Add files using upload-large-folder tool

-9FLT4oBgHgl3EQfDS7k/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:92d834160c59cfdf3c522f34308d38ce1b3f08629c74400145b78cfdba0feae0
+size 3473453

-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f9bcad0066fb6e8142864149ee76cfa475002d89a9fd6a53a9e7447e7c87a4d6
+size 1502272

-9FLT4oBgHgl3EQfvi-U/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5fb4bfc4394b3e754cb692c87366322d464d76cff84dfba62f8a9f40e285b0c9
+size 3407917

-9FLT4oBgHgl3EQfvi-U/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:460b7f85f09a2f4e40fd180f813d50196e77f3a03bf1df8004292d08133377a1
+size 107067

-NAzT4oBgHgl3EQfSvt8/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:674424785a69e8c23ec4238cf28b42183083f9bef3801925c2b87fcc6fa204d9
+size 3932205

.gitattributes

@@ -3158,3 +3158,64 @@ UdAyT4oBgHgl3EQfhfgJ/content/2301.00376v1.pdf filter=lfs diff=lfs merge=lfs -tex
 WdE3T4oBgHgl3EQfbQr_/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
 SdE0T4oBgHgl3EQfUgDE/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
 -9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf filter=lfs diff=lfs merge=lfs -text
+x9AzT4oBgHgl3EQf7_4P/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+QNAzT4oBgHgl3EQfIvuz/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+W9FRT4oBgHgl3EQfNDdW/content/2301.13508v1.pdf filter=lfs diff=lfs merge=lfs -text
+8NFQT4oBgHgl3EQfHzW5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+DdE0T4oBgHgl3EQfQQD8/content/2301.02192v1.pdf filter=lfs diff=lfs merge=lfs -text
+stA0T4oBgHgl3EQfLP-f/content/2301.02116v1.pdf filter=lfs diff=lfs merge=lfs -text
+H9E4T4oBgHgl3EQfgw3C/content/2301.05120v1.pdf filter=lfs diff=lfs merge=lfs -text
+-9FLT4oBgHgl3EQfDS7k/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+UdAzT4oBgHgl3EQflv12/content/2301.01552v1.pdf filter=lfs diff=lfs merge=lfs -text
+bdFJT4oBgHgl3EQfQSxc/content/2301.11490v1.pdf filter=lfs diff=lfs merge=lfs -text
+VdE2T4oBgHgl3EQftwjJ/content/2301.04074v1.pdf filter=lfs diff=lfs merge=lfs -text
+2NFRT4oBgHgl3EQfmzfK/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+PdE3T4oBgHgl3EQfxwsN/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf filter=lfs diff=lfs merge=lfs -text
+79E1T4oBgHgl3EQf7gUu/content/2301.03534v1.pdf filter=lfs diff=lfs merge=lfs -text
+39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf filter=lfs diff=lfs merge=lfs -text
+PdE3T4oBgHgl3EQfxwsN/content/2301.04713v1.pdf filter=lfs diff=lfs merge=lfs -text
+79E2T4oBgHgl3EQflQc0/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+C9E1T4oBgHgl3EQfpwXL/content/2301.03336v1.pdf filter=lfs diff=lfs merge=lfs -text
+89E4T4oBgHgl3EQf3A3O/content/2301.05303v1.pdf filter=lfs diff=lfs merge=lfs -text
+stA0T4oBgHgl3EQfLP-f/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+c9AyT4oBgHgl3EQfwfnU/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+6tFJT4oBgHgl3EQflyzE/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+W9FRT4oBgHgl3EQfNDdW/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+C9E1T4oBgHgl3EQfpwXL/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+vtAyT4oBgHgl3EQfafef/content/2301.00244v1.pdf filter=lfs diff=lfs merge=lfs -text
+6dE4T4oBgHgl3EQf1w1z/content/2301.05293v1.pdf filter=lfs diff=lfs merge=lfs -text
+W9FJT4oBgHgl3EQfPiyI/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+wNE2T4oBgHgl3EQfggfg/content/2301.03939v1.pdf filter=lfs diff=lfs merge=lfs -text
+HNAzT4oBgHgl3EQfxf6-/content/2301.01740v1.pdf filter=lfs diff=lfs merge=lfs -text
+PNE3T4oBgHgl3EQfxguY/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+89E4T4oBgHgl3EQf3A3O/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+4tE2T4oBgHgl3EQf6ghP/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+E9E4T4oBgHgl3EQfGwxn/content/2301.04897v1.pdf filter=lfs diff=lfs merge=lfs -text
+-9FLT4oBgHgl3EQfvi-U/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+DdE0T4oBgHgl3EQfQQD8/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf filter=lfs diff=lfs merge=lfs -text
+bdFJT4oBgHgl3EQfQSxc/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+6dE4T4oBgHgl3EQf1w1z/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+vtAyT4oBgHgl3EQfafef/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ZdE3T4oBgHgl3EQf2QtT/content/2301.04753v1.pdf filter=lfs diff=lfs merge=lfs -text
+wNE2T4oBgHgl3EQfggfg/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+uNE3T4oBgHgl3EQfNgmR/content/2301.04384v1.pdf filter=lfs diff=lfs merge=lfs -text
+DNAyT4oBgHgl3EQfSPdR/content/2301.00081v1.pdf filter=lfs diff=lfs merge=lfs -text
+39A0T4oBgHgl3EQfNf8t/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ptE1T4oBgHgl3EQfPQMl/content/2301.03024v1.pdf filter=lfs diff=lfs merge=lfs -text
+ZdE3T4oBgHgl3EQf2QtT/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+NdE0T4oBgHgl3EQfTQAI/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+-NAzT4oBgHgl3EQfSvt8/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+INE2T4oBgHgl3EQf_Qm3/content/2301.04247v1.pdf filter=lfs diff=lfs merge=lfs -text
+v9E3T4oBgHgl3EQf-wsu/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+N9E0T4oBgHgl3EQfjgF4/content/2301.02460v1.pdf filter=lfs diff=lfs merge=lfs -text
+UdAyT4oBgHgl3EQfhfgJ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ttAzT4oBgHgl3EQfr_2o/content/2301.01653v1.pdf filter=lfs diff=lfs merge=lfs -text
+E9E4T4oBgHgl3EQfGwxn/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+YNE1T4oBgHgl3EQfwAWP/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+YtFRT4oBgHgl3EQf_ziQ/content/2301.13696v1.pdf filter=lfs diff=lfs merge=lfs -text
+YNE1T4oBgHgl3EQfwAWP/content/2301.03406v1.pdf filter=lfs diff=lfs merge=lfs -text
+INE2T4oBgHgl3EQf_Qm3/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ddE4T4oBgHgl3EQfQQxz/content/2301.04980v1.pdf filter=lfs diff=lfs merge=lfs -text
+cNFIT4oBgHgl3EQfnCvP/content/2301.11312v1.pdf filter=lfs diff=lfs merge=lfs -text

