+σ2
+− + 1
+4∆2
+ˆω3
+3σ2
++ωp + ω3
+p
+0
+0
+ˆω4
+3σ4
++ + 6σ2
++ω2
+p + ω4
+p
+3σ4
+−
+3σ4
+− + 3
+2σ2
+−∆2 +
+1
+16∆4
+ˆt
+0
+0
+0
+ˆt2
+1
+σ2
++
+1
+σ2
+−
+1
+σ2
+−
+ˆt3
+0
+0
+0
+ˆt4
+3
+σ4
++
+3
+σ4
+−
+3
+σ4
+−
+ˆωˆt
+i
+i
+i
+ˆω2ˆt
+2iωp
+0
+0
+ˆωˆt2
+ωp
+σ2
++
+0
+0
+ˆω2ˆt2
+ω2
+p
+σ2
++ − 1
+−1
+∆2
+4σ2
+− − 1
+TABLE III: Expectation values of the various product of plus and minus operators on the states |ψG⟩ and |ψC⟩.
+

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+arXiv:2301.08608v1  [cs.AI]  20 Jan 2023
+On the Foundations of Cycles
+in Bayesian Networks⋆
+Christel Baier1, Clemens Dubslaff1, Holger Hermanns2,3, and Nikolai K¨afer1
+1 TU Dresden, Dresden, Germany
+2 Saarland University, Saarbr¨ucken, Germany
+3 Institute of Intelligent Software, Guangzhou, China
+Abstract. Bayesian networks (BNs) are a probabilistic graphical model
+widely used for representing expert knowledge and reasoning under un-
+certainty. Traditionally, they are based on directed acyclic graphs that
+capture dependencies between random variables. However, directed cy-
+cles can naturally arise when cross-dependencies between random vari-
+ables exist, e.g., for modeling feedback loops. Existing methods to deal
+with such cross-dependencies usually rely on reductions to BNs without
+cycles. These approaches are fragile to generalize, since their justifica-
+tions are intermingled with additional knowledge about the application
+context. In this paper, we present a foundational study regarding seman-
+tics for cyclic BNs that are generic and conservatively extend the cycle-
+free setting. First, we propose constraint-based semantics that specify
+requirements for full joint distributions over a BN to be consistent with
+the local conditional probabilities and independencies. Second, two kinds
+of limit semantics that formalize infinite unfolding approaches are intro-
+duced and shown to be computable by a Markov chain construction.
+1
+Introduction
+A Bayesian network (BN) is a probabilistic graphical model representing a set of
+random variables and their conditional dependencies. BNs are ubiquitous across
+many fields where reasoning under uncertainties is of interest [10]. Specifically,
+a BN is a directed acyclic graph with the random variables as nodes and edges
+manifesting conditional dependencies, quantified by conditional probability ta-
+bles (CPTs). The probability of any random variable can then be deduced by
+the CPT entries along all its predecessors. Here, these probabilities are indepen-
+dent of all variables that are no (direct or transitive) predecessors in the graph.
+Acyclicity is hence crucial and commonly assumed to be rooted in some sort of
+causality [23]. A classical use of BNs is in expert systems [22] where BNs ag-
+gregate statistical data obtained by several independent studies. In the medical
+⋆ This work was partially supported by the DFG in projects TRR 248 (CPEC,
+see https://perspicuous-computing.science, project ID 389792660) and EXC 2050/1
+(CeTI, project ID 390696704, as part of Germany’s Excellence Strategy), and the
+Key-Area Research and Development Program Grant 2018B010107004 of Guang-
+dong Province.
+
+2
+C. Baier et al.
+X
+Y
+X=T
+X=F
+F
+s1
+1 − s1
+T
+s2
+1 − s2
+Y
+X
+Y=T
+Y=F
+F
+t1
+1 − t1
+T
+t2
+1 − t2
+Fig. 1: A cyclic GBN with CPTs for X and Y
+domain, e.g., they can capture the correlation of certain symptoms, diseases, and
+human factors [11,15,26].
+Imagine for instance an expert system for supporting diagnosis of Covid-
+19, harvesting multiple clinical studies. One study might have investigated the
+percentage of patients who have been diagnosed with fever also having Covid-
+19, while another study in turn might have investigated among the Covid-19
+patients whether they have fever, too. Clearly, both studies investigate the de-
+pendency between fever and Covid-19, but under different conditions. Fever may
+weaken the immune system and could increase the risk of a Covid-19 infection,
+while Covid-19 itself has fever as a symptom. In case there is uniform knowledge
+about “which symptom was first” in each of the constituent studies, then dy-
+namic Bayesian networks (DBNs) [19] could be used as a model for the expert
+system, breaking the interdependence of fever and Covid-19 through a prece-
+dence relation. However, this implies either to rely only on studies where these
+temporal dependencies are clearly identified or to introduce an artificial notion
+of time that might lead to spurious results [18]. A naive encoding into the BN
+framework always yields a graph structure that contains cycles, as is the case in
+our small example shown in Fig. 1 where X and Y stand for the random variables
+of diagnosing Covid-19 and fever, respectively.
+That cycles might be unavoidable has already been observed in seminal pa-
+pers such as [22,15]. But acyclicity is crucial for computing the joint probability
+distribution of a BN, and thereby is a prerequisite for, e.g., routine inference
+tasks. Existing literature that considers cycles in BNs mainly recommends re-
+ducing questions on the probability values to properties in acyclic BNs. For
+instance, in [11] nodes are collapsed towards removing cycles, while [22] suggests
+to condition on each value combination on a cycle, generating a decomposition
+into tree-like BNs and then averaging over the results to replace cycles. Some-
+times, application-specific methods that restructure the cyclic BN towards an
+acyclic BN by introducing additional nodes [26,8] or by unrolling cycles up to a
+bounded depth [17,2] have been reported to give satisfactory results. Other ap-
+proaches either remove edges that have less influence or reverse edges on cycles
+(see, e.g., [10]). However, such approaches are highly application dependent and
+hinge on knowledge about the context of the statistical data used to construct
+the BN. Furthermore, as already pointed out by [30], they usually reduce the
+solution space of families of joint distributions to a single one, or introduce so-
+lutions not consistent with the CPTs of the original cyclic BN. While obviously
+many practitioners have stumbled on the problem how to treat cycles in BNs
+
+On the Foundations of Cycles in Bayesian Networks
+3
+and on the foundational question “What is the meaning of a cyclic BN?”, there
+is very little work on the foundations of Bayesian reasoning with cycles.
+In this paper, we approach this question by presenting general semantics for
+BNs with cycles, together with algorithms to compute families of joint distri-
+butions for such BNs. First, we investigate how the two main constituents of
+classical BNs, namely consistency with the CPTs and independencies induced
+by the graph structure, influence the joint distributions in the presence of cycles.
+This leads to constraints semantics for cyclic BNs that comprise all those joint
+distributions respecting the constraints, being either a single uniquely defined
+one, none, or infinitely many distributions. Second, we present semantics that
+formalize unfolding approaches and depend on the choice of a cutset, a set of
+random variables that break every cycle in a cyclic BN. Intuitively, such cutsets
+form the seams along which feedback loops can be unraveled. These semantics
+are defined in terms of the limit (or limit average) of a sequence of distribu-
+tions at descending levels in the infinite unfolding of the BN. We show that
+the same semantics can be defined using a Markov chain construction and sub-
+sequent long-run frequency analysis, which enables both precise computation
+of the semantics and deep insights in the semantics’ behavior. Among others,
+an immediate result is that the family of distributions induced with respect to
+the limit semantics is always non-empty. As we will argue, the limit semantics
+have obvious relations to a manifold of approaches that have appeared in the
+literature, yet they have not been spelled out and studied explicitly.
+1.1
+Notation
+Let V be a set of Boolean random variables4 over the domain B = {F, T}. We
+usually denote elements of V by X, Y, or Z. An assignment over V is a function
+b: V → B which we may specify through set notation, e.g., b = {X=T, Y=F} for
+b(X) = T and b(Y ) = F, or even more succinctly as XY . The set of all possible
+assignments over V is denoted by Asg(V). We write bU for the restriction of b to a
+subset U ⊆ V, e.g., b{X} = {X=T}, and may omit set braces, e.g., bX,Y = b{X,Y }.
+A distribution over a set S is a function µ: S → [0, 1] where �
+s∈S µ(s) = 1.
+The set of all distributions over S is denoted by Dist(S). For |S| = n, µ will
+occasionally be represented as a vector of size n for some fixed order on S. In
+the following, we are mainly concerned with distributions over assignments, that
+is distributions µ ∈ Dist(Asg(V)) for some set of random variables V. Each
+such distribution µ induces a probability measure (also called µ) on 2Asg(V).
+Thus, for a set of assignments φ ⊆ Asg(V), we have µ(φ) = �
+b∈φ µ(b). We
+are often interested in the probability of a partial assignment d ∈ Asg(U) on a
+subset U ⊊ V of variables, which is given as the probability of the set of all full
+4 We use Boolean random variables for simplicity of representation, an extension of
+the proposed semantics over random variables with arbitrary finite state spaces is
+certainly possible.
+
+4
+C. Baier et al.
+assignments b ∈ Asg(V) that agree with d on U. As a shorthand, we define
+µ(d) := µ
+�
+{b ∈ Asg(V) : bU = d}
+�
+=
+�
+b∈Asg(V)
+s.t. bU =d
+µ(b).
+The special case µ(X=T) is called the marginal probability of X. The restriction of
+µ ∈ Dist(Asg(V)) to U, denoted µ|U ∈ Dist(Asg(U)), is given by µ|U(d) := µ(d).
+For a set W disjoint from V and ν ∈ Dist(Asg(W)), the product distribution of
+µ and ν is given by (µ ⊗ ν)(c) := µ(cV) · ν(cW) for every c ∈ Asg(V ∪ W). µ is
+called a Dirac distribution if µ(b) = 1 for some assignment b ∈ Asg(V) and thus
+µ(c) = 0 for all other assignments c ̸= b. A Dirac distribution derived from a
+given assignment b is denoted by Dirac(b).
+Graph Notations. For a graph G = ⟨V, E⟩ with nodes V and directed edges
+E ⊆ V × V, we may represent an edge (X, Y ) ∈ E as X → Y if E is clear from
+context. Pre(X) := {Y ∈ V : Y → X} denotes the set of parents of a node X ∈ V,
+and Post∗(X) := {Y ∈ V : X → · · · → Y } is the set of nodes reachable from
+X. A node X is called initial if Pre(X) = ∅, and Init(G) is the set of all nodes
+initial in G. A graph G is strongly connected if each node in V is reachable from
+every other node. A set of nodes D is a strongly connected component (SCC) of
+G if all nodes in D can reach each other and D is not contained in another SCC,
+and a bottom SCC (BSCC) if no node in V \ D can be reached from D.
+Markov Chains. A discrete-time Markov chain (DTMC) is a tuple M = ⟨S, P⟩
+where S is a finite set of states and P: S × S → [0, 1] a function such that
+P(s, ·) ∈ Dist(S) for all states s ∈ S. The underlying graph GM = ⟨S, E⟩ is
+defined by E = {(s, t) ∈ S × S : P(s, t) > 0}. The transient distribution πι
+n ∈
+Dist(S) at step n is defined through the probability πι
+n(s) to be in state s after
+n steps if starting with initial state distribution ι. It satisfies (in matrix-vector
+notation) πι
+n = ι · Pn. We are also interested in the long-run frequency of state
+occupancies when n tends to infinity, defined as the Ces`aro limit lrfι : S → [0, 1]:
+lrfι(s) :=
+lim
+n→∞
+1
+n + 1
+n
+�
+i=0
+πι
+n(s).
+(LRF)
+This limit always exists and corresponds to the long-run fraction of time spent
+in each state [12]. The limit probability limn→∞ πι
+n is arguably more intuitive as
+a measure of the long-run behavior, but may not exist (due to periodicity). In
+case of existence, it agrees with the Ces`aro limit lrfι. If GM forms an SCC, the
+limit is independent of the choice of ι and the superscript can be dropped. We
+denote this limit by lrfM.
+2
+Generalized Bayesian Networks
+We introduce generalized Bayesian networks (GBNs) as a BN model that does
+not impose acyclicity and comes with a distribution over initial nodes.
+
+On the Foundations of Cycles in Bayesian Networks
+5
+Definition 1 (Generalized BN). A GBN B is a tuple ⟨G, P, ι⟩ where
+– G = ⟨V, E⟩ is a directed graph with nodes V and an edge relation E ⊆ V × V,
+– P is a function that maps all non-initial nodes X ∈ V\Init(G) paired with
+each of their parent assignments b ∈ Asg(Pre(X)) to a distribution
+P(X, b): Asg
+�
+{X}
+�
+→ [0, 1],
+– ι is a distribution over the assignments for the initial nodes Init(G), i.e.,
+ι ∈ Dist
+�
+Asg(Init(G))
+�
+.
+The distributions P(X, b) have the same role as the entries in a conditional
+probability table (CPT) for X in classical BNs: they specify the probability for
+X=T or X=F depending on the assignments of the predecessors of X. To this
+end, for X ∈ V\Init(G) and b ∈ Asg(Pre(X)), we also write Pr(X=T | b) for
+P(X, b)(X=T). In the literature, initial nodes are often assigned a marginal prob-
+ability via a CPT as well, assuming independence of all initial nodes. Differently,
+in our definition of GBNs, it is possible to specify an arbitrary distribution ι over
+all initial nodes. If needed, P can be easily extended to initial nodes by setting
+P(X, ∅) := ι|{X} for all X ∈ Init(G). Hence, classical BNs arise as a special
+instance of GBNs where the graph G is acyclic and initial nodes are pairwise
+independent. In that case, the CPTs given by P are a compact representation of
+a single unique full joint distribution dist BN(B) over all random variables X ∈ V.
+For every assignment b ∈ Asg(V), we can compute dist BN(B)(b) by the so-called
+chain rule:
+dist BN(B)(b) := ι
+�
+bInit(G)
+�
+·
+�
+X∈V\Init(G)
+Pr
+�
+bX | bPre(X)
+�
+.
+(CR)
+In light of the semantics introduced later on, we define the standard BN-semantics
+of an acyclic GBN B as the set �B�BN := {distBN(B)}, and �B�BN := ∅ if B con-
+tains cycles.
+The distribution dist BN(B) satisfies two crucial properties: First, it is con-
+sistent with the CPT entries given by P and the distribution ι, and second, it
+observes the independencies encoded in the graph G. In fact, those two properties
+are sufficient to uniquely characterize distBN(B). We briefly review the notion of
+independence and formally define CPT consistency later on in Section 3.
+Independence. Any full joint probability distribution µ ∈ Dist(Asg(V)) may in-
+duce a number of conditional independencies among the random variables in V.
+For X, Y, and Z disjoint subsets of V, the random variables in X and Y are
+independent under µ given Z if the conditional probability of each assignment
+over the nodes in X given an assignment for Z is unaffected by further condi-
+tioning on any assignment of Y. Formally, the set Indep(µ) contains the triple
+(X, Y, Z) iff for all a ∈ Asg(X), b ∈ Asg(Y), and c ∈ Asg(Z), we have
+µ(a | b, c) = µ(a | c)
+or
+µ(b, c) = 0.
+We also write (X ⊥ Y | Z) for (X, Y, Z) ∈ Indep(µ) and may omit the set
+brackets of X, Y, and Z.
+
+6
+C. Baier et al.
+d-separation. For classical BNs, the graph topology encodes independencies that
+are necessarily satisfied by any full joint distribution regardless of the CPT
+entries. Given two random variables X and Y as well as a set of observed variables
+Z, then X and Y are conditionally independent given Z if the corresponding
+nodes in the graph are d-separated given Z [6]. To establish d-separation, all
+simple undirected paths5 between X and Y need to be blocked given Z. Let W
+denote such a simple path W0, W1, . . . , Wk with W0 = X, Wk = Y, and either
+Wi → Wi+1 or Wi ← Wi+1 for all i < k. Then W is blocked given Z if and only
+if there exists an index i, 0 < i < k, such that one of the following two conditions
+holds: (1) Wi is in Z and is situated in a chain or a fork in W, i.e.,
+– Wi−1 → Wi → Wi+1 (forward chain)
+– Wi−1 ← Wi ← Wi+1 (backward chain)
+and
+Wi ∈ Z,
+– Wi−1 ← Wi → Wi+1 (fork)
+(2) Wi is in a collider and neither Wi nor any descendant of Wi is in Z, i.e.,
+– Wi−1 → Wi ← Wi+1 (collider)
+and
+Post∗(Wi) ∩ Z = ∅.
+Two sets of nodes X and Y are d-separated given a third set Z if for each X ∈ X
+and Y ∈ Y, X and Y are d-separated given Z. Notably, the d-separation criterion
+is applicable also in presence of cycles [28]. For a graph G = ⟨V, E⟩ of a GBN,
+we define the set d-sep(G) as
+d-sep(G) :=
+�
+(X, Y, Z) ∈ (2V)3 : X and Y are d-separated given Z
+�
+.
+For acyclic Bayesian networks it is well known that the independencies ev-
+ident from the standard BN semantics’ distribution include the independencies
+derived from the graph. That is, for acyclic GBNs B̸⟳ = ⟨G, P, ι⟩ where all initial
+nodes are pairwise independent under ι, we have
+d-sep(G) ⊆ Indep
+�
+dist BN(B̸⟳)
+�
+.
+For an arbitrary initial distribution, the above relation does not necessarily
+hold. However, we can still find a set of independencies that are necessarily
+observed by the standard BN semantics and thus act as a similar lower bound.
+We do so by assuming the worst case, namely that each initial node depends on
+every other initial node under ι. Formally, given a graph G = ⟨V, E⟩, we define a
+closure operation Close(·) as follows and compute the set d-sep
+�
+Close(G)
+�
+:
+Close(G) :=
+�
+V, E ∪ {(A, B) for A, B ∈ Init(G), A ̸= B}
+�
+.
+Lemma 1. Let B̸⟳ = ⟨G, P, ι⟩ be an acyclic GBN. Then
+d-sep
+�
+Close(G)
+�
+⊆ Indep
+�
+distBN(B̸⟳)
+�
+.
+As intuitively expected, the presence of cycles in G generally reduces the
+number of graph independencies, though note that also in strongly connected
+graphs independencies may exist. For example, if G is a four-node cycle with
+nodes W, X, Y, and Z, then d-sep(G) =
+�
+(W ⊥ Y | X, Z), (X ⊥ Z | W, Y )
+�
+.
+5 A path is simple if no node occurs twice in the path. “Undirected” in this context
+means that edges in either direction can occur along the path.
+
+On the Foundations of Cycles in Bayesian Networks
+7
+3
+Constraints Semantics
+For classical acyclic BNs there is exactly one distribution that agrees with all
+CPTs and satisfies the independencies encoded in the graph. This distribution
+can easily be constructed by means of the chain rule (CR). For cyclic GBNs,
+applying the chain rule towards a full joint distribution is not possible in general,
+as the result is usually not a valid probability distribution. Still, we can look for
+distributions consistent with a GBN’s CPTs and the independencies derived
+from its graph. Depending on the GBN, we will see that there may be none,
+exactly one, or even infinitely many distributions fulfilling these constraints.
+3.1
+CPT-consistency
+We first provide a formal definition of CPT consistency in terms of linear con-
+straints on full joint distributions.
+Definition 2 (Strong and weak CPT-consistency). Let B be a GBN with
+nodes V and X ∈ V. Then µ is called strongly CPT-consistent for X in B (or
+simply CPT-consistent) if for all c ∈ Asg(Pre(X))
+µ(X=T, c) = µ(c) · Pr(X=T | c).
+(Cpt)
+We say that µ is weakly CPT-consistent for X in B if
+µ(X=T) =
+�
+c∈Asg(Pre(X))
+µ(c) · Pr(X=T | c).
+(wCpt)
+Intuitively, the constraint (Cpt) is satisfied for µ if the conditional proba-
+bility µ(X=T | c) equals the entry in the CPT for X under assignment c, i.e.,
+µ(X=T | c) = Pr(X=T | c). In the weak case (wCpt), only the resulting marginal
+probability of X needs to agree with the CPTs.
+Definition 3 (Cpt and wCpt semantics).
+For a GBN B = ⟨G, P, ι⟩, the
+CPT-semantics �B�Cpt is the set of all distributions µ ∈ Dist(Asg(V)) where
+µ|Init(G) = ι and µ is CPT-consistent for every node X ∈ V\Init(G). The weak
+CPT-semantics �B�wCpt is defined analogously.
+Clearly, we have �B�Cpt ⊆ �B�wCpt for all B. The next example shows that
+depending on the CPT values, the set �B�Cpt may be empty, a singleton, or of
+infinite cardinality.
+Example 1. To find CPT-consistent distributions for the GBN from Fig. 1, we
+construct a system of linear equations whose solutions form distributions µ ∈
+Dist
+�
+Asg({X, Y })
+�
+, represented as vectors in the space [0, 1]4:
+
+
+
+
+
+
+
+
+s1
+0
+s1−1
+0
+0
+s2
+0
+s2−1
+t1
+t1−1
+0
+0
+0
+0
+t2
+t2−1
+1
+1
+1
+1
+
+
+
+
+
+
+
+
+·
+
+
+
+
+
+µXY
+µXY
+µXY
+µXY
+
+
+
+
+ =
+
+
+
+
+
+
+
+
+0
+0
+0
+0
+1
+
+
+
+
+
+
+
+
+
+8
+C. Baier et al.
+where, e.g., µXY abbreviates µ(X=T, Y=F). The first line of the matrix states
+the (Cpt) constraint for node X and the parent assignment c = {Y=F}:
+0 = s1 · µXY + 0 · µXY + (s1−1) · µXY + 0 · µXY
+µXY
+= (µXY + µXY ) · s1
+µXY
+= µY · Pr(X=T | Y=F)
+µ(X=T, c) = µ(c) · Pr(X=T | c).
+Analogously, the following three rows encode the CPT constraints for X, Y,
+and their remaining parent assignments. The last row ensures that solutions are
+indeed probability distributions satisfying �
+c µ(c) = 1.
+The number of solutions for the system now depends on the CPT entries s1,
+s2, t1, and t2. For s1 = t2 = 0 and s2 = t1 = 1, no solution exists as the first
+four equations require µ(b) = 0 for all b ∈ Asg({X, Y }), while the last equation
+ensures µXY + µXY + µXY + µXY = 1. For s1 = t1 = 0 and s2 = t2 = 1, all
+distributions with µXY = 1 − µXY and µXY = µXY = 0 are solutions. Finally,
+e.g., for s1 = t1 = 3/4 and s2 = t2 = 1/2, there is exactly one solution with
+µXY = 1/10 and µ(b) = 3/10 for the other three assignments.
+3.2
+Independence-consistency
+We extend Cpt semantics with a set of independencies that need to be observed
+by all induced distributions.
+Definition 4 (Cpt-I semantics). For a GBN B = ⟨G, P, ι⟩ and a set of inde-
+pendencies I, the CPT-I semantics �B�Cpt-I is defined as the set of all CPT-
+consistent distributions µ for which I ⊆ Indep(µ) holds.
+Technically, the distributions in �B�Cpt-I have to fulfill the following polynomial
+constraints in addition to the CPT-consistency constraints:
+µ(b) · µ(bW) = µ(b{X}∪W) · µ(bU∪W)
+(Cpt-I)
+for each independence (X ⊥ U | W) ∈ I with variable X∈V and sets of variables
+U, W ⊆ V, and for each assignment b ∈ Asg({X} ∪ U ∪ W). Note that in case
+µ(bW) > 0, (Cpt-I) is equivalent to the constraint µ(bX | bU∪W) = µ(bX | bW).
+We can now formally state the alternative characterization of the standard
+BN semantics as the unique CPT-consistent distribution that satisfies the d-
+separation independencies of the graph. For each classical BN B with acyclic
+graph G and I = d-sep(G), we have �B�BN = {distBN(B)} = �B�Cpt-I. Thus, the
+Cpt-I semantics provides a conservative extension of the standard BN semantics
+to GBNs with cycles. However, in practice, its use is limited since there might be
+no distribution that satisfies all constraints. In fact, the case where �B�Cpt-I = ∅
+is to be expected for most cyclic GBNs, given that the resulting constraint
+systems tend to be heavily over-determined.
+The next section introduces semantics that follow a more constructive ap-
+proach. We will see later on in Section 5.1 that the families of distributions
+induced by these semantics are always non-empty and usually singletons.
+
+On the Foundations of Cycles in Bayesian Networks
+9
+X
+Y
+Z
+Fig. 2: The graph of a strongly connected GBN
+X0
+Y0
+Z0
+X1
+Y1
+Z1
+X2
+Y2
+Z2
+...
+(a) Unfolding along all nodes
+Z0
+X1
+Y1
+Z1
+X2
+Y2
+Z2
+...
+(b) Unfolding along the Z nodes
+Fig. 3: Two infinite unfoldings of the graph in Fig. 2
+4
+Limit and Limit Average Semantics
+We first develop the basic ideas underling the semantics by following an example,
+before giving a formal treatment in Section 4.2.
+4.1
+Intuition
+Consider the GBN B whose graph G is depicted in Fig. 2. One way to get rid
+of the cycles is to construct an infinite unfolding of B as shown in Fig. 3a. In
+this new graph G∞, each level contains a full copy of the original nodes and
+corresponds to some n ∈ N. For any edge X → Y in the original graph, we
+add edges Xn → Yn+1 to G∞, such that each edge descends one level deeper.
+Clearly any graph constructed in this way is acyclic, but this fact alone does
+not aid in finding a matching distribution since we dearly bought it by giving
+up finiteness. However, we can consider what happens when we plug in some
+initial distribution µ0 over the nodes X0, Y0, and Z0. Looking only at the first
+two levels, we then get a fully specified acyclic BN by using the CPTs given
+by P for X1, Y1, and Z1. For this sub-BN, the standard BN semantics yields
+a full joint distribution over the six nodes from X0 to Z1, which also induces
+a distribution µ1 over the three nodes at level 1. This procedure can then be
+repeated to construct a distribution µ2 over the nodes X2, Y2, and Z2, and,
+more generally, to get a distribution µn+1 given a distribution µn. Recall that
+
+10
+C. Baier et al.
+each of those distributions can be viewed as vector of size 23. Considering the
+sequence µ0, µ1, µ2, . . . , the question naturally arises whether a limit exists, i.e.,
+a distribution/vector µ such that
+µ
+=
+lim
+n→∞ µn.
+Example 2. Consider the GBN from Fig. 1 with CPT entries s1 = t2 = 1 and
+s2 = t1 = 0, which intuitively describe the contradictory dependencies “X iff not
+Y ” and “Y iff X”. For any initial distribution µ0 = ⟨e f g h⟩, the construction
+informally described above yields the following sequence of distributions µn:
+µ0 =
+
+
+
+
+
+e
+f
+g
+h
+
+
+
+
+, µ1 =
+
+
+
+
+
+f
+h
+e
+g
+
+
+
+
+, µ2 =
+
+
+
+
+
+h
+g
+f
+e
+
+
+
+
+, µ3 =
+
+
+
+
+
+g
+e
+h
+f
+
+
+
+
+, µ4 =
+
+
+
+
+
+e
+f
+g
+h
+
+
+
+
+, . . .
+As µ4 = µ0, the sequence starts to cycle infinitely between the first four distribu-
+tions. The series converges for e = f = g = h = 1/4 (in which case the sequence
+is constant), but does not converge for any other initial distribution.
+The example shows that the existence of the limit depends on the given initial
+distribution. In case no limit exists because some distributions keep repeating
+without ever converging, it is possible to determine the limit average (or Ces`aro
+limit) of the sequence:
+˜µ
+=
+lim
+n→∞
+1
+n + 1
+n
+�
+i=0
+µi.
+The limit average has three nice properties: First, if the regular limit µ exists,
+then the limit average ˜µ exists as well and is identical to µ. Second, in our use
+case, ˜µ in fact always exists for any initial distribution µ0. And third, as we
+will see in Section 5, the limit average corresponds to the long-run frequency of
+certain Markov chains, which allows us both to explicitly compute and to derive
+important properties of the limit distributions.
+Example 3. Continuing Ex. 2, the limit average of the sequence µ0, µ1, µ2, . . . is
+the uniform distribution ˜µ = ⟨1/4 1/4 1/4 1/4⟩, regardless of the choice of µ0.
+Before we formally define the infinite unfolding of GBNs and the resulting
+limit semantics, there is one more observation to be made. To ensure that the
+unfolded graph G∞ is acyclic, we redirected every edge of the GBN B to point
+one level deeper, resulting in the graph displayed in Fig. 3a. As can be seen
+in Fig. 3b, we also get an acyclic unfolded graph by only redirecting the edges
+originating in the Z nodes to the next level and keeping all other edges on the
+same level. The relevant property is to pick a set of nodes such that for each
+cycle in the original GBN B, at least one node in the cycle is contained in the
+set. We call such sets the cutsets of B.
+
+On the Foundations of Cycles in Bayesian Networks
+11
+Definition 5 (Cutset). Let B be an GBN with graph G = ⟨V, E⟩. A subset
+C ⊆ V is a cutset for B if every cycle in G contains at least one node from C.
+Example 4. The GBN in Fig. 2 has the following cutsets: {Y }, {Z}, {X, Y },
+{X, Z}, {Y, Z}, and {X, Y, Z}. Note that {X} does not form a cutset as no
+node from the cycle Y → Z → Y is contained.
+So far we implicitly used the set V of all nodes for the unfolding, which always
+trivially forms a cutset. The following definitions will be parameterized with a
+cutset, as the choice of cutsets influences the resulting distributions as well as
+the time complexity.
+4.2
+Formal Definition
+Let Vn := {Xn : X ∈ V} denote the set of nodes on the nth level of the unfolding
+in G∞. For C ⊆ V a cutset of the GBN, the subset of cutset nodes on that level
+is given by Cn := {Xn ∈ Vn : X ∈ C}. Then a distribution γn ∈ Dist(Asg(Cn)) for
+the cutset nodes in Cn suffices to get a full distribution µn+1 ∈ Dist(Asg(Vn+1))
+over all nodes on the next level, n + 1: We look at the graph fragment Gn+1 of
+G∞ given by the nodes Cn ∪ Vn+1 and their respective edges. In this fragment,
+the cutset nodes are initial, so the cutset distribution γn can be combined with
+the initial distribution ι to act as new initial distribution. For the nodes in
+Vn+1, the corresponding CPTs as given by P can be used, i.e., Pn(Xn, ·) =
+P(X, ·) for Xn ∈ Vn. Putting everything together, we obtain an acyclic GBN
+Bn+1 = ⟨Gn+1, Pn+1, ι ⊗ γn⟩. However, GBNs constructed in this way for each
+level n > 0 are all isomorphic and only differ in the given cutset distribution γ.
+For simplicity and in light of later use, we thus define a single representative
+GBN Dissect(B, C, γ) that represents a dissection of B along a given cutset C,
+with ι ⊗ γ as initial distribution.
+Definition 6 (Dissected GBN).
+Let B = ⟨G, P, ι⟩ be a GBN with graph
+G = ⟨V, E⟩ and C ⊆ V a cutset for B with distribution γ ∈ Dist(Asg(C)). Then,
+the C-dissected GBN Dissect(B, C, γ) is the acyclic GBN ⟨GC, PC, ι ⊗ γ⟩ with
+graph GC = ⟨V ∪ C′, EC⟩ where
+– C′ := {X′ : X ∈ C} extends V by fresh copies of all cutset nodes;
+– incoming edges to nodes in C are redirected to their copies, i.e.,
+EC :=
+�
+(X, Y ′) : (X, Y ) ∈ E, Y ∈ C
+�
+∪
+�
+(X, Y ) : (X, Y ) ∈ E, Y /∈ C
+�
+;
+– the function PC uses the CPT entries given by P for the cutset nodes as
+entries for their copies and the original entries for all other nodes, i.e., we
+have PC(Y ′, a) = P(Y, a) for each node Y ′ ∈ C′ and parent assignment a ∈
+Asg(Pre(Y ′)), and PC(X, b) = P(X, b) for X ∈ V\C and b ∈ Asg(Pre(X)).
+Fig. 4 shows two examples of dissections on the GBN of Fig. 2. As any dissected
+GBN is acyclic by construction, the standard BN semantics yields a full joint
+distribution over all nodes in V ∪ C′. We restrict this distribution to the nodes
+
+12
+C. Baier et al.
+X
+Y
+Z
+X′
+Y ′
+Z′
+(a) Cutset C = {X, Y, Z}
+Z
+X
+Y
+Z′
+(b) Cutset C = {Z}
+Fig. 4: Dissections of the GBN in Fig. 2 for two cutsets
+in (V \ C) ∪ C′, as those are the ones on the “next level” of the unfolding, while
+re-identifying the cutset node copies with the original nodes to get a distribution
+over V. Formally, we define the distribution Next(B, C, γ) for each assignment
+b ∈ Asg(V) as
+Next(B, C, γ)(b) := dist BN
+�
+Dissect(B, C, γ)
+�
+(b′)
+where the assignment b′ ∈ Asg
+�
+(V\C) ∪ C′�
+is given by b′(X) = b(X) for all
+X ∈ V\C and b′(Y ′) = b(Y ) for all Y ∈ C. In the unfolded GBN, this allows
+us to get from a cutset distribution γn to the next level distribution µn+1 =
+Next(B, C, γn). The next cutset distribution γn+1 is then given by restricting the
+full distribution to the nodes in C, i.e., γn+1 = Next(B, C, γn)|C.6 Vice versa, a
+cutset distribution γ suffices to recover the full joint distribution over all nodes
+V. Again using the standard BN semantics of the dissected GBN, we define the
+distribution Extend(B, C, γ) ∈ Dist(Asg(V)) as
+Extend(B, C, γ) := dist BN
+�
+Dissect(B, C, γ)
+���
+V.
+With these definitions at hand, we can formally define the limit and limit
+average semantics described in the previous section.
+Definition 7 (Limit and limit average semantics). Let B be a GBN over
+nodes V with cutset C. The limit semantics of B w.r.t. C is the partial function
+Lim(B, C, ·) : Dist
+�
+Asg(C)
+�
+⇀ Dist
+�
+Asg(V)
+�
+from initial cutset distributions γ0 to full distributions µ = Extend(B, C, γ) where
+γ = lim
+n→∞ γn
+and
+γn+1 = Next(B, C, γn)|C.
+The set �B�Lim-C is given by the image of Lim(B, C, ·), i.e.,
+�B�Lim-C := {Lim(B, C, γ0) : γ0 ∈ Dist(Asg(C)) s.t. Lim(B, C, γ0) is defined}.
+The limit average semantics of B w.r.t. C is the partial function
+LimAvg(B, C, ·) : Dist
+�
+Asg(C)
+�
+⇀ Dist
+�
+Asg(V)
+�
+6 Recall that we may view distributions as vectors which allows us to equate distribu-
+tions over different but isomorphic domains.
+
+On the Foundations of Cycles in Bayesian Networks
+13
+X=T
+Y=T
+X=T
+Y=F
+X=F
+Y=T
+X=F
+Y=F
+3/8
+3/8
+1/8
+1/8
+1/2
+1/2
+3/4
+1/4
+1
+X=T
+Y=T
+X=T
+Y=F
+X=F
+Y=T
+X=F
+Y=F
+Fig. 5: A cutset Markov chain for a cutset C = {X, Y }
+from γ0 to distributions µ = Extend(B, C, γ) where
+γ = lim
+n→∞
+1
+n + 1
+n
+�
+i=0
+γn
+and
+γn+1 = Next(B, C, γn)|C.
+The set �B�LimAvg-C is likewise given by the image of LimAvg(B, C, ·).
+We know that the limit average coincides with the regular limit if the lat-
+ter exists, so for every initial cutset distribution γ0, we have Lim(B, C, γ0) =
+LimAvg(B, C, γ0) if Lim(B, C, γ0) is defined. Thus, �B�Lim-C ⊆ �B�LimAvg-C.
+5
+Markov Chain Semantics
+While we gave some motivation for the limit and limit average semantics, their
+definitions do not reveal an explicit way to compute their member distributions.
+In this section we introduce the (cutset) Markov chain semantics which offers
+explicit construction of distributions and is shown to coincide with the limit
+average semantics. It further paves the way for proving several properties of
+both limit semantics in Section 5.1.
+At the core of the cutset Markov chain semantics lies the eponymous cut-
+set Markov chain which captures how probability mass flows from one cutset
+assignment to the others. To this end, the Dirac distributions corresponding to
+each assignment are used as initial distributions in the dissected GBN. With the
+Next function we then get a new distribution over all cutset assignments, and
+the probabilities assigned by this distribution are used as transition probabilities
+for the Markov chain.
+Definition 8 (Cutset Markov chain).
+Let B be a GBN with cutset C. The
+cutset Markov chain CMC(B, C) = ⟨Asg(C), P⟩ w.r.t. B and C is a DTMC where
+the transition matrix P is given for all cutset assignments b, c ∈ Asg(C) by
+P(b, c) := Next
+�
+B, C, Dirac(b)
+�
+(c).
+Example 5. Fig. 5 shows the cutset Markov chain for the GBN from Fig. 1 with
+CPT entries s1 = 1/4, s2 = 1, t1 = 1/2, t2 = 0, and cutset C = {X, Y }. Exem-
+plarily, the edge at the bottom from assignment b = {X=F, Y=F} to assignment
+
+14
+C. Baier et al.
+c = {X=F, Y=T} with label 3/8 is derived as follows:
+P(b, c)
+=
+Next
+�
+B, C, Dirac(b)
+�
+(c) = dist BN
+�
+Dissect(B, C, Dirac(b))
+�
+(c′)
+=
+�
+a∈Asg(VC)
+s.t. c′⊆a
+dist BN
+�
+Dissect(B, C, Dirac(b))
+�
+(a)
+=
+�
+a∈Asg(VC)
+s.t. c′⊆a
+Dirac(b)(aX,Y ) · Pr(X′=F | aY ) · Pr(Y ′=T | aX)
+=
+Pr(X′=F | Y=F) · Pr(Y ′=T | X=F) = (1 − s1) · t1
+=
+3/8.
+Note that in the second-to-last step, in the sum over all full assignments a which
+agree with the partial assignment c′, only the assignment which also agrees with
+b remains as for all other assignments we have Dirac(b)(aX,Y ) = 0.
+Given a cutset Markov chain with transition matrix P and an initial cutset
+distribution γ0, we can compute the uniquely defined long-run frequency distri-
+bution lrfγ0 (see Section 1.1). Then the Markov chain semantics is given by the
+extension of this distribution over the whole GBN.
+Definition 9 (Markov chain semantics).
+Let B be a GBN over nodes V
+with a cutset C ⊆ V and cutset Markov chain CMC(B, C) = ⟨Asg(C), P⟩. Then
+the Markov chain semantics of B w.r.t. C is the function
+MCS(B, C, ·) : Dist
+�
+Asg(C)
+�
+→ Dist
+�
+Asg(V)
+�
+from cutset distributions γ0 to full distributions µ = Extend(B, C, lrfγ0) where
+lrfγ0 =
+lim
+n→∞
+1
+n+1
+n
+�
+i=0
+γi
+and
+γi+1 = γi · P.
+The set �B�MC-C is defined as the image of MCS(B, C, ·).
+In the following lemma, we give four equivalent characterizations of the long-
+run frequency distributions of the cutset Markov chain.
+Lemma 2. Let B be a GBN with cutset C, cutset distribution γ ∈ Dist(Asg(C)),
+and M = ⟨Asg(C), P⟩ the cutset Markov chain CMC(B, C). Then the following
+statements are equivalent:
+(a) γ = γ · P.
+(b) There exists γ0 ∈ Dist(Asg(C)) such that for γi+1 = γi · P, we have
+γ =
+lim
+n→∞
+1
+n+1
+n
+�
+i=0
+γi.
+(c) γ belongs to the convex hull of the long-run frequency distributions lrfD of
+the bottom SCCs D of M.
+(d) γ = Next(B, C, γ)|C.
+
+On the Foundations of Cycles in Bayesian Networks
+15
+Following Lemma 2, we can equivalently define the cutset Markov chain se-
+mantics as the set of extensions of all stationary distributions for P:
+�B�MC-C :=
+�
+Extend(B, C, γ) : γ ∈ Dist
+�
+Asg(C)
+�
+s.t. γ = γ · P
+�
+.
+Example 6. Continuing Ex. 5, there is a unique stationary distribution γ with
+γ = γ · P for the cutset Markov chain in Fig. 5: γ = ⟨48/121 18/121 40/121 15/121⟩.
+As in this case the cutset C = {X, Y } equals the set of all nodes V, we have
+Extend(B, C, γ) = γ and thus �B�MC-{X,Y } = {γ}.
+As shown by Lemma 2, the behavior of the Next function is captured by
+multiplication with the transition matrix P. Both the distributions in the limit
+average semantics and the long-run frequency distributions of the cutset Markov
+chain are defined in terms of a Ces`aro limit, the former over the sequence of
+distributions obtained by repeated application of Next, the latter by repeated
+multiplication with P. Thus, both semantics are equivalent.
+Theorem 1. Let B be a GBN. Then for any cutset C of B and initial distribution
+γ0 ∈ Dist(Asg(C)), we have
+MCS(B, C, γ0) = LimAvg(B, C, γ0).
+We know that Lim(B, C, γ0) is not defined for all initial distributions γ0.
+However, the set of all limits that do exist contains exactly the distributions
+admitted by the Markov chain and limit average semantics.
+Lemma 3. Let B be a GBN. Then for any cutset C of B, we have
+�B�MC-C = �B�LimAvg-C = �B�Lim-C.
+5.1
+Properties
+By the equivalences established in Theorem 1 and Lemma 3, we gain profound
+insights about the limit and limit average distributions by Markov chain analysis.
+As every finite-state Markov chain has at least one stationary distribution, it
+immediately follows that �B�MC-C—and thus �B�LimAvg-C and �B�Lim-C—is always
+non-empty. Further, if the cutset Markov chain is irreducible, i.e., the graph
+is strongly connected, the stationary distribution is unique and �B�MC-C is a
+singleton. The existence of the limit semantics for a given initial distribution γ0
+hinges on the periodicity of the cutset Markov chain.
+Example 7. We return to Example 2 and construct the cutset Markov chain
+CMC(B, C) = ⟨Asg(C), P⟩ for the (implicitly used) cutset C = {X, Y }:
+X=T
+Y=T
+X=T
+Y=F
+X=F
+Y=T
+X=F
+Y=F
+1
+1
+1
+1
+
+16
+C. Baier et al.
+The chain is strongly connected and has a period of length 4, which explains the
+observed behavior that for any initial distribution γ0, we got the sequence
+γ0, γ1, γ2, γ3, γ0, γ1, . . .
+This sequence obviously converges only for initial distributions that are station-
+ary, i.e., if we have γ0 = γ0 · P.
+The following lemma summarizes the implications that can be drawn from
+close inspection of the cutset Markov chain.
+Lemma 4 (Cardinality).
+Let B be a GBN with cutset C and cutset Markov
+chain CMC(B, C) = ⟨Asg(C), P⟩. Further, let k > 0 denote the number of bottom
+SCCs D1, . . . , Dk of CMC(B, C). Then
+1. the cardinality of the cutset Markov chain semantics is given by
+���B�MC-C
+�� =
+�
+1
+if k = 1,
+∞
+if k > 1;
+2. Lim(B, C, γ0) is defined for all γ0 ∈ Dist(Asg(C)) if all Di are aperiodic;
+3. Lim(B, C, γ) is only defined for stationary distributions γ with γ = γ · P if
+Di is periodic for any 1 ⩽ i ⩽ k.
+A handy sufficient (albeit not necessary) criterion for both aperiodicity and
+the existence of a single bottom SCC in the cutset Markov chain is the absence
+of zero and one entries in the CPTs and the initial distribution of a GBN.
+Definition 10 (Smooth GBNs).
+A GBN B = ⟨G, P, ι⟩ is called smooth iff
+all CPT entries as given by P and all values in ι are in the open interval ]0, 1[.
+Lemma 5. Let B be a smooth GBN and C a cutset of B. Then the graph of the
+cutset Markov chain CMC(B, C) is a complete digraph.
+Corollary 1. The limit semantics of a smooth GBN B is a singleton for every
+cutset C of B and Lim(B, C, γ0) is defined for all γ0 ∈ Dist(Asg(C)).
+As noted in [14], one rarely needs to assign a probability of zero (or, con-
+versely, of one) in real-world applications; and doing so in cases where some
+event is extremely unlikely but not impossible is a common modeling error. This
+observation gives reason to expect that most GBNs encountered in practice are
+smooth, and their semantics is thus, in a sense, well-behaved.
+5.2
+Relation to Constraints Semantics
+We take a closer look at how the cutset semantics relates to the CPT-consistency
+semantics defined in Section 3. CPTs of nodes outside cutsets remain unaffected
+in the dissected BNs from which the Markov chain semantics is computed. Since
+there are cyclic GBNs for which no CPT-consistent distribution exists (cf. Ex-
+ample 1) while Markov chain semantics always yields at least one solution due
+to Lemma 4, it cannot be expected that cutset nodes are necessarily CPT-
+consistent. However, they are always weakly CPT-consistent.
+
+On the Foundations of Cycles in Bayesian Networks
+17
+Lemma 6. Let B be a GBN over nodes V, C ⊆ V a cutset for B, and µ ∈
+�B�MC-C. Then µ is strongly CPT-consistent for all nodes in V\C and weakly
+CPT-consistent for the nodes in C.
+The lemma shows a way to find fully CPT consistent distributions: Consider
+there is a distribution µ ∈ �B�MC-C ∩ �B�MC-D for two disjoint cutsets C and D.
+Then by Lemma 6 the nodes in V \ C and V \ D are CPT consistent, so in fact
+µ is CPT consistent. In general, we get the following result.
+Lemma 7. Let B be a GBN over nodes V and C1, . . . , Ck cutsets of B s.t. for
+each node X ∈ V there is an i ∈ {1, . . . , k} with X /∈ Ci. Then
+�
+0⩽i⩽k
+�B�MC-Ci ⊆ �B�Cpt.
+We take a look at which independencies are necessarily observed by the
+distributions in �B�MC-C. Let γ ∈ Dist(Asg(C)) be the cutset distribution and
+let G[C] denote the graph of Dissect(B, C, γ) restricted to the nodes in V such
+that the cutset nodes in C are initial. Then by Lemma 1, the d-separation in-
+dependencies of the closure of G[C] hold in all distributions µ ∈ �B�MC-C, i.e.,
+d-sep
+�
+Close(G[C])
+�
+⊆ Indep(µ). The next lemma states that any Cpt-consistent
+distribution that satisfies these independence constraints for some cutset C also
+belongs to �B�MC-C.
+Lemma 8. Let B be a GBN with cutset C and IC = Close(G[C]). Then we have
+�B�Cpt-IC ⊆ �B�MC-C.
+Combining Lemma 7 and Lemma 8 yields the following equivalence.
+Corollary 2. For a GBN B with cutsets C1, . . . , Ck as in Lemma 7 and the
+independence set I = �
+0⩽i⩽k Close(G[Ci]), we have
+�
+0⩽i⩽k
+�B�MC-Ci = �B�Cpt-I.
+5.3
+Overview
+Fig. 6 gives an overview of the relations between all proposed semantics. Boxes
+represent the set of distributions induced by the respective semantics and arrows
+stand for set inclusion. For the non-trivial inclusions the arrows are annotated
+with the respective lemma or theorem. As an example, Cpt→wCpt states that
+�B�Cpt ⊆ �B�wCpt holds for all GBNs B. The three semantics in the top row
+parameterized with a cutset C and a distribution γ stand for the singleton set
+containing the respective function applied to γ, i.e., �B�Lim-C-γ = {Lim(B, C, γ)}.
+�
+C MC-C stands for the intersection of the Markov chain semantics for various
+cutsets as in Lemma 7, and the incoming arrow from Cpt-IC holds for the set
+of independencies IC as in Lemma 8.
+
+18
+C. Baier et al.
+Lim-C-γ
+LimAvg-C-γ
+MC-C-γ
+Lim-C
+LimAvg-C
+MC-C
+�
+C MC-C
+Cpt-IC
+Cpt
+wCpt-IC
+wCpt
+C.2
+L.6
+L.7
+L.8
+L.3
+L.3
+T.1
+Lim-C-γ
+LimAvg-C-γ
+MC-C-γ
+Lim-C
+LimAvg-C
+MC-C
+�
+C MC-C
+Cpt-IC
+Cpt
+wCpt-IC
+wCpt
+Fig. 6: Relations between different variations of limit, limit average, and Markov
+chain semantics (blue) as well as strong and weak CPT-consistency semantics
+(yellow resp. orange)
+6
+Related Work
+That cycles in a BN might be unavoidable when learning its structure is well
+known for more than 30 years [15,22]. During the learning process of BNs, cy-
+cles might even be favorable as demonstrated in the context of gene regulatory
+networks where cyclic structures induce monotonic scores [32]. That work only
+discusses learning algorithms, but does not deal with evaluating the joint dis-
+tribution of the resulting cyclic BNs. In most applications, however, cycles have
+been seen as a phenomenon to be avoided to ease the computation of the joint
+distribution in BNs. By an example BN comprising a single isolated cycle, [30]
+showed that reversing or removing edges to avoid cycles may reduce the solution
+space from infinitely many joint distributions that are (weakly) consistent with
+the CPTs to a single one. In this setting, our results on weak CPT-semantics
+also provide that wCpt cannot express conditions on the relation of variables
+like implications or mutual exclusion. This is rooted in the fact that the solution
+space of weak CPT-semantics always contains at least one full joint distribu-
+tion with pairwise independent variables. An example where reversing edges led
+to satisfactory results has been considered in [3], investigating the impact of
+reinforced defects by steel corrosion in concrete structures.
+Unfolding cycles up to a bounded depth has been applied in the setting of
+a robotic sensor system by [2]. In their use case, only cycles of length two may
+appear, and only the nodes appearing on the cycles are implicitly used as cutset
+for the unfolding. In [13], the set of all nodes is used for unfolding (correspond-
+ing to a cutset C = V in our setting) and subsequent limit construction, but
+restricted to cases where the limit exists.
+There have been numerous variants of BNs that explicitly or implicitly ad-
+dress cyclic dependencies. Dynamic Bayesian networks (DBNs) [19] extend BNs
+by an explicit notion of discrete time steps that could break cycles through
+
+On the Foundations of Cycles in Bayesian Networks
+19
+timed ordering of random variables. Cycles in BNs could be translated to the
+DBN formalism by introducing a notion of time, e.g., following [13]. Our cutset
+approach is orthogonal, choosing a time-abstract view on cycles and treating
+them as stabilizing feedback loops. Learning DBNs requires “relatively large
+time-series data” [32] and thus, may be computationally demanding. In [18] ac-
+tivator random variables break cycles in DBNs to circumvent spurious results in
+DBN reasoning when infinitesimal small time steps would be required.
+Causal BNs [23] are BNs that impose a meaning on the direction of an
+edge in terms of causal dependency. Several approaches have been proposed
+to extend causal BNs for modeling feedback loops. In [25], an equilibrium se-
+mantics is sketched that is similar to our Markov chain semantics, albeit based
+on variable oderings rather than cutsets. Determining independence relations,
+Markov properties, and joint distributions are central problems addressed for
+cyclic causal BNs [2,5,20,24,29]. Markov properties and joint distributions for
+extended versions of causal BNs have been considered recently, e.g., in directed
+graphs with hyperedges (HEDGes) [5] and cyclic structural causal models (SCMs)
+[2]. Besides others, they show that in presence of cycles, there might be multiple
+solutions for a joint distribution or even no solution at all [7]. While we consider
+all random variables to be observable, the latter approaches focus on models
+with latent variables. Further, while our focus in this paper is not on causality,
+our approach is surely also applicable to causal BNs with cycles.
+Recursive relational Bayesian networks (RRBNs) [9] allow representing prob-
+abilistic relational models where the random variables are given by relations over
+varying domains. The resulting first-order dependencies can become quite com-
+plex and may contain cycles, though semantics are given only for the acyclic
+cases by the construction of corresponding standard BNs.
+Bayesian attack graphs (BAGs) [16] are popular to model and reason about
+security vulnerabilities in computer networks. Learned graphs and thus their
+BN semantics frequently contain cycles, e.g., when using the tool MulVAL [21].
+In [27], “handling cycles correctly” is identified as “a key challenge” in security
+risk analysis. Resolution methods for cyclic patterns in BAGs [1,4,17,31] are
+mainly based on context-specific security considerations, e.g., to break cycles by
+removing edges. The semantic foundations for cyclic BNs laid in this paper do
+not require graph manipulations and decouple the probability theoretic basis
+from context-specific properties.
+7
+Conclusion
+This paper has developed a foundational perspective on the semantics of cycles in
+Bayesian networks. Constraint-based semantics provide a conservative extension
+of the standard BN semantics to the cyclic setting. While conceptually impor-
+tant, their practical use is limited by the fact that for many GBNs, the induced
+constraint system is unsatisfiable. On the other hand, the two introduced limit
+semantics echo in an abstract and formal way what practitioners have been devis-
+ing across a manifold of domain-specific situations. In this abstract perspective,
+
+20
+C. Baier et al.
+cutsets are the ingredients that enable a controlled decoupling of dependencies.
+The appropriate choice of cutsets is where, in our view, domain-specific knowl-
+edge is confined to enter the picture. Utilizing the constructively defined Markov
+chain semantics, we established key results relating and demarcating the differ-
+ent semantic notions and showed that for the ubiquitous class of smooth GBNs
+a unique full joint distribution always exists.
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+22
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+A
+Appendix
+The appendix contains the proofs omitted from the body of the submission “On
+the Foundations of Cycles in Bayesian Networks” due to space constraints.
+Lemma 1. Let B̸⟳ = ⟨G, P, ι⟩ be an acyclic GBN. Then
+d-sep
+�
+Close(G)
+�
+⊆ Indep
+�
+distBN(B̸⟳)
+�
+.
+Proof. The idea is to show that the dependencies of every possible BN structure
+for the initial distribution ι are covered by the closure operation. Let the graph
+G⋆
+ι = ⟨Init(G), E⋆⟩ be a DAG that is an I-map for ι, i.e., d-sep(G⋆
+ι ) ⊆ Indep(ι).
+Then ι factorizes according to G⋆
+ι , that is for every assignment b ∈ Asg(Init(G)),
+we have
+ι(b) =
+�
+X∈Init(G)
+ι
+�
+bX | bPreG⋆ι (X)
+�
+.
+Now consider the BN B⋆
+̸⟳ with graph G⋆ = ⟨V, E ∪ E⋆⟩ where we add the edges
+of G⋆
+ι to G. The CPTs for the nodes in V \ Init(G) are given by P whereas the
+new CPTs (according to the structure in G⋆
+ι ) for the nodes in Init(G) are derived
+from ι. Then for every assignment c ∈ Asg(V):
+distBN(B⋆
+̸⟳)(c) =
+�
+X∈V
+Pr
+�
+cX | cPre(X)
+�
+=
+�
+X∈Init(G)
+ι
+�
+cX | cPreG⋆ι (X)
+�
+·
+�
+X∈V\Init(G)
+Pr
+�
+cX | cPre(X)
+�
+= ι
+�
+cInit(G)
+�
+·
+�
+X∈V\Init(G)
+Pr
+�
+cX | cPre(X)
+�
+= distBN(B̸⟳)(c).
+As B⋆
+̸⟳ is a regular BN without an initial distribution, we have d-sep(G⋆) ⊆
+Indep(dist BN(B⋆
+̸⟳)).
+We proceed to show d-sep(Close(G)) ⊆ d-sep(G⋆). Let (X ⊥ Y | Z) ∈
+d-sep(Close(G)). Then each path from X to Y in Close(G) is blocked by the
+nodes in Z. As Close(G) contains all possible edges between the nodes Init(G)
+but G⋆ only a subset thereof, it is clear that each path in G⋆ also exists in
+Close(G). Thus, there cannot be an unblocked path from X to Y given Z in G⋆
+either, so (X ⊥ Y | Z) ∈ d-sep(G⋆). Altogether, we have
+d-sep
+�
+Close(G)
+�
+⊆ d-sep(G⋆) ⊆ Indep
+�
+dist BN(B⋆
+̸⟳)
+�
+= Indep
+�
+distBN(B̸⟳)
+�
+.
+⊓⊔
+Lemma 2. Let B be a GBN with cutset C, cutset distribution γ ∈ Dist(Asg(C)),
+and M = ⟨Asg(C), P⟩ the cutset Markov chain CMC(B, C). Then the following
+statements are equivalent:
+(a) γ = γ · P.
+
+On the Foundations of Cycles in Bayesian Networks
+23
+(b) There exists γ0 ∈ Dist(Asg(C)) such that for γi+1 = γi · P, we have
+γ =
+lim
+n→∞
+1
+n+1
+n
+�
+i=0
+γi.
+(c) γ belongs to the convex hull of the long-run frequency distributions lrfD of
+the bottom SCCs D of M.
+(d) γ = Next(B, C, γ)|C.
+Proof. (a) =⇒ (b): If we have γ = γ · P, then statement (b) is obtained by
+considering γ0 = γ, as then γi = γ for all i.
+(b) =⇒ (c): The proof of the implication relies on the following standard facts
+about finite-state Markov chains. Given a BSCC D and an arbitrary distribution
+ν0 ∈ Dist(Asg(D)), the distribution lrfD agrees with the Ces`aro limit of the
+sequence (νi)i⩾0 where νi+1 = νi · PD and PD denotes the restriction of P to
+assignments on D. That is,
+lrfD = lim
+n→∞
+1
+n+1
+n
+�
+i=0
+νi.
+Vice versa, for γ0 ∈ Dist(Asg(C)) and γi+1 = γi · P, then the Ces`aro limit γ
+of the sequence (γi)i⩾0 has the form
+γ =
+�
+D
+λ(D) · lrfD
+where D ranges over all BSCCs of M, λ(D) is the probability for reaching D in M
+with the initial distribution γ0, and all vectors lrfD are padded with zero entries
+to range over the whole state space. In particular, γ is a convex combination
+of the distributions lrfD as 0 ⩽ λ(D) ⩽ 1 and �
+D λ(D) = 1 (because every
+finite-state Markov chain almost surely reaches a BSCC).
+(c) =⇒ (a): Suppose γ = �
+D λ(D)·lrfD where 0 ⩽ λ(D) ⩽ 1, �
+D λ(D) = 1,
+and each lrfD is padded appropriately as before. Then:
+γ · P =
+�
+D
+λ(D) · lrfD · P =
+�
+D
+λ(D) · lrfD = γ
+where we use the fact that lrfD = lrfD · P.
+(a) ⇐⇒ (d): Because γ can be represented as convex combination of Dirac
+distributions as γ = �
+c∈Asg(C) γ(c) · Dirac(c), we know:
+Next(B, C, γ) =
+�
+c∈Asg(C)
+γ(c) · Next
+�
+B, C, Dirac(c)
+�
+.
+As P(c, b) = Next
+�
+B, C, Dirac(c)
+�
+(b) for any assignment b ∈ Asg(C), and assum-
+ing γ = γ · P, we get
+Next(B, C, γ)(b) =
+�
+c∈Asg(C)
+γ(c) · P(c, b) = (γ · P)(b) = γ(b).
+Conversely, assuming Next(B, C, γ)|C = γ, we yield γ = γ · P.
+⊓⊔
+
+24
+C. Baier et al.
+Lemma 3. Let B be a GBN. Then for any cutset C of B, we have
+�B�MC-C = �B�LimAvg-C = �B�Lim-C.
+Proof. We have �B�MC-C = �B�LimAvg-C by Theorem 1 and know �B�Lim-C ⊆
+�B�LimAvg-C, so it remains to show �B�MC-C ⊆ �B�Lim-C. Let µ ∈ �B�MC-C. Then
+there exists a cutset distribution γ s.t. µ = Extend(B, C, γ). We need to show
+there exists an initial distribution γ0 ∈ Dist(Asg(C)) such that γ = limn→∞ γi
+where γi+1 = Next(B, C, γi)|C. Let us choose γ0 = γ. Then we know γ0 =
+Next(B, C, γ0)|C by Lemma 2, so γi = γ0 for all i ∈ N. Thus, γ = limi→∞ γi and
+therefore µ ∈ �B�Lim-C.
+⊓⊔
+Lemma 4 (Cardinality). Let B be a GBN with cutset C and cutset Markov
+chain CMC(B, C) = ⟨Asg(C), P⟩. Further, let k > 0 denote the number of bottom
+SCCs D1, . . . , Dk of CMC(B, C). Then
+1. the cardinality of the cutset Markov chain semantics is given by
+���B�MC-C
+�� =
+�
+1
+if k = 1,
+∞
+if k > 1;
+2. Lim(B, C, γ0) is defined for all γ0 ∈ Dist(Asg(C)) if all Di are aperiodic;
+3. Lim(B, C, γ) is only defined for stationary distributions γ with γ = γ · P if
+Di is periodic for any 1 ⩽ i ⩽ k.
+Proof. (1.) By Lemma 2, every cutset distribution with γ = γ · P is a convex
+combination of the steady-state distributions for the BSCCs. Thus, for k = 1
+a unique distribution γ exists, whereas for k > 1, there are infinitely many
+real-valued distributions in the convex hull.
+(2.) A Markov chain is aperiodic if all its BSCCs are aperiodic. Aperiodicity
+suffices for the limit limn→∞ γn with γn+1 = γn · P to exist for every γ0. Then
+limn→∞ γ′
+n with γ′
+n+1 = Next(B, C, γ′
+n)|C exists as well by Lemma 2.
+(3.) Assume some BSCC D is periodic with a period of p. Then, for any γ0 ∈
+Dist(Asg(C)), γn+1 = γn · P, and νn = γn|D, we have νp·n = νn. Now consider
+γ0 and γ1 = γ0 · P. If γ0 = γ1, then γ0 = γn for all n ∈ N and γ0 = limn→∞ γn
+holds. Otherwise, if γ0 ̸= γ1, the following non-convergent sequence exists:
+ν0, ν1, . . . , νp, νp+1, . . . , ν2p, ν2p+1, . . .
+Then limn→∞ γn cannot converge either, so Lim(B, C, γ0) is undefined.
+⊓⊔
+Lemma 5. Let B be a smooth GBN and C a cutset of B. Then the graph of the
+cutset Markov chain CMC(B, C) is a complete digraph.
+Proof. The graph of CMC(B, C) = ⟨Asg(C), P⟩ is a complete digraph iff each
+entry in P is positive. Thus, for each two assignments b, c ∈ Asg(C), we need to
+
+On the Foundations of Cycles in Bayesian Networks
+25
+show P(b, c) > 0. Let Bb = Dissect(B, C, Dirac(b)). Then from Definition 8, we
+have
+P(b, c) = Next(Dirac(b), B, C)(c)
+= distBN(Bb)(c′).
+The probability dist BN(Bb)(c′) is given by the sum over all full assignments
+v ∈ Asg(V) that agree with c′ on the assignment of the cutset node copies C′.
+Further, the sum can be partitioned into those v that agree with assignment b
+on C and those that do not:
+distBN(Bb)(c′) =
+�
+v∈Asg(V)
+s.t. c′⊂v, b⊂v
+distBN(Bb)(v) +
+�
+v∈Asg(V)
+s.t. c⊂v, b̸⊂v
+dist BN(Bb)(v).
+By the definition of the standard BN-semantics, we have
+distBN(Bb)(v) = ι
+�
+vInit(G)
+�
+· Dirac(b)(vC) ·
+�
+X∈V\C
+Pr
+�
+vX | vPre(X)
+�
+.
+Now consider the second sum in the previous equation where b ̸⊂ v. For those
+assignments, Dirac(b)(vC) = 0 and thus the whole sum equals zero. For the
+first sum, we have vC = b, so Dirac(b)(vC) = 1 and we only need to consider
+the product with X ∈ V \ C and the initial distribution over Init(G). By the
+construction of Bb, the CPTs of all X ∈ V \C are the original CPTs from B, thus
+their entries all fall within the open interval ]0, 1[ by the smoothness assumption
+of B. The same holds for the value ι
+�
+vInit(G)
+�
+. Thus, the whole product resides
+in ]0, 1[ as well. Finally, note that the sum is non-empty as C′ and C are disjoint,
+so there exists at least one v ∈ Asg(V) with c ⊂ v and b ⊂ v. As a non-empty
+sum over values in ]0, 1[ is necessarily positive, we have distBN(Bb)(c′) > 0 and
+the claim follows.
+⊓⊔
+Corollary 1. The limit semantics of a smooth GBN B is a singleton for every
+cutset C of B and Lim(B, C, γ0) is defined for all γ0 ∈ Dist(Asg(C)).
+Proof. Follows from Lemma 4 and Lemma 5 because every complete graph forms
+a single bottom SCC and is necessarily aperiodic.
+⊓⊔
+Lemma 6. Let B be a GBN over nodes V, C ⊆ V a cutset for B, and µ ∈
+�B�MC-C. Then µ is strongly CPT-consistent for all nodes in V\C and weakly
+CPT-consistent for the nodes in C.
+Proof. By definition, µ = Extend(B, C, γ) for some γ ∈ Dist(Asg(C)) with
+γ = γ · P. As Extend(B, C, γ) is the standard BN semantics for the acyclic
+BN Dissect(B, C, γ) without the copies of the cutset nodes, CPT-consistency for
+the nodes in V \ C follows directly from the CPT-consistency of the standard
+semantics for acyclic BNs.
+
+26
+C. Baier et al.