29FAT4oBgHgl3EQfDxyZ/content/tmp_files/2301.08418v1.pdf.txt

@@ -0,0 +1,1578 @@
+arXiv:2301.08418v1  [math.CT]  20 Jan 2023
+Measurings of Hopf algebroids and morphisms in cyclic (co)homology
+theories
+Abhishek Banerjee *
+Surjeet Kour †
+Abstract
+In this paper, we consider measurings between Hopf algebroids and show that they induce morphisms on cyclic homology
+and cyclic cohomology. We also consider comodule measurings between SAYD modules over Hopf algebroids. These measur-
+ings induce morphisms on cyclic (co)homology of Hopf algebroids with SAYD coefficients. Finally, we obtain morphisms on
+cyclic homology induced by measurings of cyclic comp modules over operads with multiplication.
+MSC(2020) Subject Classification: 16T15, 16E40, 18D50
+Keywords: Hopf algebroids, cyclic (co)homology, SAYD modules, comp modules
+1
+Introduction
+Let k be a field, C be a k-coalgebra and A, B be k-algebras. In [20], Sweedler introduced the notion of a coalgebra measuring
+as a kind of generalized morphism between algebras. More precisely, a C-measuring from A to B consists of a k-linear map
+φ : C −→ Vectk(A, B) satisfying
+φ(c)(aa′) =
+�
+φ(c(1)(a)φ(c(2))(a′)
+φ(c)(1) = ǫ(c)1
+(1.1)
+for any a, a′ ∈ A. Here, ∆(c) = � c(1) ⊗ c(2) denotes the coproduct on C and ǫ : C −→ k denotes the counit. Since then, the
+notion of a measuring has been widely studied in the literature by several authors (see, for instance, [1], [3], [4], [9], [10], [11],
+[12],[22], [23], [24]).
+In [2], we studied how coalgebra measurings induce morphisms between Hochschild homology groups of algebras. The purpose
+of this paper is to take this idea one step further. Our aim is to consider cocommutative coalgebra morphisms between Hopf
+algebroids and show that they induce morphisms in cyclic homology and cyclic cohomology. We begin by showing that there are
+universal measurings which give an enrichment of the category HAlgk of Hopf algebroids over the category of cocommutative
+coalgebras. If U is a Hopf algebroid, the cyclic module C•(U) defining its cyclic homology groups as well as the cocyclic
+module C•(U) defining its cyclic cohomology groups are defined in [13]. We show that a cocommutative coalgebra measuring
+between two Hopf algebroids induces morphisms on their corresponding cyclic homology and cyclic cohomology groups. If a
+Hopf algebroid is commutative, we know from [15] that there is a shuffle product on its Hochschild homology groups. We show
+that a measuring between commutative Hopf algebroids induces an algebra measuring between the corresponding Hochschild
+homology rings with respct to this shuffle product. This also gives us an enrichment of commutative Hopf algebroids over
+cocommutative coalgebras.
+Thereafter, we consider comodule measurings of SAYD modules over Hopf algebroids. Accordingly, we obtain an enrichment
+of the “global category” of SAYD modules (see Theorem 5.9) over the “global category” of comodules. The cyclic homology
+*Department of Mathematics, Indian Institute of Science, Bangalore. Email: [email protected]
+†Department of Mathematics, Indian Institute of Technology, Delhi. Email: [email protected]
+1
+
+and the cyclic cohomology of a Hopf algebroid with coefficients in an SAYD module was defined in [14]. We show that a
+comodule measuring induces morphisms between cyclic (co)homology with SAYD coefficients. In the final part of the paper,
+we work with pairs of the form (O, M), where M is a cyclic unital comp module over a non-Σ operad O with multiplication
+in the sense of [16]. The cyclic homology groups of such a comp module were also defined in [16]. We consider comodule
+measurings between such pairs and show that they induce morphisms in cyclic homology.
+2
+Measurings of Hopf algebroids
+Throughout, k is a field and let Vectk be the category of k-vector spaces. Let A be a unital k-algebra. In order to define left
+and right bialgebroids, as well as Hopf algebroids in later sections, we will frequently need both the algebra A and its opposite
+algebra Aop. For this, we will often write the algebra A as AL, while Aop will often be written as AR.
+An (s, t)-ring over A consists of a unital k-algebra U along with two k-algebra morphisms s : A −→ U and t : Aop −→ U whose
+images commute in U, i.e., s(a1)t(a2) = t(a2)s(a1) for any a1, a2 ∈ U. The morphisms s and t are often referred to as source
+and target maps respectively. These morphisms introduce an (A, A)-bimodule structure on U given by left multiplication
+a1 · h · a2 := s(a1)t(a2)h
+a1, a2 ∈ A, h ∈ H
+(2.1)
+The left and right A-module structures on U in (2.1) allow us to consider the tensor product U ⊗A U. The following subspace
+of U ⊗A U is known as the Takeuchi product
+U ×A U := {� ui ⊗A u′
+i ∈ U ⊗A U | � uit(a) ⊗A u′
+i = � ui ⊗A u′
+is(a), ∀ a ∈ A}
+(2.2)
+It is well known (see, for instance, [13, § 2]) that the Takeuchi product U ×A U is a unital subalgebra of U ⊗A U.
+From now onwards, we also fix a unital k-algebra U. The multiplication on U will be denoted by µU. Since the category of
+(A, A)-bimodules is monoidal, we can consider coalgebra objects in this category. We now recall the notion of a left Hopf
+algebroid (see, for instance, [5], [13], [21]). For several closely related notions, see [18], [19].
+Definition 2.1. A left bialgebroid UL := (U, AL, sL, tL, ∆L, ǫL) over k consists of the following data:
+(1) A unital k-algebra AL
+(2) A unital k-algebra U which carries the structure of an (sL, tL) ring over AL.
+(3) A coalgebra object (U, ∆L : U −→ U ⊗AL U, ǫL : U −→ AL) in the category of (AL, AL)-bimodules satisfying the following
+conditions:
+(i) ∆L : U −→ U ⊗AL U factors through U ×A U ⊆ U ⊗AL U.
+(ii) ǫL(usL(ǫL(u′))) = ǫL(uu′) = ǫL(utL(ǫL(u′))) for all u, u′ ∈ H.
+A morphism (F, f) : (U, AL, sL, tL, ∆L, ǫL) = UL −→ U′
+L = (U′, A′
+L, s′
+L, t′
+L, ∆′
+L, ǫ′
+L) of left bialgebroids consists of a pair of
+k-algebra morphisms F : U −→ U′ and f : AL −→ A′
+L such that
+F ◦ sL = s′
+L ◦ f
+F ◦ tL = t′
+L ◦ f
+∆′
+L ◦ F = (F ⊗φ F) ◦ ∆L
+f ◦ ǫL = ǫ′
+L ◦ F
+(2.3)
+We will denote the category of left bialgebroids over k by LBialgk.
+If UL = (H, AL, sL, tL, ∆L, ǫL) is a left bialgebroid, we employ standard Sweedler notation to write ∆L(u) = � u(1) ⊗ u(2) for any
+u ∈ H and suppress the summation sign throughout. We now recall the notion of Hopf algebroid from [5, Definition 4.1].
+Definition 2.2. A Hopf algebroid U = (UL, S ) over k consists of the following data:
+(1) A left bialgebroid UL = (U, AL, sL, tL, ∆L, ǫL) over k.
+(2) An involutive anti-automorphism S : U −→ U of the k-algebra U which satisfies S ◦ tL = sL as well as
+S (u(1))(1)u(2) ⊗ S (u(1))(2) = 1H ⊗ S (u)
+S (u2)1 ⊗ S (u(2))(2)u(1) = S (u) ⊗ 1U
+(2.4)
+2
+
+as elements of U ⊗AL U, for all u ∈ U.
+A morphism (F, f) : U = (UL, S ) −→ (U′
+L, S ′) = U′ of Hopf algebroids is a morphism in LBialgk that also satisfies S ′◦F = f ◦S .
+We will denote the category of Hopf bialgebroids over k by HAlgk.
+We remark here that in this paper we will always assume the antipode on a Hopf algebroid U = (UL, S ) is involutive, i.e.,
+S 2 = id. However, this condition is not part of the original definition due to B¨ohm and Szlach´anyi in [5]. Further, it is shown in
+[5, Proposition 4.2] that a Hopf algebroid (UL, S ) is equivalent to a datum consisting of a left bialgebroid and a right bialgebroid
+connected by an antipode.
+We now recall the classical notion of a coalgebra measuring due to Sweedler [20]. Let R, R′ be k-algebras and C be a k-
+coalgebra. Then, a C-measuring from R to R′ consists of a morphism ψ : C −→ Vectk(R, R′) such that
+ψ(x)(ab) =
+�
+ψ(x(1))(a)ψ(x(2))
+ψ(x)(1R) = ǫC(x)1R′
+∀ a, b ∈ R
+(2.5)
+where the coproduct ∆C : C −→ C ⊗ C is given by ∆C(x) = � x(1) ⊗ x(2) for any x ∈ C and ǫC : C −→ k is the counit. The
+measuring as in (2.5) is said to be cocommutative if the coalgebra C is cocommutative. In this paper, we will only consider
+cocommutative measurings. By abuse of notation, if ψ : C −→ Vectk(R, R′) is a coalgebra measuring, we will often write the
+morphism ψ(x) ∈ Vectk(R, R′) simply as c : R −→ R′ for any x ∈ C.
+We are now ready to introduce the notion of measuring between Hopf algebroids.
+Definition 2.3. Let U = (U, S ) = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, S ′) = (U′, AL, s′
+L, t′
+L, ∆′
+L, ǫ′
+L, S ′) be Hopf algebroids
+over k. Let C be a cocommutative k-coalgebra. A C-measuring (Ψ, ψ) from U to U′ consists of a pair of measurings
+Ψ : C −→ Vectk(U, U′)
+ψ : C −→ Vectk(AL, A′
+L)
+(2.6)
+such that the following diagrams commute for any x ∈ C
+AL
+sL
+−−−−−−→ U
+c
+�
+�c
+A′
+L
+s′
+L
+−−−−−−→ U′
+AL
+tL
+−−−−−−→ U
+c
+�
+�c
+A′
+L
+t′
+L
+−−−−−−→ U′
+U
+S
+−−−−−−→ U
+c
+�
+�c
+U′
+S ′
+−−−−−−→ U′
+(2.7)
+U
+ǫL
+−−−−−−→ AL
+c
+�
+�c
+U′
+ǫ′
+L
+−−−−−−→ A′
+L
+U
+∆L
+−−−−−−→ U ⊗AL U
+c
+�
+�c
+U′
+∆′
+L
+−−−−−−→ U′ ⊗A′
+L U′
+(2.8)
+where the arrow c : U ⊗AL U −→ U′ ⊗A′
+L U′ is defined by setting c(u1 ⊗ u2) := x(1)(h) ⊗ x(2)(u2) for u1 ⊗ u2 ∈ U ⊗AL U.
+Before proceeding further, we need to verify the following fact.
+Lemma 2.4. For any x ∈ C, the morphism c : H ⊗AL U −→ U′ ⊗A′
+L U′ defined by setting c(u1 ⊗ u2) := x(1)(h) ⊗ x(2)(u2) for
+u1 ⊗ u2 ∈ U ⊗AL U is well-defined.
+Proof. We consider u1, u2 ∈ uL and a ∈ AL. Using the fact that Ψ : C −→ Vectk(U, U′) is a measuring and applying the
+conditions in (2.7) and (2.8), we see that
+c((u1 · a) ⊗ u2) = c(tL(a)u1 ⊗ u2)
+= x(1)(tL(a)u1) ⊗ x(2)(u2) = x(1)(tL(a))x(2)(u1) ��� x(3)(u2)
+= t′
+L(x(1)(a))x(2)(u1) ⊗ x(3)(u2)
+= t′
+L(x(2)(a))x(1)(u1) ⊗ x(3)(u2)
+(because C is cocommutative)
+= x(1)(u1) · x(2)(a) ⊗ x(3)(u2) = x(1)(u1) ⊗ x(2)(a) · x(3)(u2)
+= x(1)(u1) ⊗ s′
+L(x(2)(a))x(3)(u2) = x(1)(u1) ⊗ x(2)(sL(a))x(3)(u2)
+= x(1)(u1) ⊗ x(2)(sL(a)u2) = x(1)(u1) ⊗ x(2)(a · u2)
+= c(u1 ⊗ (a · u2))
+(2.9)
+□
+3
+
+If U = (UL, S ) and U′ = (U′
+L, S ′) are Hopf algebroids over k, we now consider the subspace
+V(U, U′) ⊆ Vectk(U, U′) × Vectk(AL, A′
+L)
+(2.10)
+given by setting
+V(U, U′) := {(F, f) | FsL = s′
+L f, FtL = t′
+L f, FS = S ′F and fǫL = ǫ′
+LF }
+(2.11)
+We note that a measuring from U to U′ by means of a cocommutative coalgebra C has an underlying morphism (Ψ, ψ) : C −→