+It remains to prove weak CPT-consistency for the cutset nodes. Let δ =
+distBN(Dissect(B, C, γ)) ∈ Dist(Asg(V ∪ C′)). Thus, µ = δ|V and γ = δ|C. Then
+for each assignment b ∈ Asg(C), we have
+µ(b) = γ(b) = (γ · P)(b) = δ(b′)
+where b′ ∈ Asg(C′) is given by b′(Y ′) = b(Y ) for all Y ∈ C. In particular, for each
+Y ∈ C:
+µ(Y=T) = δ(Y ′=T)
+Let D = Asg(Pre(Y )) where Pre(·) refers to the original scGBN. For c ∈
+Asg(C), we write Dc for the set of all assignments d ∈ D that comply with c in
+the sense that if Z ∈ C ∩ Pre(Y ) then c(Z) = d(Z). In this case, c and d can be
+combined to an assignment for C ∪Pre(Y ). Similarly, if d ∈ D, then the notation
+Asgd(C) is used for the set of assignments c ∈ Asg(C) that comply with d. Then:
+δ(Y ′=T) =
+�
+c∈Asg(C)
+δ(Y ′=T | c) · µ(c)
+=
+�
+c∈Asg(C)
+�
+d∈Dc
+δ(Y ′=T | c, d)
+�
+��
+�
+Pr(Y =T|d)
+· δ(d | c)
+� �� �
+µ(d|c)
+· δ(c)
+����
+µ(c)
+=
+�
+d∈D
+Pr(Y =T | d) ·
+�
+c∈Asgd(C)
+µ(d | c) · µ(c)
+=
+�
+d∈D
+Pr(Y =T | d) · µ(d).
+Putting everything together, we obtain:
+µ(Y =T) = δ(Y ′=T) =
+�
+d∈D
+Pr(Y =T | d) · µ(d).
+Thus, µ is weakly CPT-consistent for Y ∈ C.
+⊓⊔
+Lemma 7. Let B be a GBN over nodes V and C1, . . . , Ck cutsets of B s.t. for
+each node X ∈ V there is an i ∈ {1, . . . , k} with X /∈ Ci. Then
+�
+0⩽i⩽k
+�B�MC-Ci ⊆ �B�Cpt.
+Proof. We need to show CPT-consistency for every node under µ ∈ �
+i�B�MC-Ci.
+Let X ∈ V. Then we choose a cutset Ci s.t. X /∈ Ci and CPT consistency follows
+from Lemma 6.
+⊓⊔
+Lemma 8. Let B be a GBN with cutset C and IC = d-sep
+�
+Close(Close(G)[C])
+�
+.
+Then we have
+�B�Cpt-IC ⊆ �B�MC-C.
+
+On the Foundations of Cycles in Bayesian Networks
+27
+Proof. Let µ ∈ �B�Cpt-IC and γ = µ|C. The task is to show that γ satisfies the
+fixed point equation γ = γ · P.
+The standard BN semantics δ = dist BN(Dissect(B, C, γ)) of the dissected BN
+is the unique distribution over Asg(V ∪ C′) that
+– is CPT-consistent w.r.t. the conditional probability tables in Dissect(B, C, γ),
+– agrees with γ when restricted to the assignments for C, and
+– satisfies the conditional independencies in IC.
+Consider the distribution ˜µ ∈ Dist
+�
+Asg(V∪C′)
+�
+defined as follows for b ∈ Asg(V)
+and c′ ∈ Asg(C′):
+˜µ(b, c′) := µ(b) ·
+�
+Y∈C
+Pr
+�
+Y=c′(Y ′) | bPre(Y )
+�
+.
+Then, ˜µ satisfies the above three constraints. Hence, ˜µ = δ.
+For c ∈ Asg(C), let c′ ∈ Asg(C′) denote the corresponding assignment with
+c′(Y ′) = c(Y ) for Y ∈ C.
+(γ · P)(c) = δ(c′) = ˜µ(c′)
+=
+�
+d∈Asg(Pre(C))
+µ(d) ·
+�
+Y∈C
+Pr(Y=c′(Y ′) | d)
+�
+��
+�
+Pr(Y=c(Y )|d)
+= µ(c) = γ(c).
+Hence, γ = γ · P and µ ∈ �B�MC-C.
+⊓⊔
+

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@@ -0,0 +1,1803 @@
+Active Nematic Multipoles: Flow Responses and the Dynamics of Defects and Colloids
+Alexander J.H. Houston1 and Gareth P. Alexander1, 2, ∗
+1Department of Physics, Gibbet Hill Road, University of Warwick, Coventry, CV4 7AL, United Kingdom.
+2Centre for Complexity Science, Zeeman Building,
+University of Warwick, Coventry, CV4 7AL, United Kingdom.
+(Dated: Wednesday 1st February, 2023)
+We introduce a general description of localised distortions in active nematics using the framework
+of active nematic multipoles.
+We give the Stokesian flows for arbitrary multipoles in terms of
+differentiation of a fundamental flow response and describe them explicitly up to quadrupole order.
+We also present the response in terms of the net active force and torque associated to the multipole.
+This allows the identification of the dipolar and quadrupolar distortions that generate self-propulsion
+and self-rotation respectively and serves as a guide for the design of arbitrary flow responses. Our
+results can be applied to both defect loops in three-dimensional active nematics and to systems with
+colloidal inclusions. They reveal the geometry-dependence of the self-dynamics of defect loops and
+provide insights into how colloids might be designed to achieve propulsive or rotational dynamics,
+and more generally for the extraction of work from active nematics. Finally, we extend our analysis
+also to two dimensions and to systems with chiral active stresses.
+I.
+INTRODUCTION
+Active liquid crystals model a wide range of materials, both biological and synthetic [1–3], including cell mono-
+layers [4], tissues [5], bacteria in liquid crystalline environments [6] and bacterial suspensions [7], and synthetic
+suspensions of microtubules [8].
+Nematic and polar phases have been the focus of attention but smectic [9, 10],
+cholesteric [11, 12] and hexatic [13] phases have also been considered. Key features and motifs of the active nematic
+state include self-propelled topological defects [14–16], spontaneous flows and vortices, and on how these may be
+controlled through boundary conditions, confinement [17–19], external fields, geometry or topology. Active defects,
+in particular, have been related to processes of apoptosis in epithelial sheets [5], tissue dynamics, bacterial spreading
+and biofilm formation, and morphogenesis in Hydra [20].
+In three-dimensional active nematics the fundamental excitations are defect loops and system-spanning lines [21, 22].
+The defect loops actively self-propel [23], and self-orient [24], in addition to undergoing deformations in shape. Their
+finite extent means that they represent localised distortions to the nematic director, on scales larger than their size,
+and this facilitates a description through elastic multipoles [24]. It also invites comparison with colloidal inclusions in
+passive liquid crystals, which create localised realignments of the director and act as elastic multipoles [25–27]. These
+multipole distortions mediate interactions between colloids and allow for a means of controlling both the colloidal
+inclusions and the host material. For instance, they facilitate self-assembly and the formation of metamaterials [28, 29],
+and enable novel control of topological defects [27, 30, 31]. While there have been studies of active nematic droplets
+in a host passive liquid crystal [32, 33], colloidal inclusions in host active nematics have not been looked at previously.
+The multipole approach to describing colloidal inclusions and localised director distortions in general, offers an
+equally fruitful paradigm in active nematics. Here, we present a generic analysis of the active flows generated by
+multipole director distortions in an active nematic and predict that the presence of colloids transforms their behaviour
+similarly to the passive case. These active multipole flows represent the responses of the active nematic both to
+localised features, such as defect loops, and to colloidal inclusions. This allows us to identify those distortions which
+produce directed or rotational flows and show that such distortions may be naturally induced by colloids. We also
+characterise the response in terms of the active forces and torques that they induce. This general connection can
+serve as a guide for using colloidal inclusions as a means to control active nematics, or how to design them to engineer
+a desired response, or extract work. The properties of inclusions have been studied in scalar active matter [34], as
+have active droplets in passive nematics [35], but while there have been specific demonstrations of propulsive colloids
+[36, 37] the general responses of inclusions in active nematics have not previously been considered. Understanding
+how such responses relate to local manipulations and molecular fields in active nematics will bring both fundamental
+insights and the potential for control of active metamaterials.
+The remainder of this paper is structured as follows. In Section 2 we briefly review the equations of active nemato-
+hydrodynamics and describe the regime in which our linear multipole approach applies. In Section 3 we present these
+∗ [email protected]
+arXiv:2301.13782v1  [cond-mat.soft]  31 Jan 2023
+
+2
+multipoles as complex derivatives acting on 1/r, showing how this naturally elucidates their symmetries. In Section
+4 we show that the linear active response to a harmonic distortion is generated by the same complex derivatives
+acting on fundamental flow and pressure solutions and highlight certain examples that illustrate the self-propulsive
+and rotational dynamics that can arise. We then show in Section 5 that these phenomenological responses can be
+discerned from integrals of the active stress, allowing the identification of the distortion which produces propulsion
+along or rotation about a given axis. Sections 6 and 7 contain extensions of our approach, first to two-dimensional
+systems and then to those with chiral active stresses. Section 8 gives a discussion and summary.
+II.
+HYDRODYNAMICS OF ACTIVE NEMATICS
+We summarise the hydrodynamics of active nematics as described by their director field n and fluid velocity u. The
+fluid flow satisfies the continuity ∂iui = 0 and Stokes ∂jσij = 0 equations, with stress tensor [1–3]
+σij = −pδij + 2µDij + ν
+2
+�
+nihj + hinj
+�
++ 1
+2
+�
+nihj − hinj
+�
++ σE
+ij − ζninj.
+(1)
+Here, p is the pressure, µ is the viscosity, Dij = 1
+2(∂iuj + ∂jui) is the symmetric part of the velocity gradients, ν is
+the flow alignment parameter, hi = −δF/δni is the molecular field associated with the Frank free energy F, σE
+ij is the
+Ericksen stress, and ζ is the magnitude of the activity. The active nematic is extensile when ζ > 0 and contractile
+when ζ < 0. The director field satisfies the relaxational equation
+∂tni + uj∂jni + Ωijnj = 1
+γ hi − ν
+�
+Dijnj − ni(njDjknk)
+�
+,
+(2)
+where γ is a rotational viscosity and Ωij = 1
+2(∂iuj − ∂jui) is the antisymmetric part of the velocity gradients. We
+adopt a one-elastic-constant approximation for the Frank free energy [38]
+F =
+� K
+2
+�
+∂inj
+��
+∂inj
+�
+dV,
+(3)
+for which the molecular field is hi = K
+�
+∇2ni − ninj∇2nj
+�
+and the Ericksen stress is σE
+ij = −K∂ink ∂jnk.
+An often-used analytical approximation is to consider the active flows generated by an equilibrium director field.
+This approximation has been used previously in the theoretical description of the active flows generated by defects
+in both two [16, 39] and three dimensions [23], including on curved surfaces [40], and in active turbulence [41]. It
+may be thought of in terms of a limit of weak activity, however, even when the activity is strong enough to generate
+defects, their structure may still be close to that of equilibrium defects and the approximation remain good and the
+comparison of active defect motion and flows described in this way with full numerical simulations suggests that this
+is at least qualitatively the case. The equations can then be reduced to h = 0 for the director field and the Stokes
+equation
+− ∇p + µ∇2u = ζ∇ ·
+�
+nn
+�
+,
+(4)
+for the active flow. Here we have neglected the Ericksen stress since for an equilibrium director field it can be balanced
+by a contribution to the pressure (representing nematic hydrostatic equilibrium).
+We limit our analysis to director fields that can be linearised around a (locally) uniformly aligned state, n = ez +δn,
+with δn · ez = 0, for which the equations reduce to
+∇2δn = 0,
+(5)
+∇ · u = 0,
+(6)
+−∇p + µ∇2u = ζ
+�
+ez
+�
+∇ · δn
+�
++ ∂zδn
+�
+.
+(7)
+These correspond to elastic multipole states in the director field, which are often thought of as an asymptotic de-
+scription, however, they provide a close approximation even at only moderate distances outside a ‘core’ region that
+is the source of the multipole. To illustrate this we show in Fig. 1 a comparison between the exact director field (red
+streamlines) and linear multipole approximation (blue rods) for the most slowly varying monopole distortion created
+by uniformly rotating the director by an angle θ0 within a sphere of radius a. The agreement is close anywhere outside
+the sphere and only deviates significantly in the near-field region inside it. This is relevant to the active system as it
+is well-known that the uniformly aligned active nematic state is fundamentally unstable [42] and active nematics are
+turbulent on large enough scales. Our solutions should be interpreted as describing the behaviour on intermediate
+scales, larger than the core structure of the source but smaller than the scale on which turbulence takes over.
+
+3
+FIG. 1. Comparison of the exact director field (red streamlines) and linearised multipole approximation (blue rods) for the
+most slowly decaying monopole distortion. This is produced by uniformly rotating the director by an angle θ0 within a spherical
+volume of radius a, indicated by the grey disc; the alignment inside the sphere is indicated by the thick red line. The figure
+shows only the xz-plane in which the director rotates and in which the comparison is most strict.
+III.
+MULTIPOLE DIRECTOR DISTORTIONS
+In this section, we describe the multipole director fields satisfying (5). The far-field orientation ez gives a splitting
+of directions in space into those parallel and perpendicular to it. We complexify the perpendicular plane to give the
+decomposition as R3 ∼= C ⊕ R and convert the director deformation δn to the complex form δn = δnx + iδny. The
+real and imaginary parts of δn are harmonic, meaning that at order l they may be expressed as spherical harmonics
+1/rl+1Y l
+m or, as we shall do, as l derivatives of 1/r [43–45]. These order l multipole solutions form a 2(2l + 1)-real-
+dimensional vector space. Associated to the C ⊕ R splitting is a local symmetry group isomorphic to U(1), preserving
+ez, whose irreducible representations provide a natural basis for the vector space of multipoles at each order. We write
+the complex derivatives on C as ∂w = 1
+2(∂x − i∂y) and ∂ ¯
+w = 1
+2(∂x + i∂y) in terms of which the director deformation
+can be written
+δn =
+∞
+�
+l=0
+l
+�
+m=−l
+qlm al+1 ∂m
+¯
+w ∂l−m
+z
+1
+r ,
+(8)
+where qlm are complex coefficients and a is a characteristic length scale of the multipole, as might be set by the radius
+of a colloid. For compactness of notation it is to be understood that when m is negative ∂m
+¯
+w represents ∂|m|
+w . The
+index m denotes the topological charge of the phase winding of the spherical harmonic. This gives the spin of the
+corresponding vector field as 1 − m, where the 1 is due to a vector (δn or δn) being a spin-1 object. The multipoles
+at order l therefore have spins that range from 1 − l to 1 + l. They are illustrated up to quadrupole order in Fig. 2,
+along with a representation in terms of topological defects which we shall elaborate upon shortly. The structure of
+Fig. 2 is such that differentiation maps the distortions of one order to the next, with ∂z leaving the distortion in the
+same spin class, ∂ ¯
+w moving it one column to the left and ∂w moving it one column to the right. The operators ∂w and
+∂ ¯
+w play the same role as the raising and lowering operators in quantum mechanics and the shift by one in the spin
+values simply results from the object on which they act being a spin-1 director deformation as opposed to a spin-0
+wavefunction.
+The monopole distortions, with l = 0, result from a rotation of the director by an angle θ0 in a sphere of radius
+a [46]. They form a two-real-dimensional vector space for which a basis may be taken to be the distortions 1
+r and i 1
+r.
+These are shown at the top of Fig. 2 and can be controllably created in passive nematics using platelet inclusions [47].
+The director distortions of dipole type, with l = 1, form a six-real-dimensional vector space that splits into two-
+
+24
+FIG. 2. The multipolar director distortions up to quadrupole order. The director is shown on a planar cross-section as blue
+rods, along with a topological skeleton corresponding to the spherical harmonic, where appropriate. Defect loops are coloured
+according to wedge (blue) or twist (red-green) type and the charge of point defects is indicated through the use of opposing
+colour pairs: red (+1) and cyan (−1), yellow (+2) and blue (−2), and green (+3) and magenta (−3). Their charge is further
+indicated by a local decoration of the director with an orientation, indicated by black arrows. Each multipole order is classified
+into vertical pairs according to the spin of the distortion. For the chiral multipoles, the visualisation instead shows the director
+along some of its integral curves (orange).
+
+-1
+0
+2
+3
+Monopoles
+Dipoles
+Quadrupoles5
+real-dimensional subspaces for each value of the spin (0, 1, or 2) as
+p0 =
+�
+∂ ¯
+w
+1
+r , i ∂ ¯
+w
+1
+r
+�
+∼ − 1
+2r3
+�
+x ex + y ey, −y ex + x ey
+�
+∼ 1
+r2
+�
+Y 1
+1 , i Y 1
+1
+�
+,
+(9)
+p1 =
+�
+∂z
+1
+r , i ∂z
+1
+r
+�
+∼ − 1
+r3
+�
+z ex, z ey
+�
+∼ 1
+r2
+�
+Y 0
+1 , i Y 0
+1
+�
+,
+(10)
+p2 =
+�
+∂w
+1
+r , i ∂w
+1
+r
+�
+∼ − 1
+2r3
+�
+x ex − y ey, y ex + x ey
+�
+∼ 1
+r2
+�
+Y −1
+1
+, i Y −1
+1
+�
+.
+(11)
+For comparison, we have presented three representations for the distortions of each spin class: in terms of complex
+derivatives of 1/r, two-component vectors whose coefficients are homogenous polynomials of degree 1 and complex
+spherical harmonics. In the interest of space we have suppressed certain prefactors in the last of these, but note
+the difference in sign, and in some cases normalisation, between our representation as complex derivatives and the
+standard form of the harmonic distortions as two-component vectors [48]. The two basis functions of any spin class
+are related by a factor of i, which corresponds to a local rotation of the transverse director distortion by π
+2 . For a
+spin-s distortion this is equivalent to a global rotation by
+π
+2s, with the pair of distortions having the same character
+and simply providing a basis for all possible orientations. The exception is when s = 0, such distortions lack an
+orientation and the local rotation produces two distinct states that transform independently under rotations as a
+scalar and pseudoscalar. In the dipole case the first is the isotropic distortion recognisable as the UPenn dipole [25]
+and the second is an axisymmetric chiral distortion with the far-field character of left-handed double twist. Separating
+p0 into its isotropic and chiral components allows a decomposition of the dipole director deformations into the basis
+p = pI ⊕ pC ⊕ p1 ⊕ p2,
+(12)
+a decomposition which was presented in [49].
+Similarly, the quadrupolar distortions (l = 2) form a ten-real-dimensional vector space that splits into a sum of
+two-real-dimensional subspaces for each value of the spin
+Q−1 =
+�
+∂2
+¯
+w
+1
+r , i ∂2
+¯
+w
+1
+r
+�
+∼
+3
+4r5
+�
+(x2 − y2) ex + 2xy ey, −2xy ex + (x2 − y2) ey
+�
+∼ 1
+r3
+�
+Y 2
+2 , i Y 2
+2
+�
+,
+(13)
+Q0 =
+�
+∂2
+¯
+wz
+1
+r , i ∂2
+¯
+wz
+1
+r
+�
+∼
+3
+2r5
+�
+xz ex + yz ey, −yz ex + xz ey
+�
+∼ 1
+r3
+�
+Y 1
+2 , i Y 1
+2
+�
+,
+(14)
+Q1 =
+�
+∂2
+z
+1
+r , i ∂2
+z
+1
+r
+�
+∼ 1
+r5
+�
+(2z2 − x2 − y2) ex, (2z2 − x2 − y2) ey
+�
+∼ 1
+r3
+�
+Y 0
+2 , i Y 0
+2
+�
+,
+(15)
+Q2 =
+�
+∂2
+wz
+1
+r , i ∂2
+wz
+1
+r
+�
+∼
+3
+2r5
+�
+xz ex − yz ey, yz ex + xz ey
+�
+∼ 1
+r3
+�
+Y −1
+2
+, i Y −1
+2
+�
+,
+(16)
+Q3 =
+�
+∂2
+w
+1
+r , i ∂2
+w
+1
+r
+�
+∼
+3
+4r5
+�
+(x2 − y2) ex − 2xy ey, 2xy ex + (x2 − y2) ey
+�
+∼ 1
+r3
+�
+Y −2
+2
+, i Y −2
+2
+�
+.
+(17)
+Once again the spin-0 distortions can be further partitioned into those that transform as a scalar and pseudoscalar,
+these being the Saturn’s ring distortion [50] and a chiral quadrupole with opposing chirality in the two hemispheres,
+respectively. This yields the basis for the quadrupolar director deformations
+Q = Q−1 ⊕ QI ⊕ QC ⊕ Q1 ⊕ Q2 ⊕ Q3.
+(18)
+The well-known multipoles, the UPenn dipole and Saturn ring quadrupole, are associated to a configuration of
+topological defects in the core region and we describe now an extension of this association to all of the multipoles. In
+general, such an association is not unique, for instance, the colloidal ‘bubblegum’ configuration [51] represents the same
+far field quadrupole as the Saturn ring, however, for each multipole we can construct a representative arrangement of
+topological defects which produce it in the far field on the basis of commensurate symmetries and defects of a type and
+location corresponding to the nodal set of the harmonic. This correspondence allow us to condense the visualisation
+of complicated three-dimensional fields into a few discrete elements, suggests means by which such distortions might
+be induced and enables us to build an intuition for their behaviour in active systems through established results for
+defects [23].
+We first describe some examples, shown in Fig. 3. On the left is the spherical harmonic that describes the UPenn
+dipole, with the form ∂ ¯
+w 1
+r ∼ eiφ sin θ, visualised on a spherical surface. This has nodes at the two poles about which
+the phase has −1 winding and so we can infer similar winding of the director in the transverse plane. Supplementing
+
+6
+FIG. 3.
+The connection between spherical harmonics and nematic topological defects.
+The coloured spheres indicate the
+phase of the complex spherical harmonics with the nodal set shown in white for simplicity. A representative skeleton of the
+corresponding nematic distortion is shown in black and the red arrows indicate the winding vector of the director.
+with the far-field alignment along ez yields the familiar picture of a pair of oppositely charged hedgehog defects.
+Similarly, the Saturn ring quadrupole, described by ∂ ¯
+wz 1
+r ∼ eiφ sin 2θ, has zeros at the poles and around the equator.
+The winding about the poles is still +1, but the sign change in the lower hemisphere means that in the transverse
+plane around the south pole the vector points inwards, resulting in both point defects having topological charge +1.
+With regards to the equatorial line, since the director is everywhere radial the winding vector must be tangential to
+the defect loop, shown by the red arrows in Fig. 3. As the phase changes by π on passing from one hemisphere to the
+other the winding must be ±1 and the far-field alignment allows us to determine it to be −1. For a general multipole
+distortion of the form ∂m
+¯
+w ∂l−m
+z
+(1/r) the nodal set is the poles along with l − m lines of latitude. The phase winding
+of the spherical harmonic dictates the transverse winding of the director and, when supplemented with the far-field
+alignment, allows us to associate topological point defects with the poles. Similarly, nodal lines may be connected
+with defect loops with integer winding and a winding vector that rotates according to eimφ. In Fig. 3 we illustrate
+this for the case ∂2
+¯
+w∂3
+z(1/r) ∼ −Y 5
+2 /r6.
+We now describe briefly the correspondence for our basis of dipolar and quadrupolar distortions. As already stated,
+the isotropic scalar in p0 is the UPenn dipole, its pseudoscalar counterpart a chiral splay-free twist-bend distortion
+whose integral curves are shown in orange in Fig. 2. As a twist-bend mode it may be of particular relevance to
+extensional systems given their instability to bend distortions. The two dipoles of p1 are transverse to the far-field
+alignment, they are related to those resulting from a defect loop of wedge-twist type [21]. The distortions of p2 have
+a hyperbolic character; they describe the far field of a pair of point defects both of which have a hyperbolic structure.
+Such hyperbolic defect pairs arise in toron configurations in frustrated chiral nematics [52, 53].
+Similarly, Q0 contains the Saturn ring quadurpole as the scalar, with the pseudoscalar a pure bend chiral distortion.
+For the latter, the integral curves of the director possess opposing chirality in the two hemispheres, which could be
+generated by an appropriately coated Janus particle. The director distortion exhibits a helical perversion in the z = 0
+plane and, being a local rotation of the Saturn ring distortion, may be viewed as resulting from a pair of vortex point
+defects along with a pure twist defect loop with integer winding. This is similar to the bubblegum defect lines [51, 54]
+that appear between a colloid diad with normal anchoring, suggesting that this chiral quadrupole could be formed by
+two colloids with opposing chiral tangential anchoring.
+The spin-1 quadrupoles consist of pairs of wedge-twist defect loops. The distortions of Q2 may be associated with
+a pair of hyperbolic defects along with a defect ring with the appropriate symmetry. The harmonics of spin −1 and
+3 contain no z-derivatives and so are associated with pairs of point defects only.
+IV.
+FLOWS FROM MULTIPOLE DISTORTIONS
+In this section we calculate the active flow generated by an arbitrary director multipole. We present this initially in
+vectorial form, converting to the complex representation subsequently. As (7) is linear the responses due to the two
+components of δn are independent and so to simplify the derivation we consider only distortions in the x-component
+for now and extend to the general case afterwards. Within this restriction a generic multipole distortion at order l
+
+22
+75
+r3
+r6
+r
+Y7
+may be written as
+δnx = al∇v1 · · · ∇vl
+a
+r ,
+(19)
+where v1, . . . , vl are l directions for the differentiation. Substituting this into (7) gives the Stokes equation in the
+form
+− ∇p(x) + µ∇2u(x) = al+1ζ∇v1 · · · ∇vl
+�
+ex ∂z + ez ∂x
+�1
+r ,
+(20)
+where the use of the superscript (x) is to emphasise that we are only treating the response to distortions in the
+x-component of the director. Taking the divergence of both sides we have
+− ∇2p(x) + µ∇2∇ · u(x) = al+1ζ∇v1 · · · ∇vl∂2
+xz
+2
+r .
+(21)
+Making use of the continuity equation ∇ · u(x) = 0 in conjunction with the identity ∇2r = 2
+r we arrive at the solution
+for the pressure
+p(x) = −al+1ζ∇v1 · · · ∇vl ∂x∂zr = al+1ζ∇v1 · · · ∇vl
+xz
+r3 .
+(22)
+Substituting this back into the Stokes equation (20) we obtain
+µ∇2u(x) = al+1ζ∇v1 · · · ∇vl
+�
+ex ∂z
+�1
+r − ∂x∂xr
+�
+− ey ∂x∂y∂zr + ez ∂x
+�1
+r − ∂z∂zr
+��
+,
+(23)
+which can be integrated using the identity ∇2r3 = 12r to find
+u(x) = al+1 ζ
+4µ∇v1 · · · ∇vl
+�
+ex
+�z
+r + x2z
+r3
+�
++ ey
+xyz
+r3 + ez
+�x
+r + xz2
+r3
+��
+.
+(24)
+Both the pressure and flow solutions for a generic multipole distortion are given in terms of derivatives of a
+fundamental response to a monopole deformation, namely
+p(x) = aζ xz
+r3 ,
+(25)
+u(x) = aζ
+4µ
+�
+ex
+�z
+r + x2z
+r3
+�
++ ey
+xyz
+r3 + ez
+�x
+r + xz2
+r3
+��
+.
+(26)
+This flow response, shown as the top panel in Fig. 4, is primarily extensional in the xz-plane. Interestingly, the flow
+solution (26) does not decay with distance; this reflects the generic hydrodynamic instability of active nematics [42]
+providing a real-space local response counterpart to the usual Fourier mode analysis.
+However, the active flow
+produced by any higher multipole does decay and vanishes at large distances.
+The pressure and flow solutions in (25) and (26) are complemented by analogous ones resulting from distortions
+in the y-component of the director, obtained by simply interchanging x and y. The linearity of (7) makes these
+fundamental responses sufficient to obtain the active flow induced by an arbitrary multipole distortion through taking
+derivatives appropriate to describe the x and y components of the director, respectively.
+We now convert this description to the complex notation used in § III. This is achieved by taking the combinations
+p = p(x) − ip(y) and u = u(x) − iu(y). To see this consider the multipole distortion δn = (Lx + iLy)1/r, where the Li
+are generic real differential operators which generate the i-component of the director by acting on 1/r. This distortion
+has a conjugate partner given by i(Lx + iLy)1/r = (−Ly + iLx)1/r. Acting with this same operator on u(x) − iu(y)
+we have
+(Lx + iLy)(u(x) − iu(y)) = (Lxu(x) + Lyu(y)) − i(−Lyu(x) + Lxu(y)),
+(27)
+and can see that the flow response for our original distortion forms the real part and that for its conjugate partner
+the coefficient of −i and the same holds for the pressure response. This leads us to a complex fundamental pressure
+response
+˜p = aζ ¯wz
+r3 ,
+(28)
+
+8
+FIG. 4. The active flows due to three-dimensional nematic multipole distortions up to quadrupole order. The flows are grouped
+according to their spin, in correspondence with the distortions in Fig. 2. Green and red arrows indicate the net active force
+and torque for the relevant dipoles and quadrupoles respectively, see §V.
+and, introducing complex basis vectors ew = ex + iey and e ¯
+w = ex − iey, a complex-valued fundamental flow vector
+˜u = aζ
+4µ
+�
+ew
+¯w2z
+2r3 + e ¯
+w
+�z
+r + w ¯wz
+r3
+�
++ ez
+¯w
+r
+�
+1 + z2
+r2
+��
+.
+(29)
+We use a tilde to distinguish these fundamental responses from those that result due to a generic distortion and which
+may be found by appropriate differentiation. This provides a unified framework in which the active response to a
+generic nematic multipole can be calculated through the application of the same complex derivatives that we have
+used to describe the director distortion. The resulting active flows for distortions up to quadrupole order are shown
+
+-1
+0
+2
+3
+Monopoles
+Dipoles
+Quadrupoles9
+in Fig. 4, with their layout corresponding to that of the nematic distortions in Fig. 2 which induce them. We now
+describe some examples in more detail.
+A.
+UPenn and chiral dipole
+Typically the active responses induced by the two distortions in a spin class will, like the distortions themselves, be
+related by a global rotation such that while both are needed to form a sufficient basis, the real part essentially serves
+as a proxy for the pair. This is not true for the spin-0 distortions, due to their rotational symmetry, and so we use
+them in providing an explicit illustration of the active flow calculation. We begin with the UPenn dipole [25] and its
+partner the chiral dipole, for which the far-field transverse director is
+δn ≈ αa ∂ ¯
+w
+a
+r ,
+(30)
+where α is a dimensionless coefficient, and the corresponding derivative of the fundamental flow solution in (29) gives
+αa∂ ¯
+w˜u = ζαa2
+4µr5
+�
+ew z ¯w(4z2 + w ¯w) − e ¯
+w 3zw2 ¯w + ez 2
+�
+3z4 + (z2 + w ¯w)2��
+.
+(31)
+Taking the real part gives, after some manipulation, the flow induced by the UPenn dipole as
+u = αa R ∂ ¯
+w˜u = ζαa2
+8µ
+�
+ez
+�1
+r + z2
+r3
+�
++ er
+z
+r2
+�3z2
+r2 − 1
+��
+,
+(32)
+where er is the unit vector in the radial direction. The flow response to the conjugate distortion, the isotropic chiral
+dipole is given by
+u = −αa I ∂ ¯
+w˜u = −ζαa2
+4µ
+z
+r2 eφ,
+(33)
+with eφ the azimuthal unit vector. Both flows decay at large distances like 1/r and are highlighted in the top row of
+Fig. 5. The UPenn dipole flow has a striking net flow directed along the z-axis, reminiscent of that of the Stokeslet
+flow [55, 56] associated with a point force along ez. The chiral dipole generates an axisymmetric flow composed
+of two counter-rotating vortices aligned along ez, mirroring the circulating flows produced by spiral defects in two
+dimensions [57]. The 1/r decay of these active vortex flows is unusually slow, slower than the decay of a point torque
+in Stokesian hydrodynamics [56].
+Despite the similarity between the active flow induced by the UPenn dipole and a Stokeslet, there is a key difference
+in their angular dependence.
+In a Stokeslet, and all related squirming swimmer flows [58, 59] that result from
+derivatives of it, the terms with higher angular dependence decay more quickly such that the lowest order terms
+dominate the far field. By contrast, distortions in active nematics produce asymptotic flow fields in which all terms
+decay at the same rate regardless of their angular dependence as they all result from the same derivative of the
+fundamental flow. Thus, even if the same angular terms are present in both systems, the lowest order ones will
+dominate in the squirming case while the far field will bear the signature of the highest order in the active nematics.
+A closer point of comparison comes from the flows induced by active colloids within a passive nematic [35, 60].
+Calculation of the relevant Green’s functions [61] has shown that the anisotropy of the medium leads to a difference
+in effective viscosities such that a Stokeslet aligned along the director pumps more fluid in this direction. This fits
+with the anisotropy displayed in (32), reaffirming the similarity between the flow induced by the UPenn dipole and
+the Stokeslet.
+Considering the pressure response for these distortions in the same way we have
+αa∂ ¯
+w ˜p = ζαa2
+2r5 z(2z2 − w ¯w) = ζαa2z
+2r3
+�3z2
+r2 − 1
+�
+.
+(34)
+As this expression is purely real it comprises the response due to the UPenn dipole in its entirety; the vanishing
+of the imaginary part shows that the chiral dipole is compatible with a zero pressure solution. Our complexified
+construction allows this property to be read off immediately, since ∂ ¯
+w( ¯wzm/rn) will be real for any m and n, with
+this also resulting in the vanishing z-component of flow for the chiral dipole. Indeed, this property of pure realness is
+unchanged by the action of ∂z, it being real itself, and so extends to higher order distortions.
+
+10
+FIG. 5. The active flows induced by spin 0 dipole (top row) and quadrupole (bottom row) distortions. The flow is indicated
+by blue arrows and superposed upon integral curves of the director, shown in orange. On the left are the UPenn dipole and
+Saturn ring quadrupole and on the right their chiral counterparts.
+B.
+Saturn ring and chiral quadrupole
+Proceeding in the same fashion for the spin-0 quadrupoles, for which δn ≈ αa2∂2
+¯
+wza/r, we find that the complexified
+flow is
+αa2∂2
+¯
+wz˜u = −ζαa3
+4µr7
+�
+−ew ¯w(w2 ¯w2 + 8w ¯wz2 − 8z4) + e ¯
+w3w2 ¯w(w ¯w − 4z2)
++ez2z(w2 ¯w2 − 10w ¯wz2 + 4z2)
+�
+.
+(35)
+Taking the real part gives the flow induced by the Saturn ring quadrupole as
+u = αa2R∂2
+¯
+wz˜u = −ζαa3
+2µr6 (r4 − 12z2r2 + 15z4)er,
+(36)
+that is a purely radial flow reminiscent of a stresslet along ez, shown in the bottom left of Fig. 5. The purely radial
+nature is a result of the divergencelessness of the flow, combined with the 1/r2 decay and rotational invariance about
+ez. Working in spherical coordinates we have
+∇ · u = 1
+r2 ∂r(r2ur) +
+1
+r sin θ [∂θ(uθ sin θ) + ∂φuφ] = 0
+(37)
+All active flows induced by quadrupole distortions decay as 1/r2 and so ∂r(r2ur) = 0. The distortion is rotationally
+symmetric and achiral, meaning uφ = 0 and the condition of zero divergence reduces to
+1
+r sin θ∂θ(uθ sin θ) = 0.
+(38)
+The only non-singular solution is uθ = 0, resulting in ur being the only non-zero flow component. The corresponding
+pressure is given by
+αa2∂2
+¯
+wz ˜p = −3αa3
+2r7 (r4 − 12z2r2 + 15z4).
+(39)
+
+wz11
+Taking the imaginary part of (35) reveals the flow response of the chiral quadrupole to be
+u = −αa2I∂2
+¯
+wz˜u = ζαa3
+µr2 (3 cos2 θ − 1) sin θeφ.
+(40)
+As illustrated in Fig. 5 this is a purely azimuthal flow corresponding to rotation about the z axis and, as for the
+chiral dipole, is compatible with a zero pressure solution. The 1/r2 decay of this rotational flow is the same as that
+which results from the rotlet [55, 56], but unlike the rotlet the flow direction is not uniform. Rather, as can be seen
+in Fig. 5, there is an equatorial band of high-velocity flow accompanied by two slowly counter-rotating polar regions.
+The distribution of flow speeds is such that the net flow is along −eφ, consistent with a rotlet along −ez.
+C.
+Other multipoles
+For the remaining multipoles up to quadrupole order we do not provide the same explicit calculation but instead
+highlight the key features of the active flows they induce. In full we find that half of the dipole distortions contain
+directed components in their active flow responses. Along with the isotropic UPenn dipole which produces flow along
+ez the two spin-1 dipoles produce directed flows transverse to it. These directed flows indicated that were the source
+of the distortion free to move it would exhibit active self-propulsion. The net transverse flows for the dipoles of p1 is
+in accordance with the previously established motile nature of such defect loops [23]. A more complete description of
+the active dynamics of defect loops via their multipole distortions is presented in Section IV D and [24].
+Along with the chiral dipole, the two additional dipoles which do not generate directed flows are those with spin 2.
+These produce active flows which are extensional with the expected two-fold rotational symmetry about the z-axis.
+Direct calculation shows that the flows resulting from spin-2 distortions have zero azimuthal component. Once again,
+this observation is unaffected by z-derivatives and so holds true for the higher-order multipoles of the form ∂n
+z ∂w(1/r).
+Similarly, there are ten linearly independent quadrupoles, five of which can be seen from Fig. 4 to generate rotational
+flows. As expected, it is the four modes of Q±1 that generate rotations about transverse directions and QC that
+produces rotation around ez. For two of these, namely those in Q1, the director distortions are planar, suggesting
+a two-dimensional analogue and the potential to generate them with cogs or gears [62]. These distortions may be
+associated with a pair of opposingly oriented charge-neutral defect loops and so the rotational flow generated by these
+distortions is in accordance with their antiparallel self-propulsion.
+The quadrupoles of Q−1 are composed of pairs of point defects with topological charge +2. Using ∂2
+¯
+w
+1
+r as an
+example, the rotation can be understood by considering the splay distortions in the xz plane. The splay changes sign
+for positive and negative x, leading to antiparallel forces. The active forces are greatest in this plane, as this is where
+the transverse distortion is radial resulting in splay and bend distortions. Along ey the distortions are of twist type
+and so do not contribute to the active force. This results in the rotational flow shown in Fig. 4. The stretching of the
+flow along ez is as observed for a rotlet in a nematic environment [61].
+Although they lack the rotational symmetry of a stresslet, the flows produced by the quadrupoles of Q2 are also
+purely radial. The argument is largely the same as for the Saturn ring distortion, except that the vanishing of uφ is
+not due to rotational invariance but a property inherited from the spin-2 dipoles.
+The quadrupoles of Q3 produce extensional flows whose spin-3 behaviour under rotations about ez is commensurate
+with that of the distortions. Although they visually resemble the similarly extensional flows produced by the dipoles
+of p2, they do not share the property of a vanishing azimuthal flow component.
+D.
+Defect loops
+Of particular relevance to the dynamics of three-dimensional active nematics are charge-neutral defect loops [21,
+23, 24]. For such defect loops the director field has the planar form
+n = cos Υ
+4 ez + sin Υ
+4 ex,
+(41)
+where Υ is the solid angle function for the loop [43, 63], and is a critical point of the Frank free energy in the one-
+elastic-constant approximation [64]. This allows a multipole expansion for the director at distances larger than the
+loop size in which the multipole coefficients are determined explicitly by the loop geometry [24]
+Υ(x) = 1
+2
+�
+K
+ϵijk yj dyk ∂i
+1
+r − 1
+3
+�
+K
+ϵikl ylyk dyl ∂i∂j
+1
+r + . . . ,
+(42)
+
+12
+FIG. 6. Additonal flow solutions induced by spin-1 nematic multipoles. The nematic multipoles which induce the flows are
+shown below them as complex derivatives of 1/r. The red arrows indicate the net active torque.
+where y labels the points of the loop K and r = |x| with the ‘centre of mass’ of the loop defined to be at x = 0. The
+dipole moment vector is the projected area of the loop, while the quadrupole moment is a traceless and symmetric
+tensor with an interpretation via the first moment of area or, in the case of loops weakly perturbed from circular, the
+torsion of the curve.
+The planar form of the director field (41) corresponds to a restricted class of director deformations in which δn is
+purely real. This disrupts the complex basis we have adopted for the representation of multipoles, so that another
+choice is to be preferred.
+We may say that the planar director selects a real structure for the orthogonal plane
+C, breaking the U(1) symmetry, and the restricted multipoles should then be decomposed with respect to this real
+structure. Accordingly, the pressure and flow responses may be generated by derivatives of the fundamental responses
+for distortions in ex, (25) and (26), with these derivatives corresponding to the multipole expansion of the solid
+angle shown in (42). The details of this approach along with the consequences it has for both the self-propulsive and
+self-rotational dynamics of active nematic defect loops are given in [24].
+E.
+Technical note
+We conclude this section with a technical note on the flow solutions that we have presented. The construction
+for calculating active flow responses that we have developed in this section requires knowledge of the multipole
+as a specified set of derivatives of 1/r.
+The harmonic director components satisfy ∇2ni ∝ δ(r) and while this
+delta function does not affect the far-field director it impacts the flow solutions. Consequently, at quadrupole order
+and higher, distinct derivatives of 1
+r can produce the same multipole distortion in the director but have different
+associated active flows. As an explicit example we take the spin-1 quadrupole shown in Fig. 2, which may be written
+as n = a2∂2
+z
+a
+r ex +ez and therefore induces an active flow given by the action of a2∂2
+z on 29, as is illustrated in Fig. 4.
+However the same director distortion is captured by n = −4a2∂2
+w ¯
+w
+a
+r ex + ez, for which the corresponding active flow
+is shown in Fig. 6. A partial resolution to this ambiguity is that any non-equilibrium phenomenological features such
+as propulsion or rotation will be invariant to this choice of derivatives since, as we shall show in the following section,
+they can be expressed directly in terms of the director components. As a more complete resolution we reiterate that
+whenever an exact solution for the director is known the appropriate derivatives can be determined, as demonstrated
+earlier for defect loops [24], and so the apparent ambiguity disappears.
+V.
+ACTIVE FORCES AND TORQUES
+The directed and rotational active flow components highlighted above result in viscous stresses whose net effect
+must be balanced by their active counterparts, since the net force and torque must be zero. Consequently, these
+generic aspects of the response of an active nematic can be identified by considering the contribution that the active
+
+ww13
+stresses make to the force and torque
+f a =
+�
+ζnn · dA ≈
+�
+ζ
+�
+ex
+z δnx
+r
++ ey
+z δny
+r
++ ez
+x δnx + y δny
+r
+�
+dA,
+(43)
+τ a =
+�
+x × ζnn · dA ≈
+�
+ζ
+�
+ex
+�xy δnx
+r
++ (y2 − z2)δny
+r
+�
++ ey
+�(z2 − x2)δnx
+r
+− xy δny
+r
+�
++ ez
+z(−y δnx + x δny)
+r
+�
+dA,
+(44)
+integrating over a large sphere of radius r. These integrals depend on the surface of integration, as the active stresses
+are neither divergenceless nor compactly supported. However, a spherical surface is concordant with the multipole
+approach we are taking and the results are then independent of the radius, as a direct consequence of the orthogonality
+of spherical harmonics. From these expressions we can read off the multipole that will generate any desired active
+force or torque; dipoles generate forces and quadrupoles generate torques. When the active torque is non-zero, the
+compensating viscous torque will drive a persistent rotation of the multipole, creating an active ratchet; similarly, a
+non-zero active force will generate directed fluid flow. The above integrals therefore provide a solution to the inverse
+problem: given a particular non-equilibrium response, which distortion induces it? Hence they serve as a design guide
+for generating out of equilibrium responses in active nematics.
+If the multipole is free to move it will self-propel and rotate. The translational and rotational velocities are related
+to the viscous forces and torques by a general mobility matrix [65].
+In passive nematics, experiments [66] and
+simulations [67, 68] have found that it is sufficient to take a diagonal form for the mobility (no translation-rotation
+coupling) with separate viscosities for motion parallel, µ∥, and perpendicular, µ⊥, to the director, with typical ratio
+of viscosities µ⊥/µ∥ ∼ 1.6 [66–68]. This has the consequence that in general the force and velocity are not colinear
+U = −1
+6πa
+� 1
+µ∥
+f a
+∥ ez + 1
+µ⊥
+f a
+⊥
+�
+.
+(45)
+We again use the UPenn dipole as an example. Integrating the active stresses over a spherical surface of radius R we
+find an active force
+�
+ζnn · dA ≈ −ζαa2
+2
+� �
+ex
+xz
+R4 + ey
+yz
+R4 + ez
+� z
+R + x2 + y2
+R4
+��
+dA = −4πζαa2
+3
+ez.
+(46)
+Balancing this against Stokes drag predicts a ‘self-propulsion’ velocity for the active dipole of
+U = 2ζαa
+9µ∥
+ez.
+(47)
+For extensile activity (ζ > 0) the dipole moves ‘hyperbolic hedgehog first’ and with a speed that increases linearly
+with the core size a. This self-propulsion is in accordance with the directed component of the active flow, as can be
+seen in Fig. 5. The same self-propulsion speed along ex and ey is found for the transverse dipoles of p1, except that
+the parallel viscosity µ∥ should be replaced with µ⊥. Again, this self-propulsion agrees with the directed flow induced
+by these distortions, as calculated through the multipole approach, shown in Fig. 4 [24] and also with the results of
+both a local flow analysis and simulations [23]. The same directed motion has been observed in a related system of an
+active droplet within a passive nematic [35], with the droplet inducing a UPenn dipole in the nematic and moving in
+the direction of the hedgehog defect at a speed that grew with the droplet radius. The mechanism at play is different
+however; the motion results from directional differences in viscosity resulting from the anisotropic environment.
+To illustrate the rotational behaviour we use a member of Q1, ∂2
+z(1/r), as an example. We find an active torque
+�
+ζx × nn · dA ≈ ζαa3
+�
+1
+r6 (2z2 − x2 − y2)
+�
+xyex + (z2 − x2)ey − yzez
+�
+dA
+(48)
+= 8πζαa3
+5
+ey.
+(49)
+Balancing against Stokes drag as was done in the dipole case gives an angular velocity
+Ω = −ζα
+5µey.
+(50)
+
+14
+We note that for this and all other distortions which result in net torques the angular velocity is independent of the
+colloid size. In accordance with the relation ∂2
+z +4∂2
+w ¯
+w(1/r) = 0, the torque resulting from ∂2
+w ¯
+w(1/r) is of the opposite
+sign and a quarter the strength. The net active torques due to harmonics of Q0 and Q−1 have the directions indicated
+in Fig. 4 and half the magnitude of (49).
+Let us consider the approximate magnitude of the effects we have described. Beginning with the self-propulsion
+speed, the fluid viscosity is roughly 10−2 Pa s [17], although effects due to the elongated form of the nematogens
+could increase this by a factor of 30 or so [69, 70]. Both the activity [16] and the dipole moment constant [48] are
+of order unity, meaning the colloid would approximately cover its radius in a second. Similar approximations for the
+quadrupole give an angular velocity of about 2/3 rad s−1. For a colloid of radius 10 µm this has an associated power
+of the order of femtowatts, the same as predicted for bacterial ratchets [71].
+VI.
+TWO-DIMENSIONAL SYSTEMS AND RATCHETS
+As noted above, the planar nature of the rotational distortions in Q1 suggests the existence of two-dimensional
+analogues. In part motivated by this we now discuss the active response of multipolar distortions in two dimensions,
+again beginning with the connection between these multipoles and topological defect configurations.
+A.
+Multipoles and topological defects
+The categorisation of the harmonic distortions in two dimensions is much simpler, but we provide it here for
+completeness. Taking the asymptotic alignment to be along ey the symmetry of the far-field director is now described
+by the order 2 group {1, Ry}, with Ry reflection with axis ey, under which the monopole distortion nx ∼ A log(r/a)
+is antisymmetric. The higher-order distortions are once again generated via differentiation of the monopole, with ∂y
+leaving the symmetry under Ry unchanged and ∂x inverting it.
+It should be noted that the potential multiplicity of differential representations of harmonics that arose in three
+dimensions does not occur in two dimensions. This is because, under the assumption of a single elastic constant, the
+director angle φ may be written as the imaginary part of a meromorphic function of a single complex variable and
+this naturally defines the appropriate set of derivatives. Making z = x+iy our complex variable we write φ = I {f(z)}
+which upon performing a Laurent expansion of f(z) around z = 0 and assuming the existence of a uniform far-field
+alignment gives
+φ = I
+�
+0
+�
+n=−∞
+anzn
+�
+= I
+�
+a0 +
+∞
+�
+n=1
+(−1)n−1
+an
+(n − 1)!∂n
+z (ln z)
+�
+.
+(51)
+Hence at every order there is a one parameter family of distortions, corresponding to the phase of the an. A natural
+basis at order n is provided by {R {∂n
+z (ln z)} , I {∂n
+z (ln z)}}. This basis consists of a symmetric and anti-symmetric
+distortion under the action of Ry, the roles alternating with order, and of course correspond to the two harmonic
+functions cos nθ/rn and sin nθ/rn.
+In two dimensions the connection between defect configurations and far-field multipole distortions can be made
+concrete, and also serves as an illustration of how a particular set of derivatives is determined. For defects with
+topological charges sj at locations zj the angle that the director makes to ex is given by
+φ = φ0 +
+�
+j
+sjI
+�
+ln
+�z − zj
+a
+��
+,
+(52)
+which, upon performing a series expansion, gives
+φ = φ0 +
+�
+j
+sjI {ln(z/a)} −
+∞
+�
+n=1
+I
+��
+j sjzn
+j ¯zn�
+n|z|2n
+,
+(53)
+= φ0 +
+�
+j
+sjI {ln(z/a)} +
+∞
+�
+n=1
+(−1)nI
+��
+j sjzn
+j ∂n
+z ln z
+�
+n!
+,
+(54)
+Provided the total topological charge is zero the winding term proportional to ln w vanishes and φ0 is the far-field
+alignment. The distortions are given as a series of harmonics in which the coefficient of the nth harmonic is determined
+by a sum of zn
+j weighted by the defect charges.
+
+15
+We would like to have a basis of representative defect configurations for each harmonic distortion. However, it
+can be seen from (54) that the correspondence between arrangements of topological defects and the leading order
+nematic multipole is not one-to-one. Two defect-based representations of harmonic will prove particularly useful to
+us. The first, which we develop in this chapter, provides a representation in terms of half-integer defects on the disc
+and allows an intuition for the response to multipole distortions in active nematics through known results for such
+defects [15, 16]. The second uses the method of images to construct defect arrangements corresponding to a specific
+anchoring condition on the disc, with the same multipoles dominating the nematic distortion in the far field. This
+representation naturally lends itself to the control of induced multipoles through colloidal geometry and is explored
+fully in [62]. Nonetheless, both of these representations will be of use to us in the remainder of this chapter and as
+they are equally valid near-field representations for the asymptotic distortions that we are considering we will pass
+fairly freely between them.
+With this aforementioned half-integer representation in mind, let us consider sets of 2m defects sitting on the unit
+circle, with −1/2 defects at the mth roots of unity and +1/2 defects at the intermediate points. A useful formula here
+is the following for the sum of a given power of these roots of unity, after first rotating them all by a given angle θ
+m−1
+�
+k=0
+�
+eiθei 2π
+m k�n
+=
+�
+meinθ,
+if m|n
+0,
+otherwise .
+(55)
+The vanishing of this sum for values of n that are not multiples of m comes directly from the expression for the
+geometric sum and is a consequence of the cyclic group structure of the roots of unity. It means that the lowest order
+multipole distortion induced by such an arrangement of defects is order m and so allows a desired multipole distortion
+to be selected as the dominant far-field contribution. Explicitly, the director angle is given by
+φ = φ0 +
+�
+k odd
+I
+�
+¯zmk�
+k|z|2mk = φ0 + I {¯zm}
+|z|2m + O
+� 1
+z3m
+�
+,
+(56)
+with the approximation becoming rapidly better for higher-order multipoles due to the condition that n must be an
+odd multiple of the number of defects. Rotating the entire set of defects rigidly by an angle −π/(2m) generates the
+conjugate multipole as the dominant far-field contribution
+φ = φ0 +
+�
+k odd
+I
+�
+(−i)k¯zmk�
+k|z|2mk
+= φ0 − R {¯zm}
+|z|2m
++ O
+� 1
+z3m
+�
+,
+(57)
+with the natural interpolation between these two harmonics as the defect configuration is rigidly rotated.
+Hence we can interchange between a given harmonic distortion and a defect arrangement which has this harmonic
+as its dominant far-field contribution, with the correspondence becoming rapidly more accurate for higher orders,
+allowing us to relate the existing results for the behaviour of active defects [15, 16] to ours and vice versa. This
+correspondence is illustrated in Fig. 7. The locations of +1/2 and −1/2 defects are indicated with red and cyan dots
+respectively and the background colouring denotes the phase of the complex function � sj ln(z−zj), whose imaginary
+part provides the director angle for the given defect arrangement. The integral curves of this director field are shown
+in black and are remarkably well matched by those of the leading multipole, shown in white, despite the asymptotic
+nature of the approximation. In this context we are able to make precise the notion of a core region of a singular
+distortion, outside of which our multipole approach applies. The series in (54) is attained through a Taylor series of
+terms of the form ln(1 − 1/z), which are convergent for |z| > 1. More generally the greatest radial displacement of a
+defect defines a core radius, outside of which the multipole series converges onto the exact director angle.
+B.
+Flows from multipole distortions
+We can proceed analogously to our three-dimensional calculation in generating the active flows from a fundamental
+response in two dimensions, provided we are mindful of the logarithmic form that the monopole now has. A director
+rotation by θ0 inside a disc of radius a results in an equilibrium texture given by
+n = cos
+�θ0 log(r/R)
+log(a/R)
+�
+ey + sin
+�θ0 log(r/R)
+log(a/R)
+�
+ex,
+(58)
+which in the far field tends to a monopole distortion n ≈ ey + θ0 log(r/R)
+log(a/R) ex. Due to the logarithmic divergence of the
+fundamental harmonic in two dimensions it is necessary to normalise through a large length R such that a uniformly
+aligned far-field director is recovered.
+
+16
+FIG. 7. Representative defect configurations for nematic multipoles in two dimensions. The red and cyan dots indicate the
+locations of +1/2 and −1/2 defects respectively. The black curves are the integral curves of the corresponding director field
+and the background colour shows the phase of the complex function whose imaginary part gives the exact director angle, as
+in (52). The white lines are the integral curves of the dominant multipole, that is the leading term of (54). The multipole
+series converges onto the exact director angle outside a core region, shown as a white disc, and the leading multipole provides
+a remarkably good approximation in this region.
+Following our three-dimensional analysis we solve Stokes’ equations to linear order in nematic deformations for a
+monopole distortion. We write Stokes’ equations in terms of complex derivatives as
+2∂¯z(−p + iµω) = f,
+(59)
+where we have used that 2∂zu = ∇ · u + iω, with ω the vorticity. Hence we seek f as a ¯z-derivative, implicitly
+performing a Helmholtz derivative with the real and imaginary parts of the differentiated term corresponding to the
+scalar and vector potentials respectively. Expressing the active force in this way we have
+2∂¯z(−p + iµω) =
+ζθ0
+log(a/R)∂¯z
+�i¯z
+z
+�
+(60)
+and so
+− p + iµω =
+ζθ0
+2 log(a/R)
+i¯z
+z .
+(61)
+Reading off the pressure and vorticity, solving for the flow and converting back to Cartesians the fundamental flow
+response is now found to be
+˜u =
+ζθ0
+8µ log(a/R)
+�x2 − y2
+r2
+(−yex + xey) + 2 log
+� r
+R
+�
+(yex + xey)
+�
+,
+(62)
+˜p = −
+ζθ0
+log(a/R)
+xy
+r2 .
+(63)
+There is a clear similarity between these solutions and their three-dimensional counterparts, but while the fundamental
+flow response is still extensional it now grows linearly with distance from the distortion, with this change in scaling
+inherited by the subsequent harmonics.
+
+b)
+a
+)
+(p17
+As in the three-dimensional case we can gain general insight into the active response of a nematic by considering
+the net contribution of the active stresses to the force and torque when integrated over a large circle of radius r
+�
+ζnn · erdr ≈
+�
+ζ
+�yδnx
+r
+ex + xδnx
+r
+ey
+�
+dr,
+(64)
+�
+x × ζnn · erdr ≈
+�
+ζ (y2 − x2)δnx
+r
+dr.
+(65)
+We see that in two dimensions both dipoles will self-propel if free to move and there is a single chiral quadrupole
+which produces rotations.
+The far-field flow solutions for distortions up to dipole order are illustrated in Fig. 8, superposed over the nematic
+director.
+Both dipoles are now motile and as in the three-dimensional case they set up flows reminiscent of the
+Stokeslet.
+Vertical and horizontal self-propulsive modes may be viewed as resulting from normal and tangential
+anchoring respectively of the nematic on a disc. Interpolating between these orthogonal modes the angle of motility
+changes commensurately with the anchoring angle, such that sufficient control of the boundary conditions would allow
+for self-propulsion at an arbitrary angle with respect to the far-field alignment. This change in the dipole character
+can be represented by rigidly rotating the defect pair around the unit circle and the resulting motility is as would be
+expected from the position and orientation of the +1/2 defect [16, 72, 73]. Determining the motility induced by these
+dipolar modes is complicated by the Stokes paradox and although this can be circumvented by various means we do
+not pursue this here. If such dipolar colloids were fixed within the material they would pump the ambient fluid and
+so it should be possible to use them to produce the concentration, filtering and corralling effects observed previously
+by funneling motile bacteria [74].
+In line with our discussion at the beginning of this section, the basis quadrupoles are given by the real and
+imaginary parts of ∂2
+z, these being an achiral and chiral mode respectively, which are shown along with their flows in
+Fig. 8. The flow generated by the achiral quadrupole in Fig. 8(d) is purely radial and resembles the stresslet flow,
+unsurprising as it results from differentiating the vertical dipole in the same way as the stresslet is related to the
+Stokeslet. It is produced by a quadrupole distortion which may be associated with normal anchoring on the disc –
+its counterpart with tangential anchoring has all the charges in its representative defect configuration inverted and
+a reversed flow response. Just as for the dipole distortions, the character of the quadrupole can be smoothly varied
+through adapting the boundary condition and the topological defects which represent the harmonic rotate rigidly in
+step with the changing anchoring angle. A generic anchoring angle will produce a net active torque, maximised for an
+angle of π/4 as illustrated for the chiral quadrupole shown in Fig. 8(e). For extensile activity this distortion generates
+clockwise rotation, as can easily be justified via our representation of the far-field director structure as arising from a
+square arrangement of two +1/2 and two −1/2 defects – the dual mode with the defect charges interchanged rotates
+anticlockwise. By choosing boundary conditions such that the defects are positioned closer to the mid-line of the
+colloid the strength of the active torque can be tuned.
+VII.
+CHIRAL ACTIVE STRESSES
+Chirality is a ubiquitous trait, in living systems and liquid crystals alike. In active matter it opens a wealth of new
+phenomena, including odd viscous [75] and elastic responses [76, 77], surface waves, rotating crystals [78] and non-
+reciprocal interactions [79]. Chiral active stresses induce vortex arrays in active cholesterics [12] and have also been
+shown to be important in nematic cell monolayers where they modify collective motion, the motility of topological
+defects and generate edge currents [80, 81]. We now consider the effects of such chiral active stresses on nematic
+multipoles, both in two and three dimensions.
+A.
+Two dimensions
+For chiral stresses in two dimensions, the active stress tensor has the form σc = χJ(nn − n⊥n⊥)/2, where J is the
+complex structure defined by Jn = n⊥ and Jn⊥ = −n. The chiral active force is
+∇ · σc = χJ
+�
+∇ · (nn)
+�
+,
+(66)
+and is simply a π/2 rotation of the achiral active force. Accordingly we can modify (61) to give
+− p + iµω = −
+ζθ0
+2 log(a/R)
+¯z
+z ,
+(67)
+
+18
+FIG. 8. Distortions up to quadrupole order in two-dimensional active nematics. The active flow in white is superposed on the
+pressure field, with the integral curves of the director shown in black. (a) The fundamental monopole response is extensional
+and grows linearly with distance from the distortion. (b) and (c) show the flows induced by dipole distortions, labelled by the
+appropriate derivative of the nematic monopole, with the green arrows indicating the direction of self-propulsion that would
+result from net active forces in extensile systems. The vertical and horizontal dipoles are the far-field director responses to
+normal and tangential anchoring respectively and may also be interpreted as arising from a pair of +1/2 (cyan) and −1/2 (red)
+defects. The self-propulsion matches that expected for the +1/2 defect.
+and solve as before to find
+˜u =
+χθ0
+8µ log(a/R)
+�2xy
+r2 (−yex + xey) + 2 log
+� r
+R
+�
+(−xex + yey)
+�
+,
+(68)
+˜p =
+χθ0
+log(a/R)
+x2 − y2
+2r2
+.
+(69)
+Another way to understand the relation between achiral and chiral stresses is that, since the monopole active force
+field is spin-2, the π/2 local rotation of the active force results in a global rotation by π/4 of the force field and hence
+the fundamental flow responses. The action of this global rotation, denoted Rπ/4, may be seen by comparing the
+monopole flow responses for achiral and chiral stresses, shown in Fig. 8(a) and Fig. 9(a) respectively. For distortions
+of order n there are two basis flows, ur and ui, corresponding to the real and imaginary parts of ∂n
+z respectively.
+The rotation of the monopole response has the consequence that for achiral and chiral active stresses these flows are
+related by
+uc
+r = Rπ/4
+�
+cos
+�nπ
+4
+�
+ua
+r − sin
+�nπ
+4
+�
+ua
+i
+�
+,
+(70)
+uc
+i = Rπ/4
+�
+sin
+�nπ
+4
+�
+ua
+r + cos
+�nπ
+4
+�
+ua
+i
+�
+,
+(71)
+
+a)
+b)
+-%
+(p
+e)
+02
+hc19
+FIG. 9. Distortions up to quadrupole order in two-dimensional active nematics with purely chiral stresses. The active flow
+in white is superposed on the pressure field, with the integral curves of the director shown in black. (a) The fundamental
+monopole response is extensional and grows linearly with distance from the distortion. (b) and (c) show the flows induced
+by dipole distortions, labelled by the appropriate derivative of the nematic monopole, with the green arrows indicating the
+direction of self-propulsion that would result from net active forces in extensile systems.
+where the superscripts denote the nature of the stresses as achiral or chiral. Hence flow solutions for chiral and achiral
+stresses are related by a clockwise rotation by nπ/4 in the space of solutions followed by a rigid spatial rotation
+anticlockwise by π/4, as can be seen in Fig 9. At dipole order the chiral flow fields are rotated superpositions of
+the achiral ones, with the overall effect of chirality being to rotate the self-propulsion direction anticlockwise by π/2,
+interchanging the roles of horizontal and vertical propulsion. For a generic mixture of achiral and chiral stresses
+the direction of self-propulsion is rotated from the achiral case by an angle arctan(χ/ζ), mirroring the effect such
+stresses have on the flow profile of a +1/2 defect [80]. For the quadrupole distortions we have uc
+i = Rπ/4ua
+r and
+uc
+i = Rπ/4(−ua
+i ) = ua
+i , again swapping which distortion produces a chiral or achiral flow response.
+It is worth
+emphasising that the sign of the macroscopic rotation is not necessarily the same as the sign of the chiral stresses,
+rather it is the product of the signs of the activity and the distortion, just as for achiral stresses.
+
+a)
+b)
+(p
+22
+e)
+hc
+C20
+FIG. 10. The active flows induced by spin 0 dipole distortions with chiral active stresses. The flow is superposed upon the
+integral curves of the director, shown in orange, for the UPenn dipole (left) and chiral dipole (right).
+B.
+Three dimensions
+In three dimensions the chiral active force is χ∇×[∇ · (nn)] [12] and so, by linearity, the fundamental flow responses
+are given by the curl of those derived earlier, namely
+u(x) =
+aχ
+2µr3
+�
+−exxy + ey(x2 − z2) + ezyz
+�
+,
+(72)
+u(y) =
+aχ
+2µr3
+�
+−ex(y2 − z2) + eyxy + −ezxz
+�
+,
+(73)
+for monopole distortions in the x- and y-components respectively. Just as for achiral active stresses, we can combine
+these into a single complex fundamental flow response as u(x) − iu(y), giving
+˜u = i
+r3
+�
+− ¯w2ew + (w ¯w − 2z2)e ¯
+w + 2 ¯wzez
+�
+.
+(74)
+Since the active chiral force is a pure curl the corresponding pressure is constant.
+Owing to the additional derivative the functional behaviour of the flow responses is shifted up one order of distortion
+compared to achiral stresses, meaning dipole distortions induce rotations, although it should be noted that monopoles
+do not produce propulsive flows. The monopole flow responses are still spin-1, but since the flow response for a
+monopole distortion in nx for achiral stresses is primarily in the x − z plane, the action of curl produces a flow that is
+dominantly in the y-direction and similarly the response to a monopole distortion in ny is mainly along ex. Together
+these ingredients mean that heuristically the flow response of a given distortion with chiral active stresses will resemble
+the achiral active stress flow response of the conjugate distortion at one higher order and with the same spin, that
+is the distortion reached by the action of i∂z. This is illustrated in Fig. 10 for the spin-0 dipoles. The UPenn dipole
+induces rotation about ez while the chiral dipole produces a purely radial flow, resembling the achiral flow responses
+of the chiral quadrupole and Saturn’s ring quadurpole respectively.
+The phenomenological response can again be captured through integration of the stress tensor over a large sphere
+of radius r, just as was done for achiral active stresses. To enable us to reduce the active torque to a single boundary
+integral we use the symmetric form of the chiral active stress tensor [12], σc
+ij = [∇ × (nn)]ij + [∇ × (nn)]ji, such that
+
+d21
+to linear order in director distortions we have
+f a =
+�
+χσc · dA ≈ 0,
+(75)
+τ a =
+�
+x × χσc · dA ≈
+�
+χ
+�
+ex
+�xz ∂xδnx − 2yz∂yδnx + (y2 − z2)∂zδnx
+r
+�
++ ey
+�yz∂yδny − 2xz∂xδny + (x2 − z2)∂zδny
+r
+�
++ ez
+−2xy(∂yδnx + ∂xδny) − (x2 − y2)(∂xδnx − ∂yδny) + z(x∂zδnx + y∂zδny)
+r
+�
+dA.
+(76)
+From the first of these equations we see that, to linear order, there are no harmonic distortions which produce net
+forces in a nematic with chiral active stresses. With regard to the net active torques, the x− and y− components
+involve only δnx and δny respectively and each term yields a non-zero integral only for δni ∼ ∂z1/r, hence the two
+spin-1 dipoles produce transverse torques. Turning to the z-component, each term gives a non-zero integral only for
+δni ∼ ∂i1/r, and as the expression is symmetric under interchange of x and y we see that only the UPenn dipole
+produces torques around ez. In other words, a dipolar director distortion which produces a net active force along
+a given direction in an achiral active nematic produces a net torque around the same direction in a chiral active
+nematic.
+These results of course accord with our earlier statements regarding the spins of distortions which are
+capable of producing torques about given axes. Performing the integrals we find that in each case the net active
+torque has magnitude −12πχαa2/5. Balancing this against Stokes drag gives, using the UPenn dipole as an example,
+an angular velocity
+Ω = 3χα
+10µaez.
+(77)
+While the angular velocity in achiral active nematics is independent of the distortion size, in chiral active nematics
+it is inversely proportional to the radius, a direct consequence of the additional derivative in the active stress tensor.
+Accordingly, in chiral active nematics the rotational velocity is largest for smaller colloids.
+VIII.
+DISCUSSION
+We have introduced active nematic multipoles as a novel framework for understanding the dynamics of active
+nematics. Although only formally valid on mesoscopic lengthscales, this approach produces results for the propulsive
+dynamics of defect loops that agree with those of a local analysis [23, 24]. It also provides various testable predictions,
+for example for the axis of self-propulsion or rotation induced by a distortion or how the corresponding velocities
+would scale with the size of a colloid.
+More broadly, our results reveal self-propulsion and rotation as generic non-equilibrium responses that naturally
+arise due to colloidal inclusions in active nematics but also provide a template for the tailored design of particular
+dynamics. This provides insight into the issue of harnessing the energy of active systems to perform useful work,
+something which has been demonstrated in bacterial suspensions [71, 82] and is now receiving greater attention in
+the nematic context [36, 37, 83, 84]. Specific anchoring conditions on colloids have been investigated as a means of
+generating directed motion [36]. Our results suggest that sufficient control of the anchoring conditions would allow for
+steerable and targeted colloidal delivery [85], although there may be routes to a similar degree of dynamical control
+through colloidal geometry alone [62].
+The transformative power of colloids in passive nematics was revealed in their collective behaviour, forming crys-
+talline structures [28, 86–89] which can serve as photonic metamaterials [90]. While our predictions for the dynamics
+of individual colloids have utility in their own right, there is again considerable interest in the collective dynamics
+which might emerge [91]. Although our results are insufficient to fully address these questions, some basic points
+can nonetheless be extracted from the flow solutions. The long-range nature of the active flows suggests that the hy-
+drodynamic interactions will be dominant over elastic ones. The leading contribution to the pair-wise hydrodynamic
+interactions will be the advection of each colloid by the flow field generated by the other, and the even inversion
+symmetry of dipole flows implies that this provides a mechanism for pair-wise propulsion, even for colloids which are
+not self-propulsive themselves.
+To conclude, it has been long-established that the distinct symmetries of ±1/2 nematic defects can be directly
+related to the qualitatively different dynamics they display in active systems [15, 16]. The aim of this paper is to
+bring the insights of this symmetry-based approach to generic nematic distortions.
+
+22
+ACKNOWLEDGMENTS
+This work was supported by the UK EPSRC through Grant No. EP/N509796/1.
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+

29FQT4oBgHgl3EQf2zan/content/tmp_files/2301.13425v1.pdf.txt

@@ -0,0 +1,986 @@
+Towards Mechatronics Approach of System Design,
+Verification and Validation for Autonomous Vehicles
+Chinmay Samak∗, Tanmay Samak∗, Venkat Krovi
+Abstract—Modern-day autonomous vehicles are increasingly
+becoming complex multidisciplinary systems composed of me-
+chanical, electrical, electronic, computing and information sub-
+systems. Furthermore, the individual constituent technologies em-
+ployed for developing autonomous vehicles have started maturing
+up to a point, where it seems beneficial to start looking at the
+synergistic integration of these components into sub-systems,
+systems, and potentially, system-of-systems. Hence, this work
+applies the principles of mechatronics approach of system design,
+verification and validation for the development of autonomous
+vehicles. Particularly, we discuss leveraging multidisciplinary co-
+design practices along with virtual, hybrid and physical proto-
+typing and testing within a concurrent engineering framework
+to develop and validate a scaled autonomous vehicle using
+the AutoDRIVE ecosystem. We also describe a case-study of
+autonomous parking application using a modular probabilistic
+framework to illustrate the benefits of the proposed approach.
+Index Terms—Autonomous vehicles, mechatronics approach,
+multidisciplinary design, simulation and virtual prototyping,
+rapid prototyping, verification and validation.
+I. INTRODUCTION
+A
+UTOMOTIVE vehicles have evolved significantly over
+the course of time [1]. The gradual transition from purely
+mechanical automobiles to those with greater incorporation of
+electrical, electronic and computer-controlled sub-systems oc-
+curred in phases over the course of the past century; with each
+phase improving performance, convenience and reliability of
+these systems. Modern vehicles are increasingly adopting
+electrical, electronic, computing and information sub-systems
+along with software algorithms for low-level control as well
+as high-level advanced driver assistance system (ADAS) or
+autonomous driving (AD) features [2]. This naturally brings in
+the interplay between different levels of mechanical, electrical,
+electronic, networking and software sub-systems among a
+single vehicle system, thereby transforming them from purely
+mechanical systems, which they were in the past, to complex
+multidisciplinary systems [3]. As such, while it may have
+been justifiable for earlier ADAS/AD feature developers to
+focus on core software development, the increasing complexity
+and interdisciplinary nature of modern automotive systems can
+benefit from synergistic hardware-software co-design comple-
+mented with integrated verification and validation by following
+the mechatronics principles.
+∗These authors contributed equally.
+C. V. Samak, T. V. Samak and V. N. Krovi are with the Automation,
+Robotics and Mechatronics Laboratory (ARMLab), Department of Automo-
+tive Engineering, Clemson University International Center for Automotive
+Research (CU-ICAR), Greenville, SC 29607, USA. Email: {csamak,
+tsamak, vkrovi}@clemson.edu
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+Fig. 1. Extended V-model fostering mechatronics approach of system design,
+verification and validation for autonomous vehicles. The model depicts evolu-
+tion of a concept into a product through decomposition, design, development,
+integration and testing across component, sub-system, system and system-of-
+systems levels in a unified concurrent interdisciplinary engineering framework.
+Mechatronics engineering [4]–[6] focuses on concurrent and
+synergistic integration of mechanical, electrical and electronics
+engineering, computer science and information technology
+for development and validation of complex interdisciplinary
+systems. This ideology is derived from the fact that various
+components of a “mechatronic” system, often belonging to
+a multitude of disciplines, influence each other and hence
+have a design impact at the component, sub-system, system
+and system-of-systems levels. The resulting ”mechatronic”
+realization now builds on capabilities endowed by the vari-
+ous constituent layers. In such a milieu, the system devel-
+opment approach has also evolved from relatively ad-hoc
+to the more formal V-model [7], building on the modular
+software development and validation roadmap [8]. This model
+has evolved through several state-of-the-art progressions [9]
+and our work seeks to further formalize the adoption of
+mechatronics approach of system conceptualization, design,
+development, integration and testing for autonomous vehicles
+(refer Fig. 1).
+A recent book [10] highlights best practices for industrial
+design, development and validation of autonomous vehicles
+and notes the significant adoption of model-based design
+(MBD) for system integration and testing. However, similar
+arXiv:2301.13425v1  [cs.RO]  31 Jan 2023
+
+Freewheeling
+Front Wheel Hub
+Rear Wheel
+Drive Actuator
+Front Wheel
+Drive Actuator
+Front Monocular
+Camera
+Front Binocular
+Camera
+Rear Monocular
+Camera
+Reversed Inertial
+Measurement Unit
+Reversed LIDAR
+and AprilTag Marker
+</>
+?
+Power
+Computation
+Communication
+Others
+Lights
+Actuators
+Sensors
+Software
+Chassis
+A
+B
+Fig. 2.
+AutoDRIVE ecosystem fosters mechatronics design principles at two levels: [A] primitive reconfigurability allows permutations and combinations
+of addition, removal or replacement of selective components and sub-assemblies of the vehicle to better suit the application; [B] advanced reconfigurability
+allows complete modification of existing hardware and software architectures, and provides an opportunity for introducing new features and functionalities to
+the ecosystem.
+adoption of such streamlined workflows by academia has
+lagged behind [10]. This gap could be explained by the virtue
+of standardization (e.g., ISO 26262 [11], ISO/IEC 33061 [12],
+VDI 2221 [13], VDI 2206 [14], AUTOSAR [15], etc.) in
+industries versus the fact that majority of academic projects
+are deployed using fragmented hardware-software ecosystems
+(e.g. hobby platforms) with a key focus on developing low-cost
+initial proof-of-concept implementations. Additionally, such an
+opportunistic and potentially uninformed selection of hardware
+[16]–[18] and software [19]–[21] toolchains hinders adoption
+of co-design and concurrent engineering thinking to full extent.
+In this paper, we discuss the design philosophy and one
+of the key motivation factors behind AutoDRIVE ecosystem1
+[22], [23] – adopting and promoting mechatronics approach of
+system design, verification and validation for autonomous ve-
+hicles, with an emphasis on creating a streamlined pathway for
+seamless transition to ultimate industrial practice. This paper
+also describes a detailed case-study which demonstrates the
+methodical adoption of mechatronics approach for designing,
+developing and validating a scaled vehicle in the context of
+autonomous parking2 application using a modular probabilistic
+framework.
+II. MULTIDISCIPLINARY DESIGN
+AutoDRIVE offers an open-access, open-interface and flex-
+ible ecosystem for scaled autonomous vehicle development
+by permitting access to and alteration of hardware as well as
+software aspects of the multidisciplinary autonomous vehicle
+design, thereby making it an apt framework for demonstrat-
+ing the claims and contributions of this work. Particularly,
+AutoDRIVE ecosystem offers the following two levels of
+reconfigurability, thereby promoting hardware-software co-
+design (refer Fig. 2).
+1Webpage: https://autodrive-ecosystem.github.io
+2Video: https://youtu.be/piCyvTM2dek
+• Primitive Reconfigurability: The native vehicle of Au-
+toDRIVE ecosystem, called “Nigel”, is modular enough
+to support out-of-the-box hardware reconfigurability in
+terms of swapping and replacing selective components
+and sub-assemblies of the vehicle, in addition to flexibly
+updating the vehicle firmware and/or autonomous driving
+software stack (ADSS) to better suit the target applica-
+tion.
+• Advanced Reconfigurability: The completely open-
+hardware, open-software architecture of AutoDRIVE
+ecosystem allows modification of vehicle chassis param-
+eters (different form factors and aspect ratios), power-
+train configuration (variable driving performance), com-
+ponent mounting profiles (relocation/replacement of com-
+ponents), as well as firmware and ADSS architecture
+(software flexibility).
+The fundamental step in system design is requirement spec-
+ification, without which the design cannot be truly validated to
+be right or wrong, it can only be surprising [24]. Since Auto-
+DRIVE was intended to be a generic ecosystem for rapidly
+prototyping autonomous driving solutions, the requirement
+elicitation resulted in a superset of requirements demanded by
+the application case study discussed in this paper. Furthermore,
+with AutoDRIVE, there is always a scope for updating the
+designs of various components, sub-systems and systems for
+expanding the ecosystem. That being said, following is a
+summary of functional requirement specifications for Nigel
+as of this version of the ecosystem.
+• General design guidelines:
+– Open-source hardware and software
+– Inexpensive and user-friendly architecture
+– Manufacturing technology agnostic designs
+– Modularly reconfigurable components/sub-systems
+– Integrated and comprehensive resources and tools
+
+O
+D
+Q
+C
+0
+CddB
+A
+Chassis
+Power Electronics
+Computation
+Communication
+Software
+Sensors
+Actuators
+Lights
+PiCamera V2.1
+Robot Operating System
+NVIDIA JetPack SDK
+Throttle Feedback
+Steering Feedback
+AutoDRIVE Devkit
+RPLIDAR A1
+PiCamera V2.1
+Ethernet
+WiFi
+Jetson Nano B01
+Arduino Nano V3.0
+Firmware
+MPU-6050 IMU
+Headlights
+Taillights
+3A
+Master Switch
+10A
+Buck Converter
+11.1V 5200mAh
+LiPo Battery
+LiPo Voltage
+Checker Module
+6V DC 160 RPM 120:1 Geared Motor
+6V DC 160 RPM 120:1 Geared Motor
+Incremental Encoder
+Incremental Encoder
+20A
+Motor Driver
+Rear Wheels
+AprilTag Marker
+MG996R Servo Motor
+Steering Mechanism
+Front Wheels
+Turning Indicators
+Reverse Indicators
+Left Ticks
+Right Ticks
+INT
+Filtering
+Fusion
+I2C
+GPIO
+PWM
+USB
+Lights
+Arduino Nano V3.0
+Jetson Nano B01
+Encoders
+IMU
+Actuators
+Intensity
+Timing
+Throttle
+Steering
+SLAM
+x
+y
+1 m
+STATIC MAP
+ODOMETRY
+LOCALIZATION
+NAVIGATION
+Global
+Costmap
+Local
+Costmap
+Global
+Planner
+Local
+Planner
+Controller
+VEHICLE
+Parking
+Pose
+Throttle/Brake
+Steering
+LIDAR Scan
+Save Map
+Load Map
+TF
+TF
+Odometry
+C
+Fig. 3.
+Conceptualization and design of scaled autonomous vehicle: [A] hardware-software architecture; [B] firmware design specifications; [C] modular
+perception, planning and control architecture for autonomous parking application.
+• Perception sub-system shall offer:
+– Ranging measurements (preferably 360◦)
+– RGB visual feed (preferably front as well as rear)
+– Positional measurements/estimation
+– Inertial measurements/estimation
+– Actuation feedback measurements/estimation
+• Computation and communication sub-systems shall offer:
+– Hierarchical computation topology
+– GPU-enabled high-level edge computation platform
+– Embedded low-level computation platform
+– Vehicle-to-everything communication interface
+• Locomotion and signaling sub-systems shall offer:
+– Kinodynamically constrained drivetrain and steering
+– Standard automotive lighting and signaling
+The functional system requirements were decomposed into
+mechanical, electronics, firmware and ADSS design specifica-
+tions and carefully studied to analyze any potential trade-offs
+so as to finalize the components and ultimately come up with
+a refined system architecture design (refer Fig. 3).
+The proposed hardware-software architecture of the scaled
+autonomous vehicle system is divided into eight sub-systems
+viz. chassis, power, computation, communication, software,
+sensors, actuators and lights, each with its own share of com-
+ponents (refer Fig. 3-A). The embedded firmware architecture
+for low-level data acquisition and control is depicted in Fig.
+3-B, which links the data sources to the respective data sinks
+after processing the information.
+Finally, Fig. 3-C depicts high-level architecture of the
+autonomous parking solution described in this paper. Particu-
+larly, it is shown how this candidate autonomy solution uses
+modular algorithms for simultaneous localization and mapping
+(SLAM) [25], odometry estimation [26], localization [27],
+global [28] and local [29] path planning, and motion control.
+Implementation descriptions are necessarily brief due to the
+space limitations; however, further details can be found in this
+technical report [23].
+III. VIRTUAL PROTOTYPING AND TESTING
+Virtual prototypes help expedite the design process by
+validating the designs against system requirements through
+simulation, and suggesting design revisions at an early stage.
+The scaled autonomous vehicle system was virtually pro-
+totyped and tested in three phases. First, the mechanical
+specifications, motions and fit were carefully analyzed using
+a parametric computer aided design (CAD) assembly of the
+system in conjunction with the physical modeling approach
+for multi-body dynamic systems (refer Fig. 4-A). Parallelly,
+the electronic sub-systems were prototyped using the physical
+modeling approach, and also by adopting electronic design au-
+tomation (EDA) workflow (refer Fig. 4-B). Next, the firmware
+for low-level control (front wheel steering angle and rear wheel
+velocity) of the vehicle was verified to produce reliable results
+(within a specified tolerance of 3e-2 rad for steering angle
+and 3e-1 rad/s for wheel velocity) through model-in-the-loop
+(MIL) and software-in-the-loop (SIL) testing (refer Fig. 4-C).
+The knowledge gained through this process was used to
+update the AutoDRIVE Simulator (refer Fig. 4-D) from its
+initial version discussed in [30], [31] to the one described
+in [22], [23]. The updated simulator was then employed for
+verification and validation of individual ADSS modules and
+finally, the integrated autonomous parking solution was also
+verified using the same toolchain (refer Fig. 5-A). Particularly,
+we tested the vehicle in multiple environments, which included
+unit tests for validating the SLAM, odometry, localization,
+planning and control algorithms, followed by verification
+of the integrated pipeline with and without the addition of
+dynamic obstacles, which were absent while mapping the en-
+vironment. The autonomous navigation behavior was analyzed
+for 5 sample trials and verified to fit within an acceptable
+tolerance threshold of 2.5e-2 m; the acceptable parking pose
+tolerance was set to be 5e-2 m for linear positions in X and
+Y directions and 8.73e-2 rad for the angular orientation about
+Z-axis.
+
+A
+B
+D
+Firmware
+Model
+Vehicle
+Model
+MIL
+SIL
+Firmware
+Code
+Vehicle
+Model
+Vehicle Model
+Embedded
+Firmware
+PIL
+Embedded
+Firmware
+Real-Time
+Vehicle Model
+HIL
+VIL
+Embedded
+Firmware
+Physical
+Vehicle
+C
+E
+Fig. 4. Development and system integration of scaled autonomous vehicle: [A] mechanical assembly; [B] electronic schematic; [C] MBD workflow depicting
+MIL, SIL, PIL, HIL and VIL testing of vehicle firmware; [D] virtual prototype in AutoDRIVE Simulator; [E] physical prototype in AutoDRIVE Testbed.
+IV. HYBRID PROTOTYPING AND TESTING
+All models or virtual prototypes involve certain degrees of
+abstraction, ranging from model fidelity to simulation settings,
+and as such, cannot be treated as perfect representations
+of their real-world counterparts. Therefore, once the virtual
+prototyping and preliminary testing of the system has been
+accomplished, the next step is to prototype and validate it in
+a hybrid fashion (partly virtual and partly physical), focusing
+more on high-level system integration. This method of hybrid
+prototyping and testing is extremely beneficial since it follows
+a gradual transition from simulation to reality, thereby enabling
+a more faithful system verification framework and providing
+a room for potential design revisions even before complete
+physical prototyping is accomplished.
+The scaled vehicle system was subjected to hybrid testing
+by running processor-in-the-loop (PIL), hardware-in-the-loop
+(HIL) and vehicle-in-the-loop (VIL) tests on the embedded
+firmware for confirming minimum deviation from MIL and
+SIL results, specified by the same tolerance values of 3e-2
+rad for steering angle and 3e-1 rad/s for wheel velocity (refer
+Fig. 4-C). The performance of integrated autonomous vehicle
+system was then validated using hybrid testing in two phases.
+First, we deployed the ADSS on the physical vehicle’s
+on-board computer, which was interfaced with AutoDRIVE
+Simulator to receive live sensor feed from the virtual vehicle,
+process it and generate appropriate control commands, and
+finally relay these commands back to the simulated vehicle.
+Specifically, for the autonomous parking solution (refer Fig.
+5-A), we deployed and tested each of the SLAM, odometry,
+localization, planning and control algorithms for satisfactory
+performance. This was naturally followed by deployment and
+validation of the integrated pipeline for accomplishing reliable
+(within a specified tolerance of 2.5e-2 m) source-to-goal
+navigation (within a goal pose tolerance of 5e-2 m and 8.73e-
+2 rad) in different environments, wherein a subset of cases
+included dynamic obstacles as discussed earlier.
+Next, we collected real-world sensor data using AutoDRIVE
+Testbed and replayed it as a real-time stimulus to the ADSS
+deployed on the physical vehicle’s on-board computer run-
+ning in-the-loop with AutoDRIVE Simulator. This way, we
+increased the “real-world” component of the hybrid test and
+verified the autonomous parking solution for expected perfor-
+mance (within same tolerance values as mentioned earlier).
+Particularly, the real-world data being collected/replayed was
+occupancy-grid map of the environment built by executing
+the SLAM module on the physical vehicle, which inherently
+resulted as a unit test of this module in real-world conditions.
+The simulated vehicle had to then localize against this real-
+world map while driving in the virtual scene and navigate
+autonomously from source to goal (parking) pose, which
+further tested the robustness of the integrated pipeline against
+minor environmental variations and/or vehicle behavior.
+V. PHYSICAL PROTOTYPING AND TESTING
+Once the system confirms satisfactory performance un-
+der hybrid testing conditions, the next and final stage in
+mechatronic system development is physical prototyping and
+testing (refer Fig. 4-E). In order to physically validate the
+modular autonomy application (refer Fig. 5-B), we initially
+carried out unit tests to confirm the performance of each
+
+MPU9250
+Right Indicators
+LeftIndicators
+个个
+ArduinoNano
+JetsonNano
+RESET
+VIN
+Switch
+(Rev3.0)
+GND
+5V
+D11/MOSI
+D12/MISO
+DrivePower
+D13/SCK
+3V3
+DO/F
+D1/TX
+EncoderPower
+V
+D10
+交
+8
+8
+LED GND
+Drive GND
+1111
+Encoder Signal
+Signal
+Taillights
+(LowBeam)
+Headlights (High Beam)
+Reverse Indicators
+Drive
+Headlights 
+个个个
+SteerO
+OSteering Angle (rad)
+0.5
+MIL Test
+SIL Test
+C
+PIL Test
+HIL Test
+VIL Test
+-0.5
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+Time (s)Wheel Velocity (rad/s)
+10
+MIL Test
+SIL Test
+C
+PIL Test
+HIL Test
+VIL Test
+10
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+Time (s)24 s
+16 s
+08 s
+A
+00 s
+09 s
+18 s
+00 s
+12 s
+24 s
+08 s
+16 s
+24 s
+24 s
+08 s
+16 s
+24 s
+16 s
+08 s
+ii
+ii
+iii
+iv
+v
+00 s
+09 s
+18 s
+00 s
+12 s
+24 s
+08 s
+16 s
+24 s
+24 s
+08 s
+16 s
+Start
+Finish
+ii
+ii
+iii
+iv
+v
+C
+B
+Fig. 5. Verification and validation of scaled autonomous vehicle performance: [A] virtual/hybrid and [B] physical validation of (i) integrated system, unit
+testing of (ii) SLAM, (iii) odometry, (iv) localization, (v) planning and control modules in AutoDRIVE Simulator/Testbed; [C] repeatability/reliability analysis
+represented as mean and standard deviation of 5 trials for each deployment type with acceptable trajectory tolerance in green and parking tolerance in purple.
+of the SLAM, odometry, localization, planning and control
+algorithms followed by deployment of the integrated stack
+for autonomous parking application (refer Fig. 5-C). The
+vehicle was confirmed to exhibit a reliable (within a specified
+tolerance of 2.5e-2 m) source-to-goal navigation (within a goal
+pose tolerance of 5e-2 m and 8.73e-2 rad). Again, for testing
+the robustness of ADSS we introduced dynamic obstacles that
+were not existent while environment mapping was performed.
+VI. CONCLUSION
+In this work, we presented an extended V-model fostering
+mechatronics approach of system design, verification and
+validation for autonomous vehicles. Further, we discussed
+the design philosophy of AutoDRIVE ecosystem, which is
+to exploit and promote the mechatronics approach for au-
+tonomous vehicle development across scales and inculcate a
+habit of following it from academic education and research to
+industrial deployments. We also demonstrated the methodical
+adoption of mechatronics approach for designing, developing
+and validating a scaled autonomous vehicle in the context of
+a detailed case study pertaining to autonomous parking using
+a modular probabilistic framework; including both qualitative
+and quantitative remarks. We showed that the design, devel-
+opment as well as verification and validation of the scaled
+autonomous vehicle with regard to the aforementioned case
+study could be successfully accomplished within a stringent
+time-frame of about one month [23]. It is to be noted that al-
+though the exact timeline of any multidisciplinary project may
+vary depending upon factors such as skill set, experience and
+number of individuals involved, lead time in the supply chain,
+etc., the mechatronics approach definitely proves to be efficient
+in terms of minimizing the design-development iterations by
+the virtue of synergistic integration in a concurrent engineering
+thinking framework. This provides a room for the rectification
+of any design issues early in the development cycle, thereby
+increasing the chances of successful verification and validation
+with minimal loss of time and resources.
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+

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+ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS
+FEDERICO GLAUDO AND ANDREA CIPRIETTI
+Abstract. Let A be a multiset with elements in an abelian group. Let FS(A)
+be the multiset containing the 2|A| sums of all subsets of A.
+We study the reconstruction problem “Given FS(A), is it possible to identify
+A?”, and we give a satisfactory answer for all abelian groups. We prove that,
+up to identifying multisets through a natural equivalence relation, the function
+A �→ FS(A) is injective (and thus the reconstruction problem is solvable) if and
+only if every order n of a torsion element of the abelian group satisfies a certain
+number-theoretical property linked to the multiplicative group (Z/nZ)∗.
+The core of the proof relies on a delicate study of the structure of cyclotomic
+units. Moreover, as a tool, we develop an inversion formula for a novel discrete
+Radon transform on finite abelian groups that might be of independent interest.
+1. Introduction
+Let G be an abelian group and let A = {a1, a2, . . . , a|A|} be a finite multiset (i.e., a
+set with repeated elements) with elements in G (see Section 2.1 for a formal definition
+of multiset). Its subset sums multiset FS(A), that is, the multiset containing the
+2|A| sums over all subsets of A (taking into account multiplicities), is defined as
+FS(A) :=
+� �
+i∈I
+ai : I ⊆ {1, 2, . . . , |A|}
+�
+.
+We study the following reconstruction question:
+If one is given FS(A), is it possible to identify A?
+As we will see, this strikingly simple question features a rich structure and its solution
+spans a wide range of mathematics: from the theory of cyclotomic units, to an
+inversion formula for a novel discrete Radon transform. Before going deeper into
+the problem, let us give some background on related results in the literature.
+If, instead of FS(A), one is given the sums over all the
+�|A|
+s
+�
+subsets with fixed
+size equal to s (e.g., if s = 2, the sums over all pairs), the reconstruction problem
+has been studied in the case of a free abelian group G = Zd [SS58; GFS62]. For
+pairs (i.e. s = 2), the reconstruction is possible when the size of A is not a power of
+2 [SS58, Theorem 1 and Theorem 2]. For s-subsets with s > 2, the reconstruction
+is possible if the size of A does not belong to a finite subset of bad sizes [GFS62,
+Section 4]. See the recent survey [Fom19] for a detailed presentation of the history
+of this problem.
+It might seem that if one is only provided with the sums of s-subsets (i.e., subsets
+with size s) then the reconstruction is strictly harder than if one is provided the
+sums of all subsets. This is not true because the information is not ordered and
+1
+arXiv:2301.04635v1  [math.NT]  11 Jan 2023
+
+2
+FEDERICO GLAUDO AND ANDREA CIPRIETTI
+thus, even if we have more information, it is also harder to determine which value
+corresponds to which subset.
+Let us now go back to the reconstruction problem for FS. The first important
+observation is the following one. Given a multiset A and a subset B ⊆ A whose
+sum equals 0 (i.e. �
+b∈B b = 0), if we flip the signs of elements of B then FS does
+not change. So, if A′ := (A \ B) ∪ (−B), then FS(A) = FS(A′) (see Fig. 1 for an
+explanation).
+A
+B
+C
+A′
+−(B \ C)
+C \ B
+Figure 1.
+Proof by picture of A ∼0 A′ =⇒ FS(A) = FS(A′).
+The set C in A (highlighted in gray) and the set (C\B)∪(−(B\C))
+in A′ (highlighted in gray) have the same sum because the sum of
+the elements in B is assumed to be 0. Thus, we have a bijection
+between the subsets of A and A′ which keeps the sum unchanged,
+hence FS(A) = FS(A′).
+Hence, if we only know FS(A), the best we can hope for is to identify the equiv-
+alence class of A with respect to the following equivalence relation.
+Definition 1.1. Given two multisets A, A′ with elements in G, we say that A ∼0 A′
+if and only if A′ can be obtained from A by flipping the signs of the elements of a
+subset of A with null sum, i.e., if there exists B ⊆ A, with �
+b∈B b = 0, such that
+A′ = (A \ B) ∪ (−B).
+We have already observed that if A ∼0 A′ then FS(A) = FS(A′). If the group
+is G = Z, this turns out to be an “if and only if” (see Proposition 6.3), while if
+G = Z/2Z it is not (indeed, in Z/2Z one has FS({0, 1}) = {0, 0, 1, 1} = FS({1, 1})).
+It is natural to consider the class of abelian groups such that the double implication
+holds, i.e. the fibers of FS coincide with the equivalence classes of ∼0.
+Definition 1.2. A group G is FS-regular if, for any two multisets A, A′ with ele-
+ments in G, it holds FS(A) = FS(A′) if and only if A ∼0 A′.
+We have already observed that Z/2Z is not FS-regular; moreover, any group con-
+taining a subgroup that is not FS-regular cannot be FS-regular. The next smallest
+non-FS-regular group is elusive; in fact, it turns out that Z/nZ is FS-regular for
+n = 3, 5, 7, 9, 11, 13, 15. But Z/17Z is not FS-regular, and then Z/nZ is FS-regular
+for n = 19, 21, 23, 25, 27, 29 and not FS-regular for m = 31, 33. These small exam-
+ples suggest that the FS-regularity of G may be related to the behavior of powers
+of two in G (notice that 17, 31, 33 are adjacent to a power of two).
+Our main result is the characterization of FS-regular groups. In order to state
+our result, we need to introduce a subset of the natural numbers.
+Definition 1.3. Let OFS be the set of odd natural numbers n ≥ 1 such that (Z/nZ)∗
+is covered by {±2j : j ≥ 0}; more precisely, for each x ∈ Z relatively prime with n
+there exists j ≥ 0 such that either x − 2j or x + 2j is divisible by n.
+
+ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS
+3
+Remark. The first few elements of OFS are
+OFS = {1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 35, 37, 39, 45, 47, 49, 53, 55, . . . },
+and the first few missing odd numbers are
+(2N + 1) \ OFS = {17, 31, 33, 41, 43, 51, 57, 63, 65, 73, 85, 89, 91, 93, 97, 99, 105, . . . }.
+Let us remark that if n ∈ OFS then also all divisors of n belong to OFS. Moreover,
+one can show that if p, q, r are distinct odd primes, then pqr ̸∈ OFS, and therefore
+if n ∈ OFS then n has at most two distinct prime factors.
+We can now state our main theorem.
+Theorem 1.1 (Characterization of FS-regular groups). An abelian group G is FS-
+regular if and only if ord(g) ∈ OFS for all g ∈ G with finite order.
+As a tool in the proof of Theorem 1.1 (see Section 1.1) we define a novel discrete
+Radon transform for abelian groups and we prove an inversion formula for it. We
+refer to Section 5 for some motivation on the definition and for an in-depth discussion
+of the existing related literature. Since the invertibility of the Radon transform may
+have other applications beyond the scope of this paper, we state it here for the
+interested readers.
+Theorem 1.2 (Invertibility of the discrete Radon transform). Let n, d ≥ 1 be pos-
+itive integers. Given a function f : (Z/nZ)d → C, its discrete Radon transform
+Rf = Rn,df : Hom((Z/nZ)d, Z/nZ) × Z/nZ → C is defined as
+Rf(ψ, c) =
+�
+x: ψ(x)=c
+f(x).
+This discrete Radon transform is invertible and admits an inversion formula (see
+Definition 5.2).
+1.1. Sketch of the proof and structure of the paper. Let us briefly describe
+the strategy that the proof follows, postponing a more detailed presentation to the
+dedicated sections.
+For the negative part of the statement, it is sufficient to show that Z/nZ is
+not FS-regular if n ̸∈ OFS. For this, we construct an explicit counterexample in
+Proposition 4.1.
+Proving that if the orders belong to OFS then the group is FS-regular is more
+complicated and relies on some nontrivial properties of the units of cyclotomic fields
+and on the inversion formula for a novel discrete Radon transform on finite abelian
+groups. The proof is divided into three steps.
+Step 1: Proof for G = Z/nZ. Through the polynomial identity
+�
+s∈FS(A)
+ts ≡
+�
+a∈A
+(1 + ta)
+(mod tn − 1),
+we reduce the FS-regularity of Z/nZ to the study of the kernel of the map
+Zn ∋ x = (x0, x1, . . . , xn−1) �→
+� n−1
+�
+j=0
+(1 + ωj
+d)xj�
+d|n,
+
+4
+FEDERICO GLAUDO AND ANDREA CIPRIETTI
+where ωd ∈ C is a d-th primitive root of unity and the codomain of the map consists
+of tuples indexed by the divisors of n. Thanks to a dimensional argument, identifying
+the kernel of such map is equivalent to identifying its image, which is exactly what
+we do in Lemma 4.4. This is the hardest and most technical proof of the whole
+paper. Up to this point, we have used only that n is odd. The fact that n ∈ OFS
+is needed in the computation of the rank of the image, which relies heavily on the
+theory of cyclotomic units (see Lemma 4.2).
+This step is carried out in Section 4.
+Step 2: Z/nZ is FS-regular
+=⇒
+(Z/nZ)d is FS-regular. Take A, A′ multisets
+with elements in (Z/nZ)d such that FS(A) = FS(A′).
+Given a homomorphism
+ψ : (Z/nZ)d → Z/nZ, by linearity, it holds FS(ψ(A)) = FS(ψ(A′)), and since Z/nZ
+is FS-regular this implies that ψ(A) ∼0 ψ(A′). So, we know that ψ(A) ∼0 ψ(A′)
+for all homomorphisms ψ : (Z/nZ)d → Z/nZ. In order to deduce that A ∼0 A′,
+we introduce a novel discrete Radon transform (see Definition 5.1) and we prove an
+inversion formula (see Definition 5.2 and Theorem 1.2) which may be of indepen-
+dent interest. This allows us to reconstruct a multiset B ∈ M((Z/nZ)d) from its
+projections {φ(B) : φ ∈ Hom((Z/nZ)d, Z/nZ)}.
+This step is performed in Section 5.
+Step 3: G is FS-regular
+=⇒
+G ⊕ Z is FS-regular.
+In this step, we exploit
+crucially that Z is totally ordered. The argument is short and purely combinatorial.
+This is done in Section 6.
+Once these three steps are established, Theorem 1.1 follows naturally, as shown
+in Section 7. Let us remark here that our proof is not constructive, hence it does
+not provide an efficient algorithm to find the ∼0-equivalence class of A if FS(A) is
+known1.
+To make the paper accessible to a broad audience, in Section 2 we recall basic
+facts about multisets, abelian groups, and cyclotomic units.
+Acknowledgements. The authors are thankful to Fabio Ferri for providing valu-
+able suggestions and references about the theory of cyclotomic units, and also to
+Michele D’Adderio and Elia Bru`e for their comments and feedback on an early ver-
+sion of the manuscript. The second author is supported by the National Science
+Foundation under Grant No. DMS-1926686.
+2. Notation and Preliminaries
+2.1. Multisets. A multiset with elements in a set X is an unordered collection
+of elements of X which may contain a certain element more than once [Bli89]. For
+example, {1, 1, 2, 2, 3} is a multiset. Rigorously, a multiset A is encoded by a function
+µA : X → Z≥0 (Z≥0 denotes the set of nonnegative integers) such that µA(x)
+represents the multiplicity of the element x in A. For example, if A = {1, 1, 2, 2, 3}
+then µA(1) = 2, µA(2) = 2, µA(3) = 1.
+1The nonconstructive part of the proof is contained Section 4. In fact, we show that a certain
+map is injective by proving its surjectivity and then applying a standard dimension argument.
+This kind of reasoning does not produce an efficient way to invert the map we have proven to be
+injective.
+
+ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS
+5
+A multiset A is finite if �
+x∈X µA(x) < ∞. The cardinality of a finite multiset
+A ∈ M(X) is given by |A| := �
+x∈X µA(x).
+Given a set X, let us denote with M(X) the family of finite multisets with
+elements in X.
+Let us define the usual set operations on multisets. Notice that all of them are
+the natural generalization of the standard version when one takes into account the
+multiplicity of elements. Fix two multisets A, B ∈ M(X).
+Membership: We say that x ∈ X is an element of A, denoted by x ∈ A, if
+µA(x) ≥ 1.
+Inclusion: We say that A is a subset of B, denoted by A ⊆ B, if µA(x) ≤ µB(x)
+for all x ∈ X.
+Union: The union A∪B ∈ M(X) is defined as µA∪B(x) := µA(x)+µB(x). Hence,
+{1} ∪ {1, 2} = {1, 1, 2}.
+Cartesian product: The Cartesian product A × B ∈ M(X × X) is defined as
+µA×B((x1, x2)) = µA(x1)µB(x2).
+Difference: If A ⊆ B, the difference B \A is defined as µB\A(x) := µB(x)−µA(x).
+Pushforward: Given a function f : X → Y , the pushforward f(A) ∈ M(Y ) of the
+multiset A (denoted also by {f(a) : a ∈ A}) is defined as
+µf(A)(y) =
+�
+x∈f −1(y)
+µA(x).
+Power set: The power set of A (the family of subsets of A), denoted by P(A) ∈
+M(M(X)), is a multiset defined recursively as follows. For the empty mul-
+tiset, we have P(∅) := {∅}; otherwise let a ∈ A be an element of A and
+define
+P(A) := P(A \ {a}) ∪
+�
+A′ ∪ {a} : A′ ∈ P(A \ {a})
+�
+.
+Notice that |P(A)| = 2|A|. Whenever we iterate over the subsets of A (e.g.,
+{f(A′) : A′ ⊆ A} or �
+A′⊆A f(A′)), the iteration has to be understood over
+P(A) (hence the subsets are counted with multiplicity).
+Taking the complement is an involution of the power set, i.e., P(A) =
+{A \ A′ : A′ ∈ P(A)}, and we have the following identity for the power set
+of a union
+P(A ∪ B) = {A′ ∪ B′ : (A′, B′) ∈ P(A) × P(B)}.
+Sum (and product): If the set X is an additive abelian group, we can define the
+sum � A ∈ X of the elements of A as
+�
+A :=
+�
+x∈X
+µA(x)x.
+Analogously, if X is a multiplicative abelian group, one can define the prod-
+uct � A of the elements of A.
+2.2. Abelian Groups. Let us recall some basic facts about abelian groups that we
+will use extensively later on.
+
+6
+FEDERICO GLAUDO AND ANDREA CIPRIETTI
+Any finitely generated abelian group is isomorphic to a finite product of cyclic
+groups [Lan02, Chapter I, Section 8]. We denote with Z/nZ the cyclic group with
+n elements.
+Given some elements g1, g2, . . . , gk ∈ G of an abelian group, we denote with
+⟨g1, g2, . . . , gk⟩ the subgroup generated by such elements. Given an element g ∈ G,
+its order (which may be equal to ∞) is denoted by ord(g).
+For an abelian group G, its rank rk(G) is the cardinality of a maximal set of
+Z-independent2 elements of G. Let us list some useful properties of the rank (see
+[Lan02, Chapter I and XVI]).
+• Any finitely generated abelian group G is isomorphic to Zrk(G) ⊕ G′ where
+G′ is a finite abelian group.
+• Given two abelian groups G, H, it holds rk(G ⊕ H) = rk(G) + rk(H).
+• For a homomorphism φ : G → H of abelian groups, it holds rk(G) =
+rk(ker φ) + rk(Im φ).
+• An abelian group has null rank if and only if all elements have finite order.
+• Let G1, G2, G3 be three abelian groups and φ1 : G1 → G2, φ2 : G2 →
+G3 be two homomorphisms with full rank, i.e. rk(Im φ1) = rk(G2) and
+rk(Im φ2) = rk(G3). Then φ2 ◦ φ1 : G1 → G3 has full rank as well, i.e.
+rk(Im φ2 ◦ φ1) = rk(G3)
+• Given an abelian group G, let us denote with G ⊗ Q its tensor product (as
+a Z-module) with Q (see [Lan02, Chapter XVI]). The dimension of G ⊗ Q
+as vector space over Q coincides with rk(G).
+• For a homomorphism φ : G → H of abelian groups, let φ⊗Q : G⊗Q → H⊗Q
+be its tensorization with Q. It holds rk(Im ��) = dimQ(Im (φ ⊗ Q)).
+2.3. Units of cyclotomic fields. Given n ≥ 1, let ωn := exp(2πi/n) be the prim-
+itive n-th root of unity with minimum positive argument.
+The algebraic number field Q(ωn) is called cyclotomic field. It is well-known that
+the ring of integers of Q(ωn) coincides with Z[ωn]. Our main focus is the group of
+units of Q(ωn), that consists of the invertible elements of its ring of integers.
+For 0 < r < n and s ≥ 1 coprime with n, the element ξ := 1−ωrs
+n
+1−ωrn is a unit of
+Q(ωn). Indeed ξ = 1+ωr
+n +· · ·+ω(s−1)r
+n
+∈ Z[ωn] and, if u ∈ N is such that n divides
+us − 1, then
+ξ−1 = 1 − ωrus
+n
+1 − ωrs
+n
+= 1 + ωrs
+n + · · · + ω(u−1)rs
+n
+∈ Z[ωn].
+It turns out that these units are sufficient to generate a subgroup of finite index
+of the units of Q(ωn). The following statement follows from [Was97, Theorem 8.3
+and Theorem 4.12].
+Theorem 2.1. For any odd n ≥ 3, the multiplicative group Cn ⊆ C generated by
+�1 − ωrs
+n
+1 − ωrn
+: 0 < r < n, s ≥ 1 coprime with n
+�
+is a subgroup of finite index of the units of Q(ωn).
+2Some elements g1, g2, . . . , gk ∈ G are Z-independent if, whenever �
+i aigi = 0 for some
+a1, a2, . . . , ak ∈ Z, it holds a1 = a2 = · · · = ak = 0.
+
+ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS
+7
+Thus, applying Dirichlet’s unit Theorem (see [Mar77, Theorem 38]), we are able
+to compute the rank of Cn (since it coincides with the rank of the group of units of
+Q(ωn)).
+Corollary 2.2. For any odd n ≥ 3, we have rk(Cn) = ϕ(n)
+2
+− 1, where ϕ is Euler’s
+totient function (and Cn is defined in Theorem 2.1).
+The units of Q(ωn) satisfy a family of nontrivial relations known as distribution
+relations (see [Was97, p. 151]). We recall here the relations in the form we will need.
+Notice that 1+ωj
+n is a unit for 1 ≤ j < n because of the identity 1+ωj
+n = 1−ω2j
+n
+1−ωj
+n ∈ Cn.
+Proposition 2.3 (Distribution relations). Let n ≥ 1 be an odd integer and let p be
+one of its prime divisors3. For any 0 ≤ j < n
+p , the identity
+p−1
+�
+k=0
+(1 + ωj+kn/p
+n
+) = 1 + ωjp
+n
+holds.
+Proof. The numbers {1 + ωj+kn/p
+n
+}0≤k<p are the roots of the monic polynomial
+(t − 1)p − ωjp
+n
+∈ C[t].
+Therefore, their product equals the constant term of the
+polynomial multiplied by (−1)p, which is ((−1)p − ωjp
+n )(−1)p = 1 + ωjp
+n .
+□
+3. Definitions and basic facts
+In this section we give some fundamental definitions (some of them are already
+present in the introduction, we repeat them here for the ease of the reader) and we
+prove one basic result which will be useful multiple times in the paper.
+Definition 3.1. Let G be an additive abelian group and take A ∈ M(G). The
+subset sums multiset of A is (we adopt the notation of [TV06])
+FS(A) :=
+� �
+B : B ∈ P(A)
+�
+,
+that is, the multiset whose elements are the sums of the subsets of A.
+When studying the injectivity of FS, one soon notices that if we take a multiset
+A ∈ M(G) and we flip the sign of a subset of its elements with zero sum, obtaining
+another multiset A′ ∈ M(G), then the subset sums do not change, i.e. FS(A) =
+FS(A′). Therefore, the following definition and the results of Lemma 3.1 should
+appear natural.
+Definition 3.2. Given an additive abelian group G, we define the equivalence re-
+lations ∼ and ∼0 over M(G) as follows:
+• Given A, A′ ∈ M(G), A ∼ A′ if A′ is obtained from A by changing the sign
+of the elements of a subset of A. More formally, A ∼ A′ if and only if there
+exists B ⊆ A such that A′ = (A \ B) ∪ (−B).
+• Given A, A′ ∈ M(G), A ∼0 A′ if A′ is obtained from A by changing the sign
+of the elements of a zero-sum subset of A. More formally, A ∼0 A′ if and only
+if there exists B ⊆ A with null sum � B = 0G such that A′ = (A\B)∪(−B).
+3The identity holds, with the same proof, also without the assumption that p is prime.
+
+8
+FEDERICO GLAUDO AND ANDREA CIPRIETTI
+Notice that the relations ∼ and ∼0 are reflective and transitive.
+Lemma 3.1. Given two multisets A, A′ ∈ M(G) with elements in an abelian group
+G, we have the following statements concerning the relationship between ∼0, ∼ and
+FS.
+(1) If A ∼0 A′ then FS(A) = FS(A′).
+(2) If A ∼ A′, then there is g ∈ G such that FS(A) = FS(A′) + g.
+(3) If FS(A) = FS(A′) and A ∼ A′, then A ∼0 A′.
+(4) If FS(A) = FS(A′) + g for some g ∈ G, then there exists M(G) ∋ A′′ ∼ A′
+such that FS(A) = FS(A′′).
+Proof. The following paragraph describes a very simple bijection in a very com-
+plicated way, this is due to the formalism necessary to handle the multiplicities of
+elements in multisets. We suggest the reader to refer to the picture Fig. 1, which
+shall be much clearer than the proof itself.
+If A ∼ A′, then, by definition, there is B ⊆ A such that A′ = (A \ B) ∪ (−B). So,
+we have
+P(A′) =
+�
+C ∪ (−D) : (C, D) ∈ P(A \ B) × P(B)
+�
+=
+�
+C ∪ (−(B \ D)) : (C, D) ∈ P(A \ B) × P(B)
+�
+and therefore
+(3.1)
+FS(A′) =
+� �
+C +
+�
+D −
+�
+B : (C, D) ∈ P(A \ B) × P(B)
+�
+=
+� �
+C +
+�
+D : (C, D) ∈ P(A \ B) × P(B)
+�
+−
+�
+B
+= FS(A) −
+�
+B.
+This proves (2).
+Notice that if A ∼ A′, then Eq. (3.1) implies that FS(A) = FS(A′) is equivalent
+to � B = 0, that is equivalent to A ∼0 A′. Hence also (1) and (3) are proven.
+In order to prove (4), notice that 0G ∈ FS(A) and thus −g ∈ FS(A′); so there is
+B ⊆ A′ such that � B = −g. Let A′′ ∼ A be the multiset A′′ := (A′ \ B) ∪ (−B).
+The formula Eq. (3.1) (with A, A′ → A′, A′′) yields FS(A′′) = FS(A′) − � B =
+FS(A′) + g = FS(A) as desired.
+□
+Let us recall the definition of FS-regular groups already given in the introduction.
+Definition 3.3. An abelian group G is FS-regular if, for any A, A′ ∈ M(G), it
+holds FS(A) = FS(A′) if and only if A ∼0 A′.
+Notice that if G is FS-regular, then also its subgroups are FS-regular. Moreover,
+it is always true that A ∼0 A′ implies FS(A) = FS(A′) (see Lemma 3.1-(1)) and
+therefore the content of the FS-regularity is the opposite implication, which does
+not hold for all groups.
+
+ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS
+9
+4. FS-regularity of cyclic groups
+In this section we characterize the FS-regular finite cyclic groups; the two main
+results are Proposition 4.1 and Proposition 4.6.
+To show that if n ̸∈ OFS then Z/nZ is not FS-regular we produce an explicit
+counterexample.
+Proposition 4.1. For any n ̸∈ OFS, the group Z/nZ is not FS-regular.
+Proof. If n is even, then Z/2Z is a subgroup of Z/nZ and thus it is sufficient to
+show that Z/2Z is not FS-regular. As a counterexample to FS-regularity in Z/2Z,
+it is enough to notice that
+FS({0, 1}) = {0, 0, 1, 1} = FS({1, 1}),
+while {0, 1} ̸∼0 {1, 1} as multisets with values in Z/2Z.
+Let us now consider the case of n odd. Since n ̸∈ OFS, there exists k ∈ (Z/nZ)∗ \
+{±2j mod n}j∈N.
+Moreover, let d := ϕ(n) be so that n | 2d − 1.
+Consider the
+multisets A, A′ ∈ M(Z/nZ) defined as
+A := {20, 21, . . . , 2d−1}
+and
+A′ := k · A = {20k, 21k, . . . , 2d−1k}.
+The choice of k implies that A ∩ A′ = (−A) ∩ A′ = ∅ and, in particular, A ̸∼0 A′.
+We have that4
+FS(A) = {0, 1, 2, . . . , 2d − 1} = {0} ∪
+2d−1
+n�
+i=1
+{0, 1, . . . , n − 1}
+= {0} ∪
+2d−1
+n�
+i=1
+k · {0, 1, . . . , n − 1} = {k · 0, k · 1, k · 2, . . . , k · (2d − 1)}
+= FS(A′).
+□
+The proof that Z/nZ is FS-regular when n ∈ OFS is more involved. The rest of
+this section is devoted to establish this result by reducing it to a statement about
+the units of the cyclotomic field Q(ωn).
+Before delving into the proof, let us present the relation between the problem
+at hand and the units of the cyclotomic field Q(ωn), to clarify the importance of
+Definitions 4.1 and 4.2.
+Given two multisets A, A′ ∈ M(Z/nZ), the condition FS(A) = FS(A′) is equiva-
+lent to the polynomial identity
+�
+a∈A
+(1 + ta) =
+�
+a′∈A′
+(1 + ta′)
+(mod tn − 1),
+which is equivalent to
+n−1
+�
+j=0
+(1 + ωj
+d)µA(j)−µA′(j) = 1,
+4The unions are taken over 2d−1
+n
+copies of the same multiset and shall be interpreted in the
+multiset sense, so that the result is a multiset where each element appears 2d−1
+n
+times.
+
+10
+FEDERICO GLAUDO AND ANDREA CIPRIETTI
+for all divisors d | n (because a polynomial is divisible by tn − 1 if and only if it has
+ωd as root for all divisors d | n). Therefore, we are interested in the kernel of the
+map which takes a vector x ∈ Zn and produces the tuple, indexed by the divisors
+d | n,
+(4.2)
+� n−1
+�
+j=0
+(1 + ωj
+d)xj�
+d|n.
+Since this map is a homomorphism between abelian groups, studying its kernel is
+tightly linked to the study of its image.
+In fact, the crux of this section is the
+determination of the image of such map (see Lemma 4.4).
+The multiplicative group generated by 1 + ω0
+d, 1 + ω1
+d, . . . , 1 + ωn−1
+d
+is introduced
+in Definition 4.1, while its rank is computed in Lemma 4.2 (the assumption n ∈ OFS
+is necessary to compute the rank). Then, in Definition 4.2 we introduce the notation
+that allows studying the map mentioned in Eq. (4.2) and we go on to prove its moral
+surjectivity (i.e., its image has full rank) in Lemma 4.4 (notice that we do not need
+n ∈ OFS, n being odd suffices). Finally, in Proposition 4.6, we join all the pieces to
+obtain the desired result.
+Given an odd positive integer n, recall that, for 1 ≤ j < n, 1 + ωj
+n is a unit of
+Q(ωn) (see Section 2.3).
+Definition 4.1. Given an odd positive integer n ≥ 1, let Kn be the multiplicative
+subgroup of C generated by {1 + ωj
+n : 0 ≤ j < n}. Note that we include 1 + ω0
+n = 2
+among the generators.
+Lemma 4.2. If n ≥ 3 and n ∈ OFS, it holds rk(Kn) =
+ϕ(n)
+2 , where ϕ denotes
+Euler’s totient function. Moreover, it holds rk(K1) = 1.
+Proof. For n = 1, Kn = ⟨2⟩ ∼= Z, which has rank 1.
+Let us now consider Kn for n ≥ 3 and n ∈ OFS. Notice that all generators of Kn
+apart from the element 2 are units of Q(ωn), while the inverse of 2 is not an algebraic
+integer. Therefore, one obtains Kn ∼= ⟨2⟩ ⊕ ˜Kn, where ˜Kn := ⟨1 + ωj
+n : 1 ≤ j < n⟩.
+It remains to compute the rank of ˜Kn. We have already observed that ˜Kn is a
+subgroup of Cn (defined in the statement of Theorem 2.1). Using that n ∈ OFS we
+are going to prove that Cn is a subgroup of ˜Kn ∪ (− ˜Kn).5
+To show that Cn ⊆ ˜Kn ∪ (− ˜Kn), it is sufficient to show that all generators of Cn
+belong to ˜Kn or to − ˜Kn. Let us fix s ≥ 1 coprime with n. Since n ∈ OFS, there
+exists j ≥ 0 such that ω2j
+n = ωs
+n or ω2j
+n = ω−s
+n .
+If ω2j
+n = ωs
+n, then, for any 0 < r < n, we have
+1 − ωrs
+n
+1 − ωrn
+= 1 − ω2jr
+n
+1 − ωrn
+=
+j−1
+�
+k=0
+(1 + ω2kr
+n
+) ∈ ˜Kn.
+5One may check that −1 ̸∈ ˜
+K7, while −1 ∈ C7. So it is not true in general that Cn and ˜
+Kn
+coincide. On the other hand, for some values of n (e.g., n = 3, 5, 9) one has −1 ∈ ˜
+Kn.
+
+ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS
+11
+To handle the case ω2j
+n = ω−s
+n , let us observe that ωn =
+1+ωn
+1+ω−1
+n
+∈ ˜Kn. Therefore, for
+any 0 < r < n, we have
+1 − ωrs
+n
+1 − ωrn
+= −ωrs
+n
+1 − ω2jr
+n
+1 − ωrn
+∈ − ˜Kn.
+We have shown ˜Kn ⊆ Cn ⊆ ˜Kn ∪ (− ˜Kn) and thus rk( ˜Kn) = rk(Cn) = ϕ(n)/2 − 1
+(recall Corollary 2.2). Hence we conclude rk(Kn) = rk(⟨2⟩ ⊕ ˜Kn) = 1 + rk( ˜Kn) =
+ϕ(n)/2.
+□
+Definition 4.2. Given a positive integer n ≥ 1, for 0 ≤ j < n, let en
+j be the j-th
+canonical generator of Zn = �n−1
+j=0 Z. The index j of en
+j shall be interpreted modulo
+n, i.e., en
+j := en
+j mod n, when j ≥ n.
+For a positive divisor d of n, let πn
+d : Zn → Zd be the unique homomorphism such
+that πn
+d (en
+j ) := ed
+j (= ed
+j mod d) for all 0 < j < n.
+Let Fn : Zn → Kn be the unique group homomorphism such that Fn(en
+j ) = 1+ωj
+n
+for each 0 ≤ j < n; or equivalently
+Fn(x) = Fn(x0, . . . , xn−1) :=
+n−1
+�
+j=0
+(1 + ωj
+n)xj.
+Lemma 4.3. Let F be a field and let V be a F-vector space. Given a subset S ⊆ V ,
+we denote with ⟨S⟩F the subspace generated by the elements of S.
+Given k vectors v1, v2, . . . , vk ∈ V , for any λ ∈ F which is not a root of unity
+(i.e., λq ̸= 1 for all positive integers q ≥ 1) and for any function σ : {1, 2, . . . , k} →
+{1, 2, . . . , k}, we have
+⟨vj − λvσ(j) : 1 ≤ j ≤ k⟩F = ⟨vj : 1 ≤ j ≤ k⟩F.
+Proof. We prove the statement by induction on k. For k = 0 there is nothing to
+prove.
+If σ is not surjective then we can assume without loss of generality that σ(j) ̸= k
+for all 1 ≤ j ≤ k. Hence, we can apply the inductive hypothesis and obtain
+⟨vj − λvσ(j) : 1 ≤ j ≤ k − 1⟩F = ⟨vj : 1 ≤ j ≤ k − 1⟩F.
+Since vσ(k) ∈ ⟨vj : 1 ≤ j ≤ k − 1⟩F, we obtain
+⟨vj − λvσ(j) : 1 ≤ j ≤ k⟩F = ⟨v1, v2, . . . , vk−1, vk − λvσ(n)⟩F = ⟨vj : 1 ≤ j ≤ k⟩F,
+which is what we sought.
+If σ is surjective, then it must be a permutation. In particular there exists q ≥ 1
+such that σq(j) = j for all 1 ≤ j ≤ k. Thus, for any 1 ≤ ℓ ≤ k, we have the
+telescopic sum
+q−1
+�
+i=0
+λi�
+vσi(ℓ) − λvσ(σi(ℓ))
+�
+= (1 − λq)vℓ.
+Since 1 − λq ̸= 0 by assumption, we deduce that vℓ ∈ ⟨vj − λvσ(j) : 1 ≤ j ≤ k⟩F for
+all 1 ≤ ℓ ≤ k, which implies the statement.
+□
+Lemma 4.4. For any odd positive integer n, the image of the map (Fd ◦ πn
+d )d|n :
+Zn → ⊕d|nKd is a finite-index subgroup of ⊕d|nKd.
+
+12
+FEDERICO GLAUDO AND ANDREA CIPRIETTI
+Proof. Let us fix a divisor d of n. We are going to identify some elements of the
+kernel of Fd, which is equivalent to producing nontrivial relations in Kd. For any
+divisor p of d and any 0 ≤ j < d/p, let
+vd
+p,j := ed
+jp −
+p−1
+�
+k=0
+ed
+j+kd/p.
+Thanks to Proposition 2.3, we know that Fd(vd
+p,j) = 1 for all prime divisors p of d
+and all 0 < j < d/p. Therefore, we have identified the subspace
+Zd ⊇ Dd := ⟨vd
+p,j⟩p|d prime, 0≤j<d/p
+of the kernel of Fd. Let us identify with [ · ]Dd : Zd → Zd/Dd the projection to the
+quotient.
+We claim that Ψn := ([πn
+d ]Dd)d|n : Zn → �
+d|n Zd/Dd has full rank (i.e., the rank
+of its image coincides with the rank of its codomain). This claim implies the desired
+result since Fd is surjective for all d.
+In order to show that Ψn has full rank we consider its tensorization with Q and
+show that it is surjective as a linear map between Q-vector spaces. With a mild
+abuse of notation, we keep denoting with (ed
+j)0≤j<d the canonical basis of Qd and
+we keep denoting with Dd the Q-subspace generated by {vd
+p,j}p|d prime, 0≤j≤d/p.
+Thanks to the basic properties of the tensor product, we have (Zd/Dd) ⊗ Q =
+Qd/Dd and the tensorization Ψn ⊗ Q : Qn → �
+d|n Qd/Dd satisfies (Ψn ⊗ Q)(en
+j ) =
+([ed
+j]Dd)d|n ∈ �
+d|n Qd/Dd for all 0 ≤ j < n.
+The following commutative diagram shall clarify all the steps of the proof up to
+now.
+Qn
+�
+d|n Qd/Dd
+Zn
+�
+d|n Zd
+�
+d|n Zd/Dd
+�
+d|n Kd
+(Ψn ⊗ Q)(en
+j ) = ([ed
+j ]Dd)d|n
+(πn
+d )d|n
+Ψn
+· ⊗Q
+([ · ]Dd)d|n
+(Fd)d|n
+· ⊗Q
+.
+To prove the surjectivity of the linear map Ψn ⊗ Q : Qn → �
+d|n Qd/Dd we show
+explicitly that the canonical generators of the codomain belong to the image of the
+map.
+Given a subset S ⊆ {d ≥ 1 : d | n} and an index 0 ≤ j < n, let uS,j = (ud
+S,j)d|n ∈
+�
+d|n Qd/Dd be the element defined by
+Qd/Dd ∋ ud
+S,j :=
+�
+0
+if d ̸∈ S,
+[ed
+j]Dd
+if d ∈ S.
+The index j of uS,j should be interpreted modulo n (e.g. uS,n = uS,0).
+
+ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS
+13
+Notice that (u{d},j)d|n, 0≤j<n is a set of generators of �
+d|n Qd/Dd. Moreover, it
+holds (Ψn ⊗ Q)(en
+j ) = u{d≥1: d|n}, j.
+We say that a set S is solvable if uS,j belongs to the image of Ψn ⊗ Q for all
+0 ≤ j < n. Thanks to the previous observations, we know that {d ≥ 1 : d | n}
+is solvable and that the surjectivity of Ψn ⊗ Q is equivalent to the fact that all
+singletons {d} are solvable. Notice that if S ⊆ T ⊆ {d ≥ 1 : d | n} is solvable, then
+also T \ S is solvable. Indeed, if (Ψn ⊗ Q)(x) = uS,j and (Ψn ⊗ Q)(y) = uT,j, then
+(Ψn ⊗ Q)(y − x) = uT \S, j. Our main tool to show the solvability of a set is the
+following sub-lemma.
+Lemma 4.5. Let S ⊆ {d ≥ 1 : d | n} be a solvable subset and let p | n be a prime
+number. Let us define6 υp(S) := maxd∈S υp(d) as the maximal p-adic valuation of
+an element of S. Then, the subset {d ∈ S : υp(d) = υp(S)} is also solvable.
+Proof. Let S′ := {d ∈ S : υp(d) = υp(S)}. Let m be the minimum common multiple
+of the elements of S. Notice that υp(m) = υp(S).
+If υp(S) = 0, then S′ = S and the statement is obvious. From now on we assume
+that υp(S) > 0.
+We claim that, for any 0 ≤ j < n, it holds
+(4.3)
+uS,j − 1
+p
+p−1
+�
+k=0
+uS,j+km/p = uS′,j − 1
+puS′,jp.
+We prove Eq. (4.3) by looking at the projections of both sides onto Qd/Dd and
+considering various cases depending on the divisor d.
+• If d ̸∈ S, then d ̸∈ S′ (since S′ ⊆ S) and thus we have
+ud
+S,j − 1
+p
+p−1
+�
+k=0
+ud
+S,j+km/p = 0 = ud
+S′,j − 1
+pud
+S′,jp.
+• If d ∈ S and υp(d) < υp(S), then d |
+m
+p
+and therefore ud
+S,j+km/p =
+[ed
+j+km/p]Dd = [ed
+j]Dd = ud
+S,j.
+Since υp(d) < υp(S) implies that d ̸∈ S′,
+we deduce
+ud
+S,j − 1
+p
+p−1
+�
+k=0
+ud
+S,j+km/p = ud
+S,j − 1
+p
+p−1
+�
+k=0
+ud
+S,j = 0 = ud
+S′,j − 1
+pud
+S′,jp.
+• If d ∈ S and υp(d) = υp(S), then it holds
+(4.4)
+�
+0, m
+p mod d, 2m
+p mod d, . . . , (p−1)m
+p mod d
+�
+=
+�
+0, d
+p, 2d
+p, . . . , (p−1)d
+p
+�
+.
+To prove the latter identity, notice that for any 0 ≤ k < p, we have
+�
+k m
+p mod d
+�
+=
+�
+k m
+d mod p
+�d
+p
+and therefore the identity between sets follows from the fact that m/d is not
+divisible by p.
+6Here υp(x) denotes the p-adic valuation of a nonzero integer x, i.e. the maximum exponent
+h ≥ 0 such that ph divides x.