+V(U, U′).
+Let Coalgk denote the category of k-coalgebras. We know that the forgetful functor Coalgk −→ Vectk has a right adjoint
+C : Vectk −→ Coalgk. In other words, we have natural isomorphisms
+Vectk(C, V) � Coalgk(C, C(V))
+(2.12)
+for any k-coalgebra C and any k-vector space V.
+Proposition 2.5. Let U = (UL, S ) and U′ = (U′
+L, S ′) be Hopf algebroids over k. Then, there exists a cocommutative coalgebra
+Mc(U, U′) and a measuring (Φ, φ) from U to U′ satisfying the following universal property: given any measuring (Ψ, ψ) : C −→
+V(U, U′) with a cocommutative coalgebra C, there exists a unique morphism ξ : C −→ Mc(U, U′) of coalgebras making the
+following diagram commutative
+Mc(U, U′)
+(Φ,φ)
+� V(U, U′)
+C
+ξ
+�■■■■■■■■■■
+(Ψ,ψ)
+�✇
+✇
+✇
+✇
+✇
+✇
+✇
+✇
+✇
+(2.13)
+Proof. We set V := V(U, U′) and consider the canonical morphism π(V) : C(V) −→ V ⊆ Vectk(uL, u′
+L) × Vectk(AL, A′
+L) from
+the cofree coalgebra C(V) induced by the adjunction in (2.12). We now set Mc(U, U′) := � D, where the sum is taken over all
+cocommutative subcoalgebras of C(V) such that the restriction π(V)|D : D −→ V = V(U, U′) is a measuring. It is clear that this
+sum is still a cocommutative coalgebra, and that the restriction (Φ, φ) := π(V)|Mc(U,U′) gives a measuring from U to U′.
+In general, if (Ψ, ψ) : C −→ V = V(U, U′) is a cocommutative measuring, the adjunction in (2.12) shows that it factors through
+ξ : C −→ C(V). Then, ξ(C) ⊆ C(V) is a cocommutative coalgebra such that the restriction π(V)|ξ(C) is a measuring. By
+definition, it follows that ξ(C) ⊆ Mc(U, U′). This proves the result.
+□
+From (2.10) and (2.11) it is clear that given Hopf algebroids U = (UL, S ), U′ = (U′
+L, S ′) and U′′ = (U′′
+L, S ′′), the composition
+of morphisms induces a canonical map
+V(U, U′) ⊗ V(U′, U′′)
+◦
+−→ V(U, U′′)
+(2.14)
+We denote by CoCoalgk the category of cocommutative coalgebras over k. We know that this category is symmetric monoidal
+and our objective is to show that the category HAlgk of Hopf algebroids is enriched over CoCoalgk. For this we need the
+following result.
+Proposition 2.6. Let U = (UL, S ), U′ = (U′
+L, S ′) and U′′ = (U′′
+L, S ′′) be Hopf algebroids over k. Suppose that we have a
+measuring (Ψ, ψ) : C −→ V(U, U′) from U to U′ and a measuring (Ψ′, ψ′) : C′ −→ V(U′, U′′) from U′ to U′′. Then, the
+following
+(Ψ′, ψ′) ◦ (Ψ, ψ) : C ⊗ C′ (Ψ,ψ)⊗(Ψ′,ψ′)
+−−−−−−−−−−→ V(U, U′) ⊗ V(U′, U′′)
+◦
+−→ V(U, U′′)
+(2.15)
+determines a measuring from U to U′′.
+Proof. It is easy to verify that the compositions
+C ⊗ C′ Ψ⊗Ψ′
+−−−−→ Vectk(U, U′) ⊗ Vectk(U′, U′′)
+◦
+−−−−−−→ Vectk(U, U′′)
+C ⊗ C′ ψ⊗ψ′
+−−−−→ Vectk(AL, A′
+L) ⊗ Vectk(A′
+L, A′′
+L)
+◦
+−−−−−−→ Vectk(AL, A′′
+L)
+(2.16)
+4
+
+give coalgebra measurings from U to U′′ and from AL to A′′
+L respectively. For c ⊗ c′ ∈ C ⊗ C′ and u ∈ U, we also see that
+∆′′
+L((c ⊗ c′)(u)) = ∆′′
+L(c′(c(u)))
+= c′
+(1)(c(u)(1)) ⊗ c′
+(2)(c(u)(2))
+= c′
+(1)(x(1)(u(1))) ⊗ c′
+(2)(x(2)(u(2)))
+= (c′ ⊗ c)(1)(u(1)) ⊗ (c′ ⊗ c)(2)(u(2))
+(2.17)
+It is also clear that the morphism in (2.15) satisfies all the other conditions in Definition 2.3. This proves the result.
+□
+Theorem 2.7. The category HAlgk of Hopf algebroids is enriched over the category CoCoalgk of cocommutative k-coalgebras.
+Proof. Given Hopf algebroids U = (UL, S ) and U′ = (U′
+L, S ′), we consider the “hom object” Mc(U, U′) which lies in CoCoalgk.
+The composition of these hom objects is obtained as follows: if U, U′ and U′′ are Hopf algebroids, we obtain as in Proposition
+2.6 a measuring
+Mc(U, U′) ⊗ Mc(U′, U′′) −→ V(U, U′′)
+(2.18)
+Applying the universal property in Proposition 2.5, we now have a morphism of coalgebras Mc(U, U′) ⊗ Mc(U′, U′′) −→
+Mc(U, U′′). The unit object in CoCoalgk is k treated as a coalgebra over itself. Then, we have a unit map
+k ��→ V(U, U) ⊆ Vectk(U, U) × Vectk(AL, AL)
+t �→ (t · iduL, t · idAL)
+(2.19)
+which induces a morphism k −→ Mc(U, U) of cocommutative coalgebras. Together with the composition of hom objects in
+(2.18), we see that HAlgk is enriched over CoCoalgk.
+□
+From now onwards, we will denote by HALGk the category of Hopf algebroids enriched over the symmetric monoidal category
+CoCoalgk of cocommutative k-algebras.
+3
+Morphisms on cyclic (co)homology and Hopf-Galois maps
+Let U = (UL, S ) = (U, AL, sL, tL, ∆L, ǫL) be a Hopf algebroid over k. We now recall from [13, § 2] the cocyclic module C•(U)
+that computes the cyclic cohomology of the Hopf algebroid U. For n ≥ 1, we put
+Cn(U) := U ⊗AL ⊗ · · · ⊗AL U
+��������������������������������������
+n-times
+(3.1)
+and set C0(U) := AL. For n ≥ 1, the face maps δi : Cn(U) −→ Cn+1(U) are defined by
+δi(u1 ⊗ ... ⊗ un) :=
+
+1 ⊗ u1 ⊗ ... ⊗ un
+if i = 0
+u1 ⊗ .... ⊗ ∆Lui ⊗ ... ⊗ un
+if 1 ≤ i ≤ n
+u1 ⊗ .... ⊗ un ⊗ 1
+if i = n + 1
+(3.2)
+For n = 0, there are two maps δ0 := tL : C0(U) = AL −→ C1(U) = U and δ1 := sL : C0(U) = AL −→ C1(U) = U. The
+degeneracy maps σi : Cn(U) −→ Cn−1(U) are given by
+σi(u1 ⊗ ... ⊗ un) := u1 ⊗ ... ⊗ ǫL(ui+1) · ui+2 ⊗ ... ⊗ un
+0 ≤ i ≤ n − 1
+(3.3)
+The cyclic operator τn : Cn(U) −→ Cn(U) is defined by setting
+τn(u1 ⊗ ... ⊗ un) := (S (u1)(1) · u2) ⊗ .... ⊗ (S (u1)(n−1) · un) ⊗ S (u1)(n)
+(3.4)
+Since we have assumed that the antipode S is involutive, it follows from [13, Theorem 2.1] that C•(U) is indeed a cocyclic
+module. We will denote by HC•(U) the cyclic cohomology groups of the Hopf algebroid U by U. The Hochschild cohomology
+groups of the Hopf algebroid U will then be denoted by HH•(U).
+5
+
+Let U, U′ be Hopf algebroids and let (Ψ, ψ) : C −→ V(U, U′) be a measuring from U to U′. For each x ∈ C, we now define a
+family of morphisms
+Ψ
+n(x) : Cn(U) −→ Cn(U′)
+Ψ
+n(x)(u1 ⊗ ... ⊗ un) := x(u1 ⊗ ... ⊗ un) = x(1)(u1) ⊗ ... ⊗ x(n)(un)
+∀ n ≥ 0
+(3.5)
+We now prove the first main result of this section.
+Proposition 3.1. Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→ V(U, U′) be a measuring of Hopf algebroids.
+For each x ∈ C, the family {Ψ
+n(x) : Cn(U) −→ Cn(U′)}n≥0 gives a morphism of cyclic modules. In particular, we have induced
+morphisms
+Ψ
+•
+hoc(x) : HH•(U) −→ HH•(U)
+Ψ
+•
+cy(x) : HC•(U) −→ HC•(U)
+(3.6)
+on Hochschild and cyclic cohomologies for each x ∈ C.
+Proof. For each x ∈ C, we start by showing that Ψ
+n+1(x) ◦ δi = δ′
+i ◦ Ψ
+n(x) : Cn(U) −→ Cn+1(H ′), where δi and δ′
+i are the face
+maps on the respective cocyclic modules C•(U) and C•(U′). If i = 0 or i = n + 1, this is immediately clear from the definition
+in (3.2) and the action in (3.5). For 1 ≤ i ≤ n, we see that
+Ψ
+n+1(x) ◦ δi(u1 ⊗ ... ⊗ un)
+= Ψ
+n+1(x)(u1 ⊗ .... ⊗ ∆Lui ⊗ ... ⊗ un)
+= x(1)(u1) ⊗ ... ⊗ x(i)(ui
+(1)) ⊗ x(i+1)(ui
+(2)) ⊗ .... ⊗ x(n+1)(un)
+= x(1)(u1) ⊗ ∆L(x(i)(ui)) ⊗ ... ⊗ x(n)(un) = δ′
+i ◦ Ψ
+n(x)(u1 ⊗ ... ⊗ un)
+Next, we verify that Ψ
+n−1(x) ◦ σi = σ′
+i ◦ Ψ
+n(x), where σi and σ′
+i are the degeneracies on the respective cocyclic modules C•(U)
+and C•(U′).
+Ψ
+n−1(x) ◦ σi(u1 ⊗ ... ⊗ un)
+= Ψ
+n−1(x)(u1 ⊗ ... ⊗ ǫL(ui+1) · ui+2 ⊗ ... ⊗ un)
+= x(1)(u1) ⊗ ... ⊗ x(i+1)(ǫL(ui+1) · ui+2) ⊗ ... ⊗ x(n−1)(un)
+= x(1)(u1) ⊗ ... ⊗ x(i+1)(ǫL(ui+1)) · x(i+2)(ui+2) ⊗ ... ⊗ x(n)(un)
+= x(1)(u1) ⊗ ... ⊗ ǫL(x(i+1)(ui+1)) · x(i+2)(ui+2) ⊗ ... ⊗ x(n)(un)
+= σ′
+i ◦ Ψ
+n(x)(u1 ⊗ ... ⊗ un)
+Finally, we show that Ψ
+n(x)◦τn = τ′
+n ◦Ψ
+n(x), where τn and τ′
+n are the cyclic operators on the respective cocyclic modules C•(U)
+and C•(U′).
+Ψ
+n(x) ◦ τn(u1 ⊗ ... ⊗ un)
+= Ψ
+n(x)((S (u1)(1) · u2) ⊗ .... ⊗ (S (u1)(n−1) · un) ⊗ S (u1)(n))
+= x(1)((S (u1)(1) · u2)) ⊗ .... ⊗ x(n−1)((S (u1)(n−1) · un) ⊗ x(n)(S (u1)(n))
+= x(1)(S (u1)(1)) · x(2)(u2) ⊗ .... ⊗ x(2n−3)(S (u1)(n−1)) · x(2n−2)(un) ⊗ x(2n−1)(S (u1)(n))
+= x(1)(S (u1)(1)) · x(n+1)(u2) ⊗ .... ⊗ x(n−1)(S (u1)(n−1)) · x(2n−1)(un) ⊗ x(n)(S (u1)(n))
+= (x(1)(S (u1)))(1) · x(2)(u2) ⊗ .... ⊗ (x(1)(S (u1)))(n−1) · x(n)(un) ⊗ (x(1)(S (u1)))(n)
+= (S (x(1)(u1)))(1) · x(2)(u2) ⊗ .... ⊗ (S (x(1)(u1)))(n−1) · x(n)(un) ⊗ (S (x(1)(u1)))(n)
+= τ′
+n ◦ Ψ
+n(x)(u1 ⊗ ... ⊗ un)
+□
+We continue with a Hopf algebroid U = (UL, S ) = (U, AL, sL, tL, ∆L, ǫL). As mentioned in Section 2, we set AR := Aop
+L = Aop.
+Following [5, § 4], we also set
+sR := tL
+tR := S ◦ tL = sL
+(3.7)
+Then, U becomes an (AR, AR)-bimodule by right multiplication as follows
+a1 · h · a2 := hsR(a2)tR(a1) = htL(a2)sL(a1)
+h ∈ H, a1, a2 ∈ AR
+(3.8)
+We now consider
+S rl : H ⊗AR H −→ H ⊗AL H
+u1 ⊗ u2 �→ S (u2) ⊗ S (u1)
+S lr := S −1
+rl : H ⊗AL H −→ H ⊗AR H
+u1 ⊗ u2 �→ S (u2) ⊗ S (u1)
+(3.9)
+6
+
+as well as
+∆R := S lr ◦ ∆L ◦ S : U −→ U ⊗AR U
+ǫR := ǫL ◦ S : U −→ AR
+(3.10)
+We know from [5, § 4] that the datum UR := (U, AR, sR, tR, ∆R, ǫR) defines a right bialgebroid over k. We now adopt the Sweedler
+notation ∆R(u) := u[1] ⊗ u[2] for any u ∈ U in order to distinguish it from the left coproduct ∆L(u) = u(1) ⊗ u(2). More explicitly,
+we have
+∆R(u) = u[1] ⊗ u[2] = S (S (u)(2)) ⊗ S (S (u)(1))
+∀ u ∈ U
+(3.11)
+Now let U, U′ be Hopf algebroids and consider (F, f) ∈ V(U, U′). From the conditions in (2.11) and the definitions in (3.7) and
+(3.10), we already have
+FsR = s′
+R f
+FtR = t′
+R f
+FS = S ′F
+fǫR = ǫ′
+RF
+(3.12)
+We now need the following result.
+Lemma 3.2. Let C be a cocommutative coalgebra and (Ψ, ψ) : C −→ V(U, U′) a measuring of Hopf algebroids. Then, for
+each x ∈ C, there is a well defined morphism
+x : U −→ U ⊗AR U
+u1 ⊗ u2 �→ x(1)(u1) ⊗ x(2)(u2)
+(3.13)
+which fits into the following commutative diagram
+U
+∆R
+−−−−−−→ U ⊗AR U
+x
+�
+�x
+U′
+∆′
+R
+−−−−−−→ U′ ⊗A′
+R U′
+(3.14)
+Proof. We consider u1, u2 ∈ uL and a ∈ AR. Using the fact that Ψ : C −→ Vectk(U, U′) is a measuring and applying the
+conditions in (3.12), we see that
+c((u1 · a) ⊗ u2) = c(u1sR(a) ⊗ u2)
+= x(1)(u1sR(a)) ⊗ x(2)(u2) = x(1)(u1)x(2)(sR(a)) ⊗ x(3)(u2)
+= x(1)(u1)s′
+R(x(2)(a)) ⊗ x(3)(u2)
+= x(1)(u1) · x(2)(a) ⊗ x(3)(u2)
+= x(1)(u1) ⊗ x(2)(a) · x(3)(u2) = x(1)(u1) ⊗ x(3)(u2)t′
+R(x(2)(a))
+= x(1)(u1) ⊗ x(3)(u2)x(2)(tR(a))
+= x(1)(u1) ⊗ x(2)(u2)x(3)(tR(a))
+(as C is cocommutative)
+= x(1)(u1) ⊗ x(2)(u2tR(a)) = x(1)(u1) ⊗ x(2)(a · u2) = c(u1 ⊗ (a · u2))
+It follows that the morphism in (3.13) is well defined. It remains to verify the condition in (3.14). For u ∈ U and x ∈ C, we
+have
+c(∆R(u)) = c(S (S (u)(2)) ⊗ S (S (u)(1)))
+= x(1)(S (S (u)(2))) ⊗ x(2)(S (S (u)(1)))
+= S ′(x(1)(S (u)(2))) ⊗ S ′(x(2)(S (u)(1)))
+= S ′(x(2)(S (u)(2))) ⊗ S ′(x(1)(S (u)(1)))
+(as C is cocommutative)
+= S ′(c(S (u))(2)) ⊗ S ′(c(S (u))(1))
+= S ′(S ′(c(u))(2)) ⊗ S ′(S ′(c(u))(1)) = ∆′
+R(c(u))
+□
+We now recall from [13, § 2.3.1] the cyclic module C•(U) defining the cyclic homology of a Hopf algebroid U. For n ≥ 0, we
+set
+Cn(U) := U ⊗AR ⊗ · · · ⊗AR U
+��������������������������������������
+n-times
+(3.15)
+7
+
+and C0(U) := AR. The face maps di : Cn(U) −→ Cn−1(U) are defined by setting
+di(u1 ⊗ ... ⊗ un) :=
+
+ǫR(u1)u2 ⊗ ... ⊗ un
+if i = 0
+u1 ⊗ ... ⊗ uiui+1 ⊗ ... ⊗ un
+if i ≤ i ≤ n − 1
+u1 ⊗ ... ⊗ un−1ǫR(S (un))
+if i = n
+(3.16)
+The degeneracies si : Cn(U) −→ Cn+1(U) are defined as
+si(u1 ⊗ ... ⊗ un) :=
+� 1 ⊗ u1 ⊗ ... ⊗ un
+if i = 0
+u1 ⊗ ... ⊗ ui ⊗ 1 ⊗ ui+1 ⊗ .... ⊗ un
+if 1 ≤ i ≤ n
+(3.17)
+The cyclic operators tn : Cn(U) −→ Cn(U) are given by
+tn(u1 ⊗ ... ⊗ un) := S (u1
+(2)...un−1
+(2) un) ⊗ u1
+(1) ⊗ u2
+(1) ⊗ ... ⊗ un−1
+(1)
+(3.18)
+The Hochschild homology groups of the Hopf algebroid U will then be denoted by HH•(U) and the cyclic homology groups
+by HC•(U). We will now prove the homological counterpart for Proposition 3.1.
+Proposition 3.3. Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→ V(U, U′) be a measuring of Hopf algebroids. For
+each x ∈ C, the family
+Ψn(x) : Cn(U) −→ Cn(U′)
+u1 ⊗ ... ⊗ un �→ x(u1 ⊗ ... ⊗ un) = x(1)(u1) ⊗ ... ⊗ x(n)(un)
+(3.19)
+for n ≥ 0 gives a morphism of cyclic modules. In particular, we have induced morphisms
+Ψhoc
+• (x) : HH•(U) −→ HH•(U′)
+Ψcy
+• (x) : HC•(U) −→ HC•(U′)
+(3.20)
+on Hochschild and cyclic homologies for each x ∈ C.
+Proof. Using the properties in (3.12) and the fact that Ψ : C −→ Vectk(U, U′) is a measuring, it may easily be verified that the
+maps Ψ•(x) commute with the respective face maps and degeneracy maps on the cyclic modules C•(U) and C•(U′). Moreover,
+if tn and t′
+n are the respective cyclic operators on C•(U) and C•(U′), we have for each x ∈ C
+c(tn(u1 ⊗ ... ⊗ un))
+= c(S (u1
+(2)...un−1
+(2) un) ⊗ u1
+(1) ⊗ u2
+(1) ⊗ ... ⊗ un−1
+(1) )
+= x(1)(S (u1
+(2)...un−1
+(2) un)) ⊗ x(2)(u1
+(1)) ⊗ x(3)(u2
+(1)) ⊗ ... ⊗ x(n)(un−1
+(1) )
+= S ′(x(1)(u1
+(2))...x(n−1)(un−1
+(2) )x(n)(un)) ⊗ x(n+1)(u1
+(1)) ⊗ ... ⊗ x(2n−1)(un−1
+(1) )
+= S ′(x(2)(u1
+(2))...x(2n−2)(un−1
+(2) )x(2n−1)(un)) ⊗ x(1)(u1
+(1)) ⊗ ... ⊗ x(2n−3)(un−1
+(1) )
+= S ′(x(1)(u1)(2)...x(n−1)(un−1)(2)x(n)(un)) ⊗ x(1)(u1)(1) ⊗ ... ⊗ x(n−1)(un−1)(1)
+= t′
+n(x(1)(u1) ⊗ ... ⊗ x(n)(un))
+□
+Our final aim in this section is to show that the morphisms induced by a measuring of Hopf algebroids are well behaved with
+respect to cyclic duality. More precisely, we know from [13, § 2.3.3] that there are Hopf-Galois maps
+ξn(U) : Cn(U)
+�
+−→ Cn(U)
+u1 ⊗ ... ⊗ un �→ u1
+(1) ⊗ u1
+(2)u2
+(1) ⊗ u1
+(3)u2
+(2)u3
+(1) ⊗ ... ⊗ u1
+(n)u2
+(n−1)....un−1
+(2) un
+(3.21)
+inducing isomorphisms between C•(U) and C•(U). We now have the following result.
+Proposition 3.4. Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→ V(U, U′) be a measuring of Hopf algebroids.
+Then for each x ∈ C, the following diagram commutes
+Cn(U)
+ξn(U)
+−−−−−−→ Cn(U)
+Ψn(x)
+�
+�Ψ
+n(x)
+Cn(U)
+ξn(U)
+−−−−−−→ Cn(U)
+(3.22)
+8
+
+Proof. We put N := n(n + 1)/2. Using the fact that Ψ : C −→ Vectk(U, U′) is a measuring and that C is cocommutative we
+have
+c(ξn(U)(u1 ⊗ ... ⊗ un))
+= c(u1
+(1) ⊗ u1
+(2)u2
+(1) ⊗ u1
+(3)u2
+(2)u3
+(1) ⊗ ... ⊗ u1
+(n)u2
+(n−1)....un−1
+(2) un)
+= x(1)(u1
+(1)) ⊗ x(2)(u1
+(2))x(3)(u2
+(1)) ⊗ ... ⊗ x(N+1−n)(u1
+(n))....x(N−1)(un−1
+(2) )x(N)(un)
+= x(1)(u1
+(1)) ⊗ x(2)(u1
+(2))x(n+1)(u2
+(1)) ⊗ ... ⊗ x(n)(u1
+(n))....x(N−1)(un−1
+(2) )x(N)(un)
+= ξn(U′)(x(1)(u1) ⊗ ... ⊗ x(n)(un))
+This proves the result.
+□
+4
+Shuffle products and the enrichment of the category of commutative Hopf alge-
+broids
+We recall from Section 2 the category HALGk of Hopf algebroids over k, enriched over the symmetric monoidal category
+of CoCoalgk of cocommutative k-coalgebras.
+By a commutative Hopf algebroid, we will mean a Hopf algebroid U =
+(U, AL, sL, tL, ∆L, ǫL) such that H and AL = A = AR are commutative rings.
+Let cHALGk denote the full subcategory of HALGk consisting of commutative Hopf algebroids. Then, cHALGk is also en-
+riched over CoCoalgk. In this section, we will obtain a second enrichment of commutative Hopf algebroids in cocommutative
+coalgebras, by using the shuffle product in Hochschild homology.
+We know from [17, § 4.2] that the Hochschild homology of a commutative algebra is equipped with a shuffle product structure.
+For a commutative Hopf algebroid U = (U, AL, sL, tL, ∆L, ǫL), we now recall from [15, § 4.4.1] the (p, q)-shuffle product
+shpq(U) : Cp(U) ⊗ Cq(U) −→ Cp+q(U)
+(4.1)
+which is given by the formula (for p, q ≥ 1)
+shpq(U)((u1 ⊗ ... ⊗ up) ⊗ (up+1 ⊗ ... ⊗ up+q)) :=
+�
+σ∈S h(p,q)
+sgn(σ)(uσ−1(1) ⊗ ... ⊗ uσ−1(p+q))
+(4.2)
+Here S h(p, q) is the set of (p, q)-shuffles, i.e.,
+S h(p, q) := {σ ∈ S p+q | σ(1) < ... < σ(p); σ(p + 1) < ... < σ(p + q)}
+(4.3)
+For p = q = 0, the shuffle product is given by setting sh00(U) to be the multiplication on A. Further, one has (see [15, § 4.4.1])
+shp0(U) : Cp(U) ⊗ C0(U) −→ Cp(U)
+(u1 ⊗ ... ⊗ up) ⊗ a �→ (tL(a)u1 ⊗ ... ⊗ up)
+sh0q(U) : C0(U) ⊗ Cq(U) −→ Cq(U)
+a ⊗ (u1 ⊗ ... ⊗ up) �→ (u1 ⊗ ... ⊗ uqtL(a))
+(4.4)
+for p ≥ 1 and q ≥ 1. There is now an induced product structure shpq(U) : HHp(U) ⊗ HHq(U) −→ HHp+q(U) which makes the
+the Hochschild homology HH•(U) :=
+�
+n≥0
+HHp(U) of a commutative Hopf algebroid U into a graded algebra (see [15, § 4.4.1])
+that we denote by (HH•(U), sh(U)).
+Proposition 4.1. Let U, U′ be commutative Hopf algebroids. Let C be a cocommutative coalgebra and let (Ψ, ψ) : C −→
+V(U, U′) be a measuring of Hopf algebroids. Then, the induced K-linear map
+Ψhoc : C −→ HomK(HH•(U), HH•(U′))
+x �→ (Ψhoc
+• (x) : HH•(U) −→ HH•(U′))
+(4.5)
+gives a measuring of algebras from (HH•(U), sh(U)) to (HH•(U′), sh(U′)).
+9
+
+Proof. The unit in (HH•(U), sh(U)) is given by the class of the unit 1A ∈ A = C0(U). Since ψ : C −→ HomK(A, A′) gives in
+particular a measuring from A to A′, we have Ψhoc
+• (x)(1A) = 1A′. We now note that for any x ∈ C and p, q ≥ 1, we have
+Ψp+q(x)(shpq(U)((u1 ⊗ ... ⊗ up) ⊗ (up+1 ⊗ ... ⊗ up+q)))
+= Ψp+q(x)
+�
+�
+σ∈S h(p,q)
+sgn(σ)(uσ−1(1) ⊗ ... ⊗ uσ−1(p+q))
+�
+=
+�
+σ∈S h(p,q)
+sgn(σ)(x(1)(uσ−1(1)) ⊗ ... ⊗ x(p+q)(uσ−1(p+q)))
+=
+�
+σ∈S h(p,q)
+sgn(σ)(xσ−1(1)(uσ−1(1)) ⊗ ... ⊗ xσ−1(p+q)(uσ−1(p+q)))
+(because C is cocommutative)
+= shpq(U)((x(1)(u1) ⊗ ... ⊗ x(p)(up)) ⊗ (x(p+1)(up+1) ⊗ ... ⊗ x(p+q)(up+q)))
+(4.6)
+For p ≥ 1, we have
+Ψp(x)(shp0(U)((u1 ⊗ ... ⊗ up) ⊗ a)
+= Ψp(x)((tL(a)u1 ⊗ ... ⊗ up))
+= (x(1)(tL(a)u1) ⊗ ... ⊗ x(p)(up))
+= (x(1)(tL(a))x(2)(u1) ⊗ ... ⊗ x(p+1)(up))
+= (tL(x(p+1)(a)))x(1)(u1) ⊗ ... ⊗ x(p)(up))
+= shp0(U)((x(1)(u1) ⊗ ... ⊗ x(p)(up)) ⊗ x(p+1)(a))
+(4.7)
+We can similarly verify the case for sh0q with q ≥ 1 and for sh00. This proves the result.
+□
+Our next objective is to use Proposition 4.1 to obtain an enrichment of commutative Hopf algebroids over the category of
+cocommutative coalgebras. For that we recall the following fact: if R, R′ are k-algebras, the category of coalgebra measurings
+from R to R′ contains a final object ϕ(R, R′) : M(R, R′) −→ Vectk(R, R′) (see Sweedler [20]). Then, M(R, R′) is known as the
+universal measuring coalgebra. We let Mc(R, R′) be the cocommutative part of the coalgebra M(R, R′). Then, the restriction
+ϕc(R, R′) : Mc(R, R′) ֒→ M(R, R′) −→ Vectk(R, R′) becomes the final object in the category of cocommutative coalgebra
+measurings from R to R′ (see [9, Proposition 1.4], [10]). Further, the objects Mc(R, R′) give an enrichment of k-algebras over
+cocommutative k-coalgebras.
+We now define the enriched category
+�
+cHALGk whose objects are commutative Hopf algebroids over k and whose hom-objects
+are defined by setting
+�
+cHALGk(U, U′) := Mc((HH•(U), sh(U)), (HH•(U′), sh(U′))) ∈ CoCoalgk
+(4.8)
+for commutative Hopf algebroids U, U′. Since each (HH•(U), sh(U)) is an algebra, we also have a canonical morphism k −→
+Mc((HH•(U), sh(U)), (HH•(U), sh(U))) of cocommutative coalgebras.
+Lemma 4.2. Let U, U′ be commutative Hopf algebroids. Then, there is a canonical morphism of cocommutative coalgebras
+τ(U, U′) : Mc(U, U′) −→ Mc((HH•(U), sh(U)), (HH•(U′), sh(U′)))
+(4.9)
+Proof. By definition, (Φ, φ) : Mc(U, U′) −→ V(U, U′) is a cocommutative measuring from U to U′. By Proposition 4.1,
+this induces a measuring of algebras from (HH•(U), sh(U)) to (HH•(U′), sh(U′)). By the universal property of the universal
+cocommutative measuring coalgebra Mc((HH•(U), sh(U)), (HH•(U′), sh(U′))), we now obtain an induced morphism τ(U, U′)
+as in (4.9).
+□
+Theorem 4.3. There is a CoCoalgk enriched functor cHALGk −→
+�
+cHALGk which is identity on objects and whose mapping
+on hom-objects is given by
+τ(U, U′) : cHALGk(U, U′) = Mc(U, U′) −→ Mc((HH•(U), sh(U)), (HH•(U′), sh(U′))) =
+�
+cHALGk(U, U′)
+(4.10)
+for commutative Hopf algebroids U, U′ over k.
+10
+
+Proof. Let U, U′, U′′ be commutative Hopf algebroids. We show that the following diagram commutes
+Mc(U, U′) ⊗ Mc(U′, U′′)
+◦
+−−−−−−→
+Mc(U, U′′)
+τ(U,U′)⊗τ(U′,U′′)
+�
+�τ(U,U′′)
+Mc(HH•(U), HH•(U′)) ⊗ Mc(HH•(U′), HH•(U′′))
+◦
+−−−−−−→ Mc(HH•(U), HH•(U′′))
+(4.11)
+The top horizontal composition ◦ : Mc(U, U′) ⊗ Mc(U′, U′′) −→ Mc(U, U′′) in (4.11) is obtained from Theorem 2.7, while the
+bottom horizontal composition ◦ : Mc(HH•(U), HH•(U′)) ⊗ Mc(HH•(U′), HH•(U′′)) −→ Mc(HH•(U), HH•(U′′)) is obtained
+from the enrichment of algebras in cocommutative coalgebras.
+From Lemma 4.2 and Theorem 2.7, we note that all the maps in (4.11) are morphisms of cocommutative coalgebras. It follows
+from the property of the universal cocommutative measuring coalgebra Mc(HH•(U), HH•(U′′)) that in order to show that (4.11)
+commutes, it suffices to verify that the following two compositions are equal
+Mc(U, U′) ⊗ Mc(U′, U′′)
+τ(U,U′)⊗τ(U′,U′′)
+�
+Mc(HH•(U), HH•(U′)) ⊗ Mc(HH•(U′), HH•(U′′))
+◦
+�
+Mc(HH•(U), HH•(U′′))