+
+14
+FEDERICO GLAUDO AND ANDREA CIPRIETTI
+Exploiting Eq. (4.4) and recalling that vd
+p,j ∈ Dd, we obtain
+ud
+S,j − 1
+p
+p−1
+�
+k=0
+ud
+S, j+km/p =
+�
+ed
+j − 1
+p
+p−1
+�
+k=0
+ed
+j+km/p
+�
+Dd
+=
+�
+ed
+j − 1
+p
+p−1
+�
+k=0
+ed
+j+kd/p
+�
+Dd
+=
+�
+ed
+j − 1
+p(ed
+jp − vd
+p,j)
+�
+Dd
+=
+�
+ed
+j − 1
+ped
+jp
+�
+Dd
+= ud
+S′,j − 1
+pud
+S′,jp,
+where in the last steps we used that d ∈ S′ (which is equivalent to the
+assumptions d ∈ S and υp(d) = υp(S)).
+Since we have covered all possible cases, Eq. (4.3) is proven.
+The set S is solvable, therefore the left-hand side of Eq. (4.3) belongs to the image
+of Ψn ⊗ Q, and thus also uS′,j − 1
+puS′,jp belongs to Im (Ψn ⊗ Q) for all 0 ≤ j < n.
+Lemma 4.3, applied with vj := uS′,j, λ := 1/p, and σ(j) := (jp mod n), guarantees
+that also uS′,j belongs to the image of Ψn ⊗ Q for all 0 ≤ j < n, which proves that
+S′ is solvable as desired.
+□
+As a simple consequence of Lemma 4.5, we claim that if S is solvable, then, for
+any prime divisor p of n and for any 0 ≤ h ≤ υp(n), we have that {s ∈ S : υp(s) = h}
+is also solvable. Let us prove it by induction on h, starting from h = υp(n) and going
+backward to h = 0.
+If {s ∈ S : υp(s) = υp(n)} is empty, then it is solvable; otherwise we can apply
+Lemma 4.5 and obtain again that it is solvable. Now, we assume that {s ∈ S :
+υp(s) = h′} is solvable for h′ > h. Then, since the difference of solvable sets is
+solvable, we deduce that ˜S := {s ∈ S : υp(s) ≤ h} is solvable. If {s ∈ S : υp(s) = h}
+is empty, then it is solvable; otherwise we can apply Lemma 4.5 on the set ˜S and
+obtain again that {s ∈ S : υp(s) = h} is solvable as desired.
+We can now conclude by showing that singletons {d} are solvable for each d | n.
+This follows directly from the fact that {d ≥ 1 : d | n} is solvable and that if S
+is solvable then {s ∈ S : υp(s) = h} is solvable for all prime divisors p | n and all
+h ≥ 0.
+□
+Proposition 4.6. For any n ∈ OFS, the group Z/nZ is FS-regular.
+Proof. Let A, A′ ∈ M(Z/nZ) be two multisets such that FS(A) = FS(A′); we shall
+prove that A ∼0 A′.
+By definition of the map FS, it holds the polynomial identity in Z[t]/(tn − 1)
+n−1
+�
+j=0
+µFS(A)(j)tj ≡
+�
+s∈FS(A)
+ts ≡
+�
+a∈A
+(1 + ta) ≡
+n−1
+�
+j=0
+(1 + tj)µA(j)
+(mod tn − 1),
+Thus the condition FS(A) = FS(A′) is equivalent to
+n−1
+�
+j=0
+(1 + tj)µA(j) ≡
+n−1
+�
+j=0
+(1 + tj)µA′(j)
+(mod tn − 1).
+
+ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS
+15
+For any divisor d | n, ωd is a root of tn − 1 and therefore the latter identity implies
+n−1
+�
+j=0
+(1 + ωj
+d)µA(j) =
+n−1
+�
+j=0
+(1 + ωj
+d)µA′(j)
+which, recalling Definition 4.2, is equivalent to
+Fd
+�
+πn
+d
+�
+(µA(j) − µA′(j))0≤j<n
+��
+= 1.
+We have just shown that the vector (µA(j) − µA′(j))0≤j<n ∈ Zn belongs to the
+kernel of the map (Fd ◦ πn
+d )d|n : Zn → ⊕d|nKd. Let us now switch our attention to
+the study of such kernel.
+M(Z/nZ)
+Zn
+⊕d|nKd
+M(Z/nZ)
+Zn
+Z[t]
+(tn−1)
+⊕d|nZ[ωd]
+A�→(µA(j))0≤j<n
+FS
+x�→�n−1
+j=0 (1+tj)xj
+(Fd◦πn,d)d|n
+∼
+=
+x�→�n−1
+j=0 xjtj
+∼
+=
+[q]�→(q(ωd))d|n
+.
+Figure 2. A commutative diagram depicting the relation, ex-
+plained at the beginning of the proof of Proposition 4.6, between
+the map FS and the map (Fd ◦ πn
+d )d|n.
+Due to basic properties of the rank (see Section 2.2), we have
+rk
+�
+ker((Fd ◦ πn
+d )d|n)
+�
+= n − rk
+�
+Im ((Fd ◦ πn
+d )d|n)
+�
+= n − rk
+� �
+d|n
+Kd
+�
+= n −
+�
+d|n
+rk(Kd) = n − 1 −
+�
+1<d|n
+ϕ(d)
+2
+= n − 1
+2
+,
+where we have used Lemma 4.4 and Lemma 4.2.
+Let us now exhibit a subgroup Ln of Zn which is included in the kernel of (Fd ◦
+πn
+d )d|n (in hindsight, it coincides with such kernel). Let Ln ⊆ Zn be the subgroup7
+Ln :=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+x ∈ Zn :
+x0 = 0,
+xj + xn−j = 0 for all 1 ≤ j ≤ n − 1
+2
+,
+n−1
+2
+�
+j=1
+j · xj is divisible by n
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+7Notice that Ln is the subgroup generated by the vectors (µB(j) − µB′(j))0≤j<n for any two
+multisets B ∼0 B′.
+
+16
+FEDERICO GLAUDO AND ANDREA CIPRIETTI
+For any d | n and x ∈ Ln, we have
+Fd(πn
+d (x)) =
+n−1
+�
+j=0
+(1 + ωj
+d)xj =
+(n−1)/2
+�
+j=1
+(1 + ωj
+d)xj(1 + ω−j
+d )−xj
+(n−1)/2
+�
+j=1
+ωj·xj
+d
+= ω
+�(n−1)/2
+j=1
+j·xj
+d
+= 1,
+and this proves that Ln is a subgroup of the kernel of (Fd ◦ πn
+d )d|n.
+Notice that rk(Ln) = n−1
+2
+= rk
+�
+ker((Fd◦πn
+d )d|n)
+�
+, so for any x ∈ ker((Fd◦πn
+d )d|n)
+there exists α ≥ 1 such that αx ∈ Ln and therefore x itself must satisfy the first two
+conditions in the definition of Ln, that is
+ker((Fd ◦ πn
+d )d|n)
+�
+⊆
+�
+x ∈ Zn : x0 = 0, xj + xn−j = 0 for all 1 ≤ j ≤ n − 1
+2
+�
+.
+The latter inclusion, together with the vector (µA(j) − µA′(j))0≤j<n ∈ Zn be-
+longing to the kernel we are studying, implies
+µA(0) = µA′(0) and µA(j) + µA(n − j) = µA′(j) + µA′(n − j) for all 1 ≤ j ≤ n,
+that is equivalent to A ∼ A′. Finally, we conclude A ∼0 A′ taking advantage of
+Lemma 3.1-(3).
+□
+5. Radon transform for finite abelian groups
+In this section we will introduce a Radon transform for finite abelian groups
+and we will show an inversion formula for it.
+Then we will apply this tool to
+upgrade Proposition 4.6 to the same statement with Z/nZ replaced by (Z/nZ)d for
+an arbitrary d ≥ 1.
+Let us introduce a new discrete Radon transform.
+Definition 5.1. Let n, d ≥ 1 be positive integers. Given a function f : (Z/nZ)d →
+C, its Radon transform is the function Rf = Rn,df : Hom((Z/nZ)d, Z/nZ) ×
+Z/nZ → C given by
+Rf(ψ, c) :=
+�
+x∈(Z/nZ)d
+ψ(x)=c
+f(x),
+for all homomorphisms ψ : (Z/nZ)d → Z/nZ and all c ∈ Z/nZ.
+We named this transformation Radon transform in analogy with the continuous
+Radon transform on Rn [Hel99] which, given a function f : Rd → R, produces
+another function Rf which takes an (n−1)-affine hyperplane and returns the integral
+of f over such hyperplane. Notice that affine hyperplanes are exactly the fibers of
+linear functionals Rn → R and thus the continuous Radon transform on Rd coincides
+(up to adapting the definition to a non-discrete setting) with our definition if Z/nZ
+is replaced by R.
+Let us remark that one may restrict the discrete Radon transform to the surjective
+homomorphisms without losing information. In fact, any fiber of a non-surjective
+homomorphism (Z/nZ)d → Z/nZ can be written as the disjoint union of some
+fibers of a surjective homomorphism. If we restrict to surjective homomorphisms,
+then the fibers have size equal to nd−1 which is, essentially, the size of a “hyperplane”
+in (Z/nZ)d.
+
+ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS
+17
+One may wonder if Definition 5.1 would work even if Z/nZ was replaced ev-
+erywhere by an arbitrary finite abelian group G. Although everything would still
+hold (including the inversion formula we will see later on), it is not appropriate
+to give such a definition.
+Indeed, any finite abelian group G is a subgroup of
+(Z/nZ)k for n, k ≥ 1 (where n is the largest order of an element in G).
+Hence
+the Radon transform on Gd can be defined alternatively as the restriction of Rn,kd
+to Hom(Gd, Z/nZ)×Z/nZ; that is, by understanding Gd as a subgroup of (Z/nZ)kd
+and using the Radon transform of the latter (which uses homomorphisms with
+codomain equal to Z/nZ instead of G; notice that Z/nZ is a subgroup of G).
+In the literature, one can find many definitions of discrete Radon transform:
+• The definition given in [DG85] (and investigated in [FG87; Fil89; Vel97;
+DV04]), which boils down to the convolution with the characteristic function
+of a fixed set, is completely unrelated to ours.
+• The very general definition given in [Bol87] coincides with ours for the group
+(Z/pZ)d (p being prime) and in that work it is named (d − 1)-planes trans-
+form. The assumptions of the criterion [Bol87, Theorem 1] to establish the
+existence of an inversion formula of a Radon transform do not hold for our
+Radon transform (for example for the group (Z/4Z)2). Let us remark that
+the (d−1)-planes transform defined for Fpk does not coincide with our Radon
+transform on (Z/pkZ)d when k > 1 (in particular, proving the invertibility
+of the (d − 1)-planes transform seems to be considerably easier due to the
+larger number of symmetries).
+• The recent work [CHM18] defines a Radon transform which is almost equiva-
+lent to our discrete Radon transform on (Z/pZ)d, where p is a prime number.
+In that paper the Radon transform (which they call classical Radon trans-
+form to distinguish it from the one of Diaconis and Graham) coincides with
+the restriction of ours to the homomorphisms ψ ∈ Hom((Z/pZ)d, Z/pZ)
+such that ψ(0, 0, . . . , 0, 1) ̸= 0. Due to this restriction, they cannot establish
+a full inversion formula [CHM18, Theorem 1].
+• In the work [AI08], the authors define a discrete Radon transform on Zd
+which is equivalent to the Radon transform on Zd with our notation (if one
+allows the group to be non-finite in the definition). An inversion formula
+[AI08, Theorem 4.1] is proven for such discrete Radon transform. Joining
+the methods of [AI08] with ours, it might be possible to produce inversion
+formulas for the discrete Radon transform on groups (Z/nZ × Z)d that are
+neither finite nor torsion-free. We do not investigate this as it goes beyond
+the scope of the paper.
+• An alternative definition of discrete Radon transform for finite abelian groups
+is provided in [Ilm14]. The maximal Radon transform defined in this ref-
+erence [Ilm14, Section 7.3] computes the sum of the function f over all
+translations of maximal cyclic subgroups of G.
+It is not hard to check that, for p prime, the maximal Radon transform
+on (Z/pZ)2 coincides with ours.
+In this special case, the author proves
+the invertibility of the Radon transform [Ilm14, Lemma 3.4]. In general his
+definition does not coincide with ours and, in particular, the maximal Radon
+
+18
+FEDERICO GLAUDO AND ANDREA CIPRIETTI
+transform is not invertible in many important cases [Ilm14, Propositions 7.2,
+7.3].
+Let us introduce the concept of inversion formula for our discrete Radon transform
+(cp. [Hel99, Theorem 3.1], [Str82]). The main goal of this section is to obtain an
+inversion formula (see Theorem 1.2).
+Definition 5.2. Let n, d ≥ 1 be positive integers. We say that the Radon transform
+on (Z/nZ)d (see Definition 5.1) admits an inversion formula if there exists a function
+λ = λn,d : Hom((Z/nZ)d, Z/nZ) → Q such that
+f(x) =
+�
+ψ∈Hom((Z/nZ)d, Z/nZ)
+λ(ψ)Rf(ψ, ψ(x)),
+for all functions f : (Z/nZ)d → C and all x ∈ (Z/nZ)d.
+Let us begin with a simple but useful criterion for the existence of an inversion
+formula.
+Lemma 5.1. Let n, d ≥ 1 be positive integers. A function λ : Hom((Z/nZ)d, Z/nZ) →
+Q induces an inversion formula for the discrete Radon transform on (Z/nZ)d (see
+Definition 5.2) if and only if it satisfies, for all x ∈ (Z/nZ)d,
+(5.5)
+�
+ψ∈Hom((Z/nZ)d, Z/nZ)
+ψ(x)=0
+λ(ψ) =
+�
+1
+if x = 0,
+0
+otherwise.
+Proof. For any f : (Z/nZ)d → C, any λ : Hom((Z/nZ)d, Z/nZ) → Q and any
+x ∈ (Z/nZ)d, it holds
+�
+ψ∈Hom((Z/nZ)d, Z/nZ)
+λ(ψ)Rf(ψ, ψ(x)) =
+�
+ψ∈Hom((Z/nZ)d, Z/nZ)
+λ(ψ)
+�
+x′∈(Z/nZ)d
+ψ(x′)=ψ(x)
+f(x′)
+=
+�
+x′∈(Z/nZ)d
+f(x′)
+�
+ψ∈Hom((Z/nZ)d, Z/nZ)
+ψ(x′−x)=0
+λ(ψ).
+Thanks to this identity, it is clear that λ induces an inversion formula if and only if
+Eq. (5.5) holds.
+□
+In the next technical lemma we show that inversion formulas behave nicely with
+respect to products.
+Lemma 5.2. Let n, m, d ≥ 1 be positive integers such that n and m are coprime.
+If the discrete Radon transforms on (Z/nZ)d and on (Z/mZ)d admit inversion for-
+mulas, then also the Radon transform on (Z/nmZ)d admits an inversion formula.
+Proof. To simplify the notation, let G := Z/nZ and H := Z/mZ.
+Let ι : Hom(Gd, G)×Hom(Hd, H) → Hom(Gd⊕Hd, G⊕H) be the map such that
+ι(ψ1, ψ2)(x1, x2) = (ψ1(x1), ψ2(x2)) for all ψ1 ∈ Hom(Gd, G), ψ2 ∈ Hom(Hd, H),
+x1 ∈ Gd, x2 ∈ Hd. Since n, m are coprime the map ι is bijective.
+Since we assume that the Radon transforms on Gd and Hd admit inversion for-
+mulas, thanks to Lemma 5.1, we deduce the existence of λ1 : Hom(Gd, G) → Q and
+λ2 : Hom(Hd, H) → Q satisfying Eq. (5.5).
+
+ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS
+19
+Let λ : Hom(Gd ⊕ Hd, G ⊕ H) → Q be the function such that λ(ι(ψ1, ψ2)) =
+λ1(ψ1)λ2(ψ2) for all ψ1 ∈ Hom(Gd, G), ψ2 ∈ Hom(Hd, H). For any x1 ∈ Gd and
+x2 ∈ Hd, we have
+�
+ψ∈Hom(Gd⊕Hd, G⊕H)
+ψ(x1,x2)=(0G,0H)
+λ(ψ) =
+�
+ψ1∈Hom(Gd, G), ψ2∈Hom(Hd, H)
+ψ1(x1)=0G, ψ2(x2)=0H
+λ1(ψ1)λ2(ψ2)
+=
+�
+�
+ψ1∈Hom(Gd, G)
+ψ1(x1)=0G
+λ1(ψ1)
+��
+�
+ψ2∈Hom(Hd, H)
+ψ1(x2)=0H
+λ2(ψ2)
+�
+=
+�
+1
+if x1 = 0G and x2 = 0H,
+0
+otherwise,
+which is equivalent to the fact that the discrete Radon tranform on Gd ⊕ Hd ad-
+mits an inversion formula thanks to Lemma 5.1. This is equivalent to the desired
+statement since G ⊕ H ∼= Z/nmZ as a consequence of the coprimality of m, n.
+□
+Our next goal is to show that the Radon transform on (Z/pkZ)d (with p prime)
+admits an inversion formula (see Lemma 5.8).
+Let us begin with a sequence of technical lemmas (Lemmas 5.3 to 5.6) concerning
+the structure of (Z/pkZ)d, its automorphisms and its canonical scalar product. The
+first two statements, Lemmas 5.3 and 5.4, are special cases of known results (see
+[HR07; SS99]). For completeness, and because the proofs are much simpler compared
+to the proofs of the statements we cite, we provide a self-contained proof for both
+facts.
+Lemma 5.3. Fix a prime p and and two exponents k, d ≥ 1. Given a d × d matrix
+M ∈ (Z/pkZ)d×d, let mulM : (Z/pkZ)d → (Z/pkZ)d be the group homomorphism
+given by the multiplication with the matrix M, i.e., for all x = (x1, x2, . . . , xd) ∈
+(Z/pkZ)d,
+mulM(x) :=
+�
+d
+�
+j=1
+Mijxj
+�
+i=1,...,d.
+The group of automorphisms of (Z/pkZ)d is given by
+Aut((Z/pkZ)d) = {mulM : M ∈ (Z/pkZ)d×d so that p does not divide det(M)}.
+Proof. A homomorphism φ : (Z/pkZ)d → (Z/pkZ)d is uniquely determined by
+the images of the d generators of (Z/pkZ)d, that is by the values φ(1, 0, . . . , 0),
+φ(0, 1, 0, . . . , 0), . . . , φ(0, . . . , 0, 1). Let M ∈ (Z/pkZ)d×d be the matrix such that
+the j-th column is given by the image through φ of the j-th generator. It holds
+φ = mulM.
+It remains to prove that mulM is an automorphism (i.e., its inverse is a homo-
+morphism) if and only if det(M) is not divisible by p. Notice that mulM ◦ mulN =
+mulMN, therefore mulM is an automorphism if and only if M is invertible modulo
+pk, or equivalently det(M) is not divisible by p.
+□
+
+20
+FEDERICO GLAUDO AND ANDREA CIPRIETTI
+Lemma 5.4. Fix a prime p and two exponents k, d ≥ 1. For any 0 ≤ h ≤ k, let
+Eh ⊆ (Z/pkZ)d be the subset
+Eh := {x = (x1, . . . , xd) ∈ (Z/pkZ)d : ph divides xi for all i = 1, 2, . . . , d}.
+Moreover, let E∗
+k := Ek = {(0, . . . , 0)} ∈ (Z/pkZ)d and, for 0 ≤ h < k, E∗
+h :=
+Eh \ Eh+1.
+The orbits of the action of the automorphism group of (Z/pkZ)d are exactly
+E∗
+0, E∗
+1, . . . , E∗
+k.
+Proof. The subset Eh coincides with the elements of (Z/pkZ)d with order at most
+pk−h, hence E∗
+h coincides with the elements of (Z/pkZ)d with order equal to pk−h.
+In particular, the image of E∗
+h through an automorphism coincides with E∗
+h.
+To prove that E∗
+h is an orbit for the automorphism group, we show that given
+x = (x1, . . . , xd) ∈ E∗
+h there exists an automorphism φ ∈ Aut((Z/pkZ)d) such that
+φ(ph, 0, 0, . . . , 0) = x. By definition of E∗
+h, it holds υp(xi) ≥ h for all i = 1, 2, . . . , d
+(recall that υp denotes the p-adic valuation), and without loss of generality we may
+assume that υp(x1) = h. Consider the matrix M ∈ (Z/pkZ)d×d with the following
+entries
+M =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+x1/ph
+0
+0
+· · ·
+0
+0
+x2/ph
+1
+0
+· · ·
+0
+0
+x3/ph
+0
+1
+· · ·
+0
+0
+...
+...
+...
+...
+...
+...
+xd−1/ph
+0
+0
+· · ·
+1
+0
+xd/ph
+0
+0
+· · ·
+0
+1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+Since det(M) = x1/ph, which is not divisible by p, the classification of automor-
+phisms proven in Lemma 5.3 guarantees that φ = mulM is an automorphism which
+satisfies φ(ph, 0, . . . , 0) = x, as desired.
+□
+Lemma 5.5. Fix a prime p and two exponents k, d ≥ 1. Denote with · : (Z/pkZ)d ×
+(Z/pkZ)d → Z/pkZ the scalar product x · y := x1y1 + x2y2 + · · · + xdyd.
+For any automorphism φ ∈ Aut((Z/pkZ)d), there exists an automorphism ��t ∈
+Aut((Z/pkZ)d) such that φ(x) · y = x · φt(y) for all x, y ∈ (Z/pkZ)d.
+Proof. Thanks to Lemma 5.3, we know that there exists M ∈ (Z/pkZ)d×d such that
+φ = mulM. It can be checked that φt := mulM t, where M t is the transpose of M,
+satisfies the requirements of the statement.
+□
+Lemma 5.6. Fix a prime p and two exponents k, d ≥ 1. Recall the definitions of
+Eh and E∗
+h given in Lemma 5.4.
+Given 0 ≤ h, h′ ≤ k, for any x ∈ E∗
+h it holds
+|{y ∈ Eh′ : x · y = 0}| = p(d−1)(k−h′)+min{h, k−h′}
+and, in particular, this quantity does not depend on the specific choice of x ∈ E∗
+h.
+Proof. Thanks to Lemma 5.4, there exists an automorphism φ ∈ Aut((Z/pkZ)d)
+such that φ((ph, 0, 0, . . . , 0)) = x. Notice (recall Lemma 5.5) that x · y = 0 if and
+
+ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS
+21
+only if (ph, 0, 0, . . . , 0) · φt(y) = 0. Moreover, as a consequence of Lemma 5.4, we
+have that φt(Eh′) = Eh′. Thus, we deduce
+φt�
+{y ∈ Eh′ : x · y = 0}
+�
+= {y ∈ Eh′ : (ph, 0, 0, . . . , 0) · y = 0}.
+In particular, we have shown that the cardinality of such set does not depend on
+the specific choice of x ∈ E∗
+h and we may assume x = (ph, 0, 0, . . . , 0).
+The condition y ∈ Eh′ is equivalent to the fact that y may be expressed as y =
+ph′(z1, z2, . . . zd) with z1, z2, . . . , zd ∈ Z/pk−h′Z. The condition (ph, 0, 0, . . . , 0)·y = 0
+is equivalent to υp(z1) ≥ max(0, k−(h+h′)). To conclude, we distinguish two cases.
+• If k ≤ h+h′, then the constraint υp(z1) ≥ max(0, k −(h+h′)) is empty, and
+thus any choice of z1, z2, . . . , zd ∈ Z/pk−h′Z yields an element y = ph′z of
+{y ∈ Eh′ : (ph, 0, 0, . . . , 0) · y = 0}, thus such set has cardinality pd(k−h′) =
+p(d−1)(k−h′)+min{h,k−h′}.
+• If k ≥ h + h′, then z1 must be divisible by pk−(h+h′), while the other zi
+can be arbitrary values in Z/pk−h′Z.
+Therefore, the cardinality of the
+set {y ∈ Eh′ : (ph, 0, 0, . . . , 0) · y = 0} is p(d−1)(k−h′)+k−h′−(k−(h+h′)) =
+p(d−1)(k−h′)+min{h,k−h′}.
+□
+The only missing ingredient necessary to prove that the discrete Radon transform
+on (Z/pkZ)d admits an inversion formula is the invertibility of a certain matrix,
+which is promptly established in the following lemma.
+Lemma 5.7. Fix a prime p and two exponents k, d ≥ 1. The (k + 1) × (k + 1)
+matrix U (k) ∈ Q(k+1)×(k+1) with entries, for 0 ≤ i, j ≤ k, given by
+U (k)
+ij
+:= p(d−1)(k−j)+min{i, k−j}
+is invertible.
+Proof. It is easier to work with ˜U (k)
+ij
+:= U (k)
+i(k−j) (which is invertible if and only if U (k)
+is invertible). Indeed, defining q := pd−1, one has ˜U (k)
+ij
+= pmin{i,j}qj and therefore
+˜U (k) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+q
+q2
+· · ·
+qk−1
+qk
+1
+pq
+pq2
+· · ·
+pqk−1
+pqk
+1
+pq
+p2q2
+· · ·
+p2qk−1
+p2qk
+...
+...
+...
+...
+...
+...
+1
+pq
+p2q2
+· · ·
+pk−1qk−1
+pk−1qk
+1
+pq
+p2q2
+· · ·
+pk−1qk−1
+pkqk
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+We prove the statement by induction on k. We have ˜U (0) =
+�1�
+, which is in-
+vertible. For the inductive step, subtracting the second-to-last row of ˜U (k) from the
+last, all the entries of the last row become zero, except for the last one, which turns
+into qk(pk − pk−1) ̸= 0. Note further that the top-left k × k submatrix of ˜U (k) is
+˜U (k−1). Therefore, det ˜U (k) = qk(pk − pk−1) det ˜U (k−1), which concludes.
+□
+Lemma 5.8. Fix a prime p and two exponents k, d ≥ 1.
+The discrete Radon
+transform on (Z/pkZ)d admits an inversion formula.
+
+22
+FEDERICO GLAUDO AND ANDREA CIPRIETTI
+Proof. Notice that (Z/pkZ)d ∼= Hom((Z/pkZ)d, Z/pkZ) and the isomorphism is
+given by the map that takes y ∈ (Z/pkZ)d and produces the homomorphism (Z/pkZ)d ∋
+x → x · y (where the scalar product is defined in Lemma 5.5). Therefore, applying
+Lemma 5.1, we have that the validity of an inversion formula for the discrete Radon
+transform on (Z/pkZ)d is equivalent to the existence of a function λ : (Z/pkZ)d → Q
+such that, for all x ∈ (Z/pkZ)d,
+�
+y∈(Z/pkZ)d
+x·y=0
+λ(y) =
+�
+1
+if x = (0, 0, . . . , 0),
+0
+otherwise.
+We are going to construct a function λ with this property.
+Let U (k) ∈ Q(k+1)×(k+1) be the matrix considered in Lemma 5.7. Let V (k) ∈
+Q(k+1)×(k+1) be the matrix given by
+V (k)
+ij
+=
+�
+U (k)
+i,j
+if j = k,
+U (k)
+i,j − U (k)
+i,j+1
+if j < k.
+Since V (k) can be obtained by U (k) through Gauss moves, Lemma 5.7 implies that
+V (k) is invertible as well. Notice that, by definition of V (k), for any 0 ≤ i, j ≤ k,
+Lemma 5.6 implies that V (k)
+ij
+= |{y ∈ E∗
+j : x · y = 0}| for any x ∈ E∗
+i (recall that
+E∗
+j = Ej \ Ej+1).
+Let Λ ∈ Qk+1 be the solution of V (k)Λ = (1, 0, . . . , 0).
+Let us define λ :
+(Z/pkZ)d → Q as the function such that λ(y) := Λj when y ∈ E∗
+j .
+We show
+that this function satisfies the sought identity.
+Given x ∈ E∗
+i , we have
+�
+y∈(Z/pkZ)d
+x·y=0
+λ(y) =
+k
+�
+j=0
+|{y ∈ E∗
+j : x · y = 0}|Λj =
+k
+�
+j=0
+V (k)
+ij Λj = (V (k)Λ)i,
+which is the desired formula since the right-hand side is 1 if i = 0 (which is equivalent
+to x = (0, 0, . . . , 0) ∈ (Z/pkZ)d) and 0 otherwise.
+□
+We are ready to show the validity of an inversion formula for all instances of our
+discrete Radon transform.
+Proof of Theorem 1.2. By the classification of finite abelian groups (see Section 2.2),
+the statement follows from Lemmas 5.2 and 5.8.
+□
+Let us apply the inversion formula obtained in Theorem 1.2 to establish the FS-
+regularity of the group (Z/nZ)d when n ∈ OFS. The idea is to project through an
+homomorphism onto Z/nZ, use the FS-regularity of Z/nZ proven in Proposition 4.6,
+and then recover the FS-regularity of (Z/nZ)d thanks to the invertibility of the
+Radon transform on (Z/nZ)d.
+Proposition 5.9. For any n ∈ OFS and any d ≥ 1, the group (Z/nZ)d is FS-regular.
+Proof. For a multiset B ∈ M((Z/nZ)d), by definition of the Radon transform
+on ((Z/nZ)d (see Definition 5.1), one has RµB(ψ, c) = µψ(B)(c) (recall that µB
+denotes the multiplicity of elements in the multiset B, see Section 2.1) for any
+
+ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS
+23
+ψ ∈ Hom((Z/nZ)d, Z/nZ) and any c ∈ Z/nZ. Therefore, the inversion formula of
+Theorem 1.2 implies
+(5.6)
+µB(x) =
+�
+ψ∈Hom((Z/nZ)d, Z/nZ)
+λ(ψ)µψ(B)(ψ(x)),
+for all x ∈ (Z/nZ)d. Notice that this formula allows us to reconstruct B given all
+its projections ψ(B) onto Z/nZ.
+Take two multisets A, A′ ∈ M((Z/nZ)d) such that FS(A) = FS(A′); our goal is
+to prove that A ∼0 A′.
+For any ψ ∈ Hom((Z/nZ)d, Z/nZ), it holds FS(ψ(A)) = FS(ψ(A′)) and there-
+fore, since we have shown that Z/nZ is FS-regular in Proposition 4.6, we have
+ψ(A) ∼0 ψ(A′).
+Thus (we use only ψ(A) ∼ ψ(A′)), we deduce that for any
+ψ ∈ Hom((Z/nZ)d, Z/nZ),
+(5.7)
+µψ(A)(x) + µψ(A)(−x) = µψ(A′)(x) + µψ(A′)(−x)
+for all x ∈ (Z/nZ)d.
+Joining Eqs. (5.6) and (5.7), we obtain
+µA(x) + µA(−x) =
+�
+ψ∈Hom((Z/nZ)d, Z/nZ)
+λ(ψ)
+�
+µψ(A)(ψ(x)) + µψ(A)(−ψ(x))
+�
+=
+�
+ψ∈Hom((Z/nZ)d, Z/nZ)
+λ(ψ)
+�
+µψ(A′)(ψ(x)) + µψ(A′)(−ψ(x))
+�
+= µA′(x) + µA′(−x)
+for all x ∈ (Z/nZ)d. The latter identity is equivalent to A ∼ A′, which implies
+A ∼0 A′ thanks to Lemma 3.1-(3).
+□
+6. FS-regularity of products with Z
+In this section we show that multiplying by Z does not break the FS-regularity of
+a group (see Proposition 6.3). In order to do it, we will need two technical lemmas.
+The second one, Lemma 6.2, gives a condition equivalent to FS-regularity which
+comes handy in the proof of the main result of this section.
+Lemma 6.1. Let G be an abelian group without elements of order 2. Given three
+multisets A, A′, B ∈ M(G), if A + FS(B) = A′ + FS(B), then A = A′.
+Proof. Let us first prove the result when B = {b} is a singleton. We prove the result
+by induction on the cardinality of A.
+If |A| = 0, then ∅ = A + FS(B) = A′ + FS(B) and thus A′ = ∅.
+To handle the case |A| > 0, we begin by showing that A and A′ have a common
+element. We argue by contradiction, hence we assume that A and A′ are disjoint.
+Take any a ∈ A. We have a + b ∈ A + FS(B) = A′ + {0, b}. Since a ̸∈ A′, it
+must hold a + b ∈ A′. By repeating this argument (swapping the role of A and A′
+and replacing a with a + b) we obtain that a + 2b ∈ A. Repeating such argument k
+times, we obtain that a + kb ∈ A if k is even, and a + kb ∈ A′ if k is odd. Since A
+and A′ are finite, b must have finite order, otherwise the elements (a+kb)k∈N would
+be all distinct. Let ord(b) be the order of b; by assumption ord(b) is odd. We have
+
+24
+FEDERICO GLAUDO AND ANDREA CIPRIETTI
+the contradiction A ∋ a = a + ord(b)b ∈ A′; therefore we have proven that A and A′
+have a common element.
+Now pick ¯a ∈ A ∩ A′. It holds
+(A \ {¯a}) + FS(B) = (A + FS(B)) \ {¯a, ¯a + b}
+= (A′ + FS(B)) \ {¯a, ¯a + b} = (A′ \ {¯a}) + FS(B).
+Therefore, by the induction hypothesis, A \ {¯a} = A′ \ {¯a}, which is equivalent to
+A = A′.
+Let us now treat general multisets B. We proceed by induction on the cardinality
+of B; the case |B| = 0 is trivial and the case |B| = 1 is already established, so we
+may assume |B| > 1.
+Pick an element ¯b ∈ B. We have
+A + FS(B) = (A + FS(B \ {¯b})) + FS({¯b}),
+and likewise for A′. Applying the induction hypothesis for the three multiset A +
+FS(B\{¯b}), A′+FS(B\{¯b}), {¯b}, yields the relation A+FS(B\{¯b}) = A′+FS(B\{¯b}),
+and one more application yields the sought A = A′.
+□
+Lemma 6.2. An abelian group G is FS-regular if and only if, for all A, A′ ∈ M(G)
+such that FS(A) = FS(A′) + g for some g ∈ G, it holds A ∼ A′.
+Proof. Assume that G is FS-regular and take A, A′ ∈ M(G) such that FS(A) =
+FS(A′) + g for some g ∈ G. Applying Lemma 3.1-(4), we produce a multiset A′′ ∈
+M(G) such that A′�� ∼ A′ and FS(A) = FS(A′′); then we deduce A ∼0 A′′ because
+G is FS-regular. So, we get A ∼0 A′′ ∼ A′ which implies A ∼ A′ by transitivity.
+Let us now show the converse. Given A, A′ ∈ M(G) such that FS(A) = FS(A′),
+the condition described in the statement implies A ∼ A′ which implies A ∼0 A′
+thanks to Lemma 3.1-(3). Therefore we have proven the FS-regularity of G.
+□
+Proposition 6.3. If G is a FS-regular abelian group, then also G⊕Z is FS-regular.
+Proof. We begin by setting up some notation. For B ∈ M(G⊕Z) and z ∈ Z, define
+B<z = {(g, z′) ∈ B : z′ < z},
+B≤z = {(g, z′) ∈ B : z′ ≤ z},
+B=z = {(g, z′) ∈ B : z′ = z}.
+Let A, A′ ∈ M(G ⊕ Z) be two multisets such that FS(A) = FS(A′) + (¯g, ¯z) for
+some ¯g ∈ G and ¯z ∈ Z; we want to prove that A ∼ A′. This claim is equivalent to
+the FS-regularity of G thanks to Lemma 6.2.
+Up to changing the signs8 of A<0 and A′
+<0, we may assume that A<0 = ∅ and
+A′
+<0 = ∅. We will use repeatedly, without explicitly mentioning it, that the first
+coordinate of the elements of A and A′ is nonnegative.
+Recall that, by assumption, FS(A) = FS(A′) + (¯g, ¯z). Since (0G, 0) belongs to
+both FS(A) and FS(A′) (and the first coordinate of all the elements of both multisets
+is nonnegative), it must be ¯z = 0. So, it holds FS(A) = FS(A′) + (¯g, 0).
+8Formally, we are substituting A and A′ with ˜
+A := (A \ A<0) ∪ (−A<0) and ˜
+A′ := (A′ \ A′
+<0) ∪
+(−A′
+<0). Notice that A ∼ ˜
+A and A′ ∼ ˜
+A′.
+
+ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS
+25
+We prove, by induction on z, that A≤z ∼ A′
+≤z and FS(A≤z) = FS(A′
+≤z) + (¯g, 0).
+One can deduce A ∼ A′ by taking z sufficiently large.
+Notice that
+FS(A=0) = FS(A)=0 = FS(A′)=0 + (¯g, 0).
+By taking the projection on G of both sides of the latter identity, since G is FS-
+regular, we can apply Lemma 6.2 and get A=0 ∼ A′
+=0. This concludes the first step
+of the induction, that is z = 0 (since A=0 = A≤0 and A′
+=0 = A′
+≤0).
+For z ≥ 1, we show that A=z = A′
+=z which immediately implies, thanks to the
+inductive assumption, that A≤z ∼ A′
+≤z and FS(A≤z) = FS(A′
+≤z) + (¯g, 0).
+Given a multiset B ∈ M(G ⊕ Z) such that B<0 = ∅ (later on B will be a subset
+of A or A′), if � B = (g, z) for some g ∈ G and z ≥ 1 then either B = B<z or
+B = B=z ∪ B=0 and B=z is a singleton. Hence, one has
+FS(A)=z = FS(A<z)=z ∪ (A=z + FS(A=0)),
+FS(A′)=z = FS(A′
+<z)=z ∪ (A′
+=z + FS(A′
+=0)),
+and therefore, recalling that FS(A) = FS(A′) + (¯g, 0), we get
+(6.8)
+FS(A<z)=z ∪ (A=z + FS(A=0)) = FS(A)=z = FS(A′)=z + (¯g, 0)
+= (FS(A′
+<z)=z + (¯g, 0)) ∪ (A′
+=z + FS(A′
+=0) + (¯g, 0)).
+By inductive assumption, FS(A=0) = FS(A′
+=0) + (¯g, 0) and FS(A<z) = FS(A′
+<z) +
+(¯g, 0); hence Eq. (6.8) implies
+A=z + FS(A=0) = A′
+=z + FS(A=0)
+and we deduce A=z = A′
+=z thanks to Lemma 6.1 (since G is FS-regular it cannot
+have elements of order 2, see Proposition 4.1).
+□
+7. Proof of the Main Theorem
+The proof of the main theorem of this paper is routine work now that we have
+established Propositions 4.1, 4.6, 5.9 and 6.3.
+Proof of Theorem 1.1. If there is a torsion element g ∈ G such that ord(g) ̸∈ OFS,
+then Z/ ord(g)Z is a subgroup of G.
+Thanks to Proposition 4.1, we know that
+Z/ ord(g)Z is not FS-regular and therefore also G is not FS-regular.
+We prove the converse implication in three steps: first for groups with structure
+(Z ⊕ Z/nZ)d, then for finitely generated groups, and finally for any group.
+Let us assume that G is an abelian group such that ord(g) ∈ OFS whenever g ∈ G
+has finite order.
+Step 1: G = (Z ⊕ Z/nZ)d. The assumption on the order of the elements of G
+guarantees that n ∈ OFS. Hence, Proposition 5.9 shows that (Z/nZ)d is FS-regular.
+Thanks to Proposition 6.3, we obtain that also (Z/nZ)d ⊕ Zd is FS-regular.
+Step 2: G is finitely generated. Let n be the maximum order of an element in
+G with finite order. By assumption n ∈ OFS. The classification of finitely generated
+abelian groups (see Section 2.2) guarantees that G is a subgroup of (Z ⊕ Z/nZ)d for
+some d ≥ 1. By the previous step, we know that (Z⊕Z/nZ)d if FS-regular and thus
+also G is FS-regular (being a subgroup of an FS-regular group).
+
+26
+FEDERICO GLAUDO AND ANDREA CIPRIETTI
+Step 3: No restrictions on G. Let A, A′ ∈ M(G) be two multisets such that
+FS(A) = FS(A′); we want to prove that A ∼0 A′. Let ˜G := ⟨A ∪ A′⟩ be the group
+generated by the elements of A and A′. The condition on the orders is inherited by ˜G
+and, since ˜G is finitely generated, the previous step guarantees that ˜G is FS-regular;
+in particular A ∼0 A′ as desired.
+□
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+Department of Mathematics, University of Pisa, Pisa, Italy
+

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+1
+Comparison Between Different Designs and Realizations of
+Anomalous Reflectors
+Mostafa Movahediqomi1, Grigorii Ptitcyn1, and Sergei Tretyakov1, Fellow, IEEE
+1Department of Electronics and Nanoengineering, School of Electrical Engineering, Aalto University, 02150 Espoo, Finland
+Metasurfaces enable efficient manipulation of electromagnetic radiation. In particular, control over plane-wave reflection is one of
+the most useful features in many applications. Extensive research has been done in the field of anomalous reflectors over the past years,
+resulting in numerous introduced geometries and several distinct design approaches. Anomalously reflecting metasurfaces designed
+using different methods show different performances in terms of reflection efficiency, angular response, frequency bandwidth, etc.
+Without a comprehensive comparison between known design approaches, it is difficult to properly select the most appropriate
+design method and the most suitable metasurface geometry. Here, we consider four main approaches that can be used to design
+anomalous reflectors within the same basic topology of the structure and study the designed metasurfaces first on the level of the
+input impedance and then consider and compare the performance of the realized structures. We cover a wide range of performance
+aspects, such as the power efficiency and losses, angular response, and the scattering pattern of finite-size structures. We anticipate
+that this study will prove useful for developing new engineering methods and designing more sophisticated structures that include
+reconfigurable elements. Furthermore, we believe that this study can be considered referential since it provides comparative physical
+insight into anomalous reflectors in general.
+Index Terms—Anomalous reflectors, diffraction grating, phase gradient, surface wave, angular response, scattering parameters,
+far-field pattern.
+I. INTRODUCTION
+Wireless communication technologies constantly progress
+towards higher operational frequencies. This progress comes
+with smaller antenna sizes and, alas, at the expense of the
+need to use highly-directive and scanning antennas. Improve-
+ment of transmitters and receivers is limited, therefore com-
+munication engineers proposed to optimize the propagation
+environment using metasurfaces and metagratings
+[1]–[14],
+and reconfigurable intelligent metasurfaces (RIS) [15]–[24].
+The latter approach has gained increasing attention recently in
+communication communities. Often, reconfigurable structures
+are designed based on conventional fixed structures with the
+addition of tunable elements. Therefore, a comparison of
+known approaches to design anomalous reflectors is timely.
+There are two fundamentally different methods to realize a
+flat surface that reflects plane waves into plane waves along
+any desired direction. One is the use of periodical structures
+(diffraction gratings) whose period is chosen accordingly to
+the required angles of incidence and reflection. The other one
+is using aperiodically loaded antenna arrays whose geometrical
+period is fixed to usually λ/2 [25]. The majority of works on
+anomalous reflectors use the first approach, and here we con-
+sider various designs of periodically modulated anomalously
+reflected boundaries.
+Perhaps, the most classical approach to manipulate the
+direction of reflection from a surface is the use of phased-array
+(reflectarray) antennas [26]–[28]. Here, the phase distribution
+at the antenna aperture is tuned so that reflections from
+all antenna array elements interfere constructively along the
+desired direction of reflection. Generalizing this principle, a
+similar approach can be realized in a planar subwavelength-
+structured metasurface if the local reflection phase is made
+nonuniform over the surface, realizing a phase-gradient re-
+flector. Using this approach one can direct the reflected wave
+at will, beating the conventional law of reflection and realizing
+so-called anomalous reflection. The main drawback of this
+method is the low efficiency at large deviations from the usual
+law of reflection [2], [29]. Impedance mismatch between the
+incident and the reflected waves becomes significant, and it
+causes more scattering into parasitic propagating modes (See
+Fig. 1).
+Theoretically, the problem of reduced efficiency at large
+deflection angles can be completely solved with the use of
+active and lossy inclusions in the metasurface [1]. Worth
+mentioning, that the average power produced by the surface
+would be zero, however, some parts of it must produce energy,
+and the other parts should absorb it, which is quite impractical.
+Another possibility is to use completely passive structures,
+where auxiliary surface waves in the near-field region are
+properly tuned [1], [3] as it is shown in Fig. 1. Optimization
+of the evanescent modes can be performed in several differ-
+ent ways: based on the optimization of the input (surface)
+impedance [5], [6], grid (sheet) impedance [7]–[9], by direct
+optimization of the whole structure [10], [11], by finding an
+analytical solution [12], [13], and finally, introducing non-
+planar (power flow-conformal) structures [14]. In this paper,
+we overview and compare some of these approaches in detail
+and discuss their differences and advantages. We repeated the
+selected design methods and compared the most important
+characteristics of these works, including power efficiency,
+angular stability, far-field radiation patterns, and frequency
+bandwidth for the infinite and finite-size structures.
+The paper is organized as follows: In Sec. II the selected
+methods for comparison will be briefly introduced and the pros
+and cons for each of them will be highlighted. Then Sec. III is
+devoted to the investigation of scattering parameters for both
+ideal and realized structures and the comparison of the power
+arXiv:2301.02851v1  [physics.app-ph]  3 Jan 2023
+
+2
+efficiency of each method. The angular response of an anoma-
+lous reflector is another important aspect that is discussed
+in Sec. IV. Here, we show the behavior of reflectors when
+they are illuminated by waves at different angles. Furthermore,
+reflection and scattering by a finite-size structure in the far
+zone is important for applications, and recently it has been
+considered in several studies. We cover this issue in Sec. V.
+Finally, conclusions are formulated in Sec. VI to finalize this
+comparison and make the advantages and drawbacks of each
+approach clear.
+II. CONSIDERED DESIGN METHODS
+To provide a fair comparison, we choose methods that can
+be realized using arrays of metallic patches or strips printed on
+a grounded dielectric substrate. We select an example required
+performance: an anomalous reflection of normally incident
+plane waves with TE polarization to the 70◦-direction, at
+8 GHz. All designs are based on the same basic platform: a
+metal patch array on a grounded dielectric substrate (Fig. 1).
+The chosen example substrate is Rogers 5880 with 2.2 per-
+mittivity, 1.575 mm thickness, and 0.0002 loss tangent. For
+all designs, we split the period into 6 sub-cells that are either
+impedance strips or shaped metal strips. The use of the same
+parameters for all designs allows a meaningful comparison of
+performance.
+The initial reference design is a phase-gradient metasur-
+face, e.g. [26]–[28], [30]. The unit cells are designed in the
+conventional locally periodical approximation so that at every
+point of the reflector the reflection phase (at normal incidence)
+from an infinite array of identical cells is as required by the
+linear phase gradient rule for the desired reflection angle. It
+means that the reflection properties of a metasurface can be
+defined by the “local reflection coefficient” which is assumed
+to be controlled by adjusting the geometrical parameters of
+the unit cells. Strong coupling between the inclusions in an
+inhomogeneous array makes this approximation rather rough
+when the deflection angle is not small.
+More advanced methods that aim at overcoming the inherent
+parasitic scattering of phase-gradient reflectors we classify
+based on the degree of use of homogenized boundary con-
+ditions:
+Method 1 (input impedance method). Here, the metasurface
+is designed at the level of the equivalent input impedance,
+also known as the impenetrable impedance boundary condition
+(IBC), see Fig. 2(a). The input impedance (Zinput) relates the
+tangential components of the electric field (Et) and magnetic
+field at the interface between the metasurface structure and
+free space: (Ht)
+Et = Zinput · ˆz × Ht |z=0+ .
+(1)
+In this method, the input impedance distribution over the
+reflector surface is optimized with the goal to channel most
+of the reflected power into the desired direction. Optimization
+algorithms vary the input impedance, ensuring zero normal
+component of the Poynting vector at every point of the surface
+so that the input impedance is purely reactive [5], [6]. When
+the desired input impedance values at every point are found,
+𝜃𝑟
+−𝜃𝑟
+Normal incident 
+wave (Ei)
+Retro-reflection 
+(Er0)
+Symmetry 
+reflection (Er−1)
+Anomalous 
+reflection (Er+1)
+Fig. 1: Concept of periodical arrays acting as anomalous
+reflectors. The case with three propagating Floquet harmonics
+is illustrated. The design goal is to suppress reflections into all
+propagating modes except the desired anomalous reflection.
+the actual geometry of the structure is determined using the lo-
+cally periodic approximation. That is, the continuous reactance
+profile is discretized, and the dimensions of each unit cell are
+optimized using periodical boundary conditions, ensuring that
+the plane-wave reflection phase (at normal incidence) from
+an infinite periodical array of this cell is the same as from
+a uniform boundary with the required input reactance at this
+point.
+Method 2 (grid impedance method). This design approach is
+based on the grid (or sheet) impedance model of a patch array
+that is also known as penetrable IBC, see Fig. 2(b). In this
+method, the impedance boundary condition is used to model
+only the array of metal patches. The grid impedance (Zgrid)
+relates the surface-averaged electric field with the difference
+between the averaged tangential magnetic fields at both sides
+of the metasheet:
+Et = Zgrid · ˆz × (Ht |z=0+ −Ht |z=0−).
+(2)
+In this method, spatial dispersion of the grounded dielectric
+layer is taken into account. The optimization process in this
+case considers a more practical structure, that treats waves
+inside the substrate in a more complete way compared to
+the first method [7]–[9]. In method 2 the locally periodical
+approximation is used to design reactive sheets in contrast
+with method 1 in which it is utilized to model the whole
+metasurface volume.
+Method 3 (non-local design) accounts for all specific geo-
+metrical and electromagnetic features of the layer, not relying
+on homogenization methods. The optimization usually starts
+from some initial settings in terms of the input impedance (for
+example, in [10] it was required that the reflector if formed
+by periodically arranged regions of receiving and re-radiating
+leaky-wave antennas), but the final steps optimize the whole
+supercell of the periodical lattice instead of individual patches
+in periodical arrays. Importantly, the normal component of
+the Poynting vector along the surface is not set to zero, cor-
+responding to the effective active-lossy behavior, although the
+overall structure remains completely passive. The drawback of
+this approach lies in the need for direct optimization, which
+
+3
+𝑍input
+𝜂0
+𝜂𝑑
+𝜂0
+𝑍grid
+PEC
+Substrate
+𝜀𝑟
+𝐄𝐭 = 𝑍grid. ො𝒛 × (𝐇𝐭ห𝑧 = 0+ − 𝐇𝐭ȁ𝑧 = 0−)
+Z= 0
+d
+X
+Z
+X
+Z
+Z= 0
+𝐄𝐭 = 𝑍input. ො𝒛 × 𝐇𝐭ห𝑧 = 0+
+(a)
+(b)
+Fig. 2: Two types of IBCs: (a) input impedance, also known as
+impenetrable IBC. The left side illustrates the conceptual struc-
+ture, and the right side shows the corresponding transmission-
+line model. (b) Grid or sheet impedance is also known as
+penetrable IBC. The conceptual structure on the left consists
+of an impedance sheet placed on top of a grounded dielectric
+substrate. The equivalent transmission-line model is shown on
+the right side.
+usually requires heavy computational facilities and might also
+become time-consuming.
+Other approaches realize perfect anomalous reflection using
+arrays of loaded wires [12], [13] or non-planar structures [14].
+However, to provide an insightful comparison, we chose only
+methods that are suitable for planar structures that can be
+realized as printed circuit boards with metallic patches. Specif-
+ically, the phase-gradient sample is designed based on the
+required tangent-profile of the input impedance, for method 1
+(impenetrable IBCs) we follow [6], paper [7] for method 2
+(penetrable IBCs), and [10] for method 3 (non-local design).
+In the following sections, the scattering properties, angular
+response, as well as far-field characteristics of test finite-size
+structures for all aforementioned methods will be investigated
+and compared in detail.
+III. SCATTERING PROPERTIES
+At first, the scattering properties of all the anomalous reflec-
+tors under study will be investigated assuming infinite period-
+ical structures. Upon plane-wave illuminations, the structures
+support surface currents that are also periodic. The Floquet
+theory defines the tangential wavenumbers of modes supported
+by the surface:
+kt = kt0 + ktn = k0 sin θi + 2πn/D,
+(3)
+Where k0 is the wavenumber in free space, θi is the angle
+of the upcoming incident wave, n is an integer number that
+denotes the index of the Floquet mode, and D is the period
+of the surface pattern, determined by D = λ/(sin θr − sin θi).
+θr is the desired reflection angle. By choosing this period,
+the tangential component of the wavenumber is fixed so that
+one of the harmonics is reflected to the desired angle. Floquet
+harmonics that satisfy criterion k0 > |kt| belong to the fast-
+wave regime in the dispersion diagram and can propagate in
+free space. Other Floquet harmonics are surface waves. The
+direction of the reflection can be calculated by the following
+formula:
+sin θr = kt/k0 = (k0 sin θi + 2πn/D)/k0.
+(4)
+For the chosen design parameters (θi = 0◦, θr = 70◦, f =
+8 GHz), the period is equal to D = 1.0642λ, and the Floquet
+expansion has three propagating harmonics (k0 > |kt|): zero
+Floquet mode (0◦), −1 Floquet mode (−70◦), and +1 Floquet
+mode (+70◦), as follows from Eq. (4). The field amplitudes
+in these modes define the efficiency of power channeling from
+one mode to another.
+1) Performance of the surface-impedance models
+Initially, the ideal impedance profile is considered when
+the period is discretized to six elements. In other words, it
+is assumed that the impedance boundary condition is applied
+straight on the surface without considering actual realization
+(Fig. 2). It is noteworthy to notice that the discretization of
+the impedance profile deteriorates performance, however, it is
+spoiled in the same way for all methods. Using such discretiza-
+tion, reasonable results can be achieved rather fast. All the
+methods except the phase gradient method use optimization,
+therefore analytical closed-form formulas for the impedance
+profiles do not exist. The list of optimized impedance values
+of each unit cell in a period is presented in Table I for
+all designed methods. For both designs based on the input
+impedance model (phase-gradient and input impedance opti-
+mization), we convert the obtained input impedance profile
+to the grid impedance by using the equivalent transmission-
+line model presented in Fig. 2. The input impedance can be
+considered as that of a shunt connection of the grid impedance
+to the transmission line modeling the grounded dielectric
+substrate. For the non-local method, a pre-final optimization
+is applied here similarly to what was done in [10]. As it
+was discussed in Sec. II, for the non-local approach we can
+assume an impedance profile using repeated receiving and re-
+radiating leaky-wave sections to mimic the ideal active-lossy
+profile. Therefore, the optimization at the level of the grid
+impedance is an initial step before the final optimization for
+the whole supercell in the real structure. Eventually, the same
+configuration for all the methods enables us to complete a fair
+comparison.
+TABLE I: The impedance profile list for each unit cell (jΩ)
+Cell1
+Cell2
+Cell3
+Cell4
+Cell5
+Cell6
+Phase
+gradient
+-97.6
+-82.1
+-51.4
++2673.9
+-145.4
+-113.4
+Input
+impedance
+-291.7
+-141.3
+-114.0
+-98.0
+-83.8
+-85.3
+Grid
+impedance
+-73.8
+-1334.0
+-112.9
+-172.6
+-91.4
+-96.7
+Non-
+local
+-81.82
+-74.74
+-49.16
+-73.50
+-75.07
+-73.06
+Performance comparison of the discretized impedance pro-
+files after optimization is made using full-wave simulators,
+CST STUDIO [31] and ANSYS HFSS [32]. As it was dis-
+cussed, there are three propagating Floquet harmonics (open
+channels) in our specific example. Therefore, we can consider
+
+4
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+(h)
+No diffracted modes
+No diffracted modes
+No diffracted modes
+No diffracted modes
+Fig. 3: Power distribution between three propagating modes and scattered field distribution (bottom); (a,e) for the phase gradient,
+(b,f) input impedance, (c,g) grid impedance, (d,h) non-local design method. The horizontal black lines in the scattered field
+distribution figures illustrate the location of the metasurfaces where the IBC is applied.
+these reflectors as three-port networks. Scattering parameters
+(Sn1) can be determined numerically when the input wave
+comes from Floquet port 1 and the output wave is observed
+in the port number n. Consequently, the power efficiency is
+found as squared scattering parameters (ηn = |Sn1|2) in the
+full-wave simulators. The power efficiency for each mode
+measures the fraction of power rerouted from the incident
+wave (assuming that the incident port is 1) to the propagating
+mode n.
+Figures 3 (a-d) show ratios of power rerouted to propagating
+channels n = 0, 1, and −1. In all cases, below 7.5 GHz
+diffraction modes are not allowed, therefore all the energy
+is reflected back to the normal direction. Designs based
+on the phase-gradient, input impedance, and grid impedance
+methods show broadband behavior as compared to the non-
+local approach. The phase-gradient method does not take
+evanescent modes into account, which results in the lowest
+efficiency at the operational frequency. Power distribution and
+the corresponding field amplitudes for all methods can be
+found in Table II.
+TABLE II: Amplitude/power ratio of propagating Floquet
+modes and power efficiency level at 8 GHz
+θi
+θr
+−θr
+Phase gradient
+0.33/0.11
+1.45/0.72
+0.71/0.17
+Input impedance
+0.10/0.01
+1.67/0.95
+0.34/0.04
+Grid impedance
+0.00/0.00
+1.71/1
+0.01/0.00
+Non-local
+0.1/0.01
+1.70/0.99
+0.12/0.00
+It is noteworthy to sketch the scattered electric field distri-
+butions (Fig. 3(e-h)). Efficiency for the phase-gradient method
+is only 71.8%, and, correspondingly, Fig. 3(e) shows a field
+distribution that is distorted by fields scattered into two par-
+asitic propagating channels. For the other methods, efficiency
+is nearly perfect, however, the near-field distributions are
+different due to different methods used to optimize evanescent
+modes. It is important to note that for perfect anomalous
+reflection with ideal power efficiency, the power reflected to
+the desired direction must be equal to the power of the incident
+plane wave, and, as a result, the ratio between the amplitudes
+of the reflected and incident fields for these angles should be
+larger than one |Er| = |Ei|
+�
+cos(θi)/ cos(θr) (1.71 for our
+example case) [10].
+2) Realizations as patch arrays
+The next step is to compare actual structures designed using
+the previously obtained and discussed impedance profiles. Fol-
+lowing the procedures described in the corresponding papers,
+we design supercells formed by six unit cells based on the
+rectangular shape metal patches above the grounded dielectric
+substrate (see Fig. 4 and Table III).
+The corresponding efficiencies for all the considered meth-
+ods are shown in Fig. 5. The frequency for the best per-
+formance becomes shifted for all methods, except for the
+non-local design, where optimization of the whole supercell
+is implemented. In addition to that, dispersion and losses
+deteriorate the efficiency in different ways. The absorption
+levels as well as efficiency at the design frequency (8 GHz)
+are reported in Table IV. The remained power is scattered to
+other propagating Floquet modes that are not shown in Fig. 5.
+
+Re (E/E)
+1.5
+0.8
+1
+0.5
+0.6
+0
+2
+0.4
+-0.5
+0.2
+-1
+-1.5
+0
+-0.5
+0
+0.5
+c/ DxRe (E/E)
+1.5
+0.8
+1
+0.5
+0.6
+0
+2
+0.4
+-0.5
+0.2
+-1
+1.5
+0
+-0.5
+0
+0.5
+α/DxRe (E/E.)
+1
+1.5
+0.8
+1
+0.5
+0.6
+0
+2
+0.4
+-0.5
+0.2
+-1
+1.5
+0
+-0.5
+0
+0.5
+α/DxRe (E/E)
+1.5
+0.8
+1
+0.5
+0.6
+0
+2
+0.4
+-0.5
+0.2
+-1
+1.5
+0
+-0.5
+0
+0.5
+α/Dxn=0n=-1n=+1
+0.8
+Eficiency, In
+0.6
+0.4
+0.2
+0
+7
+7.5
+8
+8.5
+9
+Frequency (GHz)n=0n=-1n=十1
+0.8
+0.6
+0.4
+0.2
+0
+7
+7.5
+8
+8.5
+9
+Frequency (GHz)n=0n=-1n=十1
+0.8
+Efficiency, n
+0.6
+0.4
+0.2
+0
+7
+7.5
+8
+8.5
+9
+Frequency (GHz)n=0n=-1n=+1
+0.8
+Eficiency, Nn
+0.6
+0.4
+0.2
+0
+7
+7.5
+8
+8.5
+9
+Frequency (GHz)5
+Cell #1 Cell #2 Cell #3 Cell #4 Cell #5 Cell #6
+𝐷
+𝑑 =
+ൗ
+𝐷 6
+Fig. 4: The configuration of supercells utilized for the designs
+consisting of six unit cells. All the parameters of the dielectric
+substrate are given in Sec. II. The period of the array (the
+supercell size) is fixed to D = 39.9 mm, and the width of
+a single unit cell is d = D/6. The width of metal strips
+is 3.5 mm, while the strip lengths are different for different
+design methods.
+TABLE III: Lengths of metal strips for each unit cell (mm)
+Strip1
+Strip2
+Strip3
+Strip4
+Strip5
+Strip6
+Phase
+gradient
+10.8
+11.41
+13.23
+0
+9.47
+10.29
+Input
+impedance
+7.3
+9.57
+10.28
+10.78
+11.3
+11.25
+Grid
+impedance
+3.71
+10.32
+8.89
+11.03
+10.84
+11.78
+Non-
+local
+10.47
+10.91
+11.26
+12.22
+11.30
+8.88
+TABLE IV: The best-performance frequency and the corre-
+sponding efficiency versus the absorption rate and efficiency
+for the design frequency.
+Best performance
+8 GHz
+Frequency
+Efficiency
+Absorption
+Efficiency
+Phase
+gradient
+8.5 GHz
+78.2(%)
+3.9(%)
+62.7(%)
+Input
+impedance
+8.27 GHz
+86.3(%)
+3.2(%)
+56.6(%)
+Grid
+impedance
+8.2 GHz
+95.0(%)
+3.3(%)
+73.6(%)
+Non-local
+8 GHz
+96.6(%)
+3.5(%)
+96.6(%)
+IV. ANGULAR RESPONSE
+A very interesting property of anomalous reflectors which is
+often left unstudied is the angular response, i.e., performance
+of the structure for various incident angles θi, which can
+be different from the design angle of incidence. Here, we
+consider angular response for periodical arrays formed by
+repeated supercells consisting of 6 unit cells with patches
+printed on a grounded substrate and assume the periodic
+boundary condition for this analysis. To distinguish between
+the illumination angle and the incidence angle for which the
+surface was designed, we denote this design incidence angle by
+No diffracted modes
+Fig. 5: Frequency dependence of efficiency for structures
+realized with metallic rectangular patches.
+θid. Worth to note that θid together with the required reflection
+angle defines the period of the structure operating as an
+anomalous reflector for these angles. The angular response is
+studied by sweeping the incident angle θi for a fixed structure,
+designed for the angle θid. The number of propagating Floquet
+modes existing in the system is defined by the incident angle
+θi, the period of the structure D, and the frequency f (see
+Eq. (3)). The condition for the mode propagation can be
+written as follows:
+ktn < k0 → 2π
+D |n| < 2π
+λ → |n| < D
+λ ,
+(5)
+and their propagation directions can be calculated as [33], [34]:
+θtn = arctan(ktn/knn),
+(6)
+where knn is the normal component of the wavenumber for
+the nth mode, and knn =
+�
+k2
+0 − k2
+tn. If k0 > |ktn|, the
+normal component of the nth wavenumber is purely real,
+which corresponds to a propagating mode. Otherwise, the
+wavenumber is imaginary, which corresponds to a surface
+mode that propagates along the interface. Figure 6 shows that
+for this fixed operational frequency and period of the structure,
+only five propagating Floquet modes with n ∈ [−2, 2] are
+allowed in the system when the angle of incidence is changing.
+All other modes (|n| > 3) are surface modes.
+Figure 7 depicts the spatial power distribution for prop-
+agating modes versus the illumination angle at 8 GHz. At
+the angle θi = 0◦ the incident angle is equal to the design
+angle θid, therefore most of the power goes to mode +1, with
+different efficiency for each method (see Table IV). Based on
+the discussion in Refs. [33], [34], for the phase-gradient case,
+there is a retro-reflection angle (at which all the energy is
+reflected back at the angle of incidence), that can be calculated
+as θretro = arcsin[(sin θi −sin θr)/2] and is equal to −28◦ for
+the considered case. At this angle only two channels are open
+(see Fig. 6), and the angle for the other channel is −θretro.
+Ideally, 100% of the power should be scattered in the retro-
+reflection direction, however, discretization and the presence
+of losses decrease it down to 96%. Therefore, the rest of the
+power goes to the remaining channel or gets absorbed. Due
+
+Phase grad Input imp Grid impNon-local
+Efficiency, Mn
+0.5
+0
+7
+7.5
+8
+8.5
+9
+Frequency
+GHz6
+Fig. 6: Propagation angle for different Floquet modes with
+respect to the incident angle. This figure is made using Eq. 6
+when the incidence angle is swept.
+to reciprocity, the structure behaves in the same way when
+illuminated from direction −θretro. It is important to notice
+that for other design methods, retro-reflection occurs at the
+angle +70◦, when three propagating channels are open. When
+the structure is illuminated from the normal direction, most of
+the energy couples to mode n = +1, where the reflection angle
+is θr = +70◦. Therefore, channel n = −1 becomes decoupled
+from the other two, and when the structure is illuminated from
+the angle θi = −70◦, all the energy is reflected back to the
+source.
+Finally, a sweep of the incident angle reveals that the design
+method based on grid impedance is the solution that has the
+least sensitive response (see Fig. 7(c)). It means that when the
+incident angle changes between −70◦ and +70◦, the power
+couples primarily to the same modes, unlike for other methods.
+Figure 8 illustrates and reports the results of the study for
+the best performance frequency for each method. The result
+is the same for the non-local optimization approach since in
+this case, the best performance frequency matches the design
+frequency.