+�ϕc(HH•(U),HH•(U′′))
+Vectk(HH•(U), HH•(U′′))
+Mc(U, U′) ⊗ Mc(U′, U′′)
+�◦
+Mc(U, U′′)
+�τ(U,U′′)
+Mc(HH•(U), HH•(U′′))
+�ϕc(HH•(U),HH•(U′′))
+Vectk(HH•(U), HH•(U′′))
+(4.12)
+For the sake of convenience, we denote the left vertical composition in (4.12) by ψ1 and the right vertical composition by ψ2.
+We now consider x ∈ Mc(U, U′), y ∈ Mc(U′, U′′) and (u1 ⊗ ... ⊗ up) ∈ Cp(U). We see that
+ψ2(x ⊗ y)(u1 ⊗ ... ⊗ up)
+= (y ◦ x)(u1 ⊗ ... ⊗ up)
+= (y ◦ x)(1)(u1) ⊗ ... ⊗ (y ◦ x)(p)(up)
+(4.13)
+Since ◦ : Mc(U, U′) ⊗ Mc(U′, U′′) −→ Mc(U, U′′) is a morphism of coalgebras, we note that (y ◦ x)(1) ⊗ ... ⊗ (y ◦ x)(p) =
+(y(1) ◦ x(1)) ⊗ ... ⊗ (y(p) ◦ x(p)). Combining with (4.13), we see that the right vertical composition in (4.12) may be described
+explicitly as
+ψ2(x ⊗ y)(u1 ⊗ ... ⊗ up) = (y(1) ◦ x(1))(u1) ⊗ ... ⊗ (y(p) ◦ x(p))(up) = y(1)(x(1)(u1)) ⊗ ... ⊗ y(p)(x(p)(up))
+(4.14)
+On the other hand, we note that the following diagram is commutative
+Mc(U, U′) ⊗ Mc(U′, U′′)
+◦(τ(U,U′)⊗τ(U′,U′′))
+−−−−−−−−−−−−−−−−→
+Mc(HH•(U), HH•(U′′))
+(ϕc(HH•(U),HH•(U′))◦τ(U,U′))⊗
+�(ϕc(HH•(U′),HH•(U′′))◦τ(U′,U′′))
+ϕc(HH•(U),HH•(U′′))
+�
+Vectk(HH•(U), HH•(U′)) ⊗ Vectk(HH•(U′), HH•(U′′))
+◦
+−−−−−−→
+Vectk(HH•(U), HH•(U′′))
+(4.15)
+From (4.15), it follows that the left vertical composition in (4.12) may be described explicitly as
+ψ1(x ⊗ y)(u1 ⊗ ... ⊗ up)
+= y(x(u1 ⊗ ... ⊗ up))
+= y(1)(x(1)(u1)) ⊗ ... ⊗ y(p)(x(p)(up))
+(4.16)
+From (4.14) and (4.16), we see that ψ1 = ψ2 and hence the diagram (4.11) commutes. Similarly by considering the coalgebra k
+and using the fact that the p-th iterated coproduct ∆p(1) = 1 ⊗ ... ⊗ 1(p-times), we see that the following compositions are equal
+k −→ Mc(HH•(U), HH•(U))
+ϕc(HH•(U),HH•(U))
+−−−−−−−−−−−−−−−→ Vectk(HH•(U), HH•(U))
+k −→ Mc(U, U)
+τ(U,U)
+−−−−−→ Mc(HH•(U), HH•(U))
+ϕc(HH•(U),HH•(U))
+−−−−−−−−−−−−−−−→ Vectk(HH•(U), HH•(U))
+(4.17)
+11
+
+It follows from (4.17) that the following diagram commutes
+k
+�
+�●
+●
+●
+●
+●
+●
+●
+●
+●
+●
+Mc(HH•(U), HH•(U))
+Mc(U, U)
+τ(U,U)
+�❧
+❧
+❧
+❧
+❧
+❧
+❧
+❧
+❧
+❧
+❧
+❧
+❧
+(4.18)
+This proves the result.
+□
+5
+Comodule measurings for SAYD modules
+Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid. From now onwards, we set Ae := A ⊗k Aop and define
+ηL : Ae = A ⊗k Aop
+sL⊗tL
+−−−−→ U ⊗ U −→ U
+(5.1)
+where the second arrow in (5.1) is the multilplication on U. Following [14, § 2], we note that there are now four commuting
+actions of A on U which are denoted as follows
+a ⊲ u ⊳ b := sL(a)tL(b)u
+a ◮ u ◭ b := usL(b)tL(a)
+a, b ∈ A, u ∈ U
+(5.2)
+By Definition 2.1, we then have an A-coring
+∆L : U −→ U⊳ ⊗A ⊲U
+ǫL : U −→ A
+(5.3)
+The left action ◮ of A on U may be treated as a right action of Aop on U. Similarly, the right action ⊳ of A on U may be treated
+as a left action by Aop. Accordingly, we may consider the tensor product
+◮U ⊗Aop U⊳ := U ⊗k U/span{a ◮ u ⊗ v − u ⊗ v ⊳ a | u, v ∈ U, a ∈ A}
+(5.4)
+There is now a Hopf-Galois map (see [5], [14], [19])
+β(U) : ◮U ⊗Aop U⊳ −→ U⊳ ⊗A ⊲U
+u ⊗Aop v �→ u(1) ⊗A u(2)v
+(5.5)
+Since U is a Hopf algebroid, it follows (see [5, Proposition 4.2]) that the morphism β(U) in (5.5) is a bijection. Accordingly, in
+the notation of [14], [19], we write
+u+ ⊗Aop u− := β(U)−1(u ⊗A 1)
+u ∈ U
+(5.6)
+In this section, we will consider comodule measurings between stable anti-Yetter Drinfeld modules over Hopf algebroids. For
+this, we first recall the notion of comodule measuring between ordinary modules. Let R, R′ be rings and let P, P′ be modules
+over R and R′ respectively. Then, a comodule measuring from P to P′ consists of a pair of maps (see [4], [12])
+ψ : C −→ Vectk(R, R′)
+ω : D −→ Vectk(P, P′)
+(5.7)
+where C is a k-coalgebra, D is a right C-comodule, ψ : C −→ Vectk(R, R′) is a coalgebra measuring and
+ω(y)(pr) = y(pr) = y(0)(p)y(1)(r) = ω(y(0))(p)ψ(y(1))(r)
+(5.8)
+for y ∈ D, p ∈ P and r ∈ R. For U = (U, AL, sL, tL, ∆L, ǫL, S ), we will now recall the notions of U-modules, U-comodules and
+stable anti-Yetter Drinfeld modules.
+Definition 5.1. (see [14, § 2.4]) Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid. A right U-module P is a right module
+over the k-algebra U. Because of the ring homomorphism ηL : Ae −→ U, any right U-module P is also equipped with a right
+Ae-module structure (or (A, A)-bimodule structure) given by
+b ◮ p ◭ a = p(a ⊗ b) = pηL((a ⊗ 1)(1 ⊗ b)) = psL(a)tL(b)
+(5.9)
+for (a ⊗ b) ∈ Ae = A ⊗k Aop and p ∈ P.
+12
+
+Definition 5.2. (see [6], [8], [14], [18]) Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid. A left U-comodule P is a left
+comodule over the A-coring (U, ∆L : U −→ U ⊗AL U, ǫL : U −→ AL). In particular, a left U-comodule P is equipped with a left
+A-module structure (a, p) �→ ap as well as a left A-module map
+∆P : P −→ U⊳ ⊗A P
+p �→ p(−1) ⊗ p(0)
+(5.10)
+Following [14, § 2.5], we note that any left U-comodule P also carries a right A-module structure given by setting
+pa := ǫL(p(−1)sL(a))p(0)
+(5.11)
+for p ∈ P, a ∈ A. This makes any left U-comodule P into a right Ae = A ⊗k Aop-module by setting
+p(a ⊗ b) = bpa = bǫL(p(−1)sL(a))p(0)
+(5.12)
+for p ∈ P and (a ⊗ b) ∈ Ae.
+Definition 5.3. (see [14, Definition 2.7]) Let U = (U, AL, sL, tL, ∆L, ǫL, S ) be a Hopf algebroid. A stable anti-Yetter Drinfeld
+module (or SAYD module) P over U consists of the following
+(1) A right U-module structure on P denoted by (p, u) �→ pu for p ∈ P and u ∈ U.
+(2) A left U-comodule structure on P given by ∆P : P −→ U⊳ ⊗A P.
+(3) The right Ae-module structure on P induced by (5.9) coincides with the right Ae-module structure on P as in (5.12):
+psL(a)tL(b) = b ◮ p ◭ a = bǫL(p(−1)sL(a))p(0)
+(5.13)
+(4) For u ∈ U and p ∈ P, one has
+∆P(pu) = u−p(−1)u+(1) ⊗A p(0)u+(2)
+(5.14)
+(5) Stability condition: for any p ∈ P, one has p(0)p(−1) = p.
+Lemma 5.4. Let R, R′ be k-algebras and let Re = R ⊗k Rop, R′e = R′ ⊗k R′op be their respective enveloping algebras. Let C be
+a cocommutative k-coalgebra and let ψ : C −→ Vectk(R, R′) be a measuring. Then,
+ψe : C −→ Vectk(Re, R′e)
+ψe(c)(r ⊗ r′) = c(r1 ⊗ r2) = c(1)(r1) ⊗ c(2)(r2) = ψ(c(1))(r1) ⊗ ψ(c(2))(r2)
+(5.15)
+is a measuring of algebras.
+Proof. From (5.15), it is immediate that c(1 ⊗ 1) = ǫC(c)(1 ⊗ 1), where ǫC is the counit on C. Since C is cocommutative, we
+have for (r1 ⊗ r2), (r3 ⊗ r4) ∈ Re
+c((r1 ⊗ r2)(r3 ⊗ r4)) = c(r1r3 ⊗ r4r2)
+= c(1)(r1r3) ⊗ c(2)(r4r2)
+= c(1)(r1)c(2)(r3) ⊗ c(3)(r4)c(4)(r2)
+= c(1)(r1)c(3)(r3) ⊗ c(4)(r4)c(2)(r2)
+= (c(1)(r1) ⊗ c(2)(r2))(c(3)(r3) ⊗ c(4)(r4))
+= c(1)(r1 ⊗ r2)c(2)(r3 ⊗ r4)
+(5.16)
+□
+Lemma 5.5. Let U = (U, S ) = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, S ′) = (U′, AL, s′
+L, t′
+L, ∆′
+L, ǫ′
+L, S ′) be Hopf algebroids over
+k. Let P (resp. P′) be an SAYD-module over U (resp. U′). Let C be a cocommutative k-coalgebra and D be a right C-comodule.
+Suppose that we are given the following data
+Ψ : C −→ Vectk(U, U′)
+ψ : C −→ Vectk(A, A′)
+Ω : D −→ Vectk(P, P′)
+(5.17)
+such that
+13
+
+(1) (Ψ, ψ) is a measuring of Hopf algebroids from U to U′.
+(2) (Ψ, Ω) is a comodule measuring from the right U-module P to the right U′ module P′.
+Then, we have:
+(a) (ψe, Ω) is a comodule measuring from the right Ae-module P to the right A′e-module P′.
+(b) For each d ∈ D, the following morphism is well-defined
+d : U⊳ ⊗A P −→ U′
+⊳ ⊗A′ P′
+d(u ⊗A p) := Ψ(d(1))(u) ⊗A′ Ω(d(0))(p)
+(5.18)
+Proof. (a) Since C is cocommutative, we already know from Lemma 5.4 that ψe : C −→ Vectk(Ae, A′e) is a coalgebra measuring
+from Ae to A′e. We now consider (a ⊗ b) ∈ Ae = A ⊗k Aop. By (5.9), we know that p(a ⊗ b) = psL(a)tL(b). For any d ∈ D, we
+now have
+Ω(d)(p(a ⊗ b)) = Ω(d)(psL(a)tL(b))
+= Ω(d(0))(p)Ψ(d(1))(sL(a)tL(b))
+= Ω(d(0))(p)Ψ(d(1))(sL(a))Ψ(d(2))(tL(b))
+= Ω(d(0))(p)s′
+L(ψ(d(1))(a))t′
+L(ψ(d(2))(b))
+= Ω(d(0))(p)(ψ(d(1)(a)) ⊗ ψ(d(2)(b)))
+= Ω(d(0))(p)(ψe(d(1))(a ⊗ b))
+(5.19)
+(b) Since P and P′ are SAYD modules, it follows from the definition in (5.9) and the condition in (5.13) that
+ap = ptL(a)
+a′p′ = p′t′
+L(a′)
+a ∈ A, a′ ∈ A′, p ∈ P, p′ ∈ P′
+(5.20)
+where the left hand side of the equalities in (5.20) comes from the left A-module action on P (resp. the left A′-module action
+on P′) appearing in the structure map ∆P : P −→ U⊳ ⊗A P (resp. the structure map ∆′
+P′ : P′ −→ U′
+⊳ ⊗A′ P′). For a ∈ A, u ∈ U
+and p ∈ P, we now see that
+d(u ⊗A ap)
+= Ψ(d(1))(u) ⊗A′ Ω(d(0))(ap)
+= Ψ(d(1))(u) ⊗A′ Ω(d(0))(ptL(a))
+(using (5.20))
+= Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)Ψ(d(1))(tL(a))
+= Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)t′
+L(ψ(d(1))(a))
+= Ψ(d(2))(u) ⊗A′ ψ(d(1))(a)Ω(d(0))(p)
+(using (5.20))
+= Ψ(d(2))(u) ⊳ ψ(d(1))(a) ⊗A′ Ω(d(0))(p)
+= t′
+L(ψ(d(1))(a))Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)
+(using (5.2))
+(5.21)
+On the other hand, we also have
+d(u ⊳ a ⊗A p)
+= d(tL(a)u ⊗A p)
+(using (5.2))
+= Ψ(d(1))(tL(a)u) ⊗A′ Ω(d(0))(p)
+= Ψ(d(1))(tL(a))Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)
+= t′
+L(ψ(d(1))(a))Ψ(d(2))(u) ⊗A′ Ω(d(0))(p)
+(5.22)
+This proves the result.
+□
+We are now ready to introduce the notion of a comodule measuring between SAYD modules.
+Definition 5.6. Let U = (U, S ) = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, S ′) = (U′, AL, s′
+L, t′
+L, ∆′
+L, ǫ′
+L, S ′) be Hopf algebroids
+over k. Let P (resp. P′) be an SAYD-module over U (resp. U′). Let C be a cocommutative coalgebra. Then, a (right) measuring
+comodule over (C, Ψ, ψ) from P to P′ consists of the following data
+Ψ : C −→ Vectk(U, U′)
+ψ : C −→ Vectk(A, A′)
+Ω : D −→ Vectk(P, P′)
+(5.23)
+such that
+(1) (Ψ, ψ) is a measuring of Hopf algebroids from U to U′.
+14
+
+(2) (Ψ, Ω) is a comodule measuring from the right U-module P to the right U′ module P′.
+(3) For any d ∈ D, the following diagram commutes
+P
+∆P
+−−−−−−→ U⊳ ⊗A P
+d:=Ω(d)
+�
+d
+�
+P′
+∆′
+P′
+−−−−−−→ U′
+⊳ ⊗A′ P′
+(5.24)
+where the right vertical morphism is as defined in (5.18)
+We will now construct universal measuring comodules. By definition, the right comodules over a k-coalgebra C are coalgebras
+over the comonad
+⊗k C : Vectk −→ Vectk. Accordingly, the forgetful functor Comod − C −→ Vectk from the category of
+right C-comodules has a right adjoint (see, for instance, [7, § 2.4]) that we denote by RC, i.e., we have natural isomorphisms
+Vectk(D, V) � Comod − C(D, RC(V))
+(5.25)
+for any D ∈ Comod − C and V ∈ Vectk.
+Theorem 5.7. Let U = (U, AL, sL, tL, ∆L, ǫL, S ) and U′ = (U′, AL, s′
+L, t′
+L, ∆′
+L, ǫ′
+L, S ′) be Hopf algebroids over k. Let P (resp. P′)
+be an SAYD-module over U (resp. U′). Let C be a cocommutative coalgebra and (Ψ, ψ) : C −→ V(U, U′) be a measuring of
+Hopf algebroids.
+Then, there exists a measuring (C, Ψ, ψ)-comodule (QC(P, P′), Θ : QC(P, P′) −→ Vectk(P, P′)) satisfying the following property:
+given any measuring (C, Ψ, ψ)-comodule (D, Ω : D −→ Vectk(P, P′)) from P to P′, there exists a morphism χ : D −→ QC(P, P′)
+of right C-comodules such that the following diagram is commutative
+QC(P, P′)
+Θ
+� Vectk(P, P′)
+D
+χ
+�❍❍❍❍❍❍❍❍❍
+Ω
+�t
+t
+t
+t
+t
+t
+t
+t
+t
+t
+(5.26)
+Proof. We put V := Vectk(P, P′). By the adjunction in (5.25), there is a canonical morphism ρ(V) : RC(V) −→ V of vector
+spaces. We set QC(P, P′) := � Q, where the sum is taken over all right C-subcomodules over RC(V) such that the restriction
+ρ(V)|Q : Q −→ V = Vectk(P, P′) is a (C, Ψ, ψ)-comodule measuring from P to P′ in the sense of Definition 5.6. It is clear that
+Θ : ρ(V)|QC(P, P′) : QC(P, P′) −→ V = Vectk(P, P′) is a (C, Ψ, ψ)-measuring comodule.
+Additionally, given a measuring (C, Ψ, ψ)-comodule (D, Ω : D −→ Vectk(P, P′)) from P to P′, the adjunction in (5.25) gives a
+morphism χ : D −→ RC(V). But then we notice that ρ(V)|χ(D) : χ(D) −→ V is a measuring (C, Ψ, ψ)-comodule, whence it
+follows that the image χ(D) ⊆ QC(P, P′). The result is now clear.
+□
+Lemma 5.8. Let U = (U, AL, sL, tL, ∆L, ǫL, S ), U′ = (U′, AL, s′
+L, t′
+L, ∆′
+L, ǫ′
+L, S ′) and U′′ = (U′′, A′′
+L, s′′
+L, t′′
+L , ∆′′
+L, ǫ′′
+L , S ′′) be Hopf
+algebroids over k. Let P, P′ and P′′ be SAYD modules over U, U′ and U′′ respectively. Suppose that we have:
+(1) Ψ : C −→ Vectk(U, U′), ψ : C −→ Vectk(A, A′) and Ω : D −→ Vectk(P, P′) giving the data of a measuring comodule from
+P to P′.
+(2) Ψ′ : C′ −→ Vectk(U′, U′′), ψ′ : C′ −→ Vectk(A′, A′′) and Ω : D′ −→ Vectk(P′, P′′) giving the data of a measuring
+comodule from P′ to P′′.
+Then, the following
+(Ψ′, ψ′) ◦ (Ψ, ψ) : C ⊗ C′ (Ψ,ψ)⊗(Ψ′,ψ′)
+−−−−−−−−−−→ V(U, U′) ⊗ V(U′, U′′)
+◦
+−→ V(U, U′′)
+Ω′ ◦ Ω : D ⊗ D′ Ω⊗Ω′
+−−−−→ Vectk(P, P′) ⊗ Vectk(P′, P′′)
+◦−→ Vectk(P, P′′)
+(5.27)
+gives the data of a measuring comodule from P to P′′. There is also a canonical morphism of right (C ⊗ C′)-comodules
+QC(P, P′) ⊗ QC′(P′, P′′) −→ QC⊗C′(P, P′′)
+(5.28)
+15
+
+Proof. We know from Proposition 2.6 that (Ψ′, ψ′) ◦ (Ψ, ψ) : C ⊗ C′ −→ V(U, U′′) is a measuring of Hopf algebroids. It
+may also be directly verified that ((Ψ′, ψ′) ◦ (Ψ, ψ), Ω′ ◦ Ω) is a comodule measuring from the right U-module P to the right
+U′′-module P′′. To check the condition (5.24) in Definition 5.6, we observe that for any d ⊗ d′ ∈ D ⊗ D′, u ∈ U and p ∈ P:
+(d ⊗ d′)(u ⊗A p) = (d ⊗ d′)(1)(u) ⊗A′′ (d ⊗ d′)(0)(p) = d′
+(1)(d(1)(u)) ⊗A′′ d′
+(0)(d(0)(p)) = d′(d(u ⊗A p)))
+(5.29)
+Since the measurings (Ψ, ψ, Ω) and (Ψ′, ψ′, Ω′) both satisfy the condition in (5.24), it is clear that so does (Ψ′◦Ψ, ψ′ ◦ψ, Ω′ ◦Ω).
+Hence, (5.27) gives the data of a measuring comodule from P to P′′. By definition, QC(P, P′) (resp. QC′(P′, P′′)) is a measuring
+comodule from P to P′ (resp. from P′ to P′′). From (5.27) it now follows that QC(P, P′) ⊗ QC′(P′, P′′) is a measuring comodule
+from P to P′′. The morphism in (5.28) is now obtained by the universal property of QC⊗C′(P, P′′).
+□
+We now consider the “global category of comodules” Comodk whose objects are pairs (C, D), where C is a cocommutative k-
+coalgebra and D is a right C-comodule. A morphism (f, g) : (C, D) −→ (C′, D′) in Comodk consists of a k-coalgebra morphism
+f : C −→ C′ and a morphism g : D −→ D′ of C′-comodules, where D is treated as a C′-comodule by corestriction of scalars.
+It is clear that putting (C, D) ⊗ (C′, D′) := (C ⊗ C′, D ⊗ D′) makes Comodk into a symmetric monoidal category.
+Theorem 5.9. Let S AYDk be the category given by:
+(a) Objects: pairs (U, P), where U is a Hopf-algebroid and P is an S AYD-module over U
+(b) Hom-objects: for pairs (U, P), (U′, P′) ∈ S AYDk, we set
+S AYDk((U, P), (U′, P′)) := (Mc(U, U′), QMc(U,U′)(P, P′)) ∈ Comodk
+(5.30)
+Then, S AYDk is enriched over the symmetric monoidal category Comodk.
+Proof. For any (U, P) ∈ S AYDk, the scalar multiples of the identity map give a morphism k −→ Mc(U, U) of k-coalgebras,
+and along with the universal property in Theorem 5.7 give a morphism k −→ QMc(U,U)(P, P). We now consider (U, P), (U′, P′),
+(U′′, P′′) ∈ S AYDk. Applying Lemma 5.8 with C = Mc(U, U′) and C′ = Mc(U′, U′′), we obtain a morphism QMc(U,U′)(P, P′) ⊗
+QMc(U′,U′′)(P, P′) −→ QMc(U,U′)⊗Mc(U′,U′′)(P, P′′) of (Mc(U, U′) ⊗ Mc(U′, U′′))-comodules. From the proof of Theorem 2.7, we
+already have a morphism Mc(U, U′) ⊗ Mc(U′, U′′) −→ Mc(U, U′′) of k-coalgebras. Combining, we have a morphism
+S AYDk((U, P), (U′, P′)) ⊗ S AYDk((U′, P′), (U′′, P′′)) −→ (Mc(U, U′′), QMc(U,U′)⊗Mc(U′,U′′)(P, P′′))
+(5.31)
+in Comodk. In (5.31), QMc(U,U′)⊗Mc(U′,U′′)(P, P′′) is treated as a Mc(U, U′′)-module via the morphism Mc(U, U′)⊗Mc(U′, U′′) −→
+Mc(U, U′′) of k-coalgebras. From the proof of Theorem 2.7, we also know that the morphism Mc(U, U′) ⊗ Mc(U′, U′′) −→
+Mc(U, U′′) arises from the universal property of Mc(U, U′′) applied to the measuring Mc(U, U′) ⊗ Mc(U′, U′′) −→ V(U, U′) ⊗
+V(U′, U′′)
+◦−→ V(U, U′′). Hence, the canonical map QMc(U,U′)⊗Mc(U′,U′′)(P, P′′) −→ Vectk(P, P′′) gives a measuring when treated
+as a Mc(U, U′′)-comodule. The universal property of QMc(U,U′′)(P, P′′) as in Theorem 5.7 now yields a morphism
+(Mc(U, U′′), QMc(U,U′)⊗Mc(U′,U′′)(P, P′′)) −→ (Mc(U, U′′), QMc(U,U′′)(P, P′′))
+(5.32)
+in Comodk. Composing (5.32) with (5.31), we obtain the required composition of Hom-objects S AYDk((U, P), (U′, P′)) ⊗
+S AYDk((U′, P′), (U′′, P′′)) −→ S AYDk((U, P), (U′′, P′′)). This proves the result.
+□
+6
+Comodule measurings and morphisms on cyclic (co)homology
+Throughout this section, we fix the following: let U = (U, AL, sL, tL, ∆L, ǫL, S ), and U′ = (U′, A′
+L, s′
+L, t′
+L, ∆′
+L, ǫ′
+L, S ′) be Hopf
+algebroids over k. Let P and P′ be SAYD modules over U and U′ respectively. Let (Ψ, ψ) : C −→ V(U, U′) be a cocommutative
+measuring and let Ω : D −→ Vectk(P, P′) be a (C, Ψ, ψ)-measuring comodule from P to P′.
+Since U, U′ are Hopf algebroids, we have recalled in Section 5 that the morphisms β(U) : ◮U ⊗Aop U⊳ −→ U⊳ ⊗A ⊲U and
+β(U′) : ◮U′ ⊗A′op U′
+⊳ −→ U′
+⊳ ⊗A′ ⊲U′ in the notation of (5.5) are bijections. We now need the following result.
+16
+
+Lemma 6.1. For each x ∈ C, the following diagram commutes:
+U⊳ ⊗A ⊲U
+β(U)−1
+−−−−−−→
+◮U ⊗Aop U⊳
+x
+�
+x
+�
+U′
+⊳ ⊗A′ ⊲U′
+β(U′)−1
+−−−−−−→ ◮U′ ⊗A′op U′
+⊳
+(6.1)
+Here, the left vertical map is given by u1 ⊗A u2 �→ x(1)(u1) ⊗A′ x(2)(u2) and the right vertical map by u1 ⊗Aop u2 �→ x(1)(u1) ⊗A′op
+x(2)(u2).
+Proof. It is easy to see that the vertical morphisms in (6.1) are well-defined. Further, since β(U) and β(U′) are invertible, it
+suffices to check that the following diagram commutes
+◮U ⊗Aop U⊳
+β(U)
+−−−−−−→ U⊳ ⊗A ⊲U
+x
+�
+�x
+◮U′ ⊗A′op U′
+⊳
+β(U′)
+−−−−−−→ U′
+⊳ ⊗A′ ⊲U′
+(6.2)
+We now see that for u ⊗Aop v ∈ ◮U ⊗Aop U⊳ and x ∈ C, we have
+x(β(U)(u ⊗Aop v)) = x(u(1) ⊗A u(2)v) = x(1)(u(1)) ⊗A x(2)(u(2))x(3)(v) = (x(1)(u))(1) ⊗A (x(1)(u))(2)x(2)(v) = β(U′)(x(1)(u) ⊗ x(2)(v))
+This proves the result.
+□
+From Lemma 6.1, it follows in the notation of (5.6) that we have
+x(1)(u+) ⊗A′op x(2)(u−) = x(u+ ⊗Aop u−) = β(U′)−1(x(u ⊗A 1)) = x(u)+ ⊗A′op x(u)−
+(6.3)
+for each u ∈ U. We now recall from [14, Theorem 4.1] that the Hochschild homology groups HH•(U; P) (resp. the cyclic
+homology groups HC•(U; P)) of U with coefficients in the SAYD module P are obtained from the cyclic module C•(U; P) :=
+P ⊗Aop (◮U⊳)⊗Aop• with operators as follows (where ¯u := u1 ⊗Aop ⊗... ⊗Aop un, p ∈ P)
+di(p ⊗Aop ¯u) :=
+
+p ⊗Aop u1 ⊗Aop · · · ⊗Aop un−1tL(ǫL(un))
+if i = 0
+p ⊗Aop u1 ⊗Aop · · · ⊗Aop un−iun−i+1 ⊗Aop . . .
+if 1 ≤ i ≤ n − 1
+pu1 ⊗Aop u2 ⊗Aop · · · ⊗Aop un
+if i = n
+si(p ⊗Aop ¯u) :=
+
+p ⊗Aop u1 ⊗Aop · · · ⊗Aop un ⊗Aop 1
+if i = 0
+p ⊗Aop · · · ⊗Aop un−i ⊗Aop 1 ⊗Aop un−i+1 ⊗Aop . . .
+if 1 ≤ i ≤ n − 1
+p ⊗Aop 1 ⊗Aop u1 ⊗Aop · · · ⊗Aop un
+if i = n
+tn(p ⊗Aop ¯u) := p(0)u1
++ ⊗Aop u2
++ ⊗Aop · · · ⊗Aop un
++ ⊗Aop un
+− . . .u1
+−p(−1)
+(6.4)
+We now have the following result.
+Proposition 6.2. For each y ∈ D, the family
+Ωn(y) : Cn(U; P) −→ Cn(U′; P′)
+p ⊗ u1 ⊗ ... ⊗ un �→ y(p ⊗ u1 ⊗ ... ⊗ un) = y(0)(p) ⊗ y(1)(u1) ⊗ ... ⊗ y(n)(un)
+(6.5)
+for n ≥ 0 gives a morphism of cyclic modules. In particular, we have induced morphisms
+Ωhoc
+• (y) : HH•(U; P) −→ HH•(U′; P′)
+Ωcy
+• (y) : HC•(U; P) −→ HC•(U′; P′)
+(6.6)
+on Hochschild and cyclic homologies for each y ∈ D.
+17
+
+Proof. From the fact that C is cocommutative and the conditions in Definition 5.6, it is clear that the morphisms Ωn(y) are well
+defined, as well as the fact that they commute with the face maps and degeneracies appearing in the cyclic modules C•(U; P)
+and C•(U′; P′) as in (6.4). To verify that the morphisms in (6.5) also commute with the cyclic operators, we note that for
+p ⊗Aop u1 ⊗Aop ⊗... ⊗Aop un ∈ Cn(U; P)
+y(tn(p ⊗ u1 ⊗ ... ⊗ un)) = y(p(0)u1
++ ⊗ u2
++ ⊗ · · · ⊗ un
++ ⊗ un
+− . . . u1
+−p(−1))
+= y(0)(p(0))y(1)(u1
++) ⊗ y(2)(u2
++) ⊗ · · · ⊗ y(n)(un
++) ⊗ y(n+1)(un
+−) . . .y(2n)(u1
+−)y(2n+1)(p(−1))
+= y(0)(p(0))y(2)(u1
++) ⊗ y(4)(u2
++) ⊗ · · · ⊗ y(2n)(un
++) ⊗ y(2n+1)(un
+−) . . . y(3)(u1
+−)y(1)(p(−1))
+(since C is cocommutative)
+= y(0)(p)(0)y(1)(u1
++) ⊗ y(3)(u2
++) ⊗ · · · ⊗ y(2n−1)(un
++) ⊗ y(2n)(un
+−) . . . y(2)(u1
+−)y(0)(p)(−1)
+(using (5.24))
+= y(0)(p)(0)y(1)(u1)+ ⊗ y(2)(u2)+ ⊗ · · · ⊗ y(n)(un)+ ⊗ y(n)(un)− . . . y(1)(u1)−y(0)(p)(−1)
+(using (6.3))
+This proves the result.
+□
+We now come to cyclic cohomology. For this, we recall that from [14, Theorem 1.1] that the Hochschild cohomology groups
+HH•(U; P) (resp. the cyclic cohomology groups HC•(U; P)) of U with coefficients in the SAYD module P are obtained from