+V. FAR-FIELD SCATTERING FROM FINITE-SIZE
+STRUCTURES
+In the previous analysis, we considered infinite periodical
+structures excited by plane waves. Here, we study far-field
+scattering properties of finite-size structures. It is possible to
+study metasurfaces on the grid or sheet impedance levels using
+the mode-matching method for calculation of induced currents
+[7], [35] and the far-field approximation for the calculation of
+scattered fields [33], [36]. To do that, the following conditions
+have to be met:
+|r| ≫ λ,
+(7a)
+|r| ≫ L,
+(7b)
+L2/|r| ≪ λ,
+(7c)
+where |r| is the distance from the observation point to the
+center of the structure, and L is max(2a, 2b), in which a
+and b denote distances between the center of the metasur-
+face and the edges of the structure along the x and y-
+axes, respectively. Considering TE polarized incident waves
+(Ei = E0e−jk(sin θix+cos θiz)ˆy) and selecting the observation
+point in the plane of incidence (xy-plane), the normalized
+scattering pattern in spherical coordinates can be determined
+by the following expressions:
+Fr(θ) =
+1
+2 cos θi
+�
+n
+rn(θi)(cos(θrn) + cos(θ))sinc(kaefn),
+(8)
+Fsh(θ) =
+1
+2 cos θi
+(cos(θ) − cos(θi))sinc(kaef),
+(9)
+where rn(θi) are the amplitudes of excited harmonics, de-
+termined by the mode matching technique. Angle θrn shows
+the reflected angle for each harmonic, and sinc(x) is a sinc
+function. In addition, aefn and aef can be represented by
+aefn = (sin θ − sin θrn)a and aefn = (sin θ − sin θi)a, re-
+spectively. Finally, the total scattering pattern can be found as
+the sum Fsc = Fr + Fsh. Worth to mention that normalization
+is performed with respect to the maximum of the reflected
+field.
+Alternatively to the analytical approach, one can study
+finite-size structures numerically using full-wave simulators.
+The result shown in Fig. 9(a) corresponds to the analytical
+solution, and in Fig. 9(b) to the full-wave simulations. In
+both cases, the radiation pattern is calculated at 8 GHz
+for structures with the size 11.7λ × 7λ in the xy-plane.
+The discrepancy between the two radiation patterns, which
+becomes more significant for side lobes (SL), is caused by
+the neglected current distortions near the edges. Nevertheless,
+the general behavior is similar. Parameters of the patterns
+related to each method are shown in Table V.
+The beam-
+TABLE V: The normalized main lobe amplitude, the first side
+lobe amplitude in linear scale, and their directions.
+Main lobe
+Amp/angle
+corresponds to
+n=+1.
+SL Amp/angle
+corresponds to
+n=0
+SL Amp/angle
+corresponds to
+n=-1
+Phase gradient
+0.94/68◦
+0.52/-1◦
+0.49/-67◦
+Input
+impedance
+0.95/68◦
+0.45/0◦
+0.35/-67◦
+Grid
+impedance
+0.99/69◦
+0.13/-1◦
+0.33/-67◦
+Non-local
+1.00/69◦
+0.24/-2◦
+0.39/-68◦
+widths for all cases are similar and close to 9◦ (see Fig. 9).
+Moreover, as it is shown in the inset, the maximum of the
+scattered field is higher for the design methods based on grid
+impedance and non-local solution because the power efficiency
+is higher in these methods compared to the phase gradient
+design and optimization based on the input impedance. The
+most important difference corresponds to the side-lobe level
+(SLL). As it is expected, the highest side lobes occur along
+the θ = −70 and θ = 0, because there are two propagating
+Floquet harmonics along these directions.
+Figure 10 shows the scattering patterns of all the methods
+at different frequencies, where all the patterns are normalized
+
+—n=0n=-1n=+1n=-2n=+2
+50
+0
+-50
+-50
+0
+50
+Angle of incidence, :7
+(a)
+(b)
+(c)
+(d)
+Fig. 7: Power distribution among different propagating modes depending on the incident angle at frequency 8 GHz for (a)
+phase gradient, (b) input impedance optimization, (c) grid impedance optimization, and (d) non-local optimization method.
+(a)
+(b)
+(c)
+(d)
+Fig. 8: Power distribution among different propagating modes depending on the incident angle for the best-performance
+frequencies which are reported in Table IV, for (a) phase gradient, (b) input impedance optimization, (c) grid impedance
+optimization, and (d) non-local optimization method.
+
+n=0
+n=-1
+n=+1n=-2
+2n=+2
+Eficiency, Mn
+0.5
+-50
+0
+50
+Angle of incidence, :n=0
+-n=-1
+n=+1n=-2n=+2
+Efficiency, Mn
+0.5
+-50
+0
+50
+Angle of incidence, :n=0
+-n=-1
+-n=+1-n=-2
+一n=+2
+Efficiency, Mn
+0.5
+-50
+0
+50
+Angle of incidence, :n=0
+-n=-1
+n=+1n=-2
+n=+2
+Eficiency, Mn
+0.5
+-50
+0
+50
+Angle of incidence, :n=0
+-n=-1
+-n=+1
+n=-2
+n=+2
+Efficiency, Mn
+0.5
+-50
+0
+50
+Angle of incidence, :n=0n=-1n=+1n=-2n=+2
+Eficiency, Mn
+0.5
+0
+-50
+0
+50
+Angle of incidence, O:n=0
+-n=-1
+n=+1n=-2
+n=+2
+Eficiency, Nn
+0.5
+-50
+0
+50
+Angle of incidence, :n=0
+-n=-1
+-n=+1-
+-n=-2
+一n=+2
+Efficiency, Mn
+0.5
+0
+-50
+0
+50
+Angle of incidence, :8
+Fig. 9: Normalized radiation pattern in linear scale. All pat-
+terns are normalized with respect to the main lobe amplitude
+of the non-local method which has the highest gain compared
+to the other approaches. (a) analytical pattern based on the
+Huygens principle, (b) full-wave simulation in CST STUDIO.
+to the main lobe. It is important to notice that due to the
+fixed period of the structure (D = λ/(sin θr − sin θi)), by
+sweeping the frequency, the angle of reflection changes. This
+can be observed in Fig. 10. By changing the frequency, the
+scattered n = +1 Floquet harmonic scans the space from the
+desired reflection angle (+70◦) at 8 GHz to smaller angles
+at higher frequencies and larger angles at lower frequencies.
+Below 7.5 GHz there are no diffraction modes, therefore we
+plot the scattering patterns starting from 7.75 GHz, where most
+of the energy is reflected into the normal direction (see the
+blue line in Fig. 10). The red line in the figure corresponds to
+the scattering pattern at the design frequency. Eventually, the
+radiation patterns for 8.25 and 8.5 GHz are shown by yellow
+and purple lines, respectively.
+VI. CONCLUSION
+We have presented a comprehensive analysis of four main
+design methods for anomalous reflectors. In order to provide a
+meaningful comparison we chose design methods that can be
+realized within the same topology. At first, we performed an
+analysis of periodical infinite structures on the level of input
+and grid impedances. Then we proceeded to design actual
+implementations as supercells formed by six metal patches
+placed on top of a grounded dielectric substrate. Further, we
+analyzed the angular response of the designed metasurfaces
+(a)
+(b)
+(c)
+(d)
+Fig. 10: Frequency bandwidth patterns. At 7.5 GHz, which is
+not plotted here, there is no diffracted mode, and all the energy
+goes back to the specular (normal) direction. The patterns are
+plotted between 7.75 GHz to 8.5 GHz with a 0.25 GHz step.
+Designed based on (a) phase gradient, (b) input impedance
+optimization, (c) grid impedance optimization, (d) non-local
+optimization. All patterns are normalized to the main lobe
+amplitude for each case.
+and finally presented far-field radiation patterns of finite-size
+structures.
+In this work, we provide a comparative summary of the
+main features of previously introduced design methods as well
+as present an original study of a property that is frequently left
+unstudied: the angular response. This study can be considered
+referential for engineers working on reconfigurable intelligent
+surfaces, where similar design methods are utilized.
+ACKNOWLEDGMENT
+This work was supported by the European Union’s Hori-
+zon 2020 research and innovation programme under the
+Marie Skłodowska-Curie grant agreement No 956256 (project
+METAWIRELESS), and the Academy of Finland (grant
+345178).
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+Entangling microwaves with optical light
+Rishabh Sahu⋆,1, † Liu Qiu⋆,1, ‡ William Hease,1 Georg Arnold,1
+Yuri Minoguchi,2 Peter Rabl,2 and Johannes M. Fink1, §
+1Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria
+2Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1040 Vienna, Austria
+(Dated: January 10, 2023)
+Entanglement is a genuine quantum mechanical property and the key resource in currently developed quantum technologies.
+Sharing this fragile property between superconducting microwave circuits and optical or atomic systems would enable new
+functionalities but has been hindered by the tremendous energy mismatch of ∼ 105 and the resulting mutually imposed loss and
+noise. In this work we create and verify entanglement between microwave and optical fields in a millikelvin environment. Using
+an optically pulsed superconducting electro-optical device, we deterministically prepare an itinerant microwave-optical state that
+is squeezed by 0.72+0.31
+−0.25 dB and violates the Duan-Simon separability criterion by > 5 standard deviations. This establishes
+the long-sought non-classical correlations between superconducting circuits and telecom wavelength light with wide-ranging
+implications for hybrid quantum networks in the context of modularization, scaling, sensing and cross-platform verification.
+Over the past decades we have witnessed spectacular
+progress in our capabilities to manipulate and measure
+genuine quantum mechanical properties, such as quan-
+tum superpositions and entanglement, in a variety of
+physical systems. These techniques serve now as the ba-
+sis for the development of quantum technologies, where
+the demonstration of quantum supremacy with tens of
+superconducting qubits [1], an ultra-coherent quantum
+memory with nuclear spins [2], and distributed quan-
+tum entanglement over tens of kilometers using optical
+photons [3] represent just a few of the highlights that
+have already been achieved.
+Going forward, combin-
+ing these techniques [4–6] will enable the realization of
+general-purpose quantum networks, where remote quan-
+tum nodes, capable of storing and processing quantum
+information, seamlessly communicate with each other by
+distributing entanglement over optical channels [7]. As-
+pects of this approach have already been adopted to con-
+nect and entangle various quantum platforms remotely,
+involving single atoms, ions, atomic ensembles, quantum
+dots, rare-earth ions and nitrogen-vacancy centers [8].
+However, such long-distance quantum connectivity is
+considerably more difficult to achieve with other promis-
+ing platforms, such as semiconductor spin qubits or local
+cryogenic networks of superconducting circuits [9, 10],
+where no natural interface to room temperature noise-
+resilient optical photons is available.
+To overcome this limitation, a lot of effort is cur-
+rently focused on the development of coherent quantum
+transducers between microwave and optical photons [11–
+16]. Direct noiseless conversion of a quantum state typ-
+ically relies on a beam splitter process, where a strong
+driving field mediates the conversion between weak mi-
+crowave and optical signals - a deterministic approach
+with exceptionally stringent requirements on conversion
+† [email protected]
+‡ [email protected]
+§ jfi[email protected]
+⋆ These authors contributed equally to this work.
+efficiency and added classical noise that are still out of
+reach. Alternatively, the direct generation of quantum-
+correlated microwave-optical photon pairs can also be
+used as a resource for quantum teleportation and en-
+tanglement distribution in the continuous and discrete
+variable domain [17–19].
+In this paper we use an ultra-low noise cavity electro-
+optical device to generate such non-classical correlations
+in a deterministic protocol [20]. It consists of a 5 mil-
+limeter diameter, 150 µm thick lithium niobate optical
+resonator placed inside a superconducting aluminum mi-
+crowave cavity at a temperature of 7 mK, described in
+detail in Ref. [21]. As depicted in Fig. 1A, the microwave
+mode ˆae is co-localized with and electro-optically coupled
+to the optical whispering gallery modes at ωo/(2π) ≈
+193.46 THz via the Pockels effect. We match the tunable
+microwave resonance frequency ωe/(2π) to the free spec-
+tral range (FSR) of 8.799 GHz to realize a triply-resonant
+system [22, 23] with the interaction Hamiltonian,
+ˆHint = ℏg0ˆapˆa†
+eˆa†
+o + h.c.,
+(1)
+with g0 the vacuum electro-optical coupling rate, and
+ˆap (ˆao) the annihilation operator of the optical pump
+(Stokes) mode [24]. Here we have ignored the interac-
+tion with the suppressed optical anti-Stokes mode ˆat, as
+shown in Fig. 1b (see supplementary information).
+In this sideband suppressed situation, efficient two-
+mode squeezing is achieved with a strong resonant opti-
+cal pump tone, yielding the simple effective Hamiltonian,
+ˆHeff = ℏg0√¯np(ˆa†
+eˆa†
+o + ˆaeˆao), where ¯np = ⟨ˆa†
+pˆap⟩ is the
+mean intra-cavity photon number of the optical pump
+mode. Deterministic continuous-variable (CV) entangle-
+ment between the out-propagating microwave and opti-
+cal field can be generated below the parametric instability
+threshold (C < 1) in the quantum back-action dominated
+regime, where the quantum noise exceeds the microwave
+thermal noise. Here C = 4¯npg2
+0/(κeκo) is the coopera-
+tivity with the vacuum coupling rate g0/2π ≈ 37 Hz, and
+the total loss rates of the microwave and optical Stokes
+modes κe/2π ≈ 11 MHz and κo/2π ≈ 28 MHz. The re-
+quired ultra-low noise operation is achieved despite the
+arXiv:2301.03315v1  [quant-ph]  9 Jan 2023
+
+2
+!p
+^ap,in
+^ao,out
+^ae,out
+!e
+ae^
+ao^
+ao
+ap
+!p
+!p
+!e
+- FSR
+!p+ FSR
+= FSR
+= 1)
+^at
+^
+^
+ae^
+(me
+- 1)
+(mp
++ 1)
+(mp
+mp
+b
+a
+FIG. 1. Physical and conceptual mode configuration.
+a, Simulated microwave (left) and optical (right) mode dis-
+tribution with azimuthal number me = 1 and mo = 17 (for
+illustration, experimentally mo ≈ 20 000). Phase matching is
+fulfilled due to the condition mo = mp − me and entangle-
+ment is generated and verified between the out-propagating
+microwave field ˆae,out and the optical Stokes field ˆao,out. b,
+Sketch of the density of states of the relevant modes.
+Un-
+der the condition ωp − ωo = ωe the strong pump tone in ˆap
+efficiently produces entangled pairs of microwave and optical
+photons in ˆae and ˆao via spontaneous parametric downconver-
+sion. Frequency up-conversion is suppressed via hybridization
+of the anti-Stokes mode ˆat with an auxiliary mode.
+required high power optical pump due to slow heating of
+this millimeter sized device [21].
+In the following we characterize the microwave and op-
+tical output fields via the dimensionless quadrature pairs
+Xj and Pj (j = e, o for microwave and optics), which
+satisfy the canonical commutation relations [Xj, Pj] = i.
+A pair of Einstein-Podolsky-Rosen (EPR)-type opera-
+tors X+ =
+1
+√
+2 (Xe + Xo) and P− =
+1
+√
+2 (Pe − Po) are
+then constructed, and the microwave and optical output
+fields are entangled, if the variance of the joint opera-
+tors is reduced below the vacuum level, i.e.
+∆−
+EPR =
+�
+X2
++
+�
++
+�
+P 2
+−
+�
+< 1.
+This is commonly referred to as
+the Duan-Simon criterion [25, 26], which we apply to
+each near-resonant frequency component of the two-mode
+squeezed output mode (see SI).
+For efficient entanglement generation, we use a 250 ns
+long optical pump pulse (≈ 244 mW, C ≈ 0.18, ¯np ≈
+1.6 × 1010) at a 2 Hz repetition rate, cf.
+pulse 1 in
+Fig. 2a. The entangled output optical signal is filtered
+via a Fabry-Perot cavity to reject the strong pump. The
+entangled microwave output is amplified with a high-
+electron-mobility transistor amplifier. Both outputs are
+down-converted to an intermediate frequency of 40 MHz
+with two local oscillators (LO) and the four quadra-
+tures are extracted from heterodyne detection.
+Long-
+term phase stability between the two LOs is achieved via
+extracting the relative phase drift by means of a second
+phase alignment pump pulse that is applied 1 µs after
+each entanglement pulse, together with a coherent reso-
+nant microwave pulse, shown in Fig. 2(a). This generates
+a high signal-to-noise coherent optical signal via stimu-
+lated parametric down-conversion and allows for aligning
+the phase of each individual measurement.
+Figure 2b(c) shows the time-domain average power
+over one million averages for the on-resonant microwave
+(optics) signal with a spectrally under-sampled 40 MHz
+bandwidth (hence not revealing the full pulse ampli-
+tude). The two insets show the microwave (optical) signal
+from spontaneous parametric down-conversion (SPDC)
+due to pulse 1, cf. Fig. 2a, with an emission bandwidth
+of ≈ 10 MHz. The larger signals during the second half
+of the experiment are the reflected microwave pulse and
+the generated optical tone (due to pulse 2) that is used
+for LO phase alignment. The raw power measurements
+are divided by the measurement bandwidth and rescaled
+such that the off-resonant response matches the noise
+photon number Nj,add + 0.5 of the measurement setup.
+Ne,add = 13.1 ± 0.4 (2σ errors throughout the paper)
+due to loss and amplifier noise and No,add = 5.5 ± 0.2
+due to optical losses are carefully determined using noise
+thermometry of a temperature controlled 50 Ω load and
+4-port calibration, respectively (see SI). Using this pro-
+cedure ensures that the reported photon number units
+correspond to the signals at the device outputs.
+We continue the analysis in the frequency domain by
+calculating the Fourier transform of each measurement
+for three separate time intervals - before (2 µs), dur-
+ing (200 ns) and right after the entangling pump pulse
+(500 ns).
+Figure 2d shows the resulting average mi-
+crowave noise spectra for all three time intervals with
+corresponding fit curves (dashed lines) and theory (solid
+line). Before and after the pump pulse, the on-resonant
+microwave output field takes on values above the vac-
+uum level, with fitted intrinsic microwave bath occu-
+pancies ¯ne,int = 0.03 ± 0.01 and 0.09 ± 0.03, respec-
+tively. By fitting additional power dependent measure-
+ments, we independently verify that the observed noise
+floor corresponds to a waveguide bath occupancy of only
+¯ne,wg = 0.001±0.002 at the very low average pump power
+of ≈ 0.12 µW used in this experiment (see SI). The mea-
+sured noise floor therefore corresponds to the shot noise
+equivalent level Ne,add + 0.5 (gray dashed lines). Sim-
+ilarly, Fig. 2e shows the obtained average optical noise
+spectra during and after the pump, referenced to the
+measured shot noise level before the pulse. As expected,
+there is no visible increase of the optical noise level after
+the pulse.
+During the pump pulse, an approximately Lorentzian
+shaped microwave and optical power spectrum are gen-
+erated via the SPDC process. We perform a joint fit of
+the microwave and optical power spectral density during
+the pulse using a 5-mode theoretical model that includes
+
+3
+600
+400
+200
+0
+800
+1000
+13.6
+13.7
+1.5
+2.0
+2.5
+3.0
+0
+50
+100
+150
+1.5
+2.0
+2.5
+3.0
+6.05
+6.10
+!e
+!o
+!p
+!e
+!o
+Input power
+t
+Output power
+Before
+pulse
+Pulse
+1
+Entangle
+Pulse
+2
+After
+pulse
+Phase
+align
+MW
+signal
+MW
+reflection
+b
+a
+c
+0
+1
+2
+3
+t (μs)
+4
+5
+0
+1
+2
+3
+4
+5
+13.75
+14.00
+6.00
+6.25
+-20
+0
+20
+-20
+0
+20
+Before-pulse
+Theory
+In-pulse
+After-pulse
+In-pulse
+After-pulse
+Theory
+Δ!e=2π (MHz)
+Δ!o=2π (MHz)
+e
+d
+Ne,det (photons s-1 Hz-1)
+No,det (photons s-1 Hz-1)
+Ne,det (photons s-1 Hz-1)
+No,det (photons s-1 Hz-1)
+(μs)
+t
+Ne,add + 0.5
+No,add + 0.5
+FIG. 2. Measurement sequence and noise powers. a, Schematic pulse sequence of a single measurement. The optical
+pulse 1 is applied at ωp and amplifies the vacuum (and any thermal noise) in the two modes ˆae and ˆao, thus generating the
+SPDC signals. 1 µs later, a second optical pump with about 10 times lower power is applied together with a coherent microwave
+pulse at ωe. The microwave photons stimulate the optical pump to down-convert, which generates a coherent pulse in the ˆao
+mode that is used to extract slow LO phase drifts. b and c, Measured output power in the ˆae and ˆao mode in units of photons
+per second in a 1 Hz bandwidth and averaged over a million experiments. The SPDC signals are shown in the insets with
+the dashed gray lines indicating the calibrated detection noise floor Nj,add + 0.5. d, Corresponding microwave output power
+spectral density vs. ∆ωe = ω − ωe centered on resonance right before the entanglement pulse, during the pulse and right
+after the pulse, as indicated in panel a. Yellow and green dashed lines are fits to a Lorentzian function, which yields the
+microwave bath occupancies before and after the entangling pulse. Error bars represent the 2σ statistical standard error and
+the shaded regions represent the 95% confidence interval of the fit. (e), Corresponding optical output power spectral density
+vs. ∆ωo = ωo − ω during and after the entanglement pulse, both normalized to the measured noise floor before the pulse. The
+in-pulse noise spectra in panels d and e are fit jointly with theory, which yields C = 0.18 ± 0.01 and ¯ne,int = 0.07 ± 0.03.
+the effects of measurement bandwidth. In this model, the
+in-pulse microwave bath occupancy ¯ne,int = 0.07 ± 0.03
+and the cooperativity C = 0.18 ± 0.01 are the only free
+fit parameters. Here the narrowed microwave linewidth
+κe,eff/2π = 9.8 ± 1.8 MHz (taken from a Lorentzian
+fit) agrees with coherent electro-optical dynamical back-
+action [27], where κe,eff = (1 − C)κe. We conclude that
+this cavity electro-optical device is deep in the quantum
+back-action dominated regime, a prerequisite for efficient
+microwave-optics entanglement generation.
+For each frequency component the bipartite Gaussian
+state of the propagating output fields can be fully char-
+acterized by the 4 × 4 covariance matrix (CM) Vij =
+⟨δuiδuj + δujδui⟩ /2, where δui = ui − ⟨ui⟩ and u ∈
+{Xe, Pe, Xo, Po} (see SI). The diagonal elements in V cor-
+respond to the individual output field quadrature vari-
+ances in dimensionless units. These are obtained from
+the measured variances after subtracting the measured
+detection noise offsets shown in Fig. 2, i.e. Vii(∆ω) =
+Vii,meas(∆ωi) − Ni,add. The obtained CM from the data
+in Fig. 2 at ∆ω = 0 is shown in Fig. 3a in its standard
+form. It corresponds to the quantum state of the propa-
+gating modes in the coaxial line and the coupling prism
+attached to the device output, i.e. before setup losses or
+amplification incur. The non-zero off-diagonal elements
+indicate strong correlations between microwave and op-
+tical quadratures.
+The two-mode squeezed quadratures are more intu-
+itively visualized in terms of the quasi-probability Wigner
+function,
+W(u) = exp[− 1
+2uV −1uT ]
+π2�
+det(V)
+,
+(2)
+
+4
+where u = (Xe, Pe, Xo, Po). Different marginals of this
+Wigner function are shown in Fig. 3b, where the (Xe,Xo)
+and (Pe,Po) marginals show two-mode squeezing in the
+diagonal and off-diagonal directions.
+The two cross-
+quadrature marginals show a slightly different amount
+of squeezing, which is due to the statistical uncertainty
+in the measured CM.
+Figure 3c shows the amount of two-mode squeezing
+between microwave and optical photon pairs. Correla-
+tions are observed at ∆ωj = ±(ω − ωj) around the reso-
+nances due to energy conservation in the SPDC process
+(see SI). The averaged microwave quadrature variance
+(purple dots) ¯V11 = (V11+V22)/2 and the averaged optics
+quadrature variance (green dots) ¯V33 = (V33 +V44)/2 are
+shown in the top panel along with the prediction from
+our five-mode theory (solid line) and a simple fit to a
+Lorentzian function (dashed line), showing perfect agree-
+ment. Measured microwave-optical correlations (yellow
+dots) ¯V13 = (V13 − V24)/2 and the Lorentzian fit (dashed
+line) lie slightly below the theoretical prediction (solid
+line), which we assign to remaining imperfections in the
+phase stability (see SI).
+The bottom two panels of Fig. 3c show the squeezed
+and anti-squeezed joint quadrature variances ∆∓
+EPR =
+¯V11 + ¯V33 ∓ 2 ¯V13 (red and blue color respectively). We
+observe two-mode squeezing below the vacuum level, i.e.
+∆−
+EPR < 1, with a bandwidth close to the effective mi-
+crowave linewidth. The maximal on-resonant two-mode
+squeezing is ∆−
+EPR = 0.85+0.05
+−0.06 (2σ, 95% confidence) for
+∼1 million pulses with ¯V11 = 0.93, ¯V33 = 0.84 and ¯V13 =
+0.46. Hence, this deterministically generated microwave-
+optical state violates the Duan-Simon separability crite-
+rion by > 5σ. Note that this error also takes into account
+systematics in the added noise calibration used for scal-
+ing the raw data (see SI). These values correspond to a
+state purity of ρ = 1/(4
+�
+det[V ]) = 0.44 and demon-
+strate microwave-optical entanglement between output
+photons with a logarithmic negativity of EN = 0.17. The
+supplementary material contains substantial additional
+data for longer pulses and varying optical pump power,
+which corroborates the presented results and findings, al-
+beit with lower statistical significance for each individual
+pump configuration (see SI).
+Conclusions and outlook
+In conclusion, we have demonstrated deterministic
+quantum entanglement between propagating microwave
+and optical photons,thus establishing a non-classical
+communication channel between circuit quantum electro-
+dynamics and quantum photonics. Our device can read-
+ily be used for probabilistic heralding assisted protocols
+[7, 28, 29] to mitigate optical setup losses and extend the
+entanglement to room temperature fiber optics. We ex-
+pect that the pulse repetition rate can be increased by or-
+ders of magnitude with improved thermalization, higher
+microwave and optical quality factors, and electro-optic
+coupling enhancements that reduce the required pump
+power and the associated thermal load.
+Coupling effi-
+ciency improvements will allow for higher levels of two-
+mode squeezing and facilitate also deterministic entangle-
+ment distribution schemes [30], teleportation-based state
+transfer
+[20, 31] and quantum-enhanced remote detec-
+tion [32]. Being fully compatible with superconducting
+qubits in a millikelvin environment such a device will fa-
+cilitate the integration of remote superconducting quan-
+tum processors into a single coherent optical quantum
+network.
+This is not only relevant for modularization
+and scaling [33, 34], but also for efficient cross-platform
+verification of classically intractable quantum processor
+results [35].
+ACKNOWLEDGMENTS
+L.Q. acknowledges fruitful discussions with Jie Li and
+David Vitali. This work was supported by the European
+Research Council under grant agreement no.
+758053
+(ERC StG QUNNECT) and the European Union’s
+Horizon 2020 research and innovation program under
+grant agreement no. 899354 (FETopen SuperQuLAN).
+L.Q. acknowledges generous support from the ISTFEL-
+LOW programme. W.H. is the recipient of an ISTplus
+postdoctoral fellowship with funding from the European
+Union’s Horizon 2020 research and innovation program
+under the Marie Sk�lodowska-Curie grant agreement no.
+754411. G.A. is the recipient of a DOC fellowship of the
+Austrian Academy of Sciences at IST Austria. J.M.F.
+acknowledges support from the Austrian Science Fund
+(FWF) through BeyondC (F7105) and the European
+Union’s Horizon 2020 research and innovation programs
+under grant agreement No 862644 (FETopen QUAR-
+TET).
+AUTHOR CONTRIBUTIONS
+RS, WH, LQ, and GA worked on the setup. RS and
+LQ performed measurements. LQ and RS did the data
+analysis.
+LQ developed the theory with contributions
+from RS, YM and PR. RS and LQ wrote the manuscript
+with contributions from all authors. JMF supervised the
+project.
+DATA AVAILABILITY STATEMENT
+All data and code used to produce the figures in this
+manuscript will be made available on Zenodo.
+
+5
+1.0
+2.0
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+2
+2
+-2
+-2
+0
+0
+-10
+-20
+10
+20
+c
+b
+(photons s-1 Hz-1)
+(MHz)
+0
+2
+-2
+2
+-2
+0
+2
+-2
+0
+2
+-2
+0
+2
+-2
+0
+2
+-2
+0
+a
+Vij
+Pe
+Pe
+Pe
+Pe
+Pe
+Po
+Po
+Po
+Po
+Po
+Xo
+Xo
+Xo
+Xo
+Xo
+Xe
+Xe
+Xe
+Xe
+Xe
+-0.5
+0.0
+0.5
+1.0
+(V11 + V22)/2
+(V33 + V44)/2
+(V13 − V24)/2
+V
+ΔEPR
+-
+ΔEPR
++
+Δω/2π
+FIG. 3. Characterization of the two-mode squeezed state. a, Measured covariance matrix Vij in its standard form plotted
+for ∆ωj = 0 based on 925000 measurements. b, Corresponding Wigner function marginals of different output quadrature pairs
+in comparison to vacuum. The contours in blue (grey) represent the 1/e fall-off from the maximum for the measured state
+(vacuum). Middle two plots show two-mode squeezing below the vacuum level in the diagonal and off-diagonal directions.
+c, Top panel, the measured average microwave output noise ¯V11 = (V11 + V22)/2 (purple), the average optical output noise
+¯V33 = (V33 + V44)/2 (green) and the average correlations ¯V13 = (V11 − V24)/2 (yellow) as a function of the measurement
+detunings. The solid lines represent the joint theory fit and the dashed lines are individual Lorentzian fits to serve as a guide
+to eye. The middle (bottom) panel shows two-mode squeezing in red (anti-squeezing in blue) calculated from the top panels
+as ∆±
+EPR = ¯V11 + ¯V33 ± 2 ¯V13. The darker color error bars represent the 2σ statistical error and the outer (faint) 2σ error bars
+also include the systematic error in calibrating the added noise of the measurement setup.
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+Supplementary Information for: ”Entangling microwaves with optical light”
+Rishabh Sahu,1, ∗ Liu Qiu,1, ∗ William Hease,1 Georg Arnold,1 Yuri Minoguchi,2 Peter Rabl,2 and Johannes M. Fink1
+1Institute of Science and Technology Austria, am Campus 1, 3400 Klosterneuburg, Austria
+2Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1040 Vienna, Austria
+(Dated: January 10, 2023)
+CONTENTS
+Page
+I. Theory
+3
+A
+Covariance Matrix from Input-Output Theory
+. . . . . . . . . . . . . . . . . . . . . . . . .
+3
+1
+Quantum Langevin Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+2
+Input-Output-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+3
+Covariance Matrix of Filtered Output Fields . . . . . . . . . . . . . . . . . . . . . . . .
+6
+B
+Heterodyne Detection, Added Noise and Filtering . . . . . . . . . . . . . . . . . . . . . . . .
+7
+1
+Heterodyne Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+2
+Realistic Measurements: Added Noise and Gain
+. . . . . . . . . . . . . . . . . . . . . .
+8
+3
+Covariance Matrix from Realistic Heterodyne Measurements
+. . . . . . . . . . . . . . . .
+9
+C
+Entanglement Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+1
+Duan Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+2
+Logarithmic Negativity and Purity
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+II. Experimental Setup
+11
+III. Setup Characterization and calibration
+11
+A
+Microwave added noise calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+B
+Optical added noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+IV. Data treatment
+15
+A
+Time-domain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+B
+Pulse post-selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+C
+Frequency domain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+D
+Joint-quadrature correlations
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+V. Quadrature histogram raw data
+19
+VI. Non-classical correlations with 600 ns long optical pump pulses
+20
+VII. Error analysis
+20
+A
+Statistical error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+B
+Systematic error
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+22
+References
+22
+∗ These two authors contributed equally
+arXiv:2301.03315v1  [quant-ph]  9 Jan 2023
+
+2
+Introduced in Main Text
+ˆae
+microwave mode (annihilation operator)
+me
+microwave mode azimuthal number, me = 1
+ωe
+microwave cavity frequency
+κe, κe,eff, κe,in, κe,0 microwave total loss, effective total loss, waveguide coupling and intrinsic loss rates
+¯ne,int, ¯ne,wg
+microwave intrinsic and waveguide bath occupancy
+Ne,add
+added noise in the microwave detection
+ˆao, ˆap, ˆat, ˆatm
+optical Stokes, pump, anti-Stokes, and transverse-magnetic mode (annihilation operator)
+ˆae/o,out
+microwave and optical output field from the device
+mp
+optical pump mode azimuthal number, mp ∼ 20000
+κo
+optical total loss rate
+No,add
+added noise in the optical detection
+¯np
+mean photon number of the optical pump mode
+g0
+electro-optical vacuum coupling rate
+g
+photon enhanced electro-optical coupling rate (g = √¯npg0})
+C
+cooperativity ( C = 4g2/κeκo )
+Xe,Pe
+quadratures of the microwave output field
+Xo,Po
+quadratures of the optical Stokes output field
+V
+covariance matrix of the bipartite Gaussian state, Vij = ⟨∆ui∆uj + ∆uj∆ui⟩ /2, where
+∆ui = ui − ⟨ui⟩ and u ∈ {Xe, Pe, Xo, Po}.
+Nii,add
+added noise in the quadrature variances measurements, N11,add = N22,add = Ne,add,
+N33,add = N44,add = No,add
+Vii,meas
+diagonal covariance matrix elements from the calibrated measurement record, Vii
+=
+Vii,meas − Nii,add
+V11,V22, ¯V11
+quadrature variances of the microwave output field, ¯V11 = V11+V22
+2
+V33,V44, ¯V33
+quadrature variances of the optical Stokes output field, ¯V33 = V33+V44
+2
+V13,V24, ¯V13
+cross-correlation between microwave and optical quadratures, ¯V13 = V13−V24
+2
+∆∓
+EPR
+squeezed and anti-squeezed joint quadrature variance between microwave and optical output
+field, ∆∓
+EPR = ¯V11 + ¯V33 ∓ ¯V13
+Introduced in Supplementary Information
+J
+coupling rate between the optical anti-Stokes mode and TM mode
+ˆae/o,in
+input field (noise) operator for the microwave and optical mode
+ˆae/o,0
+noise operator for the microwave and optical intrinsic loss
+ηj
+external cavity coupling efficiency of individual mode, j ∈ (e, o, p, t)
+G(ω)
+spectral filter of the output field
+ˆA(Ω)
+Fourier transform of operator ˆA(t), ˆA(Ω) =
+�
+dt eiΩt ˆA(t), ˆA†(Ω) =
+�
+dt ˆA†(t)eiΩt = [ ˆA(−Ω)]†
+ˆX(ωn)
+X quadrature of the output spectral mode, ˆX(ωn) =
+1
+√
+2
+� ∞
+−∞ dω G(ωn − ω)ˆaout(ω) + h.c.
+ˆP(ωn)
+P quadrature of the output spectral mode, ˆP(ωn) =
+1
+√
+2i
+� ∞
+−∞ dω G(ωn − ω)ˆaout(ω) + h.c.
+S ˆ
+A ˆ
+B(Ω)
+Two-time correlation of two operators, S ˆ
+A ˆ
+B(Ω) =
+1
+√
+2π
+� ∞
+−∞
+�
+ˆA(t) ˆB(t′)
+�
+eiΩtdt
+∆LO
+local oscillator and signal frequency difference in heterodyne measurement, ∆LO = ωLO−ωsig
+Iout(t), Iout(ω)
+unitless output field in the equivelant heterodyne detection
+SII(ω)
+double-sided noise spectrum of the output field in the equivelant heterodyne detection
+Gdet(ω)
+frequency dependent detection gain in the heterodyne detection
+ˆIX/P,det(ωn)
+detected output photocurrent quadratures in heterodyne detection, including detection gain
+ˆIX/P,out(ωn)
+unitless output field quadratures from Iout including added noise
+D(ω)
+covariance matrix of the detected quadratures from the heterodyne measurement record
+Vmeas(ω)
+covariance matrix of the total measured output field quadratures including added noise
+ˆX+(ω)
+joint quadrature of ˆXe(ω) and ˆXo(−ω), ˆX+(ω) = ( ˆXe(ω) + ˆXo(−ω))/
+√
+2
+ˆP−(ω)
+joint quadrature of ˆPe(ω) and ˆPo(−ω), ˆP−(ω) = ( ˆPe(ω) − ˆPo(−ω))/
+√
+2
+EN
+logarithm negativity
+ρ
+state purity
+
+3
+I.
+THEORY
+A.
+Covariance Matrix from Input-Output Theory
+1.
+Quantum Langevin Equations
+Our cavity electro-optical (CEO) device consists of a millimeter-sized lithium niobate optical resonator in a 3-D
+superconducting microwave cavity at mK temperature [1]. The Pockels effect in lithium niobate allows for direct
+coupling between the microwave and optical whispering gallery modes with maximal field overlap. The optical free
+spectral range (FSR) matches the microwave cavity frequency, with microwave azimuthal mode number me = 1. As
+shown in Fig. 1 in the main text, resonant three-wave mixing between the microwave mode (ˆae) and three adjacent
+transverse-electric (TE) optical modes, i.e. Stokes (ˆao), pump (ˆap), and anti-Stokes (ˆat) mode, arises via the cavity
+enhanced electro-optical interaction [2, 3]. In addition, the anti-Stokes mode is coupled to a transverse-magnetic (TM)
+optical mode (ˆatm) of orthogonal polarization and similar frequency at rate of J [4]. This results in a total interaction
+Hamiltonian,
+ˆHI/ℏ = g0(ˆa†
+pˆaeˆao + ˆa†
+pˆa†
+eˆat) + Jˆatˆa†
+tm + h.c.,
+(1)
+with g0 the vacuum electro-optical coupling rate.
+For efficient entanglement generation, we drive the pump mode strongly with a short coherent input pulse ¯ap,in(t)
+at frequency ωp [1], which results in a time-dependent mean intra-cavity field of the pump mode ¯ap(t),
+˙¯ap =
+�
+i∆p − κp
+2
+�
+¯ap + √ηpκp¯ap,in,
+(2)
+where the pump tone is detuned from the pump mode by ∆p = ωp − ωo,p, with κp and ηp as the pump mode loss rate
+and external coupling efficiency. In our experiments, we actively lock the laser frequency to the pump mode resonance,
+with ∆p = 0.
+The presence of the strong pump field results in an effective interaction Hamiltonian,
+ˆHI,eff/ℏ = g(ˆaeˆao + ˆaeˆa†
+t) + Jˆatˆa†
+tm + h.c.,
+(3)
+with multiphoton coupling rate g = ¯apg0. This includes the two-mode-squeezing (TMS) interaction between the
+Stokes and microwave mode, and beam-splitter (BS) interaction between the anti-Stokes mode and microwave mode,
+resulting in scattered Stokes and anti-Stokes sidebands that are located on the lower and upper side of the pump
+tone by Ωe away. Microwave-optics entanglement between the microwave and optical Stokes output field can be
+achieved via spontaneous parametric down-conversion (SPDC) process due to TMS interaction [5], which is further
+facilitated by the suppressed anti-Stokes scattering due to the strong coupling between anti-Stokes and TM modes.
+We can obtain the full dynamics of the intracavity fluctuation field in the rotating frame of the scattered sidebands
+and microwave resonance, which can be described by the quantum Langevin equations (QLE),
+˙ˆae = −κe
+2 ˆae − igˆa†
+o − ig∗ˆat + √ηeκeδˆae,in +
+�
+(1 − ηe) κeδˆae,0,
+(4)
+˙ˆao =
+�
+iδo − κo
+2
+�
+ˆao − igˆa†
+e + √ηoκoδˆao,in +
+�
+(1 − ηo) κoδˆao,0,
+(5)
+˙ˆat =
+�
+iδt − κt
+2
+�
+ˆat − ig∗ˆae − iJˆatm + √κtδˆat,vac,
+(6)
+˙ˆatm =
+�
+iδtm − κtm
+2
+�
+ˆatm − iJˆat + √κtmδˆatm,vac,
+(7)
+with κj the total loss rate of the individual mode where j ∈ (e, o, t, tm), and ηk the external coupling efficiency
+of the input field where k ∈ (e, o). We note that, the optical light is only coupled to the TE modes via efficient
+prism coupling, with effective mode overlap Λ factor included in ηo for simplicity [1]. δj corresponds to the frequency
+difference between mode j and scattered sidebands, with δo = ωo,p − ωe − ωo and δt/tm = ωo,p + ωe − ωt/tm, which
+are mostly given by FSR and ωe mismatch, with additional contributions from optical mode dispersion and residual
+optical mode coupling. We note that, for resonant pumping, we have δo = −δt in the case of absent optical mode
+dispersion and residual mode coupling. In our experiments, we tune the microwave frequency to match the optical
+FSR, i.e. ωe = ωo,p − ωo.
+
+4
+The equation of motion of all relevant modes may be represented more economically in the form
+˙v(t) = M(t)v(t) + Kfin(t),
+(8)
+where we define the vectors of mode and noise operators
+v = (ˆae, ˆa†
+e, ˆao, ˆa†
+o, ˆat, ˆa†
+t, ˆatm, ˆa†
+tm)⊤,
+fin = (δˆae,0, δˆa†
+e,0, δˆae,in, δˆa†
+e,in, δˆao,0, δˆa†
+o,0, δˆao,in, δˆa†
+o,in, δˆat,vac, δˆa†
+t,vac, δˆatm,vac, δˆa†
+tm,vac)⊤,
+(9)
+as well as the matrices that encode the deterministic part of the QLE,
+M(t) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+− κe
+2
+0
+0
+−ig(t)
+−ig∗(t)
+0
+0
+0
+0
+− κe
+2
++ig∗(t)
+0
+0
+ig(t)
+0
+0
+0
+−ig(t) iδo − κo
+2
+0
+0
+0
+0
+0
+ig∗(t)
+0
+0
+−iδo − κo
+2
+0
+0
+0
+0
+−ig(t)
+0
+0
+0
+iδt − κt
+2
+0
+−iJ
+0
+0
+ig∗(t)
+0
+0
+0
+−iδt − κt
+2
+0
+iJ
+0
+0
+0
+0
+−iJ
+0
+iδtm − κtm
+2
+0
+0
+0
+0
+0
+0
+iJ
+0
+−iδtm − κtm
+2
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+(10)
+and
+K =
+�
+�
+�
+�
+�
+(1 − ηe)κe √ηeκe
+0
+0
+0
+0
+0
+0
+�
+(1 − ηo)κo √ηoκo
+0
+0
+0
+0
+0
+0
+√κt
+0
+0
+0
+0
+0
+0
+√κtm
+�
+�
+�
+� ⊗ 12,
+(11)
+which keeps track on which modes the noise acts.
+2.
+Input-Output-Theory
+In the experiment the pump field is turned on at t = 0 and kept on until τpulse. For the optical pump pulse with
+length τpulse =250 ns (600 ns, see main text), we reject a certain τdelay = 50 ns (100 ns) from the beginning of pulse
+data. Since κpτdelay ≳ 1 we may assume that after τdelay the system has approached its steady state and especially
+that the pump mode is in its steady state. Consequently we may assume that g(t > τdelay) ≃ g is constant over time.
+One important figure of merit is the multiphoton cooperativity C = 4g2/κoκe, a measure for coherent coupling versus
+the microwave and optical dissipation. Efficient entanglement generation can be achieved with complete anti-Stokes
+scattering suppression, while below the parametric instability threshold, i.e. C < 1.
+The output fields of the CEO device are
+fout(t) = (ˆae,out(t), ˆa†
+e,out(t), ˆao,out(t), ˆa†
+o,out(t))⊤,
+(12)
+which consist of a contribution which was entangled via the coherent interactions v and a contribution which has
+not interacted with the device fin. The output field fout will then propagate to the measurement device and is most
+economically represented within the framework of input-output theory [6],
+fout(t) = Lfin(t) − Nv(t),
+(13)
+where we define the matrices
+N = (NJ, 04),
+with
+NJ = Diag(√ηeκe, √ηeκe, √ηoκo, √ηoκo),
+(14)
+and
+L =
+�
+0 1 0 0 0 0
+0 0 0 1 0 0
+�
+⊗ 12.
+(15)
+As all modes have reached steady state, the correlations in the output field may be obtained by going to Fourier
+
+5
+domain. Here we commit to following convention of the Fourier transformation
+ˆA(ω) =
+1
+√
+2π
+� ∞
+−∞
+dω eiωt ˆA(t),
+(16)
+with the hermitian conjugate
+( ˆA(ω))† = A†(−ω).
+(17)
+Note that in this convention e.g. [ae(ω), a†
+e(ω′)] = δ(ω + ω′) are canonical pairs.
+In our experiments, we focus on the correlations between the output propagating spectral modes of frequencies
+ωe + ∆ωe and ωo − ∆ωo respectively for microwave and optical fields [7, 8]. We note that, due to energy conservation
+in the SPDC process, we only focus on microwave and optical photon pairs around resonances with anti-correlated
+frequencies, i.e. ∆ωe = ∆ωo = ∆ω. For this reason, we focus on the following vector of output fields in the rotating
+frame,
+fout(ω) = (ˆae,out(ω), ˆa†
+e,out(−ω), ˆao,out(−ω), ˆa†
+o,out(ω))⊤,
+(18)
+in the Fourier domain. From Eq. (8) we obtain
+v(ω) = [iωO − M]−1 · K
+�
+��
+�
+=S(ω)
+·fin(ω),
+(19)
+with
+O = Diag(1, −1, 1, 1) ⊗ σz.
+(20)
+Here we defined the vector of modes
+v(ω) = (ˆae(ω), ˆa†
+e(−ω), ˆao(−ω), ˆa†
+o(ω), ˆat(ω), ˆa†
+t(−ω), ˆatm(ω), ˆa†
+tm(−ω))⊤,
+(21)
+as well as the vector of input fields
+fin(ω) = (δˆae,0(ω), δˆa†
+e,0(−ω), δˆae,in(ω), δˆa†
+e,in(−ω), δˆao,0(−ω), δˆa†
+o,0(ω), δˆao,in(−ω), δˆa†
+o,in(ω),
+δˆat,vac(ω), δˆa†
+t,vac(−ω), δˆatm,vac(ω), δˆa†
+tm,vac(−ω))⊤
+(22)
+in the Fourier domain.
+The output fields (see Eq. (13)) of the CEO device are straight forwardly obtained since in the Fourier domain
+Eq. (13) is algebraic,
+fout(ω) = Lfin(ω) + Nv(ω) = (L + N · [iωO − M]−1 · K)fin(ω).
+(23)
+The input noise operator correlations are given by,
+⟨fin(ω)f †
+in(ω′)⟩ = Dδ(ω + ω′),
+(24)
+with
+D = Diag(¯ne,int + 1, ¯ne,int
+�
+��
+�
+bath:e
+, ¯ne,wg + 1, ¯ne,wg
+�
+��
+�
+waveguide:e
+, 1, 0
+����
+bath:o
+,
+1, 0
+����
+detector:o
+, 1, 0
+����
+bath:t
+,
+1, 0
+����
+bath:tm
+).
+(25)
+We note that, in our experiments, the microwave waveguide remains in the ground state, with ¯ne,wg = 0. The spectral
+correlations of different output field can be simply obtained analytically from
+⟨fout(ω)f †
+out(ω′)⟩ = S(ω)DS†(−ω)
+�
+��
+�
+˜
+Cff†(ω)
+δ(ω + ω′).
+(26)
+
+6
+Here we implicitly define the 4 × 4 matrix of output mode correlations with a single entry reading
+⟨ˆaout(ω)ˆbout(ω′)⟩ = ˜Cab(ω)δ(ω + ω′),
+(27)
+where the operators ˆaout(ω),ˆbout(ω) were chosen from components of fout(ω) in Eq. (16).
+3.
+Covariance Matrix of Filtered Output Fields
+We will now consider a situation where we define output field modes from a windowed Fourier transformation. Below
+we will then show that these are indeed the experimentally observed signals. We start by defining the (dimensionless)
+hermitian output field quadrature pair [8],
+ˆXα(ωn) =
+1
+√
+2T
+� T/2
+−T/2
+dτ eiωnτˆaα,out(τ) + h.c.,
+(28)
+ˆPα(ωn) =
+1
+√
+2Ti
+� T/2
+−T/2
+dτ eiωnτˆaα,out(τ) + h.c.,
+(29)
+which meets the canonical commutation relation [ ˆXα(ωn), ˆPβ(ωm)] = iδnmδαβ where α = e, o. Due to the finite
+window of the Fourier transformation, the frequencies ωn = 2π
+T n becomes discrete. The quadrature modes at discrete
+frequencies ωn can now be rewritten in terms of the (dimensionful) output fields fout(ω) from Eq. (23), which are
+defined in the continuous Fourier domain. Therefore the quadrature operators may be obtained by convolution with
+the a filter function G(ω)
+ˆXα(ωn) =
+1
+√
+2
+� ∞
+−∞
+dω G(ωn − ω)ˆaα,out(ω) + h.c.
+(30)
+ˆPα(ωn) =
+1
+√
+2i
+� ∞
+−∞
+dω G(ωn − ω)ˆaα,out(ω) + h.c.
+(31)
+Here the filter is
+G(ω) =
+1
+√
+2π
+� ∞
+−∞
+dτ eiωτ 1[0,T ](τ)
+√
+T
+=
+�
+2
+πT
+sin(ωT/2)
+ω
+,
+(32)
+which is obtained from a Fourier transformation of the unit function 1[−T/2,T/2](t) = 1(0) for |t| ≤ T/2 (|t| > T/2).
+A bipartite Gaussian state is characterized by the 4 × 4 covariance matrix (CM),
+VAB(ωn) = 1
+2⟨{δ ˆA(ωn), δ ˆB(ωn)}⟩.
+(33)
+Here we defined δ ˆA = ˆA − ⟨ ˆA⟩ an operator with zero mean ⟨δ ˆA⟩ = 0 and the quadratures from
+ˆA(ωn), ˆB(ωn) ∈ { ˆXe(ωn), ˆPe(ωn), ˆXo(−ωn), ˆPo(−ωn)}
+(34)
+and we also introduced the anti-commutator { ˆA, ˆB} = ˆA ˆB + ˆB ˆA. Note that the two-mode squeezing interaction
+results in correlation between frequency reversed pairs on the microwave ωn and the optical side −ωn. Since in our
+setting all first moments ⟨ ˆA⟩ = 0 the evaluation of the covariance matrix in Eq. (33) boils down to computing spectral
+correlations which are rewritten as
+⟨ ˆA(ωn) ˆB(ωn)⟩ =
+� ∞
+−∞
+dω
+� ∞
+−∞
+dω′ G(ωn − ω)G(ωn − ω′)⟨ ˆA(ω) ˆB(ω′)⟩
+=
+� ∞
+−∞
+dω
+� ∞
+−∞
+dω′ G(ωn − ω)G(−ωn − ω′)CAB(ω)δ(ω + ω′)
+=
+� ∞
+−∞
+dω F(ωn − ω)CAB(ω),
+(35)
+
+7
+where we used the property G(−ω) = G(ω) and defined the effective filter F(ω) = G(ω)2. Similar to Eq. (26), we
+defined the quadrature correlations
+CAB(ω) = (C(ω))AB = 1
+2
+�
+U ˜Cff †(ω)U † + (U ˜Cff †(ω)U †)⊤�
+AB .
+(36)
+Here the unitary matrix U = u ⊕ u, with
+u =
+1
+√
+2
+�
+1
+1
+−i i
+�
+,
+(37)
+corresponds to a rotation of the mode operators into quadrature operators ( ˆXα, ˆPα)⊤ = u · (ˆaα,out, ˆa†
+α,out)⊤. The
+covariance matrix of the quadrature modes at the discrete frequencies ωn is then obtained exactly by
+VAB(ωn) =
+� ∞
+−∞
+dω F(ωn − ω)CAB(ω),
+(38)
+where the quadrature correlations are convolved with an appropriate filter.
+B.
+Heterodyne Detection, Added Noise and Filtering
+1.
+Heterodyne Measurement
+Here we discuss the quadrature extractions from the equivalent linear measurement, e.g.
+balanced heterodyne
+detection, with excess added noise [9]. In the heterodyne detection, the output field ˆaoute−iωjt (j ∈ e, o) is mixed with
+a strong coherent local oscillator field ˆaLO(t) = αLOe−iωLOt at a 50:50 beam-splitter, where the output field from the
+two ports are sent to a balanced photo-detector, which results in a photon current that is proportional to
+ˆIout(t) = e−i∆LOtˆaout + ˆa†
+outei∆LOt,
+(39)
+in the limit of strong LO (αLO ≫ 1) with ∆LO = ωLO − ωj. We consider finite measurement interval of time T, on
+which we compute the windowed Fourier transformation of ˆIout(t),
+ˆIout(ωn) =
+1
+√
+T
+� T
+0
+dτ eiωnτ ˆIout(τ) =
+1
+√
+T
+� T
+0
+dτ eiωnτ(e−i∆LOτˆaout(τ) + ei∆LOτˆa†
+out(τ))
+= aout(ωn − ∆LO) + a†
+out(ωn + ∆LO),
+(40)
+where in a slight abuse of notation we define the dimensionless output fields aout(ωn). The reason why we are explicitly
+working the windowed Fourier transformation is, that despite being in a steady state during the measurement (see
+Sec. I B), the Fourier transformed data has a rather broad bandwidth (δωn = 2π
+T ∼ 5MHz for a 200 ns time window)
+due to the relatively short time of data collection T = τpulse−τdelay = 200 ns (especially for 250 ns optical pump pulse).
+In the limit of long measurement times T → ∞, the bandwidth will tend to zero and the following discussion as well
+as the results in Eq. (38) will coincide with standard Input-Output treatment in the continuous Fourier domain. In
+our experiments, we extract the quadratures of microwave and optical output field, by decomposing the heterodyne
+current spectra, in their real and imaginary parts which yields
+ˆIout(ωn) =
+1
+√
+2( ˆX(ωn − ∆LO) + ˆX(−ωn − ∆LO)
+�
+��
+�
+ˆIX,out(ωn)
++i [ ˆP(ωn − ∆LO) − ˆP(−ωn − ∆LO)]
+�
+��
+�
+ˆIP,out(ωn)
+),
+(41)
+where we define the quadrature output fields ˆaout(ωn) = ( ˆX(ωn) + i ˆP(ωn))/
+√
+2, in the same way as in Eq. (28-29).
+So far we have treated the photon current which result from a heterodyne measurement in terms of a time dependent
+hermitian operator ˆIout(t).
+In an actual experiment the heterodyne current is a real scalar I(t) quantity which
+fluctuates in time and between different experimental runs. Taking taking the (fast) Fourier transform of this current
+and decomposing it in its real and imaginary parts then yields I(ωn) = IX(ωn) + iIP (ωn). The theory of continuous
+measurements and quantum trajectories [10, 11] tells us how to connect the measured scalar currents with the current
+
+8
+operators from input-output theory [6]
+IA(ωn)IB(ωm) = 1
+2⟨{ˆIA,out(ωn), ˆIB,out(ωm)}⟩,
+(42)
+where we define the statistical average · · · over many experimental runs.
+2.
+Realistic Measurements: Added Noise and Gain
+For the vacuum, the noise spectral density for both quadratures, are obtained by
+SAA(ωn) = ⟨ ˆA(ωn) ˆA(ωn)⟩vac = 1
+2,
+(43)
+for the hermitian operator ˆA = ˆX, ˆP. Note that due to the discreteness of the Fourier domain we do not have a Dirac
+delta as opposed to Eq. (27). The noise spectrum of the heterodyne current is defined by SII(ω) ≡ I(ωn)I(ωn) =
+⟨ˆIout(ωn)ˆIout(ωn)⟩, where
+SII(ωn) = 1
+2 (SXX (ωn − ∆LO) + SP P (ωn − ∆LO) + SXX (ωn + ∆LO) + SP P (ωn + ∆LO)) .
+(44)
+Focusing on the part of the spectrum located around ∆LO,
+SII(ωn + ∆LO) = 1
+2 (SXX (ωn) + SP P (ωn) + 1) ,
+(45)
+assuming ∆LO ≫ κe, κo. This indicates the simultaneous quadratures measurements and added shot noise in the
+heterodyne measurements, even without experimental imperfections.
+So far we have focused on the ideal theory
+of the measurement and disregarded additional unknown sources of noise as well as the connection to the actually
+measured quantities. In practice, the decomposed measured quadratures contain additional uncorrelated excess noise,
+e.g. due to the added noise in the amplification or due to propagation losses [12]. We model this by phenomenologically
+adding another uncorrelated noise process from an independent thermal reservoir and then multiplying by a gain factor
+which converts the number of measured photons to the actually monitored voltage. To illustrate this we focus again
+on a single output port, and with the added noise current ˆIX/P,add(ωn) and the frequency dependent calibration gain
+Gdet(ωn), where
+ˆIX,det(ωn) =
+�
+Gdet(ωn)(ˆIX,add(ωn) + ˆIX,out(ωn)),
+(46)
+ˆIP,det(ωn) =
+�
+Gdet(ωn)(ˆIP,add(ωn) + ˆIP,out(ωn)).
+(47)
+We thus obtain the detected heterodyne noise spectral density,
+SII,det(ωn + ∆LO) =Gdet(ωn + ∆LO)[SXX (ωn) + SP P (ωn)
++ 1 + SIXIX,add(ωn + ∆LO) + SIP IP ,add(ωn + ∆LO)
+�
+��
+�
+=2Nadd
+],
+(48)
+where we define the spectra of the added noise SIOIO,add(ωn) = ⟨ˆIO,add(ωn)ˆIO,add(ωn)⟩.
+The added noise Nadd
+includes the excess vacuum noise from heterodyne measurement and the additional uncorrelated noise. Note that here
+the factor 1
+2 was absorbed in the detections gains. The gain Gdet(ωn) can be simply obtained on both microwave and
+optical side, from the cold measurements (optical pump off) with a known background. We note that, Eq. (48) lays
+the foundation of microwave and optical calibrations in our CEO device.
+In our experiments, we place the LO on opposite sites around the mode resonances, i.e.,
+∆LO,e = −ΩIF,
+∆LO,o = ΩIF,
+(49)
+where ΩIF > 0 is the intermediate frequency for down-mixing. The heterodyne output field can be obtained similar
+
+9
+to Eq. (40),
+ˆIout,e(ωn + ΩIF) =
+1
+√
+2[( ˆXe(−ωn) + ˆXe(ωn + 2ΩIF) + i(− ˆPe(−ωn) + ˆPe(ωn + 2ΩIF))],
+ˆIout,o(ωn + ΩIF) =
+1
+√
+2[ ˆXo(−ωn − 2ΩIF) + ˆXo(ωn) + i(− ˆPo(−ωn − 2ΩIF) + ˆPo(ωn))],
+(50)
+with noise spectrum given by,
+SII,e(ωn + ΩIF) = 1
+2(SXeXe (−ωn) + SPePe (−ωn)) + Ne,add,
+SII,o(ωn + ΩIF) = 1
+2(SXoXo (ωn) + SPoPo (ωn)) + No,add.
+(51)
+We note that, Eq. (50) is adopted for field quadrature extraction (including the added noise) from the heterodyne
+measurement, which reveals correlations in the quadrature histogram [cf. Fig.4 in the main text]. Despite of the
+reversed sign in the expected field quaduratures, microwave and optical output photons appear at the same frequency
+in the noise spectrum, i.e. ωn + ΩIF [cf. Fig.2 d,e in the main text].
+3.
+Covariance Matrix from Realistic Heterodyne Measurements
+Here we briefly explain the procedure of the covariance matrix reconstruction from the heterodyne measurements.
+The cross correlations of the detected heterodyne current spectra can be obtained via,
+DAB(ωn) = δIA,det(ωn + ΩIF)δIB,det(ωn + ΩIF),
+(52)
+where we define the centered current δIO,det = IO,det − IO,det, with
+IO,det(ωn) ∈ {IXe,det(ωn), IPe,det(ωn), IXo,det(ωn), IPo,det(ωn)}.
+(53)
+Similar to Eq. (42)), we can obtain
+DAB(ωn) = 1
+2⟨{δ ˆIA,det(ωn + ΩIF), δ ˆIB,det(ωn + ΩIF)}⟩
+=
+�
+GA,det(ωn + ΩIF))GB,det(ωn + ΩIF))
+�1
+2⟨{δ ˆA(ωn), δ ˆB(ωn)}⟩
+�
+��
+�
+=VAB(ωn)
++NAB,add
+�
+,
+(54)
+where we define the diagonal added noise matrix NAB,add = (Nadd)AB = NA,addδAB with the calibrated added noise
+Nadd and detection gain GA,det.
+This equation establishes how the covariance matrix of the qudrature operators [cf.
+Eq. (38)] is reconstruced from heterodyne measurements, and how they can be compared with the results from idealized
+standard input-output theory Eq. (33). For simplicity, in the main text we define the total measured covariance matrix
+including the added noise as,
+VAB,meas(ωn) = DAB(ωn)/
+�
+GA,det(ωn + ΩIF))GB,det(ωn + ΩIF)),
+(55)
+with VAB,meas(ωn) = VAB(ωn) + NAB,add.
+We note that, in principle the location of both LOs can be arbitrary. As evident in Eq. 54, our choice of the LO
+configuration, i.e. ∆LO,e = −∆LO,o = −ΩIF, offers a simple solution to the quantification of the broadband quantum
+correlations, considering the limited detection bandwidth, frequency dependent gain, or microwave cavity frequency
+shift, which may result in the loss of quantum correlations during quadrature extractions in heterodyne measurements
+due to imperfect frequency matching.
+
+10
+C.
+Entanglement Detection
+1.
+Duan Criterion
+We will now discuss how show that the photons outgoing microwave and optical photons are indeed inseparable
+or entangled. Our starting point is the covariance matrix which we defined in Eq. (33) and measured as outline in
+Eq. (54). The experimentally measured covariance matrix is of the form
+V =
+�
+Ve
+Veo
+Veo
+Vo
+�
+=
+�
+�
+��
+�
+V11
+0
+˜V13
+˜V14
+0
+V11
+˜V14 − ˜V13
+˜V13
+˜V14
+V33
+0
+˜V14 − ˜V13
+0
+V33
+�
+�
+�
+� .
+(56)
+Since there is no single mode squeezing we have V22 = V11 and V44 = V33.
+For simplicity we have omitted the
+frequency argument ωn of component. What we describe in the following will have to be repeated for every frequency
+component. The off-diagonal part in the covariance matrix which encodes the two-mode squeezing can be written as
+Veo ≃ V13(sin(θ)σx + cos(θ)σz),
+(57)
+where we define V13 = ( ˜V 2
+14 + ˜V 2
+13)1/2 and the mixing angle tan(θ) = ˜V14/ ˜V13. In our experimental setting ˜V14 maybe
+non zero e.g. due to small finite detunings δo. For the detection of inseparability, we employ the criterion introduced
+by Duan, Gidke, Cirac and Zoller [13]. This criterion states that if one can find local operations Uloc = Ue ⊗ Uo such
+that the joint amplitude variance of ˆX+ = ( ˆXe + ˆXo)/
+√
+2 break the inequality,
+∆X2
++ = ⟨U †
+loc ˆX2
++Uloc⟩ < 1/2,
+(58)
+then the state is inseparable and, thus it must be concluded that it is entangled.
+In this setting, it is enough to choose the local operations Uloc = UeUo to be a passive phase rotation on the optical
+mode only, with Ue = 1 and Uo = e−iϕˆa†
+oˆao, and phase rotation angle ϕ. In the space of covariance matrices, this
+corresponds to the (symplectic) transformation Sϕ = 12 ⊕ Rϕ, where we define the rotation matrix,
+Rϕ =
+�
+cos (ϕ)
+sin (ϕ)
+− sin (ϕ) cos (ϕ)
+�
+.
+(59)
+The local rotation of the phase V (ϕ) = SϕV S⊤
+ϕ will act on the off diagonal part of the covariance matrix as,
+Vea(ϕ) = V13(cos(θ − ϕ)σz + sin(θ − ϕ)σx).
+(60)
+With these local rotations the joint amplitude variance becomes
+∆X2
++(ϕ) = ⟨( ˆXe + ˆXo cos(ϕ) + ˆPo sin(ϕ))2⟩/2 = V11 + V33 + 2V13 cos(θ − ϕ).
+(61)
+We can similarly define the joint quadrature ˆP− = ( ˆPe − ˆPo)/
+√
+2, where ∆P 2
+−(ϕ) = ∆X2
++(ϕ). The variance of the
+joint quadratures ∆X2
++(ϕ) and ∆P 2
+−(ϕ) is minimized at the angle ϕ− = θ − π
+∆−
+EPR = ∆X2
++(ϕ−) + ∆P 2
+−(ϕ−) = 2(V11 + V33 − 2V13),
+(62)
+which corresponds to the two-mode squeezing of microwave and optical output field, and the microwave-optics entan-
+glement. In addition, the joint quadrature variance is maximized at the angle ϕ+ = θ and we obtain
+∆+
+EPR = ∆X2
++(ϕ+) + ∆P 2
+−(ϕ+) = 2(V11 + V33 + 2V13),
+(63)
+which corresponds to the anti-squeezing.