+the cocyclic module C•(U; P) := (⊲U⊳)⊗A• ⊗A P with operators as follows (where ¯u := u1 ⊗A ⊗... ⊗A un, p ∈ P)
+δi(¯u ⊗A p)
+=
+
+1 ⊗A u1 ⊗A · · · ⊗A un ⊗A p
+if i = 0
+u1 ⊗A · · · ⊗A ∆L(ui) ⊗A · · · ⊗A un ⊗A p
+if 1 ≤ i ≤ n
+u1 ⊗A · · · ⊗A un ⊗A p(−1) ⊗A p(0)
+if i = n + 1
+δi(p)
+=
+� 1 ⊗A p
+if j = 0
+p(−1) ⊗A p(0)
+if j = 1
+σi(¯u ⊗A p)
+= u1 ⊗A · · · ⊗A ǫL(ui+1) ⊗A · · · ⊗A un ⊗A p
+0 ≤ i ≤ n − 1
+τn(¯u ⊗A p)
+= u1
+−(1)u2 ⊗A · · · ⊗A u1
+−(n−1)un ⊗A u1
+−(n)p(−1) ⊗A p(0)u1
++
+(6.7)
+We now have the following result.
+Proposition 6.3. For each y ∈ D, the family
+Ω
+n(y) : Cn(U; P) −→ Cn(U′; P′)
+u1 ⊗ ... ⊗ un ⊗ p �→ y(u1 ⊗ ... ⊗ un ⊗ p) = y(1)(u1) ⊗ ... ⊗ y(n)(un) ⊗ y(0)(p)
+(6.8)
+for n ≥ 0 gives a morphism of cocyclic modules. In particular, we have induced morphisms
+Ω
+•
+hoc(y) : HH•(U; P) −→ HH•(U′; P′)
+Ω
+•
+cy(y) : HC•(U; P) −→ HC•(U′; P′)
+(6.9)
+on Hochschild and cyclic cohomologies for each y ∈ D.
+Proof. It is clear that the morphisms in (6.8) are well-defined. For y ∈ D and i = n + 1 in (6.7), we note that
+y(δn+1(u1 ⊗ · · · ⊗ un ⊗ p))
+= y(1)(u1) ⊗ . . . y(n)(un) ⊗ y(n+1)(p(−1)) ⊗ y(0)(p(0))
+= y(2)(u1) ⊗ . . . y(n+1)(un) ⊗ y(1)(p(−1)) ⊗ y(0)(p(0))
+(since C is cocommutative)
+= y(1)(u1) ⊗ . . . y(n)(un) ⊗ (y(0)(p))(−1) ⊗ y(0)(p)(0)
+(using (5.24))
+(6.10)
+Similarly, we may verify that the morphisms in (6.8) commute with the face and degeneracy maps appearing in (6.7). To show
+that they also commute with the cyclic operators appearing in (6.7), we note that for u1 ⊗ ... ⊗ un ⊗ p ∈ Cn(U; P) and y ∈ D, we
+have
+y(τn(u1 ⊗ ... ⊗ un ⊗ p)) = y(u1
+−(1)u2 ⊗A · · · ⊗A u1
+−(n−1)un ⊗A u1
+−(n)p(−1) ⊗A p(0)u1
++)
+= y(1)(u1
+−(1))y(2)(u2) ⊗A · · · ⊗A y(2n−3)(u1
+−(n−1))y(2n−2)(un) ⊗A y(2n−1)(u1
+−(n))y(2n)(p(−1)) ⊗A y(0)(p(0)u1
++)
+= y(1)(u1
+−(1))y(n+1)(u2) ⊗A · · · ⊗A y(n−1)(u1
+−(n−1))y(2n−1)(un) ⊗A y(n)(u1
+−(n))y(2n)(p(−1)) ⊗A y(0)(p(0)u1
++)
+= y(1)(u1
+−)(1)y(2)(u2) ⊗A · · · ⊗A y(1)(u1
+−)(n−1)y(n)(un) ⊗A y(1)(u1
+−)(n)y(n+1)(p(−1)) ⊗A y(0)(p(0)u1
++)
+= y(2)(u1
+−)(1)y(3)(u2) ⊗A · · · ⊗A y(2)(u1
+−)(n−1)y(n+1)(un) ⊗A y(2)(u1
+−)(n)y(n+2)(p(−1)) ⊗A y(0)(p(0))y(1)(u1
++)
+= y(1)(u1)−(1)y(2)(u2) ⊗A · · · ⊗A y(1)(u1)−(n−1)y(n)(un) ⊗A y(1)(u1)−(n)y(n+1)(p(−1)) ⊗A y(0)(p(0))y(1)(u1)+
+(using (6.3))
+= y(1)(u1)−(1)y(2)(u2) ⊗A · · · ⊗A y(1)(u1)−(n−1)y(n)(un) ⊗A y(1)(u1)−(n)y(0)(p)(−1) ⊗A y(0)(p)(0)y(1)(u1)+
+(using (5.24))
+This proves the result.
+□
+18
+
+Finally, we recall from [14, § 4.3] that there are Hopf-Galois isomorphisms relating the modules C•(U; P) and C•(U; P)
+ξn(U; P) : Cn(U; P)
+�
+−→ Cn(U; P)
+p ⊗ u1 ⊗ · · · ⊗ un �→ u1
+(1) ⊗ u1
+(2)u2
+(1) ⊗ · · · ⊗ u1
+(n)u2
+(n−1) . . .un−1
+(2) un ⊗ p
+(6.11)
+We will conclude this section by showing that the morphisms induced by comodule measurings of SAYD modules are compat-
+ible with the Hopf-Galois isomorphisms in (6.11).
+Theorem 6.4. Let U = (U, AL, sL, tL, ∆L, ǫL, S ), and U′ = (U′, A′
+L, s′
+L, t′
+L, ∆′
+L, ǫ′
+L, S ′) be Hopf algebroids over k. Let P and P′
+be SAYD modules over U and U′ respectively. Let (Ψ, ψ) : C −→ V(U, U′) be a cocommutative measuring and let Ω : D −→
+Vectk(P, P′) be a (C, Ψ, ψ)-measuring comodule from P to P′. Then, for each y ∈ D, the following diagram commutes
+Cn(U; P)
+ξn(U;P)
+−−−−−−→ Cn(U; P)
+Ωn(y)
+�
+�Ω
+n(y)
+Cn(U; P)
+ξn(U;P)
+−−−−−−→ Cn(U; P)
+(6.12)
+Proof. We set N := n(n − 1)/2. For y ∈ D and p ⊗ u1 ⊗ · · · ⊗ un ∈ Cn(U; P), we see that
+Ω
+n(y)(ξn(U; P)(p ⊗ u1 ⊗ · · · ⊗ un))
+= Ω
+n(y)(u1
+(1) ⊗ u1
+(2)u2
+(1) ⊗ · · · ⊗ u1
+(n)u2
+(n−1) . . . un−1
+(2) un ⊗ p)
+= y(1)(u1
+(1)) ⊗ y(2)(u1
+(2))y(3)(u2
+(1)) ⊗ · · · ⊗ y(N+1)(u1
+(n))y(N+2)(u2
+(n−1)) . . . y(N+n−1)(un−1
+(2) )y(N+n)(un) ⊗ y(0)(p)
+= y(1)(u1
+(1)) ⊗ y(2)(u1
+(2))y(n+1)(u2
+(1)) ⊗ · · · ⊗ y(n)(u1
+(n))y(2n−1)(u2
+(n−1)) . . .y(N+n−1)(un−1
+(2) )y(N+n)(un) ⊗ y(0)(p)
+= y(1)(u1)(1) ⊗ y(1)(u1)(2)y(2)(u2)(1) ⊗ · · · ⊗ y(1)(u1)(n)y(2)(u2)(n−1) . . .y(n−1)(un−1)(2)y(n)(un) ⊗ y(0)(p)
+= ξn(U; P)(Ωn(y)(p ⊗ u1 ⊗ · · · ⊗ un))
+(6.13)
+□
+7
+Operads with multiplication, comp modules and morphisms on cyclic homology
+We start the final section by recalling from Kowalzig [16] the following two notions.
+Definition 7.1. (see [16, Definition 2.2]) A non-Σ operad O over k consists of the following:
+(a) A collection of vector spaces O = {O(n)}n≥0.
+(b) A family of k-linear operations ◦i : O(p)⊗O(q) −→ O(p+q−1) and an identity 1 ∈ O(1) satisfying the following conditions
+(for φ ∈ O(p), ψ ∈ O(q), χ ∈ O(r))
+φ ◦i ψ
+= 0
+if p < i or p = 0
+(φ ◦i ψ) ◦ j χ
+=
+
+(φ ◦ j χ) ◦i+r−1 ψ
+if j < i
+φ ◦i (ψ ◦ j−i+1 χ)
+if i ≤ j < q + i
+(φ ◦ j−q+1 χ) ◦i ψ
+if j ≥ q + i
+φ ◦i 1
+= 1 ◦1 φ = φ
+for i ≤ p
+(c) An operad multiplication µ ∈ O(2) and a unit e ∈ O(0) such that
+µ ◦1 µ = µ ◦2 µ
+µ ◦1 e = µ ◦2 e = 1
+(7.1)
+Definition 7.2. (see [16, Definition 3.1]) A cyclic unital comp module M over an operad O with multiplication consists of the
+following data:
+(a) A collection of vector spaces M = {M(n)}n≥0.
+19
+
+(b) A family of k-bilinear operations •i : O(p) ⊗ M(n) −→ M(n − p + 1), 0 ≤ i ≤ n + 1 − p satisfying the following conditions
+for φ ∈ O(p), ψ ∈ O(q), x ∈ M(n)
+φ •i (ψ • j x) =
+
+ψ • j (φ •i+q−1 x)
+j < i
+(φ • j−i+1 ψ) •i x
+if j − p < i ≤ j
+ψ • j−p+1 (φ •i x)
+if 1 ≤ i ≤ j − p
+as well as 1 •i x = x for i = 1, 2, ..., n.
+(c) A cyclic operator t : M(n) −→ M(n) for n ≥ 1 satisfying
+t(φ •i x) = φ •i t(x)
+(7.2)
+for φ ∈ O(p), x ∈ M(n) and 0 ≤ i ≤ n − p as well as tn+1 = id.
+We take pairs (O, M) consisting of a non-linear Σ operad O and a cyclic unital comp module M over O. We now consider
+comodule measurings between such pairs
+Definition 7.3. A comodule measuring from (O, M) to (O′, M′) consists of the following:
+(a) A cocommutative coalgebra C and a family of morphisms {Φn : C −→ Vectk(O(n), O′(n))}n≥0 satisfying
+Φp+q−1(x)(φ ◦i ψ) = Φp(x(1))(φ) ◦′
+i Φq(x(2))(φ)
+Φ2(x)(µ) = ǫ(x)µ′
+Φ0(x)(e) = ǫ(x)e′
+(7.3)
+for φ ∈ O(p), ψ ∈ O(q) and any x ∈ C.
+(b) A comodule D over C and a family of morphisms {Ψn : D −→ Vectk(M(n), M′(n))}n≥0 satisfying
+Ψn−p+1(φ •i x) = Ψp(y(0))(φ) •i Ψn(y(1))(x)
+(7.4)
+for y ∈ D, φ ∈ O(p), x ∈ M(n), 0 ≤ i ≤ n + 1 − p and also
+Ψn(y)(t(x)) = t′(Ψn(y)(x))
+(7.5)
+for y ∈ D, x ∈ M(n), where t and t′ are respectively the cyclic operators on M and M′.
+We now recall from [16, Proposition 3.5] that the cyclic homology of (O, M) is obtained from the cyclic module C•(O, M) :=
+M(•) whose cyclic operators are t : M(n) −→ M(n) and whose face maps and degeneracies are given as follows:
+di(x) := µ •i x, (0 ≤ i < n)
+dn(x) := µ •0 t(x)
+sj(x) := e • j+1 x, 0 ≤ j ≤ n
+(7.6)
+The cyclic homologies of this cyclic module will be denoted by HC•(O, M). We conclude with the following result.
+Proposition 7.4. If D is a C-measuring comodule from (O, M) to (O′, M′), then each y ∈ D induces a morphism Ψcy
+• (y) :
+HC•(O, M) −→ HC•(O′, M′) on Hochschild homologies.
+Proof. We know from (7.5) that the action of any y ∈ D commutes with the cyclic operators. From the definitions in (7.6)
+and the conditions in (7.3), it is clear that the action also commutes with the degeneracies and face maps. The result is now
+clear.
+□
+References
+[1] M. Anel and A. Joyal, Sweedler theory for (co)algebras and the bar-cobar constructions, arXiv 1309.6952 (2013).
+[2] A. Banerjee and S. Kour, On measurings of algebras over operads and homology theories, Algebr. Geom. Topol. 22 (2022), no. 3, 1113–1158.
+[3] M. Batchelor, Difference operators, measuring coalgebras, and quantum group-like objects, Adv. Math. 105 (1994), no. 2, 190–218.
+[4]
+, Measuring comodules—their applications, J. Geom. Phys. 36 (2000), no. 3-4, 251–269.
+20
+
+[5] G. B¨ohm and K. Szlach´anyi, Hopf algebroids with bijective antipodes: axioms, integrals, and duals, J. Algebra 274 (2004), no. 2, 708–750.
+[6] G. B¨ohm, Galois theory for Hopf algebroids, Ann. Univ. Ferrara Sez. VII (N.S.) 51 (2005), 233–262.
+[7] G. B¨ohm, T. Brzezi´nski, and R. Wisbauer, Monads and comonads on module categories, J. Algebra 322 (2009), no. 5, 1719–1747.
+[8] T. Brzezinski and R. Wisbauer, Corings and comodules, London Mathematical Society Lecture Note Series, vol. 309, Cambridge University Press,
+Cambridge, 2003.
+[9] L. Grunenfelder and M. Mastnak, On bimeasurings, J. Pure Appl. Algebra 204 (2006), no. 2, 258–269.
+[10]
+, On bimeasurings. II, J. Pure Appl. Algebra 209 (2007), no. 3, 823–832.
+[11] M. Hyland, I. L´opez Franco, and C. Vasilakopoulou, Hopf measuring comonoids and enrichment, Proc. Lond. Math. Soc. (3) 115 (2017), no. 5, 1118–
+1148.
+[12]
+, Measuring comodules and enrichment, arXiv 1703.10137 (2017).
+[13] N. Kowalzig and H. Posthuma, The cyclic theory of Hopf algebroids, J. Noncommut. Geom. 5 (2011), no. 3, 423–476.
+[14] N. Kowalzig and U. Kr¨ahmer, Cyclic structures in algebraic (co)homology theories, Homology Homotopy Appl. 13 (2011), no. 1, 297–318.
+[15] N. Kowalzig, Batalin-Vilkovisky algebra structures on (Co)Tor and Poisson bialgebroids, J. Pure Appl. Algebra 219 (2015), no. 9, 3781–3822.
+[16]
+, Gerstenhaber and Batalin-Vilkovisky structures on modules over operads, Int. Math. Res. Not. IMRN 22 (2015), 11694–11744.
+[17] J.-L Loday, Cyclic homology, 2nd ed., Grundlehren der mathematischen Wissenschaften, vol. 301, Springer-Verlag, Berlin, 1998. Appendix E by M. O.
+Ronco; Chapter 13 by the author in collaboration with T. Pirashvili.
+[18] P. Schauenburg, Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules, Appl. Categ. Structures 6 (1998), no. 2, 193–222.
+[19]
+, Duals and doubles of quantum groupoids (×R-Hopf algebras), New trends in Hopf algebra theory (La Falda, 1999), Contemp. Math., vol. 267,
+Amer. Math. Soc., Providence, RI, 2000, pp. 273–299.
+[20] M. E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969.
+[21] M. Takeuchi, Groups of algebras over A ⊗ A, J. Math. Soc. Japan 29 (1977), no. 3, 459–492.
+[22] C. Vasilakopoulou, Enrichment of categories of algebras and modules, arXiv 1205.6450 (2012).
+[23]
+, On enriched fibrations, Cah. Topol. G´eom. Diff´er. Cat´eg. 59 (2018), no. 4, 354–387.
+[24]
+, Enriched duality in double categories: V-categories and V-cocategories, J. Pure Appl. Algebra 223 (2019), no. 7, 2889–2947.
+21
+

2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ac3f2a0f5c6c47794ac5c9f964adb8ae60eae0697c16f5e5b5f82412dddd80c8
+size 304219

2NFRT4oBgHgl3EQfmzfK/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f70a1fde2edb96a180aed17e3b2b3d62b5b3c836a9cac1db83a4d26b6e2bd850
+size 3407917

2NFRT4oBgHgl3EQfmzfK/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:2509f0f95f2907293853f173dd632605fbbe40925a03d3c037a8f88aab198664
+size 132788

39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:42b1e9f8861b6a8313395eb95bf1c1ee44134064fce833b239cc5ff90c9c5912
+size 2279290

39A0T4oBgHgl3EQfNf8t/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:944104129e123fe3bf6ca4d6ab8d992fd9ca3057d83b3fa49d1be83eac55875d
+size 6225965

39E4T4oBgHgl3EQfbQyO/content/tmp_files/2301.05071v1.pdf.txt

@@ -0,0 +1,2028 @@
+Constraining the Limitations of NEATM-like Models: A Case Study with Near-Earth
+Asteroid (285263) 1998 QE2
+Samuel A. Myers1
+, Ellen S. Howell1
+, Christopher Magri2
+, Ronald J. Vervack, Jr.3
+, Yanga R. Fernández4
+,
+Sean E. Marshall5
+, and Patrick A. Taylor6
+1 Lunar and Planetary Laboratory, University of Arizona, 1629 E. University Boulevard, Tucson, AZ 85721, USA; [email protected]
+2 University of Maine Farmington, 173 High Street, Farmington, ME 04938, USA
+3 Johns Hopkins Applied Physics Laboratory, 11100 John Hopkins Road, Laurel, MD 20723, USA
+4 University of Central Florida, 4111 Libra Drive, Orlando, FL 32816, USA
+5 Arecibo Observatory/University of Central Florida, HC-03 Box 53995, Arecibo, Puerto Rico 00612, USA
+6 National Radio Astronomy Observatory/Green Bank Observatory, 1180 Boxwood Estate Road, Charlottesville, VA 22903, USA
+Received 2022 August 10; revised 2022 November 28; accepted 2022 December 1; published 2023 January 10
+Abstract
+Near-Earth asteroids (NEAs) are a key test bed for investigations into planet formation, asteroid dynamics, and
+planetary defense initiatives. These studies rely on understanding NEA sizes, albedo distributions, and regolith
+properties. Simple thermal models are a commonly used method for determining these properties; however, they
+have inherent limitations owing to the simplifying assumptions they make about asteroid shapes and properties.
+With the recent collapse of the Arecibo Telescope and a decrease of direct size measurements, as well as future
+facilities such as LSST and NEO Surveyor coming online soon, these models will play an increasingly important
+role in our knowledge of the NEA population. Therefore, it is key to understand the limits of these models. In this
+work we constrain the limitations of simple thermal models by comparing model results to more complex
+thermophysical models, radar data, and other existing analyses. Furthermore, we present a method for placing
+tighter constraints on inferred NEA properties using simple thermal models. These comparisons and constraints are
+explored using the NEA (285263) 1998 QE2 as a case study. We analyze QE2 with a simple thermal model and
+data from both the NASA IRTF SpeX instrument and NEOWISE mission. We determine an albedo between 0.05
+and 0.10 and thermal inertia between 0 and 425J m−2 s−1/2 K−1. We find that overall the simple thermal model is
+able to well constrain the properties of QE2; however, we find that model uncertainties can be influenced by
+topography, viewing geometry, and the wavelength range of data used.
+Unified Astronomy Thesaurus concepts: Asteroids (72); Asteroid surfaces (2209); Near-Earth objects (1092)
+1. Introduction
+Asteroids were once derided by astronomers as the “vermin
+of the sky,” but they now form an important piece of our efforts
+to understand our own solar system. Understanding their sizes,
+albedo
+distributions,
+and
+regolith
+properties
+is
+key
+for
+investigations into many aspects of solar system science,
+including solar system formation, main belt asteroid orbital
+evolution, surface processes on airless bodies, and under-
+standing our meteorite collection. Near-Earth asteroids (NEAs),
+in particular, are excellent targets for these efforts owing to
+their proximity to Earth.
+In addition to understanding the albedos and regoliths of
+these objects, accurately measuring the sizes of NEAs is pivotal
+for planetary defense initiatives—the area of study focused on
+preventing catastrophic asteroid impacts with Earth. This is
+because the size of an object is directly related to the energy of
+impact (Morrison & Teller 1995), which determines the impact
+severity. Thus, observation and modeling techniques that
+provide estimates of these properties are key for understanding
+the NEA population.
+There are a few methods for obtaining size estimates and
+other physical properties from NEA observations. Radar
+images, detailed thermophysical models, and simple thermal
+models can all be used to obtain size estimates. All of these
+methods, along with light-curve measurements, can also place
+constraints on other physical properties of asteroids. Other
+methods, such as direct imaging (Dollfus 1971; Marchis et al.
+2006; Marchis & Vega 2014), stellar occultations (Millis &
+Dunham 1989; Arai et al. 2020), and spacecraft encounters
+exist (Belton et al. 1992, 1996; Veverka et al. 2000; Lauretta
+et al. 2019) but are only applicable in rare cases. Of the more
+common methods, radar images can provide a size estimate
+without other information (Ostro 1985). Radar observations
+can also be used to construct detailed models of the asteroid’s
+shape (Hudson & Ostro 1994; Magri et al. 2007, 2011; Nolan
+et al. 2013). Light-curve measurements can also produce shape
+models, although they are often less detailed than radar-derived
+shape models and do not include an absolute size scale (Ďurech
+et al. 2012 and references therein). These shape models can be
+coupled with thermal spectra to constrain other physical
+properties of the asteroid as well, such as thermal inertia or
+surface roughness (Marshall et al. 2017; Howell et al. 2018;
+Jones 2018; Hinkle et al. 2022).
+Historically, the Arecibo Telescope has been a source of
+numerous NEA radar observations. The Arecibo Telescope
+detected over 900 NEAs and made size estimates of roughly
+400 of those (Howell et al. 2020). However, with the recent
+loss of the Arecibo Telescope, there will be a lack of direct size
+and shape measurements of NEAs. (Although Goldstone is able
+to make radar measurements, it has a lower sensitivity and less
+availability for targets of opportunity.) As a result, in the future
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+https://doi.org/10.3847/PSJ/aca89d
+© 2023. The Author(s). Published by the American Astronomical Society.
+Original content from this work may be used under the terms
+of the Creative Commons Attribution 4.0 licence. Any further
+distribution of this work must maintain attribution to the author(s) and the title
+of the work, journal citation and DOI.
+1
+
+there will be a greater reliance on other methods to understand
+the
+physical
+properties
+of
+NEAs.
+These
+methods
+will
+necessarily be models, like simple thermal models, that assume
+asteroid shapes or use less well-constrained shape models.
+Simple thermal models, such as the Standard Thermal Model
+(Lebofsky et al. 1986; Lebofsky & Spencer 1989) and the
+Near-Earth
+Asteroid
+Thermophysical
+Model
+(NEATM;
+Harris 1998), are a convenient method for obtaining NEA
+sizes and physical properties in part because they are easy to
+run. They require only visible and thermal infrared data and are
+computationally
+fast.
+For
+this
+reason,
+they
+are
+already
+commonly used to analyze data collected by large survey
+missions like NEOWISE (Mainzer et al. 2011b) and Explor-
+eNEOs (Trilling et al. 2010). Due to the large volume of data
+collected by these types of surveys and the sparse amount of
+data collected on any single object, simple thermal models are
+often the only practical way to quickly interpret the data. In
+these cases, simple thermal models are used to identify both
+scientifically interesting and potentially dangerous NEAs (e.g.,
+Trilling et al. 2010).
+However, simple thermal models make simplifying assump-
+tions about the asteroid’s shape and surface that can result in
+inaccuracies and thus poor constraints of inferred NEA
+properties. This is especially relevant for determinations of
+asteroid sizes—values that are pivotal for planetary defense
+activities. Simple thermal models can only make direct
+determinations of asteroid sizes in specific cases. If absolute
+photometry in both the visible and infrared is available, size
+can be solved for directly. However, these estimates require
+assuming that the visible and infrared data were acquired at
+similar viewing geometries. This assumption is often made
+with models employing NEOWISE or ExploreNEOs observa-
+tions. Alternatively, if only normalized flux is available, then
+the size must be estimated from the modeled albedo in
+combination with the absolute magnitude, H. In this case, the
+estimates are subject to uncertainties in the magnitude (Bowell
+et al. 1989; Jurić et al. 2002; Vereš et al. 2015), as well as
+typically large error bars in the inferred albedo, producing poor
+constraints. In fact, recent work has shown that there are
+inconsistencies between sizes derived from NEOWISE data
+using these models and sizes derived using other methods
+(Howell et al. 2012; Taylor et al. 2014; Masiero et al. 2019;
+Taylor et al. 2019; Masiero et al. 2021).
+In this paper, we seek to better understand the limitations of
+simple thermal models, such as NEATM, by comparing simple
+thermal model results to more complex thermophysical models,
+radar data, and other existing analyses of a given object. We
+also present a method for placing tighter constraints on inferred
+NEA properties using these simple thermal