+
+11
+2.
+Logarithmic Negativity and Purity
+A mixed entangled state can be quantified by the logarithmic negativity,
+EN = max [0, − log (2ζ−)] ,
+(64)
+where ζ− is the smaller symplectic eigenvalue of the partially time reverse covariance matrix and can be obtained
+analytically
+ζ2
+− = S −
+�
+S2 − 4det(V )
+2
+(65)
+where we defined the Seralian invariant S = det(Ve) + detVo + 2det(Veo). Furthermore the purity of a bipartite
+Gaussian state is given by
+ρ =
+1
+4
+�
+det(V )
+,
+(66)
+with ρ = 1 for a pure state i. e. the vacuum state.
+II.
+EXPERIMENTAL SETUP
+The experimental setup is shown and described in SI Fig. 1. The laser is split into three parts, including an optical
+pulsed pump at frequency ωp, a continuous signal at ωp −FSR for the 4-port calibration (cf. SI III), and a continuous
+local oscillator (LO) at ωp − FSR + ΩIF for the optical heterodyne detection. The optical signal and pump pulse are
+sent to the optical resonator of the electro-optical device (DUT) and the reflected light (with pump pulse rejected
+by a filter cavity) is combined on a 50:50 beam splitter with the optical local oscillator with subsequent balanced
+photodetection.
+Microwave input signals are attenuated at different temperature stages of the dilution refrigerator (4 K: 20 dB,
+800 mK: 10 dB, 10 mK: 20 dB), and sent to the coupling port of the microwave cavity of the DUT. The reflected
+microwave signal is amplified and can then either be mixed with a microwave local oscillator of frequency FSR − ΩIF
+and subsequently digitized, or directly measured by a vector network analyzer or a spectrum analyzer.
+We note that, the optical LO is on the right side of optical mode, while the microwave LO is on the left side of the
+microwave mode, with ΩIF/2π = 40MHz. More details are in the caption of Supplementary Fig. 1.
+III.
+SETUP CHARACTERIZATION AND CALIBRATION
+In the main manuscript, we show results from two different sets of optical modes shown in Fig. 2.
+The main
+difference between these mode sets is the amount of suppression of the anti-Stokes scattering rate compared to Stokes
+scattering rate given by scattering ratio S, which depends on the mode hybridisation of the anti-Stokes mode [5, 14].
+The first set of optical mode (Fig. 2a) from which we show most of our main results (main text Fig. 2, 3 and 5a) has
+S =−10.3 dB on-resonance with an effective FSR = 8.799 GHz. The last power sweep shown in main text Fig. 5b
+is measured with a second set of optical modes with a lower S =−3.1 dB and a different effective FSR = 8.791 GHz
+(Fig. 2b). Despite it being the same optical resonator, the FSR for the second set of optical modes is slightly different,
+because of partial hybridisation of the optical pump mode which alters the working FSR between the optical pump
+and signal mode, see Fig. 2.
+In the following, we carefully calibrate the added noise due to the microwave detection chain at both these working
+FSRs (since microwave mode is parked at the working FSR). The added noise can be slightly different depending
+on frequency of measurement due to impedance mismatch and reflections between components in the microwave
+detection.
+A.
+Microwave added noise calibration
+In the following, we carefully calibrate the slightly different added noise in the microwave detection chain at both
+frequencies.The impedance mismatch and reflections between components in the microwave detection chain can vary
+
+12
+Dilution refrigerator
+Microwave preparation
+Optics preparation
+Optics and microwave detection
+MC1
+MS2
+MS3
+C1
+C2
+C3
+C5
+C4
+10mK
+800mK
+4K
+300K
+DC Microwave signal
+(FSR)
+DC Optical signal
+(193.5 THz  - FSR)
+Optical pump
+(193.5 THz)
+Coherent optical signal
+(193.5 THz - FSR)
+Optical LO 
+(193.5 THz -FSR - ωlo)
+Microwave LO 
+(FSR + ωlo)
+IF 
+(ωlo)
+RF lines
+DUT
+VNA
+SA
+DIGITIZER
+LNA
+RTA2
+RTA1
+Q
+I
+1
+1
+2
+90
+10
+1
+1
+2
+3
+3
+4
+PC1
+S3
+S4
+MS1
+DDG
+25
+75
+BPD
+Laser
+Lock control
+1550 nm
+SSB
+VOA1
+99
+1
+EDFA1
+EDFA2
+AOM1
+AOM2
+PC2
+PD2
+PD3
+PD4
+F3
+F1
+F2
+PD1
+S1
+S2
+ΩLO
+OSA
+PM
+HEMT
+Supplementary Fig. 1. Experimental setup for two-mode squeezing measurements. A tunable laser at frequency ωp
+is initially divided equally in two parts, i.e. the optical pump and the optical signal together with the optical local oscillator
+(LO). Light from the optical pump path is pulsed via an acousto-optic modulator (AOM1) which produces ns-pulses and shapes
+them for amplification via an Erbium-doped fiber amplifier (EDFA). The output from the EDFA is first filtered in time via
+AOM2 to remove the amplified spontaneous emission (ASE) noise and later in frequency via filter F1 (∼50 MHz linewidth with
+15 GHz FSR) to remove any noise at the optical signal frequency (the reflected power is rejected by circulator C3). The filter
+F1 is locked to the transmitted power by taking 1% of the filter transmission measured via photodiode PD3. The polarization
+of the final output is controlled via polarization controller PC1 before being mixed with the optical signal via a 90-10 beam
+splitter and sent to the dilution refrigerator (DR). The 10% output from the beam splitter is monitored on a fast detector PD2
+to measure the optical pump pulse power. The other half of the laser is again divided into two parts - 25% for the optical signal
+and 75% for the optical LO. The signal part is sent first to a variable optical attenuator VOA1 to control the power and then
+to a single sideband modulator SSB which produces the optical signal frequency at ωp − FSR and suppresses the tones at ωp
+and ωp + FSR. 1% of the optical signal is used to monitor the SSB suppression ratio via an optical spectrum analyzer OSA
+and 99% is sent to the DR after being polarization controlled via PC2. The optical LO is produced via a phase modulator PM
+and detuned by ωIF/2π = 40 MHz. As the PM produces many sidebands, the undesired sidebands are suppressed via filter F3
+(∼50 MHz linewidth with 15 GHz FSR), reflection is rejected by circulator C5. F3 is temperature-stabilized and locked to the
+transmitted power similar to F1. The optical LO is also amplified via EDFA2 before the optical balanced heterodyne. In the
+DR, the light is focused via a gradient-index (GRIN) lens on the surface of the prism and coupled to the optical whispering
+gallery mode resonator (WGMR) via evanescent coupling. Polarization controllers PC1 and PC2 are adjusted to efficiently
+couple to the TE modes of the optical WGMR. The output light is sent in a similar fashion to the collection grin lens. Outside
+the DR, the optical pump is filtered via filter F2 (similar to F3). The reflected light from F2 is redirected via C4 to be measured
+with PD1 which produces the lock signal for the laser to be locked to optical WGMR. The filtered signal is finally mixed with
+the optical LO and measured with a high speed balanced photo-diode BPD (400 MHz). The electrical signal from the BPD is
+amplified via RTA1 before getting digitized. On the microwave side, the signal is sent from the microwave source S3 which is
+connected to the DDG for accurately timed pulse generation (or from the VNA for microwave mode spectroscopy) to the fridge
+input line via the microwave combiner (MC1). The input line is attenuated with attenuators distributed between 4 K and 10 mK
+accumulating to 50 dB in order to suppress room temperature microwave noise. Circulator C1 and C2 shield the reflected tone
+from the input signal and lead it to the amplified output line. The output line is amplified at 4 K by a HEMT-amplifier and
+then at room temperature again with a low noise amplifier (LNA). The output line is connected to switch MS1 and MS2, to
+select between an ESA, a VNA or a digitizer measurement via manual downconversion using MW LO S4 (40 MHz detuned).
+Lastly, microwave switch MS3 allows to swap the device under test (DUT) for a temperature T50 Ω controllable load, which
+serves as a broad band noise source in order to calibrate the microwave output line’s total gain and added noise.
+the added noise slightly as a function of frequency. This added noise and corresponding gain due to a series of amplifiers
+
+13
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+a
+b
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+|Soo|2 
+|Soo|2 
+ωp
++FSR
+ωp
+-FSR
+ωp
+ωp
++FSR
+ωp
+-FSR
+ωp
+Supplementary Fig. 2. Optical mode spectra in reflection. Normalized reflection intensity |Soo|2 spectra of optical modes
+ˆao, ˆap and ˆat in red, green and blue respectively. a (b) shows the optical mode spectra of the first (second) set of modes with
+the anti-Stokes and Stokes scattering ratio S = −10.3 dB (−3.1 dB). The dashed line marks the effective FSR between the
+pump mode ˆap and the optical mode ˆao. The participation anti-Stokes optical mode ˆat is suppressed for this effective FSR as
+marked by the dashed line over the blue mode.
+and cable losses in the microwave detection chain is calibrated using a combination of a 50 Ω load, a thermometer
+and a resistive heater that are thermally connected. The microwave detection chain is identical for the signals from
+the 50 Ω load and the microwave cavity reflection, except for a small difference in cable length which we adjust for.
+To calibrate the detection chain, we heat the 50 Ω load with the resistive heater and record the amplified noise
+spectrum P50Ω(ω) as a function of temperature of 50 Ω load T50Ω. The output noise detected over a bandwidth B,
+P50Ω, as a function of T50Ω is given as,
+P50Ω = ℏωeGB
+�1
+2 coth
+�
+ℏωe
+2kBT50Ω
+�
++ Ne,add
+�
+,
+(67)
+with ωe the center microwave frequency, Ne,add (G) the added noise (gain) of the microwave detection chain, and kB
+the Boltzmann constant.
+A bandwidth of 11 MHz is selected around the region of interest to calculate Ne,add and G. For ωe = 8.799 GHz, we
+show the detected noise Ne,det = P50Ω/(ℏωeGB) as a function of T50Ω in SI Fig. 3 along with a fit using Eq. 67, with
+two fitting parameters G and Ne,add. We note that, at T50Ω = 0 K, Ne,det = Ne,add + 0.5. Table 1 (third row) shows
+the obtained added noise and gain for two frequencies of interest, i.e. ωe/2π = 8.799 GHz and ωe/2π = 8.791 GHz.
+Next, we consider the difference in cable losses between the 50 Ω load and the microwave cavity, which are in-
+dependently determined by measuring the microwave reflection from the microwave cavity and from the microwave
+switch directly before it. Including the cable losses, the effective added noise increases while the gain decreases for
+the reflected microwave detection, shown in Table 1 (fourth row).
+Finally, we consider an additional error due to the temperature sensor inaccuracy of 2.5%. Although this does not
+change the final Ne,add and G, it increases the uncertainty as shown in Table 1 (fifth row). The error calculated in
+this section contributes to the systematic error reported in the main text.
+
+14
+Supplementary Tab. 1. The added noise and gain in microwave detection chain (1σ errors shown)
+8.799 GHz
+8.791 GHz
+Detection Chain
+Ne,add
+G (dB)
+Ne,add
+G (dB)
+50Ω load
+(with fitting error)
+11.74 ± 0.08 66.67 ± 0.02 11.76 ± 0.09 66.72 ± 0.03
+MW cavity
+(including cable loss)
+13.09 ± 0.09 66.20 ± 0.02 13.16 ± 0.10 66.23 ± 0.03
+MW cavity
+(including temperature sensor uncertainty) 13.09 ± 0.33 66.20 ± 0.12 13.16 ± 0.34 66.23 ± 0.12
+0
+0.5
+1.0
+1.5
+13
+12
+14
+15
+16
+Experiment
+Fit
+(K)
+Tf
+Ne,det (photons s-1 Hz-1)
+Supplementary Fig. 3. Characterization of the added noise in the microwave detection chain. Measured output
+noise from a 50 Ω calibration load as a function of its temperature Tf. The measured noise is plotted in units of photons as
+N 50Ω
+det = P50Ω/(ℏωeGB). The dashed line at the bottom represents the fitted vacuum noise level in addition to the added noise.
+The red line and shaded region represents the fit and the 95% confidence interval around it.
+B.
+Optical added noise
+Optical added noise is calculated via 4-port calibration of our device [14]. In this calibration, we measure the
+coherent response of our device through its 4 ports - optical input/output and microwave input/output. Sending
+an optical (or microwave) signal to the DUT in combination with a strong pump leads to stimulated parametric
+down-conversion (StPDC) process, which generates an amplified microwave (optical) coherent signal. We measure the
+4 S-parameters of our device - microwave reflection (S11), optics reflection (S22), microwave to optics transmission
+(S21) and optics to microwave transmission (S12). The mean transduction efficiency between microwave and optics
+of the DUT is then calculated as,
+η =
+�
+S12S21
+S11S22
+.
+(68)
+We use the transduction efficiency and Ne,add in the microwave detection chain from Sec. III A to calculate the
+optical added noise. Ne,add is firstly used to calculate the effective microwave detection gain (different from the one
+in Sec. III A, because the microwave detection line used for the 4-port calibration uses analog downconversion and
+digitization, while the thermal calibration uses SA, see Fig. 1). The microwave gain, along with the (off-resonant)
+microwave reflection measurement, is used to calculate the microwave input loss.
+We can obtain the microwave
+signal power at the DUT, which allows us to calculate the output optical power of the DUT using the transduction
+efficiency. In conjunction with the measured output power at the end of the detection chain, the losses in the optical
+detection path and hence, the effective added noise with respect to the optical port of the DUT can be calculated.
+The calculated optical added noise is No,add = 5.54 ± 0.21(7.42 ± 0.22) for ωe = 8.799 GHz (8.791 GHz).
+
+15
+IV.
+DATA TREATMENT
+In this section, we describe all the steps for the data treatment in detail, which includes the time domain analysis
+(Sec. IV A), the pulse post-selection due to setup drift (Sec. IV B), the frequency domain analysis (Sec. IV C), and
+the quadrature correlations (Sec. IV D).
+A.
+Time-domain analysis
+Both microwave and optical signals are detected via heterodyne detection by mixing with a strong local oscillator
+that is ∼40 MHz detuned from respective mode resonance. The output heterodyne signals are digitized using a digitizer
+at 1 GigaSamples/second. First, we digitally downconvert the digitized data at ωIF = 40 MHz. This yields the two
+quadratures IXe/o,det(t) and IPe/o,det(t) of the microwave or optical output signal record with 40 MHz resolution
+bandwidth (using 25 ns time resolution).
+Supplementary Fig.
+4 shows the calibrated output power (I2
+Xe/o,out +
+I2
+Pe/o,out) [cf.
+Eq. 50] and the phase (arctan(IXe/o,out/IPe/o,out)) from a single pulse sequence.
+This includes the
+stochastic SPDC signals from a strong pump pulse, and the coherent StPDC signal from a weaker pump pulse
+together with a coherent microwave signal for calibration purposes. The SPDC signal produced by the first strong
+pulse is labeled by the shaded region for one single pulse, and the averaged output power over many pulses is shown
+in main text Fig. 2. The coherent microwave reflection and stimulated parametric downconverted optical signal are
+adopted to obtain the phases during the pulse. We record this measured phase in both signal outputs during the
+second optical pump pulse for phase-drift correction in later post processing.
+0
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+1.0
+0
+200
+400
+0
+50
+100
+150
+200
+600
+800
+1000
+1200
+0.5
+0.0
+0.5
+-1.0
+1.0
+0.5
+0.0
+0.5
+-1.0
+a
+b
+(µs)
+Phase (rad/π)
+Phase (rad/π)
+ (photons/s/Hz)
+Ne,
+ (photons/s/Hz)
+No,
+t
+(µs)
+t
+Supplementary Fig. 4. Downconverted output signal for a single measured pulse sequence. a (b) show the measured
+microwave (optical) output signal downconverted at 40 MHz. The shaded part in each case shows the region of the SPDC
+signal (the first optical pump pulse). For a single pulse, the SNR of a SPDC signal is too small to be seen. However, during
+the second optical pump pulse, a coherent response is seen in both signal outputs where the phase can be measured with high
+SNR for each single shot.
+In order to determine the accuracy of a phase correction for the first pump pulse based on the phase measurement
+during the second pump pulse, we send a continuous microwave signal during both pump pulses and recorded the
+phase of the converted optical pulse during the first and the second optical pump pulse. Supplementary Fig. 5 shows
+the phase difference between the first and second optical pump pulse for 2500 trials along with a normal distribution
+fit.
+The fit variance for the distribution is 0.17 rad.
+On a similar set of model data, applying a random phase
+variation of 0.17 rad results in about 1.5-2.0% loss of correlations [cf. Sec. IV D], whereas, we observe about 6-8% loss
+of correlations in the experiments. The imperfection in phase correction does not completely explain the decreased
+quantum correlations, which might be due to other experimental instabilities, especially the optical pump laser lock.
+
+16
+0.0
+0.2
+0.4
+0.6
+0.8
+0.0
+0.2
+-0.2
+0.4
+-0.4
+0.6
+-0.6
+1.0
+Probability density
+Phase difference (radians)
+Supplementary Fig. 5. Accuracy of phase correction scheme. The histogram shows the difference in the measured phase
+between the first and second optical pump pulse. Since we correct the phase in the first pump pulse based on the measured
+optical phase of the second optical pump pulse, the difference shows the limitations of this method. The grey dashed line is a
+normal distribution fit with variance 0.17 rad.
+B.
+Pulse post-selection
+In our experiments, we use three temperature-stabilized optical filters, which may drift slowly in time. Two of
+them are used in the optical heterodyne detection. In the signal path, one filter (F2 in Fig.1) is used to filter the
+optical signal while reject the strong optical pump. In the LO path, one filter (F3 in Fig.1) is used to obtain a clean
+optical LO tone, which is genearted by an electro-optic phase modulator (which produces multiple sidebands) and
+then amplified using an EDFA (which produces excess amplified spontaneous emissions).
+The slow filter drifts can be identified from the amplitude of the coherent optical signal produced via stimulated
+parametric downconversion during the second optical pump pulse, which drops due to either the decreased transmission
+after F2 or the reduced LO power after the F3. This is evident in the histogram of the converted optical power during
+the second optical pump pulse as shown in Fig. 6a. The histogram is not symmetric and has a tail at the lower end.
+To filter out the instances of drifted heterodyne detection, we select a threshold (in this case marked by a dashed
+line in Fig 6) and remove all pulses below the selected threshold along with 20 neighboring pulses (10 s in total time)
+before and after such instance. These numbers are chosen according to the filter drift and the filter temperature lock
+time-scales. After such filtering, usually about 10% of the data is removed and the histogram of the converted optical
+power during the second optical pump pulse becomes symmetric as shown in Fig. 6b.
+C.
+Frequency domain analysis
+As already mentioned in the main text, with the help of time-domain analysis, we select three different time-snippets
+to analyze the data in the frequency domain - before-pulse, on-pulse and post-pulse defined with respect to the first
+optical pump pulse. The main challenge in processing the data in frequency domain is the proper normalization of
+the measured output spectrum [cf. Eq. 48]. The microwave reflection baseline is not flat because of slight impedance
+mismatches between different components in the microwave detection chain, with similar optical heterodyne shot noise
+floor due to the frequency dependent balanced detector gain. In addition, we observe slight shift of a few millivolts each
+time in the digitizer measurements when a new measurement is launched and the digitizer is reinitialized. Combined
+with the fact that the amplifier gain in the microwave detection chain as well as the optical heterodyne gain (due to
+optical LO power drift) may drift over a long time, an in-situ calibration of vacuum noise level is needed.
+In case of microwave, we need to first correct for the microwave reflection baseline distortion from impedance
+mismatch and then correct for the signal level shift caused by the digitizer. For the distorted baseline, we separately
+measure the microwave output spectrum when the microwave cavity is in its ground state (thermalized to 7 mK at
+mixing chamber). This measurement is shown in Supplementary Fig 4a (gray) along with the measured before-pulse
+(cyan), on-pulse (purple) and after-pulse microwave noise spectrum (orange). Dividing the measured spectra with
+the cold cavity spectrum reveals a flat baseline Lorentzian noise spectra, however with an offset due to the digitizer
+drift. To correct for this offset, we perform an in-situ vacuum noise calibration using the off-resonance (waveguide)
+noise in the before-pulse microwave noise spectrum. An independent measurement of the microwave waveguide noise
+as a function of the average optical pump power (averaged over the full duty cycle) is shown in Supplementary Fig
+
+17
+50
+0
+100
+150
+200
+0.00
+0.01
+0.02
+50
+0
+100
+150
+200
+0.00
+0.01
+0.02
+a
+b
+Optics quanta (photons s-1 Hz-1)
+Optics quanta (photons s-1 Hz-1)
+Count (Normalized)
+Count (Normalized)
+Supplementary Fig. 6. Histogram of the converted optical power. The measured coherent optical power during the
+second optical pump pulse depends on the optical heterodyne gain and the received optical signal power. Both of these values
+can drift depending on the experimental setup’s stability. a shows the normalized histogram of this measured optical power
+over all the collected pulses. The histogram has a tail on the lower end owing to the times when the heterodyne setup drifted.
+Filtering the points which do not meet a selected threshold (shown by the grey dashed line in a), we remove the instances where
+the setup had drifted and the optical heterodyne detection efficiency was compromised. The same histogram after removing
+such points is shown in b.
+8. The error bars (2σ deviation) result from the microwave detection chain gain and the measurement instrument
+drift. The power law fit reveals that the microwave waveguide noise grows almost linearly with average optical pump
+power, and only deviates significantly from 0 for optical pump power >3 µW. As we work with average optical pump
+powers of ≪1 µW, we can safely assume the microwave waveguide noise to be zero. Therefore, we use the off-resonant
+waveguide noise for before-pulse microwave noise spectrum as an in-situ vacuum noise calibration.
+In case of optics, the optical detection is shot-noise limited, and the excess LO noise at the optical signal frequency
+is suppressed by more than 40 dB using filter F1 in Fig.1. We use the before-pulse optical noise spectrum as the
+vacuum noise level and normalize the optical on-pulse spectrum directly with the before-pulse in-situ calibration.
+Supplementary Fig 4b shows noise spectrum (without normalization) of the optical off-pulse (cyan), on-pulse (green),
+and the after-pulse (yellow). The signal during the optical pump pulse is clearly visible, and the noise level is identical
+before and after the optical pulse.
+The normalized noise spectra for both microwave and optics are shown in main text Fig. 2d and 2e., where we can
+obtain the normalization gain [cf. Eq. 48].
+D.
+Joint-quadrature correlations
+The detected output quadratures including excess added noise, i.e.
+ˆIXe,out(∆ω),
+ˆIPe,out(∆ω),
+ˆIXo,out(∆ω),
+ˆIPo,out(∆ω), can be obtained from the real and imagrinary parts in the discret Fourier transform of the photocurrent
+by normalizing to the detection gain [cf. Eq. 50].
+Similar to Sec. I C 1, we can define the joint detected quadratures, by applying phase rotation on the optical ones,
+ˆIX,+(∆ω, φ) =
+ˆIXe,out(∆ω) +
+�
+ˆIXo,out(∆ω) cos φ − ˆIPo,out(∆ω) sin φ
+�
+√
+2
+,
+ˆIP,−(∆ω, φ) =
+ˆIPe,out(∆ω) −
+�
+ˆIXo,out(∆ω) sin φ + ˆIPo,out(∆ω) cos φ
+�
+√
+2
+.
+(69)
+To verify the non-classical correlation between the unitless quadrature variables for output microwave and optics
+field, i.e.
+ˆXe(∆ω) & ˆXo(−∆ω) and ˆPe(∆ω) & ˆPo(−∆ω), we can calculate the phase dependent joint quadrature
+
+18
+40
+20
+60
+59
+60
+61
+0.30
+0.32
+0.16
+0.18
+0.20
+0.22
+0.24
+0.26
+0.28
+62
+63
+64
+65
+a
+b
+ (MHz)
+40
+20
+60
+MW output (nW/2MHz)
+Opt output (µW/2MHz)
+Off-pulse
+In-pulse
+Off-pulse
+In-pulse
+After-pulse
+After-pulse
+Cold cavity
+ω/2π
+ (MHz)
+ω/2π
+Supplementary Fig. 7.
+Spectra of output signals.
+The microwave reflection baseline is not flat due to an impedance
+mismatch between different components in the microwave detection chain.
+As a result, the output power measured from
+amplified vacuum noise (from the cold microwave cavity) is not flat (shown in gray in a). Additionally, the digitizer in our
+setup has a different noise level each time it is started. As a result, the cold cavity baseline has an extra offset with respect to
+all other measurements. a also shows the measured output spectra for time region before (during, after) the first optical pulse
+shown in cyan (purple, orange). Similarly, b shows the output spectra for the optical output before (during, after) the first
+optical pulse in cyan (green, orange).
+100
+101
+0.0
+0.2
+0.4
+Experiment
+0.01Pavg
+0.97
+Pavg (µW)
+Wavegide noise (photons/s/Hz)
+Supplementary Fig. 8. Microwave waveguide noise as a function of the average optical pump power. The error bars
+represent 2σ error. The solid line is a power law fit. We find the power law is actually quite close to a linear function.
+variance [cf. Eq. 61],
+�
+ˆX2
++(∆ω, φ)
+�
+=
+�
+ˆI2
+X,+(∆ω, φ)
+�
+− Ne,add + No,add
+2
+,
+�
+ˆP 2
+−(∆ω, φ)
+�
+=
+�
+ˆI2
+P,−(∆ω, φ)
+�
+− Ne,add + No,add
+2
+.
+(70)
+For ∆ω = 0, we plot the joint quadrature variance as a function of local oscillator phase in Fig. 9 (a). The shaded
+region represent the 2σ statistical error in the calculated joint quadrature variances. We note that, the statistical 1σ
+error of the variance for a Gaussian distributed data is given by
+√
+2σ2/
+√
+N − 1, where N is the length of the dataset.
+The obtained resonant ∆EPR(0, φ) is shown in Fig. 9(b). The minimum and maximum of ∆EPR(φ) over the local
+oscillator phase are defined as min[∆EPR] = ∆−
+EPR and max[∆EPR] = ∆+
+EPR. ∆−
+EPR < 1 indicates non-classical joint
+correlations and squeezing below vacuum levels.
+
+19
+-1.0
+-0.5
+0.0
+0.5
+1.0
+0.5
+1.0
+1.0
+2.0
+1.5
+2.5
+a
+b
+Quanta
+Phase (rad/π)
+-1.0
+-0.5
+0.0
+0.5
+1.0
+Phase (rad/π)
+ΔEPR
+X+
+P+
+Supplementary Fig. 9. Joint quadrature correlations and ∆EPR.
+a. Joint quadratures at resonance X+(∆ω = 0) and
+P+(∆ω = 0) are plotted as a function of the local oscillator phase φ. b. ∆EPR as a function of φ. The shaded region in both
+plots represents the 2σ statistical error.
+The broadband phase that minimizes ∆EPR(∆ω, φ), i.e. φmin(∆ω), reveals the difference in arrival times (group
+delay) between the microwave and optical signal output (Supplementary Fig. 10a). After fixing the inferred time delay
+between the in-pulse arrival time of the microwave and optical signal, φmin becomes independent of frequency detuning
+from the mode resonances. Thus, we adjust for the differences in arrival times by ensuring that the slope of φmin with
+respect to detuning ∆ω is minimized for all datasets we analyze, utilizing the broadband quantum correlations.
+-1.0
+-0.5
+0.0
+0.5
+1.0
+a
+b
+0
+-20
+-30
+-10
+20
+10
+30
+(MHz)
+-1.0
+-0.5
+0.0
+0.5
+1.0
+0
+-20
+-30
+-10
+20
+10
+30
+ (rad/π)
+φ
+ (rad/π)
+φ
+∆ω/2π
+(MHz)
+∆ω/2π
+Supplementary Fig. 10. Effect of time delay between the microwave and optics signals. The plots show the local
+oscillator phase φmin which minimizes ∆EPR(∆ω, φ) as a function of detuning frequency ∆ω. a (b) shows the case when the
+time difference of arrival between the microwave and optics signals was 25 ns (≈ 0 ns). The solid lines are the linear fit to the
+experimental data.
+V.
+QUADRATURE HISTOGRAM RAW DATA
+Fig. 11 shows the normalized difference of the two-variable quadrature histograms obtained during and before the
+optical pump pulse based on the data shown in Figs. 2 and 3 of main text. These unprocessed histograms directly
+show the phase insensitive amplification in each channel as well as the correlations in (Xe,Xo) and (Pe,Po). Note
+however that - in contrast to the analysis in the main text - taking this difference does not lead to a valid phase space
+representation since also the vacuum noise of 0.5 together with the output noise of 0.026 ± 0.011 photons (due to the
+residual microwave bath occupancy right before the pulse) are subtracted, hence the negative values.
+
+20
+20
+10
+0
+-10
+-20
+20
+10
+0
+-10
+-20
+20
+10
+0
+-10
+-20
+1.0
+0.5
+-1.0
+-0.5
+0.0
+10
+0
+-10
+10
+0
+-10
+10
+0
+-10
+-20
+-10
+0
+10
+-10
+0
+10
+-10
+0
+10
+-10
+0
+10
+20
+-20
+-10
+0
+10
+20
+-20
+-10
+0
+10
+20
+Pe
+Pe
+Pe
+Po
+Po
+Po
+Xo
+Xo
+Xo
+Xe
+Xe
+Xe
+Supplementary Fig. 11. Quadrature histogram raw data. Normalized difference of the two-variable quadrature histograms
+obtained during and before the optical pump pulse based on the data shown in Figs. 2 and 3 of the main text.
+VI.
+NON-CLASSICAL CORRELATIONS WITH 600 ns LONG OPTICAL PUMP PULSES
+Before experimenting with 250 ns long optical pump pulses, we used 600 ns long optical pump pulses. A sample
+measurement with a 600 ns is shown in Fig. 12a similar to Fig. 3c of main text. Compared to 250 ns long pulses, the
+main difference lies in the fact that ∆−
+EPR in the middle panel exhibits a double-dip shape because the correlations
+¯V13 have a wider bandwidth than the emitted noise spectra ( ¯V11 and ¯V33), which are narrowed due to dynamical back-
+action [5]. Since in the measurement the correlations don’t clearly overwhelm the emitted noise, interference between
+two Lorentzian functions of different widths (dashed line) leads to the specific shape of ∆−
+EPR. Theory confirms this
+even though the shown theory curve (solid red line) does not exhibit the specific line-shape due to higher expected
+correlations compared to the experimentally observed values. These results indicate that ¯ne,int due to a 600 ns optical
+pump pulse is large enough to prevent a clear observation of squeezing over the full bandwidth below the vacuum
+level (∆−
+EPR < 1). As a result, we switched to 250 ns optical pump pulses with higher statistics as shown in the main
+text.
+We also repeated the measurement with 600 ns long pulses with different optical pump powers. Fig. 12b shows the
+measured pump power dependence with each data point based on 170000-412500 individual measurements each with
+a 2 Hz repetition rate. The microwave mode thermal bath occupancy ¯ne,int changes little as a function of the peak
+optical pump power at the device and is approximated with a constant function (solid maroon line in the top panel).
+The on-resonance mean CM elements scale with cooperativity and are in excellent agreement with theory (solid lines)
+based on the ¯ne,int. The on-resonance squeezing ∆−
+EPR does not change significantly with cooperativity since both
+excess noise and correlations scale together with cooperativity. The anti-squeezing ∆+
+EPR scales up with cooperativity
+as expected. All but one measured mean values are below the vacuum level and three power settings show a > 2σ
+significance for entanglement. Note that this power sweep was conducted on a different set of optical modes with a
+different amount of anti-Stokes sideband suppression (see section III).
+VII.
+ERROR ANALYSIS
+The covariance matrix of the output field quadratures V (ω) can be directly calculated from the extracted microwave
+and optical quadratures from frequency domain analysis [cf. Eq. 54] We simply rotate the optical quadratures with the
+phase that minimized the joint quadrature variance, and obtain the covariance matrix in the normal form. We note
+that, the error in calculating the covariance matrix comes from two sources - the statistical error due to finite number
+of pulses, and the systematic error in the vacuum noise level calibration. The detailed error analysis is described in
+the following subsections. We note that, the uncertainty in all the reported numbers in the main text corresponds to
+2 standard deviation.
+
+21
+0
+-10
+-20
+1.0
+2.0
+3.0
+4.0
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.2
+1.0
+1.4
+10
+20
+a
+b
+Interpolation
+In-pulse
+After-pulse
+Before-pulse
+0.1
+0.0
+1.0
+0.5
+0.0
+0.8
+1.0
+1.0
+2.0
+3.0
+0.2
+0.3
+140
+160
+180
+200
+220
+240
+Optical pump power (mW)
+0.10
+0.12
+0.14
+0.16
+0.18
+0.20
+0.22
+Cooperativity
+(V11 + V22)/2
+(V33 + V44)/2
+(V13 − V24)/2
+(V11 + V22)/2
+(V33 + V44)/2
+(V13 − V24)/2
+V
+(photons s-1 Hz-1)
+(photons s-1 Hz-1)
+(MHz)
+V
+ΔEPR
+-
+ΔEPR
++
+ΔEPR
+-
+¯ne,int
+ΔEPR
++
+Δω/2π
+Supplementary Fig. 12. Non-classical correlations vs. optical pump power for 600 ns long pump pulses. a, Top
+panel, the average microwave output noise ¯V11 (purple), the optical output noise ¯V33 (green) and correlations ¯V13 (yellow) as
+a function detuning based on 412500 measurements with a 2 Hz repetition rate. The solid lines represent the joint theory
+with fit parameters C = 0.22 and in-pulse microwave thermal bath occupancy ¯ne,int = 0.19 ± 0.03. The dashed lines are
+individual Lorentzian fits to serve as a guide to the eye. ∆−
+EPR (∆+
+EPR) in the middle (bottom) panel shown in red (blue) color
+are calculated from the top panel data and fits as described in the main text. The darker color error bars represent the 2σ
+statistical error and the outer (faint) error bars also include systematic errors. b, Power dependence of CM elements. The
+top panel shows the microwave mode thermal bath occupancy ¯ne,int for before-pulse, after-pulse and in-pulse regimes (marked
+in Fig. 2A) as a function of the peak optical pump power at the device and the corresponding cooperativity. The in-pulse
+¯ne,int is obtained by the joint theory fit and approximated with a constant function (solid line). The middle panel shows
+the on-resonance mean CM elements based on the ¯ne,int from the top panel. The bottom two panels show the on-resonance
+squeezing ∆−
+EPR and anti-squeezing ∆+
+EPR calculated from the middle panel along with theory (solid lines). The darker color
+error bars represent the 2σ statistical error and the outer (faint) error bars also include systematic errors. All measured mean
+values are below the vacuum level and three power settings show a > 2σ significance for entanglement.
+A.
+Statistical error
+The error in the calculation of bivariate variances comes from the statistical uncertainties, arising from finite number
+of observations of a random sample. This error is the major component of our total error in diagonal covariance matrix
+elements. The error in calculating the variance of a sample distribution sampled from a Gaussian variable follows the
+Chi-squared distribution and is given as,
+Var(σ2) =
+2σ2
+N − 1,
+(71)
+where, σ2 is the variance of sample distribution and N is its size.
+In addition, the error in the covariance from a bivariate variable is given by the Wishart distribution [15]. For a
+general bivariate covariance matrix Σ given as,
+Σ =
+�
+σ2
+11
+ρσ11σ22
+ρσ11σ22
+σ2
+22
+�
+,
+(72)
+
+22
+the variance of the covariance matrix is given by,
+Var(Σ) =
+1
+N − 1
+�
+2σ4
+11
+(1 + ρ2)σ2
+11σ2
+22
+(1 + ρ2)σ2
+11σ2
+22
+2σ4
+22
+�
+.
+(73)
+B.
+Systematic error
+Although, the systematic error in our measurements are not as significant, they still are a noticeable source of
+error. Here the error in calculating the covariance matrix results form the error in the estimation of the vacuum
+noise levels. More specifically, the error in determining the added noise due to the microwave and optical detection
+chain, as discussed in Sec. III. Propagating these systematic errors through the covariance matrix analysis is non-
+trivial, since calculating the error in variance of erroneous quantities is challenging. Therefore, we use a worst-case
+scenario approach to calculate the total error including the statistical error and the systematic error. We repeat
+the full analysis, including the statistical errors, for the lower and upper bound of the uncertainty range from the
+systematic errors for the microwave and optical added noise levels. Repeating the analysis expands the error bars in
+the calculated quantities. We take the extremum of all the error bars from all the repetitions of analysis to get the
+total error bar. We show both statistical error and the total error in the main text.
+REFERENCES
+[1] W. Hease, A. Rueda, R. Sahu, M. Wulf, G. Arnold, H. G. Schwefel, and J. M. Fink, PRX Quantum 1, 020315 (2020).
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+G. Leuchs, and H. G. L. Schwefel, Optica 3, 597 (2016).
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+[15] J. Wishart, Biometrika 20A, 32 (1928).
+

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+Boltzmann equation and its cosmological applications
+Seishi Enomoto∗, Yu-Hang Su, Man-Zhu Zheng, and Hong-Hao Zhang†
+School of Physics, Sun Yat-sen University, Guangzhou 510275, China
+Abstract
+We review the derivation of the Boltzmann equation and its cosmological applica-
+tions. Our paper derives the Boltzmann equation by the language of quantum field
+theory without any assumption of the finite temperature system. We also introduce
+two examples of cosmological applications, dark matter abundance and baryogenesis,
+with techniques in their calculations.
+1
+Introduction
+The early Universe, composed of hot plasma, evolves in time while maintaining thermal
+equilibrium at most epochs, and thus the evolution is described simply by thermodynam-
+ics. The cosmologically important events, therefore, are focused on the various turning
+epochs in which some particle species depart from the thermal bath as seen in, e.g., Big
+Bang nucleosynthesis (BBN) and recombination confirmed by both the theory and obser-
+vations, and the dark matter (DM) relic abundance and the baryogenesis scenario derived
+from some hypotheses. Qualitatively, the turning epochs can be estimated by comparing
+the reaction rate and the spatial expansion rate of the Universe, but more quantitative
+treatment to ensure accurate predictions is required by the present cosmological obser-
+vations. The Boltzmann equation is a powerful tool for following the out-of-equilibrium
+dynamics in detail, and can describe the evolution of the particle distribution due to the
+spatial expansion and the momentum exchanges through interactions. Since the Boltz-
+mann equation provides the abstract relation between the macroscopic and the microscopic
+evolution of the particle distributions, it is applicable to various situations, and hence, the
+solving equations differ for each system.
+The most successful application of the Boltzmann equation is for the theory of BBN
+[1, 2, 3, 4, 5, 6, 7] (see also, e.g., [8, 9, 10]), which describes how the initial protons and
+neutrons form the light elements at the final. As the result, the abundances of the light
+elements are predicted, and they are well consistent with the present observations. The
+other well-motivated and formulated example is the Kompaneets equation [11, 12], which
+is derived from the Boltzmann equation in the photon-electron system. The evolution
+equation is applied to the various analysis in cosmological or astronomical situations,
+e.g., the last scattering surface in the recombination epoch, the Sunyaev-Zel’dovich effect
+[13, 14, 15, 16] that is a distortion of the cosmic microwave background radiation by hot
+electrons in galaxies, etc.
+On the other side, exploring Beyond the Standard Model (BSM), there are many
+studies on the DM candidate realizing the present density parameter through the relic
+abundance, which can be predicted using the Boltzmann equation.
+The most famous
+and simple scenario for DM candidates is described by weakly interacting massive par-
+ticles (WIMPs) [17, 18], but the current observation requires a more extended scenario
+or the other mechanism, e.g., feebly interacting massive particles (FIMPs) [19], strongly
+interacting massive particles (SIMPs) [20], processing into forbidden channels [21, 22], co-
+annihilations [21], co-scattering [23, 24], zombie [25, 26], and inverse decays [27, 28], etc.
+∗[email protected]
+†[email protected]
+1
+arXiv:2301.11819v1  [hep-ph]  27 Jan 2023
+
+(See also [29] for their summarized analysis.) The more complicated models are consid-
+ered, the more technical treatment in the Boltzmann equation is required. As the other
+topic for exploring BSM with the Boltzmann equation, there are also many studies for
+baryogenesis scenarios explaining the origin of (baryonic) matter-antimatter asymmetry
+in our Universe, e.g., GUT baryogenesis [30, 31, 32], leptogenesis [33], electroweak baryo-
+genesis [34, 35], and Affleck-Dine baryogenesis [36, 37], etc. The analysis tends to be more
+complicated than the case of DM abundance and requires more technical treatment.
+The studies using the Boltzmann equation have been more popular and important as
+some approaches to confirm the current physics in detail and explore BSM in the early
+Universe. We aim to provide a clearer understanding of the Boltzmann equation and its
+techniques with some cosmological examples. This paper is organized as follows. First,
+we derive the Boltzmann equation by the language of the quantum field theory in section
+2. Next, we demonstrate application examples how the Boltzmann equation is applied for
+the DM abundance in section 3 and the baryogenesis scenario in section 4 with simple toy
+models. Finally, we summarize our discussion in section 5.
+2
+Boltzmann equation
+The Boltzmann equation is a quite powerful tool to describe the evolution of particles in
+Cosmology. Although the equation is applicable to various situations, the used formulae
+are also dependent on the circumstance. In this section, we derive the evolution formula
+of the distribution function f = f(xµ, pµ) from the basic statement
+L[f] = C[f]
+(1)
+where L and C are called the Liouville operator and the collision operator, respectively.
+The Liouville operator describes the variation of the distribution of a particle along a
+dynamical parameter, and the collision operator describes the source of the variation
+through the microscopic processes.
+2.1
+Liouville operator
+We define the Liouville operator to describe the variation of the distribution along the
+geodesic line parametrized by the affine parameter λ. Using the momentum relations
+pµ = dxµ
+dλ ,
+pµpµ = m2
+(2)
+where pµ is a four-momentum and m is the mass, and the geodesic equation
+0 = dpµ
+dλ + Γµ
+νρpνpρ
+(3)
+where Γµ
+νρ is the affine connection, the Liouville operator can be written as
+L[f] ≡
+df
+dλ
+����
+geodesic line
+=
+dxµ
+dλ ∂µf + dpµ
+dλ
+∂f
+∂pµ
+(4)
+=
+pµ∂µf − Γµ
+νρpνpρ ∂f
+∂pµ .
+(5)
+In the FLRW space-time, gµν = diag(1, −a2, −a2, −a2) where a = a(t) is the scale
+factor, the distribution function is spatially homogeneous and isotropic: f = f(t, E).
+Then the concrete representation of the Liouville operator term can be written as
+L[f] = E ˙f − H|⃗p|2 ∂f
+∂E
+(6)
+2
+
+where H = ˙a/a is the Hubble parameter and (⃗p)i ≡ api is the physical momentum.
+2.2
+Collision operator
+We define the collision operator C[f] as the variation rate by the microscopic processes:
+C[f] ≡
+df
+dλ
+����
+microscopic process
+=
+E df
+dt
+����
+microscopic process
+.
+(7)
+In order to evaluate the variation of the distribution function, we assume the followings.
+First, the process can be described by the quantum field theory. Second, the microscopic
+process can be evaluated on the Minkowski space since the gravitational effect is already
+evaluated in the Liouville operator part.
+With the above assumption, the distribution function f can be regarded as the expec-
+tation value of all possible occupation numbers with their probabilities, as we will see later.
+Also the probabilities can be evaluated by the quantum field theory on the flat space. To
+derive the concrete representation of (7), we need to construct the corresponding quantum
+state and then evaluate the transition probability.
+2.2.1
+Eigenstate for occupation number
+At first, we construct a multi-particle state |{n}⟩ in order to include all information about
+the particle occupations. We impose |{n}⟩ to be the eigenstate satisfying
+ˆna(⃗k)|{n}⟩ = na(⃗k)|{n}⟩,
+⟨{n}|{n}⟩ = 1,
+(8)
+where ˆna(⃗k) and na(⃗k) are the occupation operator and its corresponding occupation
+number for species a ∈ {n} in the unit phase space, respectively. Here the occupation
+operator is defined by
+ˆna(⃗k) ≡ 1
+V a(a)†
+⃗k
+a(a)
+⃗k ,
+(9)
+where V ≡
+�
+d3x = (2π)3δ3(⃗k = 0) is a volume of the system, and a(a)
+⃗k
+is an annihilation
+operator for species a which satisfies
+[a(a)
+⃗k , a(b)†
+⃗p
+] = δab · (2π)3δ3(⃗k − ⃗p),
+(others) = 0.
+(10)
+In the case of the fermionic species, the commutation relations are replaced with the anti-
+commutation relations. Then one can obtain the representation of the eigenstate |{n}⟩
+by
+|{n}⟩ ≡
+�
+a∈{n}
+�
+��
+⃗p
+1
+√na! ·
+√
+V na (a(a)†
+⃗p
+)na
+�
+� |0⟩.
+(11)
+Note that the occupation number na(⃗k) must be an integer.
+Furthermore, it is convenient to define the increased/decreased state from |{n}⟩ for
+3
+
+later discussion. We define them by1
+|{n};⃗k(+1)
+a
+⟩ =
+1
+√1 ± na
+√
+V
+a(a)†
+⃗k
+|{n}⟩,
+( + : bosons,
+− : fermions ) ,
+(14)
+|{n};⃗k(−1)
+a
+⟩ =
+1
+√na
+√
+V
+a(a)
+⃗k |{n}⟩.
+(15)
+The coefficients are chosen to be unit vectors
+⟨{n};⃗k(±1)
+a
+|{n};⃗k(±1)
+a
+⟩ = 1.
+(16)
+These increased/decreased states also become the eigenstate of the occupation operator:
+ˆna(⃗k)|{n};⃗k(+1)
+a
+⟩ =
+�
+1 ± na(⃗k)
+�
+|{n};⃗k(+1)
+a
+⟩,
+( + : bosons,
+− : fermions ) ,
+(17)
+ˆna(⃗k)|{n};⃗k(−1)
+a
+⟩ = ±
+�
+na(⃗k) − 1
+�
+|{n};⃗k(−1)
+a
+⟩,
+( + : bosons,
+− : fermions ) .
+(18)
+2.2.2
+Transition probability
+Using the eigenstates discussed in the previous section, let us consider the transition
+probability of the process
+A, B, · · · → X, Y, · · ·
+(19)
+in the background in which other particles ({n}) exist.
+For simplicity, we consider a
+case that each species in the process are different. Taking the initial state as |{n}⟩ in
+order to begin the given occupation numbers, the final state through the process (19)
+can be represented as |{n};⃗k(−1)
+A
+,⃗k(−1)
+B
+, · · · ,⃗k(+1)
+X
+,⃗k(+1)
+Y
+, · · · ⟩. The probability from the
+infinite past (in-state) to the infinite future (out-state) on the background particles can
+be evaluated by2
+P(A, B, · · · → X, Y, · · · ){n}
+=
+�
+a=A,B,··· ,X,Y,···
+�
+V
+�
+d3⃗ka
+(2π)3
+�
+ga
+�
+×
+���⟨{n};���k(−1)
+A
+,⃗k(−1)
+B
+, · · · ,⃗k(+1)
+X
+,⃗k(+1)
+Y
+, · · · | ˆS|{n}⟩
+���
+2
+(20)
+where ga denotes the internal degrees of freedom for species a, and ˆS is the S-matrix oper-
+ator. The S-matrix element describing the process (19) without the background particles
+can be represented by the invariant scattering amplitude as
+inv⟨kX, kY , · · · | ˆS|kA, kB, · · · ⟩inv
+=
+iM(kA, kB, · · · → kX, kY , · · · )
+×(2π)4δ4(kA + kB + · · · − kX − kY − · · · )(21)
+1For the bosonic state, the N-increased/decreased state can be defined by
+|{n};⃗k(+N)
+a
+⟩
+=
+�
+na!
+(na + N)!
+1
+V N · (a(a)†
+⃗k
+)N|{n}⟩,
+(12)
+|{n};⃗k(−N)
+a
+⟩
+=
+�
+(na − 1)!
+(na + N − 1)!
+1
+V N · (a(a)
+⃗k )N|{n}⟩.
+(13)
+2If the initial or the final state includes N of the same species, the extra factor
+1
+N! for each duplicated
+species is needed.
+4
+
+where
+|ka, kb, · · · ⟩inv ≡
+�
+2Eka2Ekb · · · a†
+⃗kaa†
+⃗kb · · · |0⟩
+(22)
+is a Lorentz invariant particle state. The representation (21) indicates that the S-matrix
+operator includes
+ˆS
+⊃
+�
+a=A,B,··· ,X,Y,···
+��
+d3⃗k′
+a
+(2π)3
+1
+�
+2E′a
+�
+ga
+�
+× a†
+⃗k′
+Xa†
+⃗k′
+Y · · · a⃗k′
+Aa⃗k′
+B · · ·
+×iM(k′
+A, k′
+B, · · · → k′
+X, k′
+Y , · · · ) · (2π)4δ4(k′
+A + k′
+B + · · · − k′
+X − k′
+Y − · · · ).(23)
+Using the above expression, the S-matrix element on the particle background can be
+written as
+⟨{n};⃗k(−1)
+A
+,⃗k(−1)
+B
+, · · ·⃗k(+1)
+X
+,⃗k(+1)
+Y
+, · · · | ˆS|{n}⟩
+(24)
+=
+�
+a=A,B,··· ,X,Y,···
+��
+d3⃗k′
+a
+(2π)3
+1
+�
+2E′
+A
+�
+ga
+�
+×iM(k′
+A, k′
+B, · · · → k′
+X, k′
+Y , · · · ) · (2π)4δ4(k′
+A + k′
+B + · · · − k′
+X − k′
+Y − · · · )
+×⟨{n};⃗k(−1)
+A
+,⃗k(−1)
+B
+, · · ·⃗k(+1)
+X
+,⃗k(+1)
+Y
+, · · · |a†
+⃗k′
+Xa†
+⃗k′
+Y · · · a⃗k′
+Aa⃗k′
+B · · · |{n}⟩
+(25)
+=
+1
+√2EAV · 2EBV · · · · 2EXV · 2EY V · · · ·
+×iM(kA, kB, · · · → kX, kY , · · · ) · (2π)4δ4(kA + kB + · · · − kX − kY − · · · )
+×
+�
+nAnB · · · (1 ± nX)(1 ± nY ) · · ·
+(26)
+where +/− is for bosonic/fermionic particles of the produced species X, Y, · · · . As substi-
+tuting the above form into (20), one can obtain
+P(A, B, · · · → X, Y, · · · ){n}
+=
+�
+a=A,B··· ,X,Y,···
+��
+d3⃗ka
+(2π)3
+1
+2Eka
+�
+ga
+�
+× |M(kA, kB, · · · → kX, kY , · · · )|2
+×(2π)4δ4(kA + kB + · · · − kX − kY − · · · ) · V T
+×nAnB · · · (1 ± nX)(1 ± nY ) · · ·
+(27)
+where T =
+�
+dt = 2πδ(E = 0) is the transition time scale.
+Although the expression of the probability is derived, the result (27) is constructed by
+the exact information of quanta represented by the microscopic occupation numbers per a
+unit phase space na that is an integer. Since it is impossible to know the exact quantum
+state, the statistical average should be considered. The probability to realize the state
+|{n}⟩ can be represented by
+P{n} ≡
+�
+a∈{n}
+�
+⃗k
+p(na(⃗k)),
+∞
+�
+na=0
+p(na(⃗k)) = 1
+(28)
+where p(na(⃗k)) is a probability to be the occupation na on the momentum ⃗k. Multiplying
+(28) into (27) and summing over by each occupation numbers3, we can obtained the
+3This procedure is equivalent to
+|{n}⟩⟨{n}| →
+�
+{n}
+P{n}|{n}⟩⟨{n}|
+(29)
+in (20), that is, the initial state is considered by the density operator.
+5
+
+statistical probability as
+⟨P(A, B, · · · → X, Y, · · · ){n}⟩
+≡
+�
+{n}
+P{n} · P(A, B, · · · → X, Y, · · · ){n}
+=
+�
+a=A,B,··· ,X,Y,···
+��
+d3⃗ka
+(2π)3
+1
+2Ea
+�
+ga
+�
+× |M(kA, kB, · · · → kX, kY , · · · )|2
+×(2π)4δ4(kA + kB + · · · − kX − kY − · · · ) · V T
+×fAfB · · · (1 ± fX)(1 ± fY ) · · ·
+(30)
+where we denoted
+fa ≡
+∞
+�
+na=0
+p(na(⃗k)) na(⃗k).
+(31)
+The important thing is that the expectation value fa can be interpreted as the distribution
+function even though the exact forms of both the probability p(na(⃗k)) and the relating
+occupation na(⃗k) are unknown.
+Using the result of the total probability (30), one can also define the partial probability,
+as an example, for the species A of the momentum ⃗kA by
+pA(A, B, · · · → X, Y, · · · )
+≡
+d⟨P(A, B, · · · → X, Y, · · · ){n}⟩
+V d3⃗kA
+(2π)3
+�
+gA
+(32)
+=
+T
+2EA
+�
+a̸=A
+��
+d3⃗ka
+(2π)3
+1
+2Ea
+�
+ga
+�
+× |M(kA, kB, · · · → kX, kY , · · · )|2
+×(2π)4δ4(kA + kB + · · · − kX − kY − · · · )
+×fAfB · · · (1 ± fX)(1 ± fY ) · · · .
+(33)
+2.2.3
+Expression of collision term
+The variation of the distribution ∆f through the microscopic process can be evaluated by
+∆fφ ∼
+�
+all processes
+∆Nφ · [−pφ(φ, A, B, · · · → X, Y, · · · ) + pφ(X, Y, · · · → φ, A, B, · · · )] (34)
+where φ is the focusing species, ∆Nφ is a changing number of the quantum φ in the process
+φ, A, B, · · · ↔ X, Y, · · · (∆Nφ = 1 in this case), and pφ is the partial transition probability
+6
+
+for φ derived in (33). Finally, the collision term can be evaluated as
+C[fφ]
+∼
+Eφ
+∆fφ
+∆t
+����
+microscopic process
+(35)
+=
+−1
+2
+�
+all processes
+�
+a̸=φ
+��
+d3⃗ka
+(2π)3
+1
+2Ea
+�
+ga
+�
+×(2π)4δ4(kA + kB + · · · − kX − kY − · · · )
+×∆Nφ
+�
+|M(kφ, kA, kB, · · · → kX, kY , · · · )|2
+×fφfAfB · · · (1 ± fX)(1 ± fY ) · · ·
+− |M(kX, kY , · · · → kφ, kA, kB, · · · )|2
+×fXfY · · · (1 ± fφ)(1 ± fA)(1 ± fB) · · · ] .
+(36)
+In the above derivation, we set ∆t = T. We derived the above result with the single-particle
+state for all species for simplicity. In the case of including the N-duplicated species in
+φ, A, B, · · · or X, Y, · · · , one needs to multiply an extra factor 1/N! for the species.
+Note that all the squared amplitudes in (36) must be regarded as the subtracted state
+in which the contribution of on-shell particles in the intermediate processes is subtracted
+in order to avoid the double-counting of the processes. Such a situation will be faced in
+which the leading contributions of the amplitude consist of the loop diagrams or higher
+order of couplings, e.g., the baryogenesis scenario as we discuss later.
+2.3
+Full and integrated Boltzmann equation
+Eqs. (6) and (36) lead the full Boltzmann equation for a species φ on the FLRW space-time
+as
+˙fφ − H |⃗kφ|2
+Eφ
+∂fφ
+∂Eφ
+=
+− 1
+2Eφ
+�
+all processes
+�
+a̸=φ
+��
+d3⃗ka
+(2π)3
+1
+2Ea
+�
+ga
+�
+×(2π)4δ4(kA + kB + · · · − kX − kY − · · · )
+×∆Nφ
+�
+|M(kφ, kA, kB, · · · → kX, kY , · · · )|2
+×fφfAfB · · · (1 ± fX)(1 ± fY ) · · ·
+− |M(kX, kY , · · · → kφ, kA, kB, · · · )|2
+×fXfY · · · (1 ± fφ)(1 ± fA)(1 ± fB) · · · ] .(37)
+Eqs. (37) for all species describe the detail evolution of the distribution functions, but
+they are not useful to solve because of a lot of variables. To simplify the equations, the
+integrated Boltzmann equation is useful and convenient. The momentum integral of the
+left hand side of (37) leads
+�
+d3⃗kφ
+(2π)3
+�
+gφ
+(LHS of (37))
+=
+˙nφ + 3Hnφ
+(38)
+where
+nφ ≡
+�
+d3⃗kφ
+(2π)3
+�
+gφ
+fφ
+(39)
+7
+
+is the number density of φ. Note in the integration (38) that the variables t and |⃗k| =
+√
+E2 − m2 are independent and a property of the total derivative
+�
+d3⃗k
+(2π)3
+|⃗k|2
+E
+∂
+∂E (· · · ) =
+�
+d3⃗k
+(2π)3 ⃗k · ∂
+∂⃗k
+(· · · ) = −3
+�
+d3⃗k
+(2π)3 (· · · )
+(40)
+is used. As the result, the number of the dynamical variables are reduced from (#species)×(#t)×
+(#E) to (#species)×(#t), while the right hand side of the integrated (37) still includes
+the E-dependent distribution functions. An useful approximation is to apply the Maxwell-
+Boltzmann similarity distribution4
+fa, fMB
+a
+≪ 1
+and
+fa(t, Ea) ∼
+na(t)
+nMB
+a
+(t)fMB
+a
+(t, Ea),
+(41)
+where
+fMB
+a
+= exp
+�
+−Ea − µa
+Ta
+�
+,
+nMB
+a
+=
+�
+d3⃗kφ
+(2π)3
+�
+gφ
+fMB
+a
+(42)
+are the Maxwell-Boltzmann distribution and its number density, respectively. Supposing
+the identical temperature for all species Tφ = TA = · · · ≡ T, and substituting (41) into
+the right hand side of (38) and assuming the chemical equilibrium
+µφ + µA + µB + · · · = µX + µY + · · ·
+(43)
+where µa is the chemical potential of species a, one can obtain the integrated Boltzmann
+equation as
+˙nφ + 3Hnφ
+=
+�
+d3⃗kφ
+(2π)3
+�
+gφ
+(RHS of (37))
+=
+−
+�
+all processes
+∆Nφ [nφnAnB · · · × ⟨R(φ, A, B, · · · → X, Y, · · · )⟩
+−nMB
+φ
+nMB
+A nMB
+B
+· · ·
+nXnY · · ·
+nMB
+X nMB
+Y
+· · · × ⟨R(¯φ, ¯A, ¯B, · · · → ¯X, ¯Y , · · · )⟩
+�
+(44)
+where the bar “ ¯ ” denotes its anti-particle state,
+R(A, B, · · · → X, Y, · · · )
+≡
+1
+2EA2EB · · ·
+� d3kX
+(2π)3
+d3kY
+(2π)3 · · ·
+1
+2EX2EY · · ·
+×
+�
+gA,gB,··· ,gX,gY ,··· |M(kA, kB · · · → kX, kY , · · · )|2
+�
+gA,gB,···
+×(2π)4δ4(kA + kB + · · · − kX − kY − · · · )
+(45)
+is a reaction rate integrated over the final state, and
+⟨R(A, B, · · · → X, Y, · · · )⟩
+≡
+1
+nMB
+A nMB
+B
+· · ·
+�
+d3kA
+(2π)3
+d3kB
+(2π)3 · · · fMB
+A
+fMB
+B
+· · ·
+×
+�
+gA,gB,···
+R(A, B, · · · → X, Y, · · · ) (46)
+4The well-used approximation fa ∼
+na
+nMB
+a
+f MB
+a
+can be justified in the case that species a is in the kinetic
+equilibrium through interacting with the thermal bath. See appendix A for detail. The case in deviating
+from the kinetic equilibrium is discussed in section 2.4.
+8
+
+is the thermally averaged reaction rate by the Maxwell-Boltzmann distributions of the
+species appearing in the initial state. We used the property of the CPT-invariance of the
+amplitude
+M(X, Y, · · · → A, B, · · · ) = M( ¯A, ¯B, · · · → ¯X, ¯Y , · · · )
+(47)
+to derive the last term in (44).
+Although eq. (44) describing the evolution of number densities is obtained by integra-
+tion of (37) directly, in general, the other evolution equations of the statistical quantities
+Qa(t) ≡
+�
+d3⃗ka
+(2π)3
+�
+ga
+fa(⃗ka)qa(t, Ea)
+(48)
+can also be derived through the same procedure with the corresponding coefficient qa(t, Ea),
+e.g., the energy density Q = ρ for q = Ea, and the pressure Q = P for q = |⃗k|2/3Ea. Such
+equations help to extract more detailed thermodynamic variables, e.g., to determine the
+independent temperatures for each species, as we will see in the next subsection.
+2.4
+Temperature parameter
+The integrated Boltzmann equation (44) is quite useful and can be applied to many
+situations. However, it might not be suitable for some situations in which the kinetic
+equilibrium is highly violated because the formula is based on the approximation by the
+Maxwell-Boltzmann similarity distribution, which is justified by the kinetic equilibrium of
+the target particles with the thermal bath as discussed in appendix A. Although to obtain
+the most appropriate solution is to solve the full Boltzmann equation, it takes a lot of
+costs to the calculation. In this section, we introduce an alternative method based on the
+integrated Boltzmann equation.
+Instead of using the similarity distribution (41), we introduce more generalized simi-
+9
+
+larity distribution by5 6
+fa(t, Ea) ∼ na(t)
+nneq
+a
+(t)fneq
+a
+(t, Ea),
+fneq
+a
+(t, Ea) ≡ exp
+�
+−Ea − µa
+Ta(t)
+�
+.
+(54)
+Here nneq
+a
+is the number density evaluated by the “non-equilibrium” Maxwell-Boltzmann
+distribution fneq
+a
+that is parametrized by the temperature parameter Ta(t). In general, the
+temperature parameter is independent of the thermal bath temperature T(t).
+Especially as the property of the Maxwell-Boltzmann distribution form, the tempera-
+ture parameter can be expressed by the ratio of the pressure and the number density
+Ta =
+P neq
+a
+nneq
+a
+=
+Pa
+na
+(55)
+because of
+P neq
+a
+=
+�
+d3⃗k
+(2π)3
+�
+ga
+fneq
+a
+|⃗ka|2
+3Ea
+=
+�
+d3⃗k
+(2π)3
+�
+ga
+Ta
+3
+�
+−⃗pa ·
+∂
+∂⃗pa
+�
+fneq
+a
+=
+Ta
+�
+d3⃗p
+(2π)3
+�
+ga
+fneq
+a
+= nneq
+a
+Ta
+(56)
+and
+Pa =
+�
+d3⃗k
+(2π)3
+�
+ga
+fa
+|⃗ka|2
+3Ea
+= na
+nneq
+a
+�
+d3⃗p
+(2π)3
+�
+ga
+fneq
+a
+|⃗ka|2
+3Ea
+= na
+nneq
+a
+P neq
+a
+.
+(57)
+Therefore, the evolution equation for the temperature parameter can be derived from the
+pressure’s one which can be constructed from the original full Boltzmann equation (37)
+multiplied by |⃗ka|2/3Ea.
+After the integration by the momentum, one can obtain the
+5The normalization factor na(t)/nneq
+a
+(t) can be regarded as a corresponding quantity to the chemical
+potential parameter ˜µa(t):
+na(t)
+nneq
+a
+(t)
+=
+exp
+� ˜µa(t) − µa
+Ta(t)
+�
+.
+(49)
+6In the case of the Bose-Einstein/Fermi-Dirac type distribution
+fa(t, Ea) =
+�
+e(Ea−˜µa(t))/Ta(t) ∓ 1
+�−1
+( − : boson,
++ : fermion)
+(50)
+where Ta(t) and ˜µa(t) are the temperature and the chemical potential parameters respectively, the tem-
+perature parameter can be represented as
+˜Ta(t) =
+1
+ρa(t) + Pa(t)
+�
+d3⃗ka
+(2π)3
+⃗k2
+a
+3 fa(t, Ea) (1 ± fa(t, Ea))
+( + : boson,
+− : fermion)
+(51)
+with energy density ρa and pressure Pa evaluated by (50). The above representation is consistent with (55)
+in the nonrelativistic limit: fa ≪ 1, Ea ∼ ma, and ρa ∼ mana ≫ Pa. The chemical potential parameter
+can be obtained by
+˜µa(t) = ρa(t) + pa(t) − Ta(t)sa(t)
+na(t)
+(52)
+where sa(t) is the entropy density defined by
+sa(t) =
+�
+d3⃗ka
+(2π)3 [±(1 ± fa) ln(1 ± fa) − fa ln fa] .
+( + : boson,
+− : fermion)
+(53)
+10
+
+coupled equations for species φ as
+˙nφ + 3Hnφ
+= −
+�
+all processes
+∆Nφ [nφnAnB · · · × ⟨R(φ, A, B, · · · → X, Y, · · · )⟩neq
+−nXnY · · · × ⟨R(X, Y, · · · → φ, A, B, · · · )⟩neq] ,
+(58)
+nφ ˙Tφ + Hnφ
+�
+2Tφ −
+�
+|⃗kφ|4
+3E3
+φ
+�neq�
+= −
+�
+all processes
+∆Nφ
+�
+nφnAnB · · · ×
+��
+|⃗kφ|2
+3Eφ
+− Tφ
+�
+R(φ, A, B, · · · → X, Y, · · · )
+�neq
+−nXnY · · · ×
+�
+⟨RTφ(X, Y, · · · → φ, A, B, · · · )⟩neq
+−Tφ⟨R(X, Y, · · · → φ, A, B, · · · )⟩neq)] ,
+(59)
+where R is a rate defined in (45) and
+RTφ(X, Y, · · · → φ, A, B, · · · )
+=
+1
+2EX2EY · · ·
+�
+d3⃗kφ
+(2π)3
+d3⃗kA
+(2π)3
+d3⃗kB
+(2π)3 · · ·
+×
+�
+gφ,gA,gB,··· ,gX,gY ,··· |M(kX, kY , · · · → kφ, kA, kB, · · · )|2
+�
+gX,gY ,···
+· |⃗kφ|2
+3Eφ
+×(2π)4δ4(kφ + kA + kB + · · · − kX − kY − · · · )
+(60)
+is a “temperature weighted” rate, and
+�
+|⃗kφ|4
+3E3
+φ
+�neq
+=
+1
+nneq
+φ
+�
+d3⃗kφ
+(2π)3
+�
+gφ
+fneq
+φ
+· |⃗kφ|4
+3E3
+φ
+,
+(61)
+��R(a, b, · · · → i, j, · · · )⟩neq
+=
+1
+nneq
+a
+nneq
+b
+· · ·
+�
+d3⃗ka
+(2π)3
+d3⃗kb
+(2π)3 · · ·
+�
+ga,gb,···
+×fneq
+a
+fneq
+b
+· · · R(a, b, · · · → i, j, · · · ),
+(62)
+are the thermally averaged quantities by the non-equilibrium distribution fneq including
+only the initial species a, b, · · · , not the final species i, j, · · · . Solving the coupled equations
+(58) and (59) for all the species can be expected to obtain more accurate results than
+the former integrated Boltzmann equation (44). Following the evolution in practice, the
+combined quantity
+y =
+mφTφ
+s2/3
+∝
+Tφ
+T 2
+(63)
+instead of the solo Tφ, where s is the entropy density, is convenient for the non-relativistic
+φ because of the asymptotic behavior Tφ(t) ∝ a(t)−2 ∝ T(t)2 after freezing out.
+3
+Application to DM abundance
+One of the cosmological application of the Boltzmann equation is for the estimation of the
+DM abundance. Because DM is stable, the main process changing the particle number is
+not decay/inverse-decay but the 2-2 annihilation/creation scatterings
+χ, ¯χ ↔ ψ, ¯ψ
+(64)
+11
+
+where χ is a DM and ψ are a standard model particle. Since the rate in the 2-2 scattering
+can be represented by the annihilation cross section as
+R(χ, ¯χ → ψ, ¯ψ) = σv
+(65)
+where v is the Møller velocity7 for the pair of the DM particles, the dynamics can be solve
+as the annihilation cross section is given. Assuming the symmetric DM nχ = n¯χ and the
+thermal distribution for the standard model particles nψ = n ¯ψ = nMB
+ψ , the Boltzmann
+equation (44) for the DM leads a simple form
+˙nχ + 3Hnχ = −
+�
+n2
+χ − (nMB
+χ
+)2�
+⟨σv⟩.
+(67)
+Instead of the particle number to follow its evolution by time, it is convenient to use the
+yield Yχ ≡ nχ/s with a dynamical variable x ≡ mχ/T, where s = 2π2
+45 heff(T)T 3 is the
+entropy density and heff(T) ∼ 100 for T ≳ 100 GeV is the effective degrees of freedom
+defined by the entropy density. In the case of no creation/annihilation process, the yield
+Yχ becomes a constant since the number and the entropy in the comoving volume is
+conserved. With these variables, the Boltzmann equation (67) can be represented as
+Y ′
+χ = −(1 + δh)s⟨σv⟩
+xH
+�
+Y 2
+χ − (Y MB
+χ
+)2�
+(68)
+where we denote ′ ≡ d/dx, and
+δh ≡
+T
+3heff
+dheff
+dT .
+(69)
+Since the adiabatic parameter δh tends to be negligible in the almost era of the thermal
+history8, we set δh = 0 in the later discussion for simplicity. Moreover, we denote
+Y MB
+χ
+≡
+nMB
+χ
+s
+∼
+gχ
+heff
+45
+25/2π7/2 x3/2e−x
+(x ≫ 1),
+(70)
+where gχ is the degrees of freedom for the DM particle.
+3.1
+Relic abundance in freeze-out
+As a simple and reasonable setup, we assume that the DM particles χ are in thermal
+equilibrium initially. Then, the dynamics described by (68) can be explained as follow. At
+first, the system is in the thermal equilibrium due to the stronger scattering effect than
+the spatial expansion9, but the yield has a small deviation from the thermal value due to
+7The definition with the 4-momenta is given by
+v12 =
+�
+(k1 · k2)2 − m2
+1m2
+2
+k0
+1k0
+2
+,
+(66)
+which can be identical to the relative velocity only in case of the parallel 3-momenta; ⃗k1 · ⃗k2 = ±|⃗k1||⃗k2|.
+8If the DM mass scale is around O(10) GeV, the freeze-out occurs around the QCD transition scale
+T ∼ O(100) MeV, in which |δh| ∼ O(1). Thus, there is a few percent level contribution from the adiabatic
+parameter δh even in the WIMP model. See Refs. [38, 39, 40, 41, 42, 43, 44, 45] for the determination of
+that parameter in detail.
+9If the interaction rate becomes lower than the Hubble rate at the relativistic regime x ≲ 1, the
+abundance freezes out with the massless abundance (hot relic):
+Y∞ ∼ Yhot = 45ζ(3)
+2π4
+gχ
+heff(Tf )
+×
+�
+1
+(boson)
+3/4
+(fermion)
+(71)
+where ζ(3) = 1.202 · · · .
+12
+
+the expansion effect as
+Yχ ∼ Y MB
+χ
++ ∆(x),
+∆(x) =
+xH
+s⟨σv⟩
+−Y ′
+χ
+Y MB
+χ
++ Y ∼
+xH
+2s⟨σv⟩ ≪ Y MB
+χ
+(72)
+as long as nMB⟨σv⟩ ≫ xH. The deviation ∆ continues growing in later time, and finally
+the evolution of the yield freezes out because the expansion rate exceeds the scattering
+rate. The freeze-out occurs when ∆(xf) = cY MB(xf), c ∼ O(1). The freeze-out time
+x = xf and the final abundance Y∞ = Y (x = ∞) can be estimated by [8, 46]
+xf
+=
+ln
+�
+c(c + 2)
+√
+90
+(2π)3
+gχ
+�
+geff(Tf)mχMplσn
+�
+−
+�
+n + 1
+2
+�
+ln xf,
+(73)
+=
+ln
+�
+c(c + 2)
+√
+90
+(2π)3
+gχ
+�
+geff(Tf)mχMplσn
+�
+−
+�
+n + 1
+2
+�
+ln
+�
+ln
+�
+c(c + 2)
+√
+90
+(2π)3
+gχ
+�
+geff(Tf)mχMplσn
+��
++ · · · ,
+(74)
+Y∞
+=
+(n + 1)
+�
+45
+π
+gχ
+�
+geff(Tf)
+xn+1
+f
+Mplmχσn
+(75)
+where Mpl = 1.22 × 1019 GeV is the Planck mass, geff(T) is the effective degrees of
+freedom defined by the energy density ρ = π2
+30geff(T)T 4, and Tf = mχ/xf is the freeze-
+out temperature. In the derivation of the analytic results (73) and (75), the temperature
+dependence of the cross section is approximated by the most dominant part as
+⟨σv⟩ = σnx−n,
+(76)
+where σn is a constant10.
+Especially, n = 0 and 1 correspond to s-wave and p-wave
+scattering, respectively.
+Although a numerical factor c still has uncertainty, choosing
+c(c + 2) = n + 1 leads to better analysis for the final abundance Y∞ within 5% accuracy
+for xf ≳ 3 [8].
+As an example, let us consider a WIMP model. Choosing the parameters as mχ = 120
+GeV, n = 1, σn = α2
+W
+m2χ , αW =
+1
+30, gχ = 2, heff = geff = 90, one can obtain the analytic
+results
+xf = 23.2,
+Y∞ = 3.81 × 10−12.
+(77)
+The actual evolution is depicted in Figure 1.
+Finally, we need to mention the validity of the approximated results (73) and (75).
+Their behaviors can deviate easily if the master equation (67) includes the significant ex-
+tra processes by other species or the singular behavior of the cross sections. Especially it
+is known some exceptional cases; (i) mutual annihilations of multiple species (coannihi-
+lations), (ii) annihilations into heaver states (forbidden channels), (iii) annihilations near
+a pole in the cross section [21], and (iv) simultaneous chemical and kinetic decoupling
+(coscattering) [23]. In these cases, the analysis should be performed more carefully. See
+[21, 22, 23, 24, 47, 48, 49] as their example cases, and also [50, 51, 52] as examples of the
+evaluation with the temperature parameter.
+10See also Appendix B for the actual analysis of the thermally averaged cross section.
+13
+
+ 1x10-13
+ 1x10-12
+ 1x10-11
+ 1x10-10
+ 1x10-9
+ 1x10-8
+ 15
+ 20
+ 25
+ 30
+ 35
+ 40
+ 45
+Yχ
+YχMB
+Ylow
+Yhigh
+Y
+x
+Evolution of yields
+Figure 1: The numerical plots of the evolution for each yield with parameters mχ = 120
+GeV, n = 1, σn = α2
+W
+m2χ , αW =
+1
+30, gχ = 2, heff = geff = 90. The red and the blue lines
+show the actual evolution of Yχ and the thermal yield Y MB
+χ
+, respectively. The dashed
+lines of green and purple show the approximated solutions Ylow ≡ Y MB
+χ
++ ∆ and Yhigh ≡
+Y∞
+�
+1 − Y∞
+n+1
+s⟨σv⟩
+H
+�−1
+, respectively.
+3.2
+Constraint on relic abundance
+The relic abundance for the stable particles through the freeze-out of their annihilation
+processes, as similar to χ particles discussed in the above, are restricted by the cosmological
+observation results. An useful parameter relating to the relic abundance is the density
+parameter defined by
+Ωχ ≡
+ρχ
+3M2
+pl
+8π H2
+∼ 16π3
+135
+heff(T)T 3
+M2
+plH2
+· mχYχ.
+(78)
+Since the yield maintains the constant after the freeze-out unless the additional entropy
+production occurs in the later era, one can estimate the present density parameter of
+χ with the present values. The current observation through the the Cosmic Microwave
+Background [53] provides T0 = 2.726 K, heff(T0) = 3.91, H0 = 100h2 km/s/Mpc, h =
+0.677, therefore one can estimate to
+Ωχ,now ∼
+mχYχ
+3.64h2 × 10−9 GeV.
+(79)
+Because the present density parameter for the cold matter component is observed as
+Ωch2 = 0.119 and it must be larger than the χ’s component, one can obtain a bound as
+mχYχ < 4.36 × 10−10 GeV.
+(80)
+The set of parameters shown in (77) is seemingly suitable for the above constraint with
+a bit of the modification.
+However, that would fail by taking into account the direct
+detections of the DM that focuses on the process of χ, ψ ↔ χ, ψ, where ψ is a standard
+model particle. If the annihilation process occurs through a similar interaction to the
+14
+
+electroweak gauge interaction, the cross section for χ-ψ elastic scattering also relates to
+the same gauge interaction. One can estimate σχψ→χψ ∼ G2
+F m2
+χ ∼ 10−36 cm2, but it is
+already excluded by the direct detection [54].
+3.3
+Relic abundance in freeze-in
+The discussion and the result in the previous subsections are based on the freeze-out
+scenario in which the DM particles are in thermal equilibrium initially. However, it is not
+satisfied if the interaction between the DM particles and the thermal bath is too small,
+so-called FIMP (feebly interacting massive particle) scenario [19, 55]. In this situation,
+the yield of DM evolves from zero through the thermal production from the thermal bath.
+Although the DM never reaches the thermal equilibrium, the yield freezes in with a non-
+thermal yield at last.
+We discuss here the relic abundance by the freeze-in scenario in two cases of the pair-
+creation of DM by (1) scattering from thermal scattering and (2) decay from a heavier
+particle.
+3.3.1
+Pair-creation by scattering
+In the case that DM-pair (χ¯χ) is produced by thermal pair particles (ψ ¯ψ), the Boltzmann
+equation is given by (68) as
+Y ′
+χ
+∼
+s⟨σv⟩
+xH (Y MB
+χ
+)2,
+(81)
+where we approximated Yχ ≪ Y MB
+χ
+and the adiabatic degrees δh ∼ 0 until the freezing-in.
+For simplicity, we consider the simple interaction described by
+Lint = λ(χ��χ)(ψ†ψ).
+(82)
+where χ is the bosonic DM, ψi labeled i are the massless bosons in the thermal bath, and
+y is a coupling constant. The thermally averaged cross section is given by
+⟨σv⟩
+=
+g2
+χg2
+ψ
+(nMB
+χ
+)2
+λ2
+(2π)5 T
+� ∞
+4m2χ
+ds
+�
+s − 4m2χ K1(√s/T).
+(83)
+where gχ and gψ are the degrees of freedom for each species. Therefore, one can estimate
+the final yield at x = mχ/T = ∞ as
+Yχ(∞)
+∼
+� ∞
+0
+dx s⟨σv⟩
+xH (Y MB
+χ
+)2
+(84)
+=
+3π2
+128 · g2
+χg2
+ψ
+λ2
+(2π)5 ·
+m4
+χ
+H(T = mχ) s(T = mχ).
+(85)
+This result implies that the yield freezing occurs around the earlier stage x ∼ O(1) because
+(85) can be regarded as Yχ(∞) ∼
+nMB
+χ
+⟨σv⟩
+H
+Y MB
+χ
+���
+x∼1.
+Applying the obtained relic abundance (85) to the relation of the present density
+parameter (79), one can obtain the required strength of the coupling as
+λ = 1.0 × 10−12 ·
+1
+gχgψ
+·
+�geff(T = mχ)
+100
+�1/4 �heff(T = mχ)
+100
+�1/2
+·
+�Ωχ,nowh2
+0.119
+�1/2
+. (86)
+15
+
+3.3.2
+Pair-creation by decay
+The other possible freeze-in scenario is due to the pair production from a heavier particle:
+σ → χ¯χ [19, 56]. The original Boltzmann equation for DM is given by
+dYχ
+dxσ
+=
+(1 + δh)Γσ→χ¯χ
+xσH
+K1(xσ)
+K2(xσ)
+�
+Yσ −
+� Yχ
+Y MB
+χ
+�2
+Y MB
+σ
+�
+(87)
+∼
+Γσ→χ¯χ
+xσH
+K1(xσ)
+K2(xσ)Y MB
+σ
+(88)
+where Γσ→χ¯χ is a decay constant and xσ ≡ mσ/T. We also approximated Yσ ∼ Y MB
+σ
+,
+Yχ ≪ Y MB
+χ
+, and δh ∼ 0 in the second line. Therefore, the final yield can be estimated as
+Yχ(∞)
+∼
+� ∞
+0
+dxσ
+Γσ→χ¯χ
+xσH
+K1(xσ)
+K2(xσ)Y MB
+σ
+(89)
+=
+3gσ
+4π ·
+Γσ→χ¯χ
+H(T = mσ)
+m3
+σ
+s(T = mσ).
+(90)
+where gσ is the degrees of freedom for σ. If the decay constant can be represented by the
+coupling constant y as
+Γσ→χ¯χ = gχ · y2
+8πmσ,
+(91)
+the required magnitude of the coupling with the relation formula to the density parameter
+(79) can be estimated as
+y2 ∼ 2.7 × 10−24 · mσ
+mχ
+·
+1
+gσgχ
+·
+�geff(T = mσ)
+100
+�1/2 heff(T = mσ)
+100
+· Ωχ,nowh2
+0.119
+.
+(92)
+4
+Application to baryogenesis
+The other popular application of the Boltzmann equation in cosmology is the baryogenesis
+scenario that describes the dynamical evolution of the baryon number in the Universe from
+zero at the beginning to the non-zero at present. The present abundance of the baryons
+can be estimated from (79). Replacing χ’s mass mχ into the nucleon mass mN = 939
+MeV and using the present density parameter for the baryon Ωbh2 = 0.0224 [53], one can
+obtain
+YB,now = 8.69 × 10−11.
+(93)
+There are three conditions suggested by A. D. Sakharov [57] in order to develop the
+baryon abundance from YB = 0 to non-zero: (1) baryon number (B) violation, (2) C
+and CP violation, (3) non-equilibrium condition. Their brief reasons are as follows. The
+B violation is trivial by definition. If the baryon number violating processes conserve
+C or CP, their anti-particle processes happen with the same rate.
+As the result, the
+net baryon number is always zero.
+Even if the processes violate the baryon number,
+C and CP, the thermal equilibrium reduces the baryon asymmetry due to their inverse
+processes. Especially, the Boltzmann equation provides a powerful tool to quantify the
+third condition.
+To see how to construct the Boltzmann equations for the baryogenesis, let us consider
+with a toy model11 as shown in Table 1. The model includes Majorana-type of chiral
+11Replacing X, ψ, φ into the right-handed neutrino, left-handed neutrino, Higgs doublet in the standard
+model, respectively, one can obtain the type-I seesaw model that can realize the well-known leptogenesis
+scenario [33]. However, the correspondence is not complete: the type-I seesaw model includes the gauge
+interactions that induces ∆L = 1 scattering process.
+16
+
+Species
+Particle statistic
+#B
+Xa
+Chiral fermion (Majorana)
+−
+ψi
+Chiral fermion
+b
+φ
+Complex scalar
+0
+Process
+∆B
+Xa → ψi, φ
+b
+Xa → ¯ψi, ¯φ
+−b
+ψi, ψj → ¯φ, ¯φ
+−2b
+ψi, φ → ¯ψj, ¯φ
+−2b
+Table 1: Left: the matter contents and their baryon number. All the anti-particles have
+the opposite sign of the baryon number. Right: Possible processes up to the 4-body B-
+violating interactions and their variation of the baryon number. The bar (¯) on each species
+denotes the anti-particle. In addition to the shown processes here, their inverse processes
+are also possible. Although there are elastic scatterings Xa, ψi, → Xb, ψj, Xa, φ → Xb, φ,
+and ψi, φ → ψj, φ, we omited them because they do not change the baryon number.
+fermions Xa for a = 1, · · · , NX, baryonic chiral fermions ψi for i = 1, · · · , Nψ with the
+common baryon number b, and a non-baryonic complex scalar φ.
+For simplicity, the
+baryonic fermions ψi and the scalar φ are massless and they are always in the thermal
+equilibrium. Because Xa are the Majorana fermion, Xa and ¯Xa can be identified. Thus,
+once we set the fundamental interaction to provide a decay/inverse-decay processes Xa ↔
+ψi, φ, their anti-particle processes Xa ↔ ¯ψi, ¯φ also exist. These 3-body interactions also
+induce the B-violating 2-2 scatterings exchanging Xa fermions.
+4.1
+Mean net baryon number
+At first, we consider only the decay processes for simplicity. This situation is realized
+when Xa particles start to decay after the scattering processes freeze out. Here we define
+the mean net baryon number by
+ϵa
+=
+�
+f
+∆Bf
+�
+rXa→f − r ¯
+Xa→ ¯f
+�
+(94)
+where the summation runs for all decay processes, ∆Bf and rXa→f are the generated
+baryon number through the process of Xa → f and its branching ratio, respectively. The
+physical meaning of the mean net baryon number ϵa is an average of the produced baryon
+number by a single quantum of Xa. In the case of our toy model, this quantity can be
+represented as
+ϵa
+=
+b
+�
+i
+�
+rXa→ψi,φ − rXa→ ¯ψi,¯φ
+�
+(95)
+This result reflect the requirements of B-violation and C, CP violation.
+If the decay
+processes are B-conserving b = 0 or C, CP conserving processes rXa→ψi,φ = rXa→ ¯ψi,¯φ, the
+mean net baryon number is vanished.
+Supposing that only a single flavour X1 survives and all of X1 particles decay into
+the baryonic fermions ψi, the generated baryon number can be estimated by YB ∼ ϵ1YX1.
+Especially, the baryon abundance can be maximized if X1 particles are the hot relic YX1 ∼
+Yhot ∼
+45
+2π4
+gX1
+heff(Tf):
+YB ∼ 45
+2π4 ·
+ϵ1gX1
+heff(Tf),
+(96)
+where gX1 = 2 is the degrees of freedom of the Majorana-type fermion X1.
+17
+
+4.2
+Boltzmann equations in baryogenesis scenario
+Although we considered quite simplified situation in the previous subsection, in reality,
+the situation is more complicated since the system includes the dynamical decay/inverse
+decay and scattering processes. In order to quantify the actual evolution of the baryon
+abundance including the scattering effects, we need to construct the Boltzmann equations
+in this system and solve them.
+For simplicity, we suppose again that only a single flavour X1 affects to the evolu-
+tion of the net baryon number.
+Using the definition of the net baryon density nB =
+b �
+i
+�
+nψi − n ¯ψi
+�
+, the evolution of the system including the processes in Table 1 are de-
+scribed by12
+˙nX1 + 3HnX1
+=
+−
+�MX1
+EX1
+�
+ΓX1(nX1 − nMB
+X1 ) + · · · ,
+(97)
+˙nB + 3HnB
+=
+ϵ1
+�MX1
+EX1
+�
+ΓX1
+�
+nX1 − nMB
+X1
+�
+− 2ΓS nB + · · · ,
+(98)
+where MX1 (EX1) is the mass (energy) of X1, ΓX1 ∼ �
+i
+�
+ΓX1→ψiφ + ΓX1→ ¯ψi,¯φ
+�
+is the
+total width of X1 and
+ϵ1 ≡ b ·
+�
+i
+�
+ΓX1→ψiφ − ΓX1→ ¯ψi,¯φ
+�
+ΓX1
+(99)
+is the mean net baryon number corresponding to (95), and
+ΓS
+=
+nMB
+φ
+⟨σψφ→ ¯ψ ¯φv⟩ + nMB
+ψ ⟨σψψ→¯φ¯φv⟩
+(100)
+is the reaction rates through the B-violating scatterings up to the tree level. The omitted
+parts “· · · ” denote the sub-leading processes in terms of the order of couplings. Eq. (97)
+describes the dissipation of Xa, and it converts to the baryon with the rate ϵa and flows
+into the baryon sector. However, the produced baryons also wash themselves out through
+the B-violating scattering processes due to the last term in (98). Therefore, the smaller
+B-violating scattering effect is favored for remaining the more net baryons as long as Xa
+can be thermalized enough at the initial.
+To solve the equations of motion (97) and (98), it is convenient to use the yields
+YX1 = nX1/s, YB = nB/s and the variable x = MX1/T as
+Y ′
+X1
+=
+−γD(YX1 − Y MB
+X1 ),
+(101)
+Y ′
+B
+=
+ϵ1γD(YX1 − Y MB
+X1 ) − 2γSYB,
+(102)
+where
+Y MB
+X1
+=
+nMB
+X1
+s
+=
+45
+4π4
+gX1
+heff
+x2K2(x),
+(103)
+γD
+=
+1
+x · ΓX1
+H(T)
+�mX1
+EX1
+�
+=
+�
+45
+4π3geff
+Mpl
+MX1
+· K1(x)
+K2(x)
+ΓX1
+T ,
+(104)
+γS
+=
+1
+x ·
+ΓS
+H(T)
+=
+�
+45
+4π3geff
+Mpl
+MX1
+· ΓS(T)
+T
+,
+(105)
+with the n-th order of the modified Bessel function Kn(x). We assumed the adiabatic evo-
+lution of the relativistic degrees h′
+eff/heff ∼ 0 to obtain (101) and (102). The dimensionless
+12See appendix B for the treatment of the thermally averaged quantities. And also see Appendix C for
+the detail of the derivation of the equations, especially, the treatment of the real intermediate state (RIS)
+to avoid the double-counting.
+18
+
+parameters γD,S are the reaction rates normalized by the Hubble parameter. In general,
+γD is proportional to x2 (x1) at the limit of x ≪ 1 (x ≫ 1), whereas the behavior of γS
+depends on the detail of the interaction as we will see its concrete form with an example
+model later.
+Eqs.(101) and (102) can provide the analytic form of YB as
+YB(∞)
+=
+−ϵ1
+� ∞
+0
+dx Y ′
+X1(x) exp
+�
+−2
+� ∞
+x
+dx′ γS(x′)
+�
+(106)
+Especially in the weakly scattering case,
+� ∞
+0 dx γS ≲ 1, one can approximate the above
+result as
+YB(∞)
+∼
+−ϵ1
+� ∞
+0
+dx Y ′
+X1(x) = ϵ1Yhot.
+(107)
+The physical interpretation is that the whole X1 particles existing from the beginning
+can convert to the net baryons without any wash-out process in this case. Hence the
+approximated result does not depend on the detail of the decay process γD. The result
+(107) is consistent with the former estimation in (96). On the other hand, the strongly
+scattering case causes the wash-out process significantly, and thus the final net baryon
+abundance is strongly suppressed from the result of (107).
+To see the concrete evolution dynamics, we consider the following interaction
+Lint
+=
+−
+�
+a,i
+yaiφXaψi + (h.c.)
+(108)
+with the Yukawa coupling yai and the two-component spinors Xa and ψi. This interaction
+leads the concrete representation of the decay width and the scattering rate as
+ΓX1
+=
+˜αMX1,
+(109)
+ΓS
+=
+T · 8˜α2
+πgψ
+˜γS(x)
+(110)
+where we denoted
+˜α
+=
+�
+i
+gψigφ
+|y1i|2
+32π ,
+(111)
+˜γS
+=
+1
+8
+� ∞
+0
+dz K1(z)
+�
+2
+�
+z4
+z2 + x2 +
+x2z2
+z2 + 2x2 ln
+�
+1 + z2
+x2
+��
++
+x2z4
+(z2 − x2)2 + ˜α2x4 + 2
+�
+z2 − x2 ln
+�
+1 + z2
+x2
+��
++
+4x2(z2 − x2)
+(z2 − x2)2 + ˜α2x4
+�
+z2 −
+�
+z2 + x2�
+ln
+�
+1 + z2
+x2
+���
+(112)
+∼
+�
+1
+(x ≪ 1)
+8/x2
+(x ≫ 1) ,
+(113)
+and gψ ≡ �
+i gψi = Nψgψi. Here gψi and gφ are the degrees of freedom of the chiral fermion
+ψi and the scalar φ, not including their anti-particle state. The asymptotic behaviors for
+each reaction rate are governed by
+γD(x) ∝
+� x2
+(x ≪ 1)
+x1
+(x ≫ 1) ,
+γS(x) ∝
+� (constant)
+(x ≪ 1)
+x−2
+(x ≫ 1) .
+(114)
+19
+
+1e-06
+1e-04
+1e-02
+1e+00
+1e+02
+1e+04
+1e+06
+ 0.01
+ 0.1
+ 1
+ 10
+ 100
+For decays (γD)
+For scatterings (γS)
+MX1 = 1016 GeV
+MX1 = 1015 GeV
+MX1 = 1014 GeV
+MX1 = 1013 GeV
+MX1 = 1013 GeV
+MX1 = 1014 GeV
+MX1 = 1015 GeV
+MX1 = 1016 GeV
+γ = Γ/xH
+x = MX1/T
+Evolution of rate of reactions
+1e-12
+1e-10
+1e-08
+1e-06
+1e-04
+1e-02
+1e+00
+ 0.01
+ 0.1
+ 1
+ 10
+ 100
+MX1 = 1016 GeV
+MX1 = 1015 GeV
+MX1 = 1014 GeV
+MX1 = 1013 GeV
+Analytic (hot relic decay)
+YB/ε1
+x = MX1/T
+Evolution of YB/ε1
+Figure 2: The numerical plots of the evolution of the interaction rates (upper) and YB/ϵ1
+(lower) for each mass of X. The numerical parameters are chosen as gX1 = 2, gψi = gφ = 1,
+Nψ = 3, ˜α = 0.01, heff = geff = 100, and assumed the thermal distribution for X1 and
+YB = 0 at the initial.
+The solid lines in red, yellow, green, and blue correspond to
+MX1 = 1016, 1015, 1014, and 1013 GeV, respectively. In the upper figure, “For decays” and
+“For scatterings” depict γD(x) and γS(x), respectively. In the lower figure, the dashed
+line in purple shows the approximated solution (96) due to the decay of the hot relic,
+YB/ϵ1 ∼ Yhot = 45
+2π ·
+gX1
+heff .
+20
+
+The actual behavior of γD and γS with concrete parameters are shown in the upper side
+of Figure 2. The asymptotic behaviors at x ≪ 1 and x ≫ 1 are consistent with (114). The
+enhancement structures for each γS seen around x ∼ O(1) are induced by the resonant
+process through the on-shell s-channel shown in (112).
+The lower side in Figure 2 shows the evolution of YB/ϵ1, which is the numerical result
+from the coupled equations (101) and (102). The result shows that the heavier mass of X
+can generate more the net baryon number because the reaction rates are reduced for the
+heavier case, and hence the generated baryons can avoid the wash-out process. Especially,
+the plot for MX1 = 1016 GeV leads the close result to the hot relic approximation (107),
+whereas the plot for MX1 = 1013 GeV shows the dumping by the wash-out effect at the
+late stage. The milder decrease at the middle stage is caused by the decay of X particles
+that supplies the net baryons to compensate for the wash-out effect.
+Finally, the obtained yield of the net baryon number YB should be compared with
+the current bound (93), YB,now ∼ 10−10. Since the mean net baryon number can roughly
+be estimated by ϵ1 ∼ ˜α2 sin2 θCP where θCP is a CP phase in the considered model, one
+can obtain the constraint from the current observation as YB,now/ϵ1 ≳ 10−10/˜α2 ∼ 10−6,
+where we used ˜α = 0.01. Therefore, one can find that MX1 ≳ 1014 GeV is allowed by
+compared with the lower plot in Figure 2.
+5
+Summary
+In this paper we have demonstrated the derivation of the Boltzmann equation from the
+microscopic point of view with the quantum field theory, in which the transition probabil-
+ity has been constructed with the statistically averaged quantum states. Although both
+results of the full and the integrated Boltzmann equation (37) and (44) are consistent with
+the well-known results, our derivation ensures that especially the full Boltzmann equation
+is widely applicable even in the non-equilibrium state since the derivation does not as-
+sume any distribution type nor the temperature of the system. Especially the integrated
+Boltzmann equation (44) is quite convenient and applicable for wide situations. In the
+particular case that the kinetic equilibrium cannot be ensured, the coupled equations with
+the temperature parameter (58) and (59) are better for following the dynamics.
+As the application examples of the (integrated) Boltzmann equation in cosmology, we
+have reviewed two cases, the relic abundance of the DM and the baryogenesis scenario.
+For the former case, we have shown the Boltzmann equation and its analysis. The analytic
+results (73) and (75) are quite helpful for estimating the final relic abundance of the DM
+and its freeze-out epoch. For the latter case, we have derived the Boltzmann equation
+with a specific model and show the numerical analysis. The final net baryon number can
+be estimated by the analytic result (107) in the case of the weakly interacting system,
+whereas that is strongly suppressed by the wash-out effect in the case of the strongly
+interacting system.
+The Boltzmann equation is a powerful tool for following the evolution of the particle
+number or other thermal quantities, and thus it will be applied for many more situations
+in future and will open a new frontier of the current physics. We hope this paper helps
+you to use the Boltzmann equation and its techniques thoughtfully.
+Acknowledgment
+We thank Chengfeng Cai, Yi-Lei Tang, and Masato Yamanaka for useful discussions and
+comments. This work is supported in part by the National Natural Science Foundation of
+21
+
+China under Grant No. 12275367, and the Sun Yat-Sen University Science Foundation.
+A
+Validity of the Maxwell-Boltzmann similarity approxi-
+mation
+Although the approximation of the distribution function by the Maxwell Boltzmann simi-
+larity distribution is used well in many situations, such approximation is not always valid.
+In this appendix, we show that the approximation is valid if the focusing species is in the
+kinetic equilibrium through interacting with the thermal bath.
+Let us consider the situation of the particle number conserving process a(k1)+b(k2) ↔
+a(k3)+b(k4), where a and b denote the particle species. If this process happens fast enough
+and the species b maintains the thermal distribution, the condition of the detailed balance
+leads
+0
+=
+fa(t, E1)fMB
+b
+(t, E2) − fa(t, E3)fMB
+b
+(t, E4)
+(115)
+=
+� fa(t, E1)
+fMB
+a
+(t, E1) −
+fa(t, E3)
+fMB
+a
+(t, E3)
+�
+fMB
+a
+(t, E1)fMB
+b
+(t, E2),
+(116)
+where we assumed the common temperature to the thermal bath and the energy conserva-
+tion law: fMB
+a
+(t, E1)fMB
+b
+(t, E2) = fMB
+a
+(t, E3)fMB
+b
+(t, E4). Because the above relation must
+be satisfied by arbitrary energy, one can obtain
+fa(t, E)
+fMB
+a
+(t, E)
+= C(t)
+(117)
+where C(t) is a function which is dependent on time but independent of the energy.
+The function C(t) can be determined by integrating over the momentum of fa(t, E) =
+C(t)fMB
+a
+(t, E1), i.e., na(t) = C(t)nMB
+a
+(t). Finally, one can obtain the desired form of the
+distribution:
+fa(t, E) = C(t)fMB
+a
+(t, E) =
+na(t)
+nMB
+a
+(t)fMB
+a
+(t, E).
+(118)
+B
+Formulae for thermal average by Boltzmann-Maxwell dis-
+tribution
+In this section, we summarize the convenient formulae used in the various thermally av-
+eraged quantities by the Maxwell-Boltzmann distribution, especially for number density,
+decay rate, and cross section.
+B.1
+Number density and modified Bessel function
+The number density with the Maxwell-Boltzmann distribution is given by
+nMB
+=
+g
+�
+d3k
+(2π)3 fMB
+(119)
+=
+g
+2π2 m2TK2(m/T)eµ/T
+(120)
+=
+g ×
+�
+�
+�
+�
+�
+�
+�
+1
+π2 T 3eµ/T + · · ·
+(T ≫ m)
+�mT
+2π
+�3/2
+e−(m−µ)/T
+�
+1 + 15T
+8m + · · ·
+�
+(T ≪ m)
+,
+(121)
+22
+
+where Kn is the n-th order of the modified Bessel function given by
+Kn(x)
+=
+� ∞
+0
+dθ e−x cosh θ cosh nθ
+(122)
+=
+�
+�
+�
+�
+�
+�
+�
+Γ(n)
+2
+�2
+x
+�n
++ · · ·
+(0 < x ≪ √1 + n)
+� π
+2x e−x
+�
+1 + 4n2 − 1
+8x
++ · · ·
+�
+(x ≫ 1)
+(123)
+Especially, the following relations are helpful in analysis:
+Kn(x)
+=
+x
+2n (Kn+1(x) − Kn−1(x)) ,
+(124)
+d
+dx (xnKn(x))
+=
+−xnKn−1(x).
+(125)
+B.2
+Thermally averaged decay rate
+The rate defined in (45) for the single initial species relates to the decay rate, R(A →
+X, Y, · · · ) = mA
+2EA ΓA→X,Y,···, where
+ΓA→X,Y,···
+=
+1
+2mA
+� d3kX
+(2π)3
+d3kY
+(2π)3 · · ·
+1
+2EX2EY
+· · ·
+×(2π)4δ4(kA − kX − kY − · · · )
+× 1
+gA
+�
+gA,gX,gY ,···
+|M(A → X, Y, · · · )|2
+(126)
+is the partial width for the process A → X, Y, · · · . The factor mA/EA in R corresponds to
+the inverse Lorentz gamma factor describing the life-time dilation. The thermal average
+of the rate is given by
+⟨R(A → X, Y, · · · )⟩
+=
+1
+2ΓA→X,Y,···
+�mA
+EA
+�
+,
+(127)
+�mA
+EA
+�
+=
+gA
+nMB
+A
+�
+d3kA
+(2π)3
+mA
+EA
+fMB
+A
+(128)
+=
+K1(mA/T)
+K2(mA/T)
+(129)
+=
+�
+�
+�
+mA
+2T + · · ·
+(T ≫ mA)
+1 − 3T
+2mA
++ · · ·
+(T ≪ mA)
+.
+(130)
+B.3
+Thermal averaged cross section
+The rate averaged by the initial 2-species relates to the scattering rate,
+R(A, B → X, Y, · · · )
+=
+σv
+(131)
+=
+1
+2EA2EB
+� d3kX
+(2π)3
+d3kY
+(2π)3 · · ·
+1
+2EX2EY
+· · ·
+×(2π)4δ4(kA + kB − kX − kY − · · · )
+×
+1
+gAgB
+�
+gA,gB,gX,gY ,···
+|M(A → X, Y, · · · )|2,
+(132)
+23
+
+where σ = σ(s) is the cross section for the process A, B → X, Y, · · · dependent on the
+Mandelstam variable s = (kA + kB)2, and v is the Møller velocity
+v =
+�
+(kA · kB)2 − m2
+Am2
+B
+EAEB
+=
+�
+(s − (mA + mB)2)(s − (mA − mB)2)
+2EAEB
+.
+(133)
+The thermal average of the rate can be obtained by
+⟨R(A, B → X, Y, · · · )⟩
+=
+⟨σv⟩
+(134)
+=
+gAgB
+nMB
+A nMB
+B
+�
+d3kA
+(2π)3
+d3kB
+(2π)3 σv · fMB
+A
+fMB
+B
+(135)
+=
+gAgB
+nMB
+A nMB
+B
+� ∞
+0
+d|⃗kA| d|⃗kB|
+� π
+0
+dθ ·
+1
+4π2
+|⃗kA|2|⃗kB|2 sin θ
+EAEB
+×σ(s) ·
+�
+(s − (mA + mB)2)(s − (mA − mB)2)
+× exp
+�
+−EA + EB
+T
++ µA + µB
+T
+�
+,
+(136)
+where the integral variable θ denotes the angle between ⃗kA and ⃗kB, i.e., ⃗kA · ⃗kB =
+|⃗kA||⃗kB| cos θ.
+In order to perform the integral in (136), it is convenient to change the integral variables
+(|⃗kA|, |⃗kB|, θ) to (E+, E−, s), where E± ≡ EA ± EB [39]. The Jacobian is given by
+�����
+∂(|⃗kA|, |⃗kB|, θ)
+∂(E+, E−, s)
+�����
+=
+EAEB
+4|⃗kA|2|⃗kB|2 sin θ
+.
+(137)
+The integral region can be obtained from the expression of the Mandelstam variable,
+s = m2
+A + m2
+B + 2
+�
+EAEB + |⃗kA||⃗kB| cos θ
+�
+,
+(138)
+which leads
+(s − m2
+A − m2
+B − 2EAEB)2 ≤ 4|⃗kA|2|⃗kB|2 = 4(E2
+A − m2
+A)(E2
+B − m2
+B).
+(139)
+The above inequality is equivalent to
+�
+E− − m2
+A − m2
+B
+s
+E+
+�2
+≤ (E2
++ − s)
+�
+1 − (mA + mB)2
+s
+� �
+1 − (mA − mB)2
+s
+�
+(140)
+Therefore, the integral region can be obtained as
+e− ≤ E− ≤ e+,
+(141)
+E+ ≥ √s,
+(142)
+s ≥ (mA + mB)2,
+(143)
+where
+e± ≡ m2
+A − m2
+B
+s
+E+ ±
+�
+(E2
++ − s)
+�
+1 − (mA + mB)2
+s
+� �
+1 − (mA − mB)2
+s
+�
+.
+(144)
+24
+
+Using the above results, the integral (136) can be performed as
+⟨σv⟩
+=
+gAgB
+nMB
+A nMB
+B
+1
+2(2π)4 e(µA+µB)/T
+×
+� ∞
+(mA+mB)2 ds · σ(s) · (s − (mA + mB)2)(s − (mA − mB)2)
+× T
+√sK1(√s/T)
+(145)
+=
+1
+4m2
+Am2
+BT
+� ∞
+mA+mB
+d√s · σ(s) · (s − (mA + mB)2)
+×(s − (mA − mB)2) ·
+K1(√s/T)
+K2(mA/T)K2(mB/T),
+(146)
+where we used the representation of the number density (120).
+Especially in the case of the non-relativistic limit, mA, mB ≫ T, it is convenient to
+use the representation13
+�
+σv(s)
+≡
+σ(s) · vNR(s),
+vNR(s) ≡
+�
+s − (mA + mB)2
+mAmB
+(147)
+and the replacement of the integral variable s to y defined by
+√s
+=
+mA + mB + Ty.
+(148)
+Since the integral parameter y corresponds to ⃗k2/mT naively, we can expect that the
+significant integral interval is on y ≲ O(1). Then (146) can be approximated as
+⟨σv⟩
+∼
+2
+√π
+�
+1 − 15T
+8mA
+− 15T
+8mB
++
+3T
+8(mA + mB) + · · ·
+�
+×
+� ∞
+0
+dy ·
+�
+(�
+σv)0 + Ty · (�
+σv)′
+0 + · · ·
+�
+×e−y · √y
+�
+1 + Ty
+2mA
++ Ty
+2mB
+−
+Ty
+4(mA + mB) + · · ·
+�
+(149)
+=
+(�
+σv)0 + 3
+2T
+�
+−3
+4
+� 1
+mA
++
+1
+mB
+�
+(�
+σv)0 + (�
+σv)′
+0
+�
++ O(T 2),
+(150)
+where we used the asymptotic expansion (123) and the Taylor series around √s = mA +
+mB,
+�
+σv = (�
+σv)0 + (√s − mA − mB) · (�
+σv)′
+0 + · · ·
+(151)
+(�
+σv)0 ≡ �
+σv(√s = mA + mB),
+(�
+σv)′
+0 ≡
+d �
+σv
+d√s
+����√s=mA+mB
+.
+(152)
+C
+Derivation of the Boltzmann equations in baryogenesis
+scenario
+In this section, we demonstrate the derivation of the Boltzmann equation in the baryoge-
+nesis scenario with the processes listed in Table 1. Indeed, the straightforward derivation
+13The Lorentz-invariant “velocity” vNR behaves as vNR ∼
+���
+⃗kA
+mA −
+⃗kB
+mB
+��� at the non-relativistic limit. Note
+that
+lim
+vNR→0 �
+σv remains non-zero (s-wave contribution) in general.
+25
+
+of the Boltzmann equations leads to the over-counting problem in the amplitudes. For
+example, once a contribution of the decay/inverse-decay process X ↔ ψ, φ is included
+in the Boltzmann equation, the straightforward contribution from the scattering process
+¯ψ, ¯φ ↔ ψ, φ is over-counted because such process can be divided into ¯ψ, ¯φ ↔ X and
+X ↔ ψ, φ if the intermediate state X is on-shell. Therefore, in general, one must regard
+the straightforward contribution in the scattering processes as the subtracted state of the
+real intermediated state (RIS) from the full contribution [8, 58]:
+|M|2
+Boltzmann eq.
+=
+|M|2
+subtracted ≡ |M|2
+full − |M|2
+RIS.
+In a case of the scattering process ¯ψ, ¯φ → ψ, φ, the full amplitude part can be represented
+as
+iM( ¯ψ, ¯φ → ψ, φ)full
+∼
+iM(X → ψ, φ) ·
+i
+s − M2
+X + iMXΓX
+· iM( ¯ψ, ¯φ → X) (153)
+where s is the Mandelstam variable, MX and ΓX are X’s mass and total decay width,
+respectively. On the other hand, the RIS part can be evaluated as the limit of the narrow
+width by
+��M( ¯ψ, ¯φ → ψ, φ)
+��2
+RIS
+=
+lim
+ΓX→0
+��M( ¯ψ, ¯φ → ψ, φ)
+��2
+full
+(154)
+=
+lim
+ΓX→0 |M(X → ψ, φ)|2
+1
+(s − M2
+X)2 + (MXΓX)2 |M( ¯ψ, ¯φ → X)|2
+∼
+|M(X → ψ, φ)|2 ·
+π
+MXΓX
+δ(s − M2
+X) · |M( ¯ψ, ¯φ → X)|2.
+(155)
+In the last line, the narrow width approximation is applied. Since the contribution of both
+amplitudes in (155) is the order of ΓX, the RIS contribution is also the order of ΓX in
+total. Therefore, RIS part in the scattering process contributes to the decay/inverse-decay
+process.
+Taking into account the above notice, we derive the Boltzmann equation. For simplic-
+ity, we suppose that only a single flavour X1 affects to the evolution of the net baryon
+number. Because of no scattering processes associated with Xa, the equation governing
+nXa is simply written as
+˙nX1 + 3HnX1
+=
+� d3kX1
+(2π)3
+d3kψi
+(2π)3
+d3kφ
+(2π)3
+1
+2EX12Eψi2Eφ
+(2π)4δ4(kX1 − kψi − kφ)
+× 1
+gX1
+�
+gX1,gψi,gφ
+�
+−fX1|M(X1 → ψiφ)|2 + fψifφ|M(ψiφ → X1)|2
+−fX1|M(X1 → ¯ψi ¯φ)|2 + f ¯ψif¯φ|M( ¯ψi ¯φ → X1)|2�
++ · · ·
+(156)
+=
+−
+�
+i
+⟨ΓX1⟩(nX1 − nMB
+X1 ) + · · ·
+(157)
+where
+ΓX1
+=
+ΓX1→ψiφ + ΓX1→ ¯ψi ¯φ + · · ·
+(158)
+is the total decay width of X1 and
+⟨ΓX1⟩
+≡
+1
+nMB
+X1
+�
+gX1
+� d3kX1
+(2π)3
+MX1
+EX1
+ΓX1fMB
+X1
+(159)
+=
+ΓX1 · K1(MX1/T)
+K2(MX1/T)
+∼ ΓX1 ×
+� MX1/2T
+(MX1 ≪ T)
+1
+(MX1 ≫ T)
+(160)
+26
+
+is the thermally averaged width, Kn(x) is the modified Bessel function. To derive (157),
+we assumed the universal distributions for ψi (fψi ∼
+1
+Nψ fψ, fψ ≡ �
+i fψi) and ignored the
+chemical potentials in the thermal distributions (fMB
+ψ
+= fMB
+¯ψ
+, fMB
+φ
+= fMB
+¯φ
+). Besides,
+we assumed φ is always in the thermal equilibrium (fφ = fMB
+φ
+).
+On the other hand,
+ψi’s equation should be derived with the consideration of the subtracted state in some
+scattering processes to avoid the over-counting of the decay/inverse-decay processes:
+˙nψ + 3Hnψ
+=
+�
+i
+� d3kX1
+(2π)3
+d3kψi
+(2π)3
+d3kφ
+(2π)3
+1
+2EX12Eψi2Eφ
+(2π)4δ4(kX1 − kψi − kφ)
+×
+�
+gX1,gψi,gφ
+�
+fX1|M(X1 → ψiφ)|2 − fψifφ|M(ψiφ → X1)|2�
++
+�
+i,j
+� d3kψi
+(2π)3
+d3kψj
+(2π)3
+d3kφ1
+(2π)3
+d3kφ2
+(2π)3
+1
+2Eψi2Eψj2Eφ12Eφ2
+�
+gψi,gψj ,gφ1,gφ2
+×
+�
+(2π)4δ4(kψi + kψj − kφ1 − kφ2)
+×
+�
+−fψifψj|M(ψiψj → ¯φ1 ¯φ2)|2 + f¯φ1f¯φ2|M(¯φ1 ¯φ2 → ψiψj)|2�
++(2π)4δ4(kψi + kφ1 − kψj − kφ2)
+×
+�
+−fψifφ1|M(ψiφ1 → ¯ψj ¯φ2)|2
+sub + f ¯ψjf¯φ2|M( ¯ψj ¯φ2 → ψiφ1)|2
+sub
+��
++ · · ·
+(161)
+=
+�
+nX1⟨ΓX1→ψφ⟩ − nMB
+X1
+nψ
+nMB
+ψ
+⟨ΓX1→ ¯ψ ¯φ⟩
+�
+−(nψ)2⟨σψψ→¯φ¯φv⟩ + (nMB
+ψ )2⟨σ ¯ψ ¯ψ→φφv⟩
+−nMB
+φ
+�
+nψ⟨σψφ→ ¯ψ ¯φv⟩ − n ¯ψ⟨σ ¯ψ ¯φ→ψφv⟩
+�
++
+�
+nψ
+nMB
+ψ
+⟨(ΓX1→ ¯ψ ¯φ)2⟩
+ΓX1
+− n ¯ψ
+nMB
+ψ
+⟨(ΓX1→ψφ)2⟩
+ΓX1
+�
+nMB
+X1
++ · · ·
+(162)
+where we used the notations nψ ≡ �
+i nψi, ΓXa→ψφ ≡ �
+i ΓXa→ψiφ, and
+⟨σψφ→ ¯ψ ¯φv⟩
+≡
+1
+nMB
+ψ nMB
+φ
+· gψgφ
+� d3kψi
+(2π)3
+d3kφ
+(2π)3 σψφ→ ¯ψ ¯φv · fMB
+ψi fMB
+φ
+(163)
+⟨σψψ→¯φ¯φv⟩
+≡
+1
+(nMB
+ψ )2 · g2
+ψ
+� d3kψi
+(2π)3
+d3kψj
+(2π)3 σψψ→¯φ¯φv · fMB
+ψi fMB
+ψj
+(164)
+with gψ ≡ �
+i gψi are the thermally averaged cross sections.
+The fourth line in (162)
+corresponds to the RIS contribution that makes the thermal balance to the first line,
+while the processes in the third line includes the resonant structure as seen in (153). With
+the expression of the net baryon number density nB = b(nψ − n ¯ψ), one can finally obtain
+the equation for the net baryons using (162) as
+˙nB + 3HnB
+=
+ϵ1⟨ΓX1⟩
+�
+nX1 − nMB
+X1
+�
+−nMB
+φ
+�
+2bnMB
+ψ
+�
+⟨σψφ→ ¯ψ ¯φv⟩ − ⟨σ ¯ψ ¯φ→ψφv⟩
+�
++nB
+�
+⟨σψiφ→ ¯ψ ¯φv⟩ + ⟨σ ¯ψi ¯φ→ψφv⟩
+��
+−nMB
+ψ
+�
+2bnMB
+ψ
+�
+⟨σψψ→¯φ¯φv⟩ − ⟨σ ¯ψ ¯ψ→φφv⟩
+�
++nB
+�
+⟨σψψ→¯φ¯φv⟩ + ⟨σ ¯ψ ¯ψ→φφv⟩
+��
++ · · · .
+(165)
+27
+
+where ϵ1 is the mean net number defined in (99), and we used the approximation
+nψ + n ¯ψ ∼ 2nMB
+ψ
+≫ |nB| = b|nψ − n ¯ψ|.
+(166)
+Note that the tree level contribution of cross sections and their anti-state are same in
+general. Therefore, the terms in second and the fourth lines in (165) are cancelled in the
+leading order, respectively.
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+A Greedy Sensor Selection Algorithm for
+Hyperparameterized Linear Bayesian Inverse Problems
+Nicole Aretza, Peng Chenb, Denise D. Degenc, Karen Veroyd
+aOden Institute for Computational Engineering and Sciences, University of Texas at Austin, 201 E 24th St,
+Austin, TX 78712, USA
+bSchool of Computational Science and Engineering, Georgia Institute of Technology, 756 W Peachtree St
+NW, Atlanta, GA 30308, USA
+cComputational Geoscience, Geothermics, and Reservoir Geophysics, RWTH Aachen University,
+Mathieustr. 30, 52074 Aachen, Germany
+dCenter for Analysis, Scientific Computing and Applications, Department of Mathematics and Computer
+Science, Eindhoven University of Technology, 5612 AZ Eindhoven, The Netherlands
+Abstract
+We consider optimal sensor placement for a family of linear Bayesian inverse prob-
+lems characterized by a deterministic hyper-parameter. The hyper-parameter describes
+distinct configurations in which measurements can be taken of the observed physical
+system. To optimally reduce the uncertainty in the system’s model with a single set of
+sensors, the initial sensor placement needs to account for the non-linear state changes
+of all admissible configurations. We address this requirement through an observabil-
+ity coefficient which links the posteriors’ uncertainties directly to the choice of sensors.
+We propose a greedy sensor selection algorithm to iteratively improve the observability
+coefficient for all configurations through orthogonal matching pursuit. The algorithm
+allows explicitly correlated noise models even for large sets of candidate sensors, and
+remains computationally efficient for high-dimensional forward models through model
+order reduction. We demonstrate our approach on a large-scale geophysical model
+of the Perth Basin, and provide numerical studies regarding optimality and scalability
+with regard to classic optimal experimental design utility functions.
+1. Introduction
+In the Bayesian approach to inverse problems (c.f. [1]), the uncertainty in a param-
+eter is described via a probability distribution. With Bayes’ Theorem, the prior belief in
+a parameter is updated when new information is revealed such that the posterior distri-
+bution describes the parameter with improved certainty. Bayes’ posterior is optimal in
+the sense that it is the unique minimizer of the sum of the relative entropy between the
+posterior and the prior, and the mean squared error between the model prediction and
+the experimental data. The noise model drives, along with the measurements, how the
+posterior’s uncertainty is reduced in comparison to the prior. A critical aspect – espe-
+arXiv:2301.12019v1  [math.NA]  27 Jan 2023
+
+cially for expensive experimental data1 – is how to select the measurements to improve
+the posterior’s credibility best. The selection of adequate sensors meeting individual
+applications’ needs is, therefore, a big goal of the optimal experimental design (OED)
+research field and its surrounding community. We refer to the literature (e.g., [3, 4, 5])
+for introductions.
+The analysis and algorithm presented in this work significantly extend our initial
+ideas presented in [6] in which we seek to generalize the 3D-VAR stability results
+from [7] to the probabilistic Bayesian setting. Our proposed algorithm is directly re-
+lated to the orthogonal matching pursuit (OMP) algorithm [8, 9] for the parameterized-
+background data-weak (PBDW) method and the empirical interpolation method (EIM)
+([10, 11]). Closely related OED methods for linear Bayesian inverse problems over
+partial differential equations (PDEs) include [12, 13, 14, 15, 16, 17], mostly for A- and
+D-OED and uncorrelated noise. In recent years, these methods have also been extended
+to non-linear Bayesian inverse problems, e.g., [18, 19, 20, 21, 22], while an advance to
+correlated noise has been made in [23]. In particular, [21, 22] use similar algorithmic
+approaches to this work by applying a greedy algorithm to maximize the expected in-
+formation gain. Common strategies for dealing with the high dimensions imposed by
+the PDE model use the framework in [24] for discretization, combined with parameter
+reduction methods (e.g., [25, 26, 27, 28, 29, 30, 31]) and model order reduction (MOR)
+methods for uncertainty quantification (UQ) problems (e.g., [32, 33, 34, 35, 36]).
+In this paper, we consider inverse problem settings, in which a deterministic hyper-
+parameter describes anticipated system configurations such as material properties or
+loading conditions. Each configuration changes the model non-linearly, so we obtain
+a family of possible posterior distributions for any measurement data. Supposing data
+can only be obtained with a single set of sensors regardless of the system’s configu-
+ration, the OED task becomes to reduce the uncertainty in each posterior uniformly
+over all hyper-parameters. This task is challenging for high-dimensional models since
+1) each configuration requires its own computationally expensive model solve, and 2)
+for large sets of admissible measurements, the comparison between sensors requires
+the inversion of the associated, possibly dense noise covariance matrix. By building
+upon [6], this paper addresses both challenges and proposes in detail a sensor selection
+algorithm that remains efficient even for correlated noise models.
+The main contributions are as follows: First, we identify an observability coeffi-
+cient as a link between the sensor choice and the maximum eigenvalue of each poste-
+rior distribution. We also provide an analysis of its sensitivity to model approximations.
+Second, we decompose the noise covariance matrix for any observation operator to al-
+low fast computation of the observability gain under expansion with additional sensors.
+Third, we propose a sensor selection algorithm that iteratively constructs an observa-
+tion operator from a large set of sensors to increase the observability coefficient over
+all hyper-parameters. The algorithm is applicable to correlated noise models, and re-
+quires, through the efficient use of MOR techniques, only a single full-order model
+1For instance, for projects harvesting geothermal energy, the development costs (e.g., drilling, stimula-
+tion, and tests) take up 50 − 70% of the total budget ([2]). As each borehole can cost several million dollars,
+it is essential to plan their location carefully.
+2
+
+evaluation per selected sensor.
+While the main idea and derivation of the observability coefficient are similar to
+[6], this work additionally features 1) an analysis of the observability coefficient re-
+garding model approximations, 2) explicit computational details for treating correlated
+noise models, and 3) a comprehensive discussion of the individual steps in the sen-
+sor selection algorithm. Moreover, the proposed method is tested using a large-scale
+geophysical model of the Perth Basin.
+This paper is structured as follows: In Section 2 we introduce the hyper-parameterized
+inverse problem setting, including all assumptions for the prior distribution, the noise
+model, and the forward model. In Section 3, we then establish and analyze the con-
+nection between the observability coefficient and the posterior uncertainty.
+We fi-
+nally propose our sensor selection algorithm in Section 4 which exploits the presented
+analysis to choose sensors that improve the observability coefficient even in a hyper-
+parameterized setting. We demonstrate the applicability and scalability of our approach
+on a high-dimensional geophysical model in Section 5 before concluding in Section 6.
+2. Problem setting
+Let X be a Hilbert space with inner product ⟨·, ·⟩X and induced norm ∥x∥2
+X :=
+⟨x, x⟩X. We consider the problem of identifying unknown states xtrue(θ) ∈ X of a single
+physical system under changeable configurations θ from noisy measurements
+d(θ) ≈ [ℓ1(xtrue(θ)), . . . , ℓK(xtrue(θ))]T ∈ RK.
+The measurements are obtained by a set of K unique sensors (or experiments) ℓ1, . . . , ℓK ∈
+X′. Our goal is to choose these sensors from a large sensor library L ⊂ X′ of options
+in a way that optimizes how much information is gained from their measurements for
+any configurations θ.
+Hyper-parameterized forward model
+We consider the unknown state xtrue to be uniquely characterized by two sources of
+information:
+• an unknown parameter utrue ∈ RM describing uncertainties in the governing
+physical laws, and
+• a hyper-parameter (or configuration2) θ ∈ P ⊂ Rp describing dependencies on
+controllable configurations under which the system may be observed (such as
+material properties or loading conditions) where P is a given compact set en-
+closing all possible configurations.
+For any given u ∈ RM and θ ∈ P, we let xθ(u) ∈ X be the solution of an abstract model
+equation Mθ(xθ(u); u) = 0 and assume that the map u → xθ(u) is well-defined, linear,
+2We call θ interchangeably hyper-parameter or configuration to either stress its role in the mathematical
+model or physical interpretation.
+3
+
+and uniformly continuous in u, i.e.
+∃ ¯η > 0 :
+η(θ) := sup
+u∈RM
+∥xθ(u)∥X
+∥u∥Σ−1
+pr
+< ¯η
+∀ θ ∈ P.
+(1)
+Remark 1. Although we assumed that utrue lies in the Euclidean space RM, any other
+linear space can be considered via an affine transformation onto an appropriate basis
+(see [12, 37]). For infinite-dimensional spaces, we first discretize with appropriate
+treatment of the adjoint operator (c.f. [24]).
+Remark 2. By keeping the model equation general, we stress the applicability of our
+approach to a wide range of problems. For instance, time-dependent states can be
+treated by choosing X as a Bochner space or its discretization (c.f. [38]). We also
+do not formally restrict the dimension of X, though any implementation relies on the
+ability to compute xθ(u) with sufficient accuracy. To this end, we note that the analysis
+in Section 3.2 can be applied to determine how discretization errors affect the observ-
+ability criterion in the sensor selection.
+Following a probabilistic approach to inverse problems, we express the initial un-
+certainty in utrue = utrue(θ) of any xtrue = xθ(utrue) in configuration θ through a random
+variable u with Gaussian prior µpr = N
+�
+upr, Σpr
+�
+, where upr ∈ RM is the prior mean
+and Σpr ∈ RM×M is a symmetric positive definite (s.p.d.) covariance matrix. The latter
+defines the inner product ⟨·, ·⟩Σ−1
+pr and its induced norm ∥ · ∥Σ−1
+pr through
+⟨u, v⟩Σ−1
+pr := uTΣ−1
+pr ˜u,
+∥u∥2
+Σ−1
+pr := ⟨u, u⟩Σ−1
+pr ,
+∀ u, v ∈ RM.
+(2)
+With these definitions, the probability density function (pdf) for µpr is
+πpr(u) =
+1
+�
+(2π)M det Σpr
+exp
+�
+−1
+2∥u − upr∥2
+Σ−1
+pr
+�
+.
+For simplicity, we assume {utrue(θ)}θ∈P to be independent realizations of u such that
+we may consider the same prior for all θ without accounting for a possible history of
+measurements at different configurations.
+Sensor library and noise model
+For taking measurements of the unknown states {xtrue(θ)}θ, we call any linear func-
+tional ℓ ∈ X′ a sensor, and its application to a state x ∈ X its measurement ℓ(x) ∈ R.
+We model experimental measurements dℓ ∈ R of the actual physical state xtrue as
+dℓ = ℓ(xtrue) + εℓ where εℓ ∼ N(0, cov(εℓ, εℓ)) is a Gaussian random variable. We
+permit noise in different sensor measurements to be correlated with a known covari-
+ance function cov. In a slight overload of notation, we write cov : L × L → R,
+cov(ℓi, ℓj) := cov(εℓi, εℓj) as a symmetric bilinear form over the sensor library. Any
+ordered subset S = {ℓ1, . . . , ℓK} ⊂ L of sensors can then form a (linear and continuous)
+observation operator through
+L := [ℓ1, . . . , ℓK]T : X → RK,
+Lx := [ℓ1(x), . . . , ℓK(x)]T .
+4
+
+The experimental measurements of L have the form
+d = �ℓ1(xtrue) + εℓ1, . . . , ℓK(xtrue) + εℓK
+�T = Lxtrue + ε
+with
+ε = �εℓ1, . . . , εℓK
+�T ∼ N(0, σ2ΣL),
+(3)
+where σ2ΣL is the noise covariance matrix defined through
+ΣL ∈ RK×K,
+such that
+�
+σ2ΣL
+�
+i, j := cov(ℓj, ℓi) = cov(εℓj, εℓi)
+(4)
+with an auxiliary scaling parameter3 σ2 > 0. We assume that the library L and the noise
+covariance function cov have been chosen such that ΣL is s.p.d. for any combination
+of sensors in L. This assumption gives rise to the L-dependent inner product and its
+induced norm
+�
+d, ˜d
+�
+Σ−1
+L := dTΣ−1
+L ˜d,
+∥d∥2
+Σ−1
+L := ⟨d, d⟩Σ−1
+L ,
+∀ d, ˜d ∈ RK.
+(5)
+Measured with respect to this norm, the largest observation of any (normalized) state
+is thus
+γL := sup
+∥x∥X=1
+∥Lx∥Σ−1
+L = sup
+x∈X
+∥Lx∥Σ−1
+L
+∥x∥X
+.
+(6)
+We show in Section 4.1 that γL increases under expansion of L with additional sensors
+despite the change in norm, and is therefore bounded by γL ≤ γL.
+We also define the parameter-to-observable map
+GL,θ : RM → RK,
+such that
+GL,θ (u) := Lxθ(u).
+(7)
+With the assumptions above – in particular the linearity and uniform continuity (1) of
+x in u – the map GL,θ is linear and uniformly bounded in u. We let GL,θ ∈ RK×M
+denote its matrix representation with respect to the unit basis {em}M
+m=1. The likelihood
+of d ∈ RK obtained through the observation operator L for the parameter u ∈ RM and
+the system configuration θ is then
+ΦL
+�
+d
+��� u, θ
+�
+:=
+1
+�
+2K det ΣL
+exp
+�
+− 1
+2σ2
+���d − GL,θ (u)
+���2
+Σ−1
+L
+�
+.
+Note that GL,θ and GL,θ may depend non-linearly on θ.
+Posterior distribution
+Once noisy measurement data d ≈ Lxtrue(θ) is available, Bayes’ theorem yields the
+posterior pdf as
+πL,θ
+post(u | d) =
+1
+Z(θ) exp
+�
+− 1
+2σ2
+���GL,θ (u) − d
+���2
+Σ−1
+L − 1
+2∥u − upr∥2
+Σ−1
+pr
+�
+∝ πpr(u)·ΦL
+�
+d
+��� u, θ
+�
+,
+(8)
+3We introduce σ2 here as an additional variable to ease the discussion of scaling in Section 13. However,
+we can set σ2 = 1 without loss of generality (w.l.o.g.).
+5
+
+with normalization constant
+Z(θ) :=
+�
+Rp exp
+�
+− 1
+2σ2
+���GL,θ (u) − d
+���2
+Σ−1
+L
+�
+dµpr.
+Due to the linearity of the parameter-to-observable map, the posterior measure µL,θ
+post is
+a Gaussian
+µL,θ
+post = N(uL,θ
+post(d), ΣL,θ
+post)
+with known (c.f. [1]) mean and covariance matrix
+uL,θ
+post(d) = ΣL,θ
+post
+� 1
+σ2 GT
+L,θΣ−1
+L d + Σ−1
+pr upr
+�
+∈ RM,
+(9)
+ΣL,θ
+post =
+� 1
+σ2 GT
+L,θΣ−1
+L GL,θ + Σ−1
+pr
+�−1
+∈ RM×M.
+(10)
+The posterior µL,θ
+post thus depends not only on the choice of sensors, but also on the con-
+figuration θ under which their measurements were obtained. Therefore, to decrease the
+uncertainty in all possible posteriors with a single, θ-independent observation operator
+L, the construction of L should account for all admissible configurations θ ∈ P under
+which xtrue may be observed.
+Remark 3. The linearity of xθ(u) in u is a strong assumption that dictates the Gaussian
+posterior. However, in combination with the hyper-parameter θ, our setting here can
+be re-interpreted as the Laplace-approximation for a non-linear state map θ �→ x(θ)
+(c.f. [39, 21, 40]). The sensor selection presented here is then an intermediary step for
+OED over non-linear forward models.
+3. The Observability Coefficient
+In this section, we characterize how the choice of sensors in the observation op-
+erator L and its associated noise covariance matrix ΣL influence the uncertainty in the
+posteriors µL,θ
+post, θ ∈ P. We identify an observability coefficient that bounds the eigen-
+values of the posterior covariance matrices ΣL,θ
+post, θ ∈ P with respect to L, and facilitates
+the sensor selection algorithm presented in Section 4.
+3.1. Eigenvalues of the Posterior Covariance Matrix
+The uncertainty in the posterior πL,θ
+post for any configuration θ ∈ P is uniquely char-
+acterized by the posterior covariance matrix ΣL,θ
+post, which is in turn connected to the
+observation operator L through the parameter-to-observable map GL,θ and the noise co-
+variance matrix ΣL. To measure the uncertainty in ΣL,θ
+post, the OED literature suggests a
+variety of different utility functions to be minimized over L in order to optimize the sen-
+sor choice. Many of these utility functions can be expressed in terms of the eigenvalues
+6
+
+λθ,1
+L ≥ · · · ≥ λθ,M
+L
+> 0 of ΣL,θ
+post, e.g.,
+A-OED:
+trace(ΣL,θ
+post) =
+M
+�
+m=1
+λθ,m
+L
+(mean variance)
+D-OED:
+det(ΣL,θ
+post) =
+M
+�
+m=1
+λθ,m
+L
+(volume)
+E-OED:
+λmax(ΣL,θ
+post) = λθ,1
+L
+(spectral radius).
+In practice, the choice of the utility function is dictated by the application. In E-optimal
+experimental design (E-OED), for instance, posteriors whose uncertainty ellipsoids
+stretch out into any one direction are avoided, whereas D-OED minimizes the overall
+volume of the uncertainty ellipsoid regardless of the uncertainty in any one parameter
+direction. We refer to [3] for a detailed introduction and other OED criteria.
+Considering the hyper-parameterized setting where each configuration θ influences
+the posterior uncertainty, we seek to choose a single observation operator L such that
+the selected utility function remains small for all configurations θ ∈ P, e.g., for E-OED,
+minimizing
+min
+ℓ1,...,ℓK∈L max
+θ∈P λmax(ΣL,θ
+post)
+such that
+L = [ℓ1, . . . , ℓK]T
+guarantees that the longest axis of each posterior covariance matrix ΣL,θ
+post for any θ ∈ P
+has the same guaranteed upper bound. The difficulty here is that the minimization over
+P necessitates repeated, cost-intensive model evaluations to compute the utility func-
+tion for many different configurations θ. In the following, we therefore introduce an
+upper bound to the posterior eigenvalues that can be optimized through an observabil-
+ity criterion with far fewer model solves. The bound’s optimization indirectly reduces
+the different utility functions through the posterior eigenvalues.
+Recalling that ΣL,θ
+post is s.p.d., let {ψm}M
+m=1 be an orthonormal eigenvector basis of
+ΣL,θ
+post, i.e. ψT
+mψn = δm,n and
+ΣL,θ
+postψm = λθ,m
+L ψm
+m = 1, . . . , M.
+(11)
+Using the representation (10), any eigenvalue λθ,m
+L
+can be written in the form
+1
+λθ,m
+L
+= ψT
+m
+�
+ΣL,θ
+post
+�−1 ψm = ψT
+m
+� 1
+σ2 GT
+L,θΣ−1
+L GL,θ + Σ−1
+pr
+�
+ψm = 1
+σ2
+���GL,θ (ψm)
+���2
+Σ−1
+L + ∥ψm∥2
+Σ−1
+pr .
+(12)
+Since ψm depends implicitly on L and θ through (11), we cannot use this representation
+directly to optimize over L. To take out the dependency on ψm, we bound ∥ψm∥2
+Σ−1
+pr ≥
+1
+λmax
+pr
+in terms of the maximum eigenvalue of the prior covariance matrix Σpr. Likewise, we
+define
+βG(θ) := inf
+u∈RM
+���GL,θ (u)
+���Σ−1
+L
+∥u∥Σ−1
+pr
+= inf
+u∈RM
+∥Lxθ(u)∥Σ−1
+L
+∥u∥Σ−1
+pr
+,
+(13)
+7
+
+as the minimum ratio between an observation for a parameter u relative to the prior’s
+covariance norm. From (12) and (13) we obtain the upper bound
+λθ,m
+L
+=
+�����������
+1
+σ2
+���GL,θ (ψm)
+���2
+Σ−1
+L
+∥ψm∥2
+Σ−1
+pr
++ 1
+�����������
+−1
+∥ψm∥−2
+Σ−1
+pr ≤
+� 1
+σ2 βG(θ)2 + 1
+�−1
+λmax
+pr .
+Geometrically, this bound means that the radius λθ,1
+L of the outer ball around the pos-
+terior uncertainty ellipsoid is smaller than that of the prior uncertainty ellipsoid by at
+least the factor
+� 1
+σ2 βG(θ)2 + 1
+�−1. By choosing L to maximize minθ βG(θ), we therefore
+minimize this outer ball containing all uncertainty ellipsoids (i.e., for any θ ∈ P). As
+expected, the influence of L is strongest when the measurement noise is small such that
+data can be trusted (σ2 ≪ 1), and diminishes with increasing noise levels (σ2 ≫ 1).
+3.2. Parameter Restriction
+An essential property of βG(θ) is that βG(θ) = 0 if K < M, i.e., the number of sen-
+sors in L is smaller than the number of parameter dimensions. In this case, βG(θ) cannot
+distinguish between sensors during the first M − 1 steps of an iterative algorithm, or in
+general when less than a total of M sensors are supposed to be chosen. For medium-
+dimensional parameter spaces (M ∈ O(10)), we mitigate this issue by restricting u to
+the subspace span{ϕ1, . . . , ϕmin{K,M}} ⊂ RM spanned by the first min{K, M} eigenvec-
+tors of Σpr corresponding to its largest eigenvalues, i.e., the subspace with the largest
+prior uncertainty. For high-dimensional parameter spaces or when the model Mθ has a
+non-trivial null-space, we bound βG(θ) further
+βG(θ) = inf
+u∈RM
+∥Lxθ(u)∥Σ−1
+L
+∥xθ(u)∥X
+∥xθ(u)∥X
+∥u∥Σ−1
+pr
+≥ inf
+x∈Wθ
+∥Lx∥Σ−1
+L
+∥x∥X
+inf
+u∈RM
+∥xθ(u)∥X
+∥u∥Σ−1
+pr
+= βL|W(θ) η(θ)
+(14)
+where we define the linear space Wθ of all achievable states
+Wθ := {xθ(u) ∈ X : u ∈ RM}
+and the coefficients
+βL|W(θ) := inf
+x∈Wθ
+∥Lx∥Σ−1
+L
+∥x∥X
+,
+η(θ) := inf
+u∈RM
+∥xθ(u)∥X
+∥u∥Σ−1
+pr
+.
+(15)
+The value of η(θ) describes the minimal state change that a parameter u can achieve
+relative to its prior-induced norm ∥u∥Σ−1
+pr . It can filter out parameter directions that have
+little influence on the states xθ(u). In contrast, the observability coefficient βL|W(θ)
+depends on the prior only implicitly via Wθ; it quantifies the minimum amount of
+information (measured with respect to the noise model) that can be obtained on any
+state in Wθ relative to its norm. Future work will investigate how to optimally restrict
+the parameter space based on η(θ) before choosing sensors that maximize βL|W(θ).
+Existing parameter reduction approaches in a similar context include [28, 41, 42, 27].
+In this work, however, we solely focus on the maximization of βG(θ) and, by extension,
+βL|W(θ) and henceforth assume that M is sufficiently small and η := infθ∈P η(θ) > 0 is
+bounded away from zero.
+8
+
+3.3. Observability under model approximations
+To optimize the observability coefficient βG(θ) or βL|W(θ), it must be computed for
+many different configurations θ ∈ P. The accumulating computational cost motivates
+the use of reduced-order surrogate models, which typically yield considerable com-
+putational savings versus the original full-order model. However, this leads to errors
+in the state approximation. In the following, we thus quantify the influence of state
+approximation error on the observability coefficients βG(θ) and βL|W(θ). An analysis
+of the change in posterior distributions when the entire model Mθ is substituted in the
+inverse problem can be found in [1].
+Suppose a reduced-order surrogate model ˜
+Mθ(˜xθ(u); u) = 0 is available that yields
+for any configuration θ ∈ P and parameter u ∈ RM a unique solution ˜xθ(u) ∈ X such
+that
+∥xθ(u) − ˜xθ(u)∥X ≤ εθ ∥xθ(u)∥X
+with accuracy
+0 ≤ εθ ≤ ε < 1.
+(16)
+Analogously to (13) and (15), we define the reduced-order observability coefficients
+˜βG(θ) := inf
+u∈RM
+∥L˜xθ(u)∥Σ−1
+L
+∥u∥Σ−1
+pr
+,
+˜βL|W(θ) := inf
+u∈RM
+∥L˜xθ(u)∥Σ−1
+L
+∥˜xθ(u)∥X
+(17)
+to quantify the smallest observations of the surrogate states. For many applications,
+it is possible to choose a reduced-order model whose solution can be computed at a
+significantly reduced cost such that ˜βG(θ) and ˜βL|W(θ) are much cheaper to compute
+than their full-order counterparts βG(θ) and βL|W(θ). Since the construction of such a
+surrogate model depends strongly on the application itself, we refer to the literature
+(e.g., [43, 44, 45, 46, 47]) for tangible approaches.
+Recalling the definition of γL in (6), we start by bounding how closely the surrogate
+observability coefficient ˜βL|W(θ) approximates the full-order βL|W(θ).
+Proposition 1. Let η(θ) > 0 hold, and let ˜xθ(u) ∈ X be an approximation to xθ(u) such
+that (16) holds for all θ ∈ P, u ∈ RM. Then
+(1 − εθ) ˜βL|W(θ) − γLεθ ≤ βL|W(θ) ≤ (1 + εθ) ˜βL|W(θ) + γLεθ.
+(18)
+Proof. Let u ∈ RM \ {0} be arbitrary. Using (16) and the (reversed) triangle inequality,
+we obtain the bound
+∥˜xθ(u)∥X
+∥xθ(u)∥X
+≥ ∥xθ(u)∥X − ∥xθ(u) − ˜xθ(u)∥X
+∥xθ(u)∥X
+≥ 1 − εθ.
+(19)
+Note here that η(θ) > 0 implies ∥xθ(u)∥X > 0 so the quotient is indeed well defined.
+The ratio of observation to state can now be bounded from below by
+∥Lxθ(u)∥Σ−1
+L
+∥xθ(u)∥X
+≥
+∥L˜xθ(u)∥Σ−1
+L
+∥xθ(u)∥X
+−
+∥L(xθ(u) − ˜xθ(u))∥Σ−1
+L
+∥xθ(u)∥X
+≥ ∥˜xθ(u)∥X
+∥xθ(u)∥X
+∥L˜xθ(u)∥Σ−1
+L
+∥˜xθ(u)∥X
+− γL
+∥xθ(u) − ˜xθ(u)∥X
+∥xθ(u)∥X
+≥ (1 − εθ)
+∥L˜xθ(u)∥Σ−1
+L
+∥˜xθ(u)∥X
+− γLεθ
+≥ (1 − εθ)˜βL|W(θ) − γLεθ,
+9
+
+where we have applied the reverse triangle inequality, definition (6), the bounds (16),
+(19), and definition (17) of ˜βL|W(θ). Since u is arbitrary, the lower bound in (18) follows
+from definition (13) of βL|W(θ). The upper bound in (18) follows analogously.
+For the observability of the parameter-to-observable map GL,θ and its approxima-
+tion u �→ L˜xθ(u), we obtain a similar bound. It uses the norm η(θ) of xθ : u �→ xθ(u) as
+a map from the parameter to the state space, see (1).
+Proposition 2. Let ˜xθ(u) ∈ X be an approximation to xθ(u) such that (16) holds for all
+θ ∈ P, u ∈ RM. Then
+˜βG(θ) − γLη(θ)εθ ≤ βG(θ) ≤ ˜βG(θ) + γLη(θ)εθ.
+(20)
+Proof. Let u ∈ RM \ {0} be arbitrary. Then
+∥Lxθ(u)∥Σ−1
+L ≥ ∥L˜xθ(u)∥Σ−1
+L − ∥L(xθ(u) − ˜xθ(u))∥Σ−1
+L
+≥ ∥L˜xθ(u)∥Σ−1
+L − γL ∥xθ(u) − ˜xθ(u)∥X
+≥ ∥L˜xθ(u)∥Σ−1
+L − γLεθ ∥xθ(u)∥X
+≥ ∥L˜xθ(u)∥Σ−1
+L − γLεθη(θ)∥u∥Σ−1
+pr ,
+where we have used the reverse triangle inequality, followed by (6), (16), and (1). We
+divide by ∥u∥Σ−1
+pr and take the infimum over u to obtain
+βG(θ) = inf
+u∈RM
+∥Lxθ(u)∥Σ−1
+L
+∥u∥Σ−1
+pr
+≥ inf
+u∈RM
+∥L˜xθ(u)∥Σ−1
+L
+∥u∥Σ−1
+pr
+− γL η(θ) εθ = ˜βG(θ) − γL η(θ) εθ.
+The upper bound in (20) follows analogously.
+If εθ is sufficiently small, Propositions 1 and 2 justify employing the surrogates
+˜βL|W(θ) and ˜βG(θ) instead of the original full-order observability coefficients βL|W(θ)
+and βG(θ). This substitution becomes especially necessary when the computation of
+xθ(u) is too expensive to evaluate βL|W(θ) or βG(θ) repeatedly for different configura-
+tions θ.
+Another approximation step in our sensor selection algorithm relies on the identifi-
+cation of a parameter direction v ∈ RM with comparatively small observability, i.e.
+∥Lxθ(v)∥Σ−1
+L
+∥v∥Σ−1
+pr
+≈ inf
+u∈RM
+∥Lxθ(u)∥Σ−1
+L
+∥u∥Σ−1
+pr
+= βG(θ)
+or
+∥Lxθ(v)∥Σ−1
+L
+∥xθ(v)∥X
+≈ inf
+x∈Wθ
+∥Lx∥Σ−1
+L
+∥x∥X
+= βL|W(θ).
+The ideal choice would be the infimizer of respectively βG(θ) or βL|W(θ), but its compu-
+tation involves M full-order model evaluations (c.f. Section 4.2). To avoid these costly
+computations, we instead choose v as the infimizer of the respective reduced-order
+observability coefficient. This choice is justified for small εθ < 1 by the following
+proposition:
+10
+
+Proposition 3. Let η(θ) > 0 hold, and let ˜xθ(u) ∈ X be an approximation to xθ(u) such
+that (16) holds for all θ ∈ P, u ∈ RM. Suppose v ∈ arg infu∈RM ∥u∥−1
+Σ−1
+pr ∥L˜xθ(u)∥Σ−1
+L , then
+βG(θ) ≤
+∥Lxθ(v)∥Σ−1
+L
+∥v∥Σ−1
+pr
+≤ βG(θ) + 2γLη(θ)εθ.
+(21)
+Likewise, if v ∈ arg infu∈RM ∥˜xθ(u)∥−1
+X ∥L˜xθ(u)∥Σ−1
+L , then
+βL|W(θ) ≤
+∥Lxθ(v)∥Σ−1
+L
+∥xθ(v)∥X
+≤ 1 + εθ
+1 − εθ
+�βL|W(θ) + γLεθ
+� + γLεθ.
+(22)
+Proof. For both (21) and (22) the lower bound follows directly from definitions (13)
+and (15). To prove the upper bound in (21), let v ∈ arg infu∈RM ∥u∥−1
+Σ−1
+pr ∥L˜xθ(u)∥Σ−1
+L .
+Following the same steps as in the proof of Proposition 2, we can then bound
+∥Lxθ(v)∥Σ−1
+L
+∥v∥Σ−1
+pr
+≤
+∥L˜xθ(v)∥Σ−1
+L
+∥v∥Σ−1
+pr
++
+∥L(xθ(v) − ˜xθ(v))∥Σ−1
+L
+∥v∥Σ−1
+pr
+≤ ˜βG(θ) + γLη(θ)εθ.
+The upper bound in (21) then follows with Proposition 2.
+To prove the upper bound in (22), let v ∈ arg infu∈RM ∥˜xθ(u)∥−1
+X ∥L˜xθ(u)∥Σ−1
+L . Then
+∥Lxθ(v)∥Σ−1
+L
+∥xθ(v)∥X
+≤
+∥L˜xθ(v)∥Σ−1
+L
+∥˜xθ(v)∥X
+∥˜xθ(v)∥X
+∥xθ(v)∥X
++
+∥L(xθ(v) − ˜xθ(v))∥Σ−1
+L
+∥xθ(v)∥X
+≤ (1 + ε) ˜βL|W(θ) + γLεθ.
+The result then follows with Proposition 1.
+4. Sensor selection
+In the following, we present a sensor selection algorithm that iteratively increases
+the minimal observability coefficient minθ∈P βG(θ) and thereby decreases the upper
+bound for the eigenvalues of the posterior covariance matrix for all admissible system
+configurations θ ∈ P. The iterative approach is relatively easy to implement, allows a
+simple way of dealing with combinatorial restrictions, and can deal with large4 sensor
+libraries.
+4.1. Cholesky decomposition
+The covariance function cov connects an observation operator L to its observabil-
+ity coefficients βG(θ) and βL|W(θ) through the noise covariance matrix ΣL. Its inverse
+enters the norm ∥·∥Σ−1
+L and the posterior covariance matrix ΣL,θ
+post. The inversion poses
+a challenge when the noise is correlated, i.e., when ΣL is not diagonal, as even the
+expansion of L with a single sensor ℓ ∈ L changes each entry of Σ−1
+L . In naive compu-
+tations of the observability coefficients and the posterior covariance matrix, this leads
+4For instance, in Section 5.3 we apply the presented algorithm to a library with KL = 11, 045 available
+sensor positions.
+11
+
+Algorithm 1: CholeskyExpansion
+Input: observation operator L = [ℓ1, . . . , ℓK]T, noise covariance matrix ΣL,
+Cholesky matrix CL, new sensor ℓ ∈ X′
+L ← [ℓ1, . . . , ℓK, ℓ]T
+// operator expansion
+if K = 0 then
+ΣL ← (cov(ℓ, ℓ)) , CL ←
+� √cov(ℓ, ℓ)
+�
+∈ R1×1
+// first sensor
+else
+v ← [cov(ℓ1, ℓ), . . . , cov(ℓK, ℓ)]T ∈ RK
+// matrix expansion
+w ← C−1
+L v ∈ RK, s ← cov(ℓ, ℓ), c ← s − wTw ∈ R
+ΣL ←
+� ΣL
+v
+vT
+s
+�
+, CL ←
+� CL
+0
+wT
+c
+�
+∈ R(K+1)×(K+1)
+return L, ΣL, CL
+to M dense linear system solves of order O((K + 1)3) each time the observation oper-
+ator is expanded. In the following, we therefore expound on how Σ−1
+L changes under
+expansion of L to exploit its structure when comparing potential sensor choices.
+Suppose L = [ℓ1, . . . , ℓK]T has already been chosen with sensors ℓk ∈ X′, but shall
+be expanded by another sensor ℓ to
+[L, ℓ] := [ℓ1, . . . , ℓK, ℓ]T : X → RK+1.
+Following definition (4), the noise covariance matrix Σ[L,ℓ] of the expanded operator
+[L, ℓ] has the form
+Σ[L,ℓ] =
+� ΣL
+vL,ℓ
+vT
+L,ℓ
+vℓ,ℓ
+�
+=
+� CL
+0
+cT
+L,ℓ
+cℓ,ℓ
+� � CT
+L
+cL,ℓ
+0
+cℓ,ℓ
+�
+,
+where CLCT
+L = ΣL ∈ RK×K is the Cholesky decomposition of the s.p.d. noise covariance
+matrix ΣL for the original observation operator L, and vL,ℓ, cL,ℓ ∈ RK, vℓ,ℓ, cℓ,ℓ ∈ R are
+defined through
+�vL,ℓ
+�
+i := cov(ℓi, ℓ),
+cL,ℓ := C−1
+L vL,ℓ,
+vℓ,ℓ := cov(ℓ, ℓ),
+cℓ,ℓ :=
+�
+vℓ,ℓ − cT
+L,ℓcL,ℓ.
+Note that Σ[L,ℓ] is s.p.d. by the assumptions posed on cov in Section 2; consequently,
+cℓ,ℓ is well-defined and strictly positive. With this factorization, the expanded Cholesky
+matrix C[L,ℓ] with C[L,ℓ]CT
+[L,ℓ] = Σ[L,ℓ] can be computed in O(K2), dominated by the
+linear system solve with the triangular CL for obtaining cL,ℓ. It is summarized in Algo-
+rithm 1 for later use in the sensor selection algorithm.
+Using the Cholesky decomposition, the inverse of Σ[L,ℓ] factorizes to
+Σ−1
+[L,ℓ] =
+� CT
+L
+cL,ℓ
+0
+cℓ,ℓ
+�−1 � CL
+0
+cT
+L,ℓ
+cℓ,ℓ
+�−1
+=
+� C−T
+L
+rL,ℓ
+0
+1/cℓ,ℓ
+� � C−1
+L
+0
+rT
+L,ℓ
+1/cℓ,ℓ
+�
+,
+12
+
+Algorithm 2: ObservabilityGain
+Input: observation operator L = [ℓ1, . . . , ℓK]T, Cholesky matrix CL, sensor
+candidate ℓ ∈ X′, state x ∈ X
+d ← Lx, z ← C−1
+L d
+// preparation
+if K = 0 then
+return ℓ(xK)2/cov(ℓ, ℓ)
+// one sensor only
+else
+v ← [cov(ℓ1, ℓ), . . . , cov(ℓK, ℓ)]T ∈ RK
+// general case
+w ← C−1
+L v ∈ RK
+return (ℓ(xK)−wT z)
+2
+cov(ℓ,ℓ)−wT w
+where
+rL,ℓ := − 1
+cℓ,ℓ
+C−T
+L cL,ℓ = − 1
+cℓ,ℓ
+C−T
+L C−1
+L vL,ℓ = − 1
+cℓ,ℓ
+Σ−1
+L vL,ℓ.
+For an arbitrary state x ∈ X, the norm of the extended observation [L, ℓ](x) =
+�
+LxT, ℓ(x)
+�T ∈
+RK+1 in the corresponding norm ∥·∥Σ−1
+[L,ℓ] is hence connected to the original observation
+Lx ∈ RK in the original norm ∥·∥Σ−1
+L via
+∥ [L, ℓ](x) ∥2
+Σ−1
+[L,ℓ] =
+� Lx
+ℓ(x)
+�T � ΣL
+vL,ℓ
+vT
+L,ℓ
+vℓ,ℓ
+�−1 � Lx
+ℓ(x)
+�
+=
+� Lx
+ℓ(x)
+�T � C−T
+L
+rL,ℓ
+0
+1/cℓ,ℓ
+� � C−1
+L
+0
+rT
+L,ℓ
+1/cℓ,ℓ
+� � Lx
+ℓ(x)
+�
+=
+�
+C−1
+L Lx
+rT
+L,ℓLx + ℓ(x)/cℓ,ℓ
+�T �
+C−1
+L Lx
+rT
+L,ℓLx + ℓ(x)/cℓ,ℓ
+�
+= (Lx)TC−T
+L C−1
+L Lx + (rT
+L,ℓLx + ℓ(x)/cℓ,ℓ)2
+= ∥Lx∥2
+Σ−1
+L + (rT
+L,ℓLx + ℓK+1(x)/cℓ,ℓ)2
+≥ ∥Lx∥2
+Σ−1
+L .
+(23)
+We conclude from this result that the norm ∥Lx∥Σ−1
+L of any observation, and therefore
+also the continuity coefficient γL defined in (6), is increasing under expansion of L
+despite the change in norms. For any configuration θ, the observability coefficients
+βG(θ) and βL|W(θ) are thus non-decreasing when sensors are selected iteratively.
+Given a state x ∈ X and an observation operator L, we can determine the sen-
+sor ℓK+1 ∈ L that increases the observation of x the most by comparing the increase
+(rT
+L,ℓLx + ℓ(x)/cℓ,ℓ)2 for all ℓ ∈ L. Algorithm 2 summarizes the computation of this
+observability gain for use in the sensor selection algorithm (see Section 4.3). Its gen-
+eral runtime is determined by K + 1 sensor evaluations and two linear solves with the
+triangular Cholesky matrix CL in O(K2). When called with the same L and the same
+13
+
+state x for different candidate sensors ℓ, the preparation step must only be performed
+once, which reduces the runtime to one sensor evaluation and one linear system solve
+in all subsequent calls. Compared to computing ∥ [L, ℓ](x) ∥2
+Σ−1
+[L,ℓ] for all KL candidate
+sensors in the library L, we save O(KLK2).
+4.2. Computation of the observability coefficient
+We next discuss the computation of the observability coefficient βG(θ) for a given
+configuration θ and observation operator L.
+Let Σpr = UTDprU be the eigenvalue decomposition of the s.p.d. prior covariance
+matrix with U = �ϕ1, . . . , ϕM
+� ∈ RM×M, ϕ j ∈ RM orthonormal in the Euclidean inner
+product, and Dpr = diag(λ1
+pr, . . . , λM
+pr) a diagonal matrix containing the eigenvalues
+λ1
+pr ≥ · · · ≥ λM
+pr > 0 in decreasing order. Using the eigenvector basis {ϕm}M
+m=1, we define
+the matrix
+M(θ) := �Lxθ(ϕ1), . . . , Lxθ(ϕM)� ∈ RK×M
+(24)
+featuring all observations of the associated states xθ(ϕ j) for the configuration θ. The
+observability coefficient βG(θ) can then be computed as the square root of the minimum
+eigenvalue λmin of the generalized eigenvalue problem
+M(θ)TC−T
+L C−1
+L M(θ)umin = λminD−1
+pr umin.
+(25)
+Note that (25) has M real, non-negative eigenvalues because the matrix on the left is
+symmetric positive semi-definite, and Dpr is s.p.d. (c.f. [48]). The eigenvector umin
+contains the basis coefficients in the eigenvector basis {ϕm}M
+m=1 of the “worst-case” pa-
+rameter, i.e. the infimizer of βG(θ).
+Remark 4. For computing βL|W(θ), we exchange the right-hand side matrix D−1
+pr in
+(25) with the X-inner-product matrix for the states xθ(ϕ1), . . . , xθ(ϕM).
+The solution of the eigenvalue problem can be computed in O(M3), with an addi-
+tional O(MK2 + M2K) for the computation of the left-hand side matrix in (25). The
+dominating cost is hidden in M(θ) since it requires KM sensor observations and K full-
+order model solves. To reduce the computational cost, we therefore approximate βG(θ)
+with ˜βG(θ) by exchanging the full-order states xθ(ϕ j) in (24) with their reduced-order
+approximations ˜xθ(ϕ j). The procedure is summarized in Algorithm 3.
+Remark 5. If K < M, Algorithm 3 restricts the parameter space, as discussed in Sec-
+tion 3.2, to the span of the first K eigenvectors ϕ1, . . . , ϕK encoding the least certain
+directions in the prior. A variation briefly discussed in [8] in the context of the PBDW
+method to prioritize the least certain parameters even further is to only expand the
+parameter space once the observability coefficient on the subspace surpasses a prede-
+termined threshold.
+4.3. Sensor selection
+In our sensor selection algorithm, we iteratively expand the observation operator
+L and thereby increase the observability coefficient L for all θ ∈ P. Although this
+14
+
+Algorithm 3: SurrogateObservability
+Input: configuration θ ∈ P, observation operator L = [ℓ1, . . . , ℓK]T with
+K > 0, Cholesky matrix CL
+N ← min{M, K}
+// parameter restriction
+M ← �L˜xθ(ϕ1), . . . , L˜xθ(ϕN)�, S ←
+��
+xθ(ϕi), xθ(ϕ j)
+�
+X
+�N
+i, j=1 // matrix setup
+Find (λmin, umin) of
+�
+C−1
+L M
+�T �
+C−1
+L M
+�
+umin = λminSumin
+// eigenvalue
+problem
+return
+√
+λmin, umin
+procedure cannot guarantee finding the maximum observability over all sensor com-
+binations, the underlying greedy searches are well-established in practice, and can be
+shown to perform with exponentially decreasing error rates in closely related settings,
+see [49, 8, 50, 51, 52]. In each iteration, the algorithm performs two main steps:
+• A greedy search over a training set Ξtrain ⊂ P to identify the configuration
+θ ∈ Ξtrain for which the observability coefficient βG(θ) is minimal;
+• A data-matching step to identify the sensor in the library that maximizes the
+observation of the “worst-case” parameter at the selected configuration θ.
+The procedure is summarized in Algorithm 4. It terminates when Kmax ≤ KL sensors
+have been selected.5 In the following, we explain its computational details.
+Preparations
+In order to increase βG(θ) uniformly over the hyper-parameter domain P, we con-
+sider a finite training set, Ξtrain ⊂ P, that is chosen to be fine enough to capture the θ-
+dependent variations in xθ(u). We assume a reduced-order model is available such that
+we can compute approximations ˜xθ(ϕm) ≈ xθ(ϕm) for each θ ∈ Ξtrain and 1 ≤ m ≤ M
+within an acceptable computation time while guaranteeing the accuracy requirement
+(16). If necessary, the two criteria can be balanced via adaptive training domains (e.g.,
+[53, 54]).
+Remark 6. If storage allows (e.g., with projection-based surrogate models), we only
+compute the surrogate states once and avoid unnecessary re-computations when up-
+dating the surrogate observability coefficients ˜βG(θ) in each iteration.
+As a first “worst-case” parameter direction, u0, we choose the vector ϕ1 with the
+largest prior uncertainty. Likewise, we choose the “worst-case” configuration θK ∈ P
+as the one for which the corresponding state ˜xθ(ϕ1) is the largest.
+5This termination criterion can easily be adapted to prescribe a minimum value of the observability
+coefficient. This value should be chosen with respect to the observability βG(L) achieved with the entire
+sensor library.
+15
+
+Algorithm 4: SensorSelection
+Input: sensor library L ⊂ X′, training set Ξtrain ⊂ P, maximum number of
+sensors Kmax ≤ |KL|, surrogate model ˜
+Mθ, covariance function
+cov : L × L → R
+Compute Σpr = �ϕ1, . . . , ϕM
+�T Dpr
+�ϕ1, . . . , ϕM
+�
+// eigenvalue
+decomposition
+For all θ ∈ Ξtrain, 1 ≤ m ≤ M, compute ˜xθ(ϕm)
+// preparation
+K ← 0, θ0 ← arg maxθ∈Ξtrain ∥˜xθ(ϕ1)∥X, u0 ← ϕ1
+// initialization
+while K < Kmax do
+Solve full-order equation MθK(xK, uK) for xK
+// "worst-case" state
+ℓK+1 ← arg maxℓ∈L ObservabilityGain(L, CL, ℓ)
+// sensor
+selection
+L, ΣL, CL ← CholeskyExpansion(L, ΣL, CL, ℓK+1)
+// expansion
+K ← K + 1
+for θ ∈ Ξtrain do
+˜βL|W(θ), umin(θ) ← SurrogateObservability(θ, L, CL) // update
+coefficients
+θK ← arg minθ∈Ξtrain ˜βL|W(θ)
+// greedy step
+uK ← �min{M,K}
+m=1
+[umin(θK)]m ϕm
+return L, CL
+Data-matching step
+In each iteration, we first compute the full-order state xK = xθK(uK) at the “worst-
+case” parameter uK and configuration θK. We then choose the sensor ℓK+1 which most
+improves the observation of the “worst-case” state xK under the expanded observation
+operator [LT, ℓK+1]T and its associated norm. We thereby iteratively approximate the
+information that would be obtained by measuring with all sensors in the library L. For
+fixed θK and in combination with selecting x to have the smallest observability in Wθ,
+we arrive at an algorithm similar to worst-case orthogonal matching pursuit (c.f. [8, 9])
+but generalized to deal with the covariance function cov in the noise model (3).
+Remark 7. We use the full-order state xθK(uK) rather than its reduced-order approxi-
+mation in order to avoid training on local approximation inaccuracies in the reduced-
+order model. Here, by using the “worst-case” parameter direction uK, we only require
+a single full-order solve per iteration instead of the M required for setting up the entire
+posterior covariance matrix ΣL,θ
+post.
+Greedy step
+We train the observation operator L on all configurations θ ∈ Ξtrain by varying for
+which θ the “worst-case” state is computed. Specifically, we follow a greedy approach
+16
+
+where, in iteration K, we choose the minimizer θK of βG(θ) over the training domain
+Ξtrain, i.e., the configuration for which the current observation operator L is the least
+advantageous. The corresponding “worst-case” parameter uK is the parameter direction
+for which the least significant observation is achieved. By iteratively increasing the
+observability at the “worst-case” parameters and hyper-parameters, we increase the
+minimum of βG(θ) throughout the training domain.
+Remark 8. Since the computation of ˜βG(θ) requires as many reduced-order model
+solves as needed for the posterior covariance matrix over the surrogate model, it is
+possible to directly target an (approximated) OED utility function in the greedy step
+in place of ˜βL|W(θ) without major concessions in the computational efficiency. The
+OMP step can then still be performed for the “worst-case” parameter with only one
+full-order model solve, though its benefit for the utility function should be evaluated
+carefully.
+Runtime
+Assuming the dominating computational restriction is the model evaluation to solve
+for xθ(u) – as is usually the case for PDE models – then the runtime of each iteration in
+Algorithm 4 is determined by one full-order model evaluation, and KL sensor measure-
+ments of the full-order state. Compared to computed the posterior covariance matrix
+for the chosen configuration, the OMP step saves N − 1 full-order model solves.
+The other main factor in the runtime of Algorithm 4 is the |Ξtrain|M reduced-order
+model evaluations with KL sensor evaluations each that need to be performed in each
+iteration (unless they can be pre-computed). The parameter dimension M not only en-
+ters as a scaling factor, but also affects the cost of the reduced-order model itself since
+larger values of M generally require larger or more complicated reduced-order models
+to achieve the desired accuracy (16). In turn, the computational cost of the reduced-
+order model indicates how large Ξtrain may be chosen for a given computational budget.
+While some cost can be saved through adaptive training sets and models, overall, this
+connection to M stresses the need for an adequate initial parameter reduction as dis-
+cussed in Section 3.2.
+5. Numerical Results
+We numerically confirm the validity of our sensor selection approach using a geo-
+physical model of a section of the Perth Basin in Western Australia. The basin has
+raised interest in the geophysics community due to its high potential for geothermal
+energy, e.g., [55, 56, 57, 58, 59]. We focus on a subsection that spans an area of
+63 km × 70 km and reaches 19 km below the surface. The model was introduced in
+[60] and the presented section of the model was discussed extensively in the context
+of MOR in [61, 62]. In particular, the subsurface temperature distribution is described
+through a steady-state heat conduction problem with different subdomains for the geo-
+logical layers, and local measurements may be obtained through boreholes. The bore-
+hole locations need to be chosen carefully due to their high costs (typically several
+million dollars, [63]), which in turn motivates our application of Algorithm 4. For
+demonstration purposes, we make the following simplifications to our test model: 1)
+17
+
+Figure 1: Schematic overview of the Perth Basin section including (merged) geological layers, depths for
+potential measurements, and configuration range for thermal conductivity θ on each subdomain. The bounds
+are obtained from the reference values (c.f. [60, 61]) with a ±50% margin. Adapted from [61].
+We neglect radiogenic heat production; 2) we merge geological layers with similar
+conductive behaviors; and 3) we scale the prior to emphasize the influence of different
+sensor measurements on the posterior. All computations were performed in Python
+3.7 on a computer with a 2.3 GHz Quad-Core Intel Core i5 processor and 16 GB of
+RAM. The code will be available in a public GitHub repository for another geophysical
+test problem.6
+5.1. Model Description
+We model the temperature distribution xθ with the steady-state PDE
+−∇ (θ∇xθ) = 0
+in Ω := (0, 0.2714) × (0, 0.9) × (0, 1) ⊂ R3,
+(26)
+where the domain Ω is a non-dimensionalized representation of the basin, and θ : Ω →
+R>0 the local thermal conductivity. The section comprises three main geological layers
+Ω = �
+i=1,2,3 Ωi, each characterized by different rock properties, i.e. thermal conductiv-
+ity θ|Ωi ≡ θi shown in Figure 1. We consider the position of the geological layers to be
+fixed as these are often determined beforehand by geological and geophysical surveys
+but allow the thermal conductivity to vary. In a slight abuse of notation, this lets us
+identify the field θ with the vector
+θ = (θ1, θ2, θ3) ∈ P := [0.453, 1.360] × [0.448, 1.343] × [0.360, 1.081].
+in the hyper-parameter domain P.
+We impose zero-Dirichlet boundary conditions at the surface7, and zero-Neumann
+(“no-flow”) boundary conditions at the lateral faces of the domain. The remaining
+boundary ΓIn corresponds to an area spanning 63 km × 70 km area in the Perth basin
+19 km below the surface. At this depth, local variations in the heat flux have mostly
+stabilized which makes modeling possible, but since most boreholes – often originat-
+ing from hydrocarbon exploration – are found in the uppermost 2 km we treat it as
+6The Perth Basin Model is available upon request from the third author.
+7Non-zero Dirichlet boundary conditions obtained from satellite data could be considered via a lifting
+function and an affine transformation of the measurement data (see [62]).
+18
+
+Geology
+Thermal Conductivity
+Creteaous-
+2
+with 1 E [0.453,1.360], [0refl1 = 0.9065
+-380 m
+Yarragadee
+Eneabba-
+-760 m
+0.2 
+with 02 E [0.448,1.343], [0rerl2 = 0.9855
+Lesueur
+D.1
+-1140 m
+Permian
+.
+with 3 E [0.360,1.081], [0refl3 = 0.7205
+0.2
+Basement
+-1520 m
+0.4
+0.6
+-1900 m
+= 0.2443
+0.8
+X1uncertain. Specifically, we model it as a Neumann boundary condition
+n · ∇xθ = u · p
+a.e. on ΓIn := {0} × [0, 0.9] × [0, 1]
+where n : ΓIn → R3 is the outward pointing unit normal on Ω, p : ΓIn → R5 is a vector
+composed of quadratic, L2(ΓIn)-orthonormal polynomials on the basal boundary that
+vary either in north-south or east-west direction, and u ∼ πpr = N(upr, Σpr) is a random
+variable. The prior is chosen such that the largest uncertainty is attributed to a constant
+entry in p, and the quadratic terms are treated as the most certain with prior zero. This
+setup reflects typical geophysical boundary conditions, where it is most common to
+assume a constant Neumann heat flux (e.g., [61]), and sometimes a linear one (e.g.,
+[60]). With the quadratic functions, we allow an additional degree of freedom than
+typically considered.
+The problem is discretized using a linear finite element (FE) basis of dimension
+132,651. The underlying mesh was created with GemPy ([64]) and MOOSE ([65]).
+Since the FE matrices decouple in θ, we precompute and store an affine decomposition
+using DwarfElephant ([61]). Given a configuration θ and a coefficient vector u for the
+heat flux at ΓIn, the computation of a full-order solution xθ(u) ∈ X then takes 2.96 s on
+average. We then exploit the affine decomposition further to construct a reduced basis
+(RB) surrogate model via a greedy algorithm (c.f. [49, 66]). Using the inner product8
+⟨x, φ⟩X :=
+�
+Ω ∇x · ∇φdΩ and an a posteriori error bound ∆(θ), we prescribe the relative
+target accuracy
+max
+u∈RM
+∥xθ(u) − ˜xθ(u)∥X
+∥˜xθ(u)∥X
+≤ max
+u∈RM
+∆(θ)
+∥˜xθ(u)∥X
+< ε := 1e − 4
+(27)
+to be reached for 511,000 consecutively drawn, uniformly distributed samples of θ.
+The training phase and final computational performance of the RB surrogate model are
+summarized in Figure 2. The speedup of the surrogate model (approximately a factor
+of 3,000 without error bounds) justifies its offline training time, with computational
+savings expected already after 152 approximations of βG(θ).
+For taking measurements, we consider a 47 × 47 grid over the surface to represent
+possible drilling sites. At each, a single point evaluation9 of the basin’s temperature
+distribution may be made at any one of five possible depths as shown in Figure 1. In
+total, we obtain a set L ⊂ Ω of 11, 045 admissible points for measurements. We model
+the noise covariance between sensors ℓχ, ℓ˜χ ∈ L at points χ, ˜χ ∈ Ω via
+cov(ℓχ, ℓ˜χ) := a + b − y(h)
+with the exponential variogram model
+y(h) := a + (b − a)
+�3
+2 max{h
+c, 1} − 1
+2 max{h
+c, 1}3
+�
+8Note that ⟨·, ·⟩X is indeed an inner product due to the Dirichlet boundary conditions.
+9Point evaluations are standard for geophysical models because a borehole (diameter approximately 1 m)
+is very small compared to the size of the model.
+19
+
+0
+10
+20
+30
+40
+50
+RB dimension
+10
+3
+10
+2
+10
+1
+100
+norm of the error
+max relative error bound
+true relative error
+Reduced-order model
+RB dimension
+83
+training time
+37.58 min
+training accuracy
+1e-4
+RB solve
+0.97 ms
+�→ speedup
+3,058
+RB error bound
+4.78 ms
+�→ speedup
+515
+Figure 2: Training of the RB surrogate model for the Perth Basin section. On the left: Maximum relative
+error bound (27) in the course of the greedy algorithm, computed over the training set Ξtrain together with the
+true relative error at the corresponding configuration θ. On the right: Performance pointers for the obtained
+RB model after (27) was reached; online computation times and speedups are averages computed over 1000
+randomly drawn configurations θ.
+where h2 := (χ2 − ˜χ2)2 + (χ3 − ˜χ3)2 is the horizontal distance between the points and
+a := 2.2054073480730403
+(sill)
+b := 1.6850672040263555
+(nugget)
+c := 20.606782733391228
+(range)
+The covariance function was computed via kriging (c.f. [67]) from the existing mea-
+surements [68]. With this covariance function, the noise between measurements at any
+two sensor locations is increasingly correlated the closer they are on the horizontal
+plane. Note that for any subset of sensor locations, the associated noise covariance ma-
+trix remains regular as long as each sensor is placed at a distinct drilling location. We
+choose this experimental setup because measurements in typical geothermal data sets
+are often made at the bottom of a borehole (“bottom hole temperature measurements”)
+within the first 2 km below the surface.
+5.2. Restricted Library
+To test the feasibility of the observability coefficient for sensor selection, we first
+consider a small sensor library (denoted as L5×5 below) with 25 drilling locations po-
+sitioned on a 5 × 5 grid. We consider the problem of choosing 8 pair-wise different,
+unordered sensor locations out of the given 25 positions; this is a combinatorial prob-
+lem with 1,081,575 possible combinations.
+Sensor selection
+We run Algorithm 4, using the RB surrogate model and a training set Ξtrain ⊂ P
+with 512,000 configurations on an 80 × 80 × 80 regular grid on P. When new sensors
+are chosen, the surrogate observability coefficient ˜βG(θ) increases monotonously with
+a strong incline just after the initial M = 5 sensors, followed by a visible stagnation
+(see Figure 3a) as is often observed for similar OMP-based sensor selection algorithms
+20
+
+5
+6
+7
+8
+number of sensors
+10
+4
+10
+3
+10
+2
+10
+1
+observability coefficient
+mean observability coefficient
+min observability coefficient
+improvement with next sensor
+observability coefficient, fixed config.
+(a) Observability during sensor selection
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+observability coefficient
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+max. observability 
+A-OED 
+D-OED 
+E-OED 
+proposal 
+proposal, fixed 
+(b) Histogram of βG(θref)
+Figure 3: Observability coefficient for different methods when choosing 8 out of 25 sensor locations. Left:
+Minimum and mean over θ of ˜βG(θ) as well as βG(θref) obtained in the course of running Algorithm 4 once
+for 512,000 configurations and once for the training set {θref}. Right: Distribution of βG(θref) over all possible
+sensor combinations with indicators for the A-, D-, and E-optimal choices, the combination with maximum
+observability, and the sensors chosen by the Algorithm 4 with Ξtrain-training (“proposal”, purple, marked
+“x”) and θref-training (“proposal, fixed”, turquoise, marked “+”). Note that the height of the indicator line
+was chosen solely for readability.
+(e.g., [8, 69, 70, 7]). Algorithm 4 terminates in 7.93 min with a minimum reduced-order
+observability of ˜βG(θ) = 7.3227e-2 and an average of 1.0995e-1. At the reference
+configuration θref, the full-order observability coefficient is βG(θref) = 1.0985, slightly
+below the reduced-order average. We call this training procedure “Ξtrain-training” here-
+after and denote the chosen sensors as “Ξtrain-trained sensor set” in the subsequent text
+and as “proposal” in the plots.
+In order to get an accurate understanding of how the surrogate model ˜xθ(u) and the
+large configuration training set Ξtrain influence the sensor selection, we run Algorithm 4
+again, this time restricted on the full-order FE model xθref(u) at only the reference con-
+figuration θref. The increase in βG(θref) in the course of the algorithm is shown in Fig-
+ure 3a. The curve starts significantly above the average for Ξtrain-training, presumably
+because conflicting configurations cannot occur, e.g., when one sensor would signifi-
+cantly increase the observability at one configuration but cause little change in another.
+However, in the stagnation phase, the curve comes closer to the average achieved with
+Ξtrain-training. The computation finishes within 12.53 s, showing that the long runtime
+before can be attributed to the size of Ξtrain. The final observability coefficient with 8
+sensors is βG(θref) = 1.2647e-1, above the average over ˜βG(θ) achieved training on
+Ξtrain. We call this training procedure “θref-training” hereafter, and the sensor configu-
+ration “θref-trained” in the text or “proposal, fixed config.” in the plots.
+Comparison at the reference configuration
+For comparing the performance of the Ξtrain- and θref-trained sensor combinations,
+we compute – at the reference configuration θref – all 1,081,575 posterior covariance
+matrices Σθref,L
+post for all unordered combinations L of 8 distinct sensors in the sensor li-
+brary L5×5. For each matrix, we compute the trace (A-OED criterion), the determinant
+(D-OED criterion), the maximum eigenvalue (E-OED criterion), and the observability
+coefficient βG(θref). This lets us identify the A-, D-, and E-optimal sensor combina-
+21
+
+0
+0.05
+0.1
+0.15
+0.2
+Observability Coefficient
+10
+0.25
+100
+100.25
+100.5
+100.75
+101.0
+101.25
+101.5
+101.75
+posterior trace
+A-optimal sensor combination
+proposal
+proposal, fixed configuration
+maximum observability coefficient
+all 25 sensors
+best
+sensor selection
+0.587 %
+proposal
+0.022 %
+proposal, fixed config.
+1.778 %
+max. observability
+Figure 4: Distribution of trace(ΣL,θ
+post) for θ = θref over all 1,081,575 combinations for choosing 8 out of the
+25 sensor locations. On the left: distribution of trace(ΣL,θ
+post) against the observability coefficient βG(θref).
+Note that the marginal distribution of the horizontal axis is provided in Figure 3b. On the right: histogram
+of trace(ΣL,θ
+post) (marginal distribution for the plot on the left) with for the different sensor combinations (in
+percent out of 1,081,575 combinations). The plots include markers for the A-optimal sensor choice, the
+sensors chosen by Algorithm 4 with Ξtrain-training (“proposal”) and with {θref}-training (“proposal, fixed
+configuration”), the sensor combination with maximum observability βG(θref), and when all 25 sensors are
+included.
+tions. The total runtime for these computations is 4 min – well above the 12.53 s of
+θref-training. The (almost) 8 min for Ξtrain-training remain reasonable considering it is
+trained on |Ξtrain| = 512, 000 configurations and not only θref.
+A histogram for the distribution of βG(θref) is given in Figure 3b with markers for
+the values of the A-, D-, and E-optimal choices and the Ξtrain- and θref-trained ob-
+servation operators. Out of these five, the D-optimal choice has the smallest value,
+since the posterior determinant is influenced less by the maximum posterior eigenvalue
+and hence the observability coefficient. In contrast, both the A- and E-optimal sen-
+sor choices are among the 700 combinations with the largest βG(θref) (this corresponds
+to the top 0.065%). The θref-trained sensors have similar observability and are even
+among the top 500 combinations. For the Ξtrain- trained sensors, the observability co-
+efficient is smaller, presumably because Ξtrain-training is not as optimized for θref. Still,
+it ranks among the top 0.705 % of sensor combinations with the largest observability.
+In order to visualize the connection between the observability coefficient βG(θref)
+and the classic A-, D-, and E-OED criteria, we plot the distribution of the posterior
+covariance matrix’s trace, determinant, and maximum eigenvalue over all sensor com-
+binations against βG(θ) in Figures 4, 5, 6. Overall we observe a strong correlation
+between the respective OED criteria and βG(θref): It is the most pronounced in Figure
+6 for E-optimality, and the least pronounced for D-optimality in Figure 5. For all OED
+22
+
+0
+0.05
+0.1
+0.15
+0.2
+Observability Coefficient
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Posterior determinant
+D-optimal sensor combination
+proposal
+proposal, fixed configuration
+maximum observability coefficient
+all 25 sensors
+best
+sensor selection
+0.252 %
+proposal
+0.081 %
+proposal, fixed config.
+12.923 %
+max. observability
+Figure 5: Distribution of the posterior determinant det(ΣL,θ
+post) for θ = θref. See Figure 4 for details about the
+plot structure.
+0
+0.05
+0.1
+0.15
+0.2
+Observability Coefficient
+10
+0.5
+100
+100.5
+101
+101.5
+maximum posterior eigenvalue
+E-optimal sensor combination
+proposal
+proposal, fixed configuration
+maximum observability coefficient
+all 25 sensors
+best
+sensor selection
+1.679 %
+proposal
+0.001 %
+proposal, fixed config.
+4.080 %
+max. observability
+Figure 6: Distribution of the maximum eigenvalue of the posterior covariance matrix ΣL,θ
+post for θ = θref. See
+Figure 4 for details about the plot structure. Note that the θref-trained sensor combination has the 101-st
+smallest maximum posterior eigenvalue among all 1,081,575 possibilities.
+23
+
+design criterion
+training
+pctl
+A-OED
+D-OED
+E-OED
+θref
+Ξtrain
+99-th
+3.5835
+81.8508
+1.1724
+2.2223
+9.2512
+95-th
+2.2747
+26.8430
+0.3601
+0.7846
+4.0374
+75-th
+0.5141
+3.8600
+0.0532
+0.1419
+0.8106
+50-th
+0.1527
+1.4641
+0.0159
+0.0438
+0.2354
+25-th
+0.0414
+0.3669
+0.0035
+0.0068
+0.0621
+Figure 7: Ranking in βG(θref) of the A-, D-, E- optimal and the θref- and Ξtrain-trained sensor choices for
+all possible combinations of choosing 8 unordered sensors in the library. Left: Boxplots obtained over 200
+random sensor libraries. Right: worst-case ranking (in percent) of the corresponding percentiles (“pctl”).
+criteria, the correlation becomes stronger for smaller scaling factors σ2 and weakens
+for large σ2 when the prior is prioritized (plots not shown). This behavior aligns with
+the discussion in Section 3.1 that βG(θ) primarily targets the largest posterior eigenvalue
+and is most decisive for priors with higher uncertainty.
+Comparison for different libraries
+We finally evaluate the influence of the library L5×5 on our results. To this end,
+we randomly select 200 sets of new measurement positions, each consisting of 25
+drilling locations with an associated drilling depth. For each library, we run Algorithm
+4 to choose 8 sensors, once with Ξtrain-training on the surrogate model, and once with
+the full-order model at θref only. For comparison, we then consider in each library
+each possible combination of choosing 8 unordered sensor sets and compute the trace,
+determinant, and maximum eigenvalue of the associated posterior covariance matrix at
+the reference configuration θref together with its observability coefficient. This lets us
+identify the A-, D-, and E-optimal sensor combinations.
+Figure 7 shows how βG(θref) is distributed over the 200 libraries, with percentiles
+provided in the adjacent table. For 75% of the libraries, the A- and E-optimal, and the
+Ξtrain- and θref-trained sensor choices rank among the top 1% of combinations with the
+largest observability. Due to its non-optimized training for θref, the Ξtrain-trained sensor
+set performs slightly worse than what is achieved with θref-training, but still yields
+a comparatively large value for βG(θref). In contrast, overall, the D-optimal sensor
+choices have smaller observability coefficients, presumably because the minimization
+of the posterior determinant is influenced less by the maximum posterior eigenvalue.
+The ranking of the Ξtrain- and θref-trained sensor configurations in terms of the pos-
+terior covariance matrix’s trace, determinant, and maximum eigenvalue over the 200
+libraries is given in Figure 8. Both perform well and lie for 75% of the libraries within
+the top 1% of combinations. As the ranking is performed for the configuration param-
+eter θref, the θref-trained sensor combination performs better, remaining in 95% of the
+libraries within the top 5% of sensor combinations.
+24
+
+A-OED
+D-OED
+E-OED
+obs. coef.
+100
+101
+102
+103
+104
+105
+106
+ranking
+proposal, flexible configuration
+opt.
+top 10
+0.01%
+0.1%
+1%
+5%
+25%
+100%
+percentile
+pctl
+A-OED
+D-OED
+E-OED
+βG(θref)
+99-th
+3.9240
+6.2372
+10.8391
+9.2512
+95-th
+1.9093
+3.1544
+4.5583
+4.0374
+75-th
+0.3083
+0.7718
+0.9185
+0.8106
+50-th
+0.0664
+0.2361
+0.2763
+0.2354
+25-th
+0.0177
+0.0536
+0.0596
+0.0621
+A-OED
+D-OED
+E-OED
+obs. coef.
+100
+101
+102
+103
+104
+105
+106
+ranking
+proposal, fixed configuration
+opt.
+top 10
+0.01%
+0.1%
+1%
+5%
+25%
+100%
+percentile
+pctl
+A-OED
+D-OED
+E-OED
+βG(θref)
+99-th
+2.5261
+2.9752
+11.1534
+2.2223
+95-th
+1.0134
+1.8324
+2.8458
+0.7846
+75-th
+0.1155
+0.4698
+0.3549
+0.1419
+50-th
+0.0224
+0.1212
+0.0687
+0.0438
+25-th
+0.0041
+0.0181
+0.0138
+0.0068
+Figure 8: Ranking of the posterior covariance matrix Σθref,L
+post
+in terms of the A-, D-, E-OED criteria and
+the observability coefficient βG(θref) when the observation operator GL,θ is chosen with Algorithm 4 and
+Ξtrain-training (top) or θref-training (bottom). The ranking is obtained by comparing all possible unordered
+combinations of 8 sensors in each sensor library. On the left: Boxplots of the ranking over 200 sensor
+libraries; on the right: ranking (in percent) among different percentiles.
+25
+
+(a) upmost layer, Ξtrain-training
+(b) upmost layer, θref-training
+(c) lowest layer, Ξtrain-training
+(d) lowest layer, θref-training
+Figure 9: Sensor positions chosen by Algorithm 4 from a grid of 47 × 47 available horizontal positions
+with available 5 depths each, though only the lowest (bottom) and upmost (top) layers were chosen. The
+underlying plot shows cuts through the full-order solution xθ(u) at θ = θref. Left: Ξtrain-training with the
+RB surrogate model on a training set Ξtrain ⊂ P with 10,000 random configurations; runtime 14.19 s for 10
+sensors. Right: θref-training with full-order model at reference parameter; runtime 15.85 s for 10 sensors.
+5.3. Unrestricted Library
+We next verify the scalability of Algorithm 4 to large sensor libraries by permitting
+all 2,209 drilling locations, at each of which at most one measurement may be taken
+at any of the 5 available measurement depths. Choosing 10 unordered sensors yields
+approximately 7.29e+33 possible combinations. Using the RB surrogate model from
+before, we run Algorithm 4 once on a training grid Ξtrain ⊂ P consisting of 10,000
+randomly chosen configurations using only the surrogate model (runtime 14.19 s), and
+once on the reference configuration θref using the full-order model (runtime 15.85 s) for
+comparison. We terminate the algorithm whenever 10 sensors are selected. Compared
+to the training time on L5×5 before, the results confirm that the size of the library itself
+has little influence on the overall runtime but that the full-order computations and the
+size of Ξtrain relative to the surrogate compute dominate.
+The sensors chosen by the two runs of Algorithm 4 are shown in Figure 9. They
+share many structural similarities:
+• Depth: Despite the availability of 5 measurement depths, sensors have only been
+chosen on the lowest and the upmost layers with 5 sensors each. The lower sen-
+sors were chosen first (with one exception, sensor 3 in θref-training), presumably
+26
+
+z
+1
+0.8
+0.6
+0.4
+0.2
+0
+0.9
+0.9
+P
+9
+0.8
+0.8
+Temperature
+0.7
+0.7
+0.18
+0.6
+0.6
+10
+0.16
+0.5
+-0.5
+Y
+Y
+0.4
+-0.4
+E0.14
+0.3
+0.3
+E0.12
+0.2
+0.2
+4
+E0.10
+0.1
+0.1
+0
+0.8
+0.6
+0.4
+0.2
+1
+0
+zz
+1
+0.8
+0.6
+0.4
+0.2
+0
+0.9
+0.9
+P
+0.8
+0.8
+Temperature
+0.7
+0.7
+0.18
+0.6
+0.6
+0.16
+0.5
+-0.5
+Y
+5.2
+6
+Y
+0.4
+-0.4
+E0.14
+0.3
+0.3
+E0.12
+0.2
+-0.2
+E0.10
+9
+0.1
+0.1
+8
+0
+0.8
+0.6
+0.4
+0.2
+1
+0
+zz
+1
+0.8
+0.6
+0.4
+0.2
+0
+0.9
+0.9
+3
+0.8
+-0.8
+Temperature
+0.7
+0.7
+0.91
+0.6
+0.6
+6
+0.81
+0.5
+-0.5
+Y
+Y
+0.4-
+0.4
+E0.71
+0.3
+-0.3
+E0.61
+0.2
+0.2
+E0.50
+0.1
+0.1
+-0
+-0
+0.8
+0.6
+0.4
+0.2
+0
+1
+zz
+1
+0.8
+0.6
+0.4
+0.2
+0
+0.9
+0.9
+0.8
+0.8
+Temperature
+0.7
+0.7
+0.91
+0.6
+0.6
+0.81
+0.5
+-0.5
+5
+Y
++
+Y
+0.4
+0.4
+E0.71
+0.3
+0.3
+E0.61
+0.2
+0.2
+2
+E0.50
+3
+0.1
+0.1
+-0
+-0
+0.8
+0.6
+0.4
+0.2
+0
+1
+zbecause the lower layer is closer to the uncertain Neumann boundary condition
+and therefore yields larger measurement values.
+• Pairing Each sensor on the lowest layer has a counterpart on the upmost layer
+that has almost the same position on the horizontal plane. This pairing targets
+noise sensitivity: With the prescribed error covariance function, the noise in two
+measurements is increasingly correlated the closer the measurements lie horizon-
+tally, independent of their depth coordinate. Choosing a reference measurement
+near the zero-Dirichlet boundary at the surface helps filter out noise terms in the
+lower measurement.
+• Organization On each layer, the sensors are spread out evenly and approxi-
+mately aligned in 3 rows and 3 columns. The alignment helps distinguish be-
+tween the constant, linear, and quadratic parts of the uncertain Neumann flux
+function in north-south and east-west directions.
+Figure 10 (left side) shows the increase in the observability coefficients ˜βG(θ) (for
+Ξtrain-training) and βG(θref) (for θref-training) over the number of chosen sensors. We
+again observe a strong initial incline followed by stagnation for the Ξtrain-trained sen-
+sors, whereas the curve for θref-training already starts at a large value to remain then
+almost constant. The latter is explained by the positions of the first 5 sensors in Fig-
+ure 9 (right), as they are already spaced apart in both directions for the identification of
+quadratic polynomials. In contrast, for Ξtrain-training, the “3 rows, 3 columns” structure
+is only completed after the sixth sensor (c.f. Figure 9, left). With 6 sensors, the observ-
+ability coefficients in both training schemes have already surpassed the final observ-
+ability coefficients with 8 sensors in the previous training on the smaller library L5×5.
+The final observability coefficients at the reference parameter θref are βG(θref) = 0.4042
+for θref-training, and βG(θref) = 0.3595 for Ξtrain-training.
+As a final experiment, we compare the eigenvalues of the posterior covariance ma-
+trix ΣL,θref
+post for the Ξtrain- and θref-trained sensors against 50,000 sets of 10 random sen-
+sors each. We confirm that all 50,000 sensor combinations comply with the combina-
+torial restrictions. Boxplots of the eigenvalues are provided in Figure 10 (right side).
+The eigenvalues of the posterior covariance matrix with sensors chosen by Algorithm
+4 are smaller10 than all posterior eigenvalues for the random sensor combinations.
+6. Conclusion
+In this work, we analyzed the connection between the observation operator and
+the eigenvalues of the posterior covariance matrix in the inference of an uncertain pa-
+rameter via Bayesian inversion for a linear, hyper-parameterized forward model. We
+identified an observability coefficient whose maximization decreases the uncertainty in
+the posterior probability distribution for all hyper-parameters. To this end, we proposed
+10Here we compare the largest eigenvalue of one matrix to the largest eigenvalue of another, the second
+largest to the second largest, and so on.
+27
+
+5
+6
+7
+8
+9
+10
+number of sensors
+10
+3
+10
+2
+10
+1
+100
+observability coefficient
+mean observability coefficient
+min observability coefficient
+improvement with next sensor
+observability coefficient, fixed config.
+1
+2
+3
+4
+5
+posterior eigenvalue (largest to smallest)
+10
+3
+10
+2
+10
+1
+100
+101
+random combinations with 10 sensors each
+proposal
+proposal, fixed config
+Figure 10: Left: Observabity coefficients during sensor selection with Ξtrain- and θref-training for a library
+with 11,045 measurement positions and combinatorial restrictions. Shown are 1) the minimum and mean
+surrogate observability coefficient ˜βG(θ) over a training set with 10,000 random configurations with final
+values minθ ˜βG(θ) = 0.4160 and meanθ ˜βG(θ) = 0.6488, and 2) the full-order observability coefficient βG(θref)
+when training on the reference parameter θref alone (final value βG(θref) = 0.4042). Right: Boxplots for the
+5 eigenvalues of the posterior covariance matrix ΣL,θ
+post over 50,000 sets of 10 sensors chosen uniformly from
+a 5 × 47 × 47 grid with imposed combinatorial restrictions. The eigenvalues are compared according to their
+order from largest to smallest. Indicated are also the eigenvalues for the Ξtrain-trained (purple, “x”-marker)
+and θref-trained (turquoise, “+”-marker) sensors from Figure 9.
+a sensor selection algorithm that expands an observation operator iteratively to guaran-
+tee a uniformly large observability coefficient for all hyper-parameters. Computational
+feasibility is retained through a reduced-order model in the greedy step and an OMP
+search for the next sensor that only requires a single full-order model evaluation. The
+validity of the approach was demonstrated on a large-scale heat conduction problem
+over a section of the Perth Basin in Western Australia. Future extensions of this work
+are planned to address 1) high-dimensional parameter spaces through parameter reduc-
+tion techniques, 2) the combination with the PBDW inf-sup-criterion to inform sensors
+by functionalanalytic means in addition to the noise covariance, and 3) the expansion
+to non-linear models through a Laplace approximation.
+Acknowledgments
+We would like to thank Tan Bui-Thanh, Youssef Marzouk, Francesco Silva, An-
+drew Stuart, Dariusz Ucinski, and Keyi Wu for very helpful discussions, and Florian
+Wellmann at the Institute for Computational Geoscience, Geothermics and Reservoir
+Geophysics at RWTH Aachen University for providing the Perth Basin Model. This
+work was supported by the Excellence Initiative of the German federal and state gov-
+ernments and the German Research Foundation through Grants GSC 111 and 33849990/GRK2379
+(IRTG Modern Inverse Problems). This project has also received funding from the
+European Research Council (ERC) under the European Union’s Horizon 2020 re-
+search and innovation programme (grant agreement n° 818473), the US Department
+of Energy (grant DE-SC0021239), and the US Air Force Office of Scientific Research
+(grant FA9550-21-1-0084). Peng Chen is partially supported by the NSF grant DMS
+#2245674.
+28
+
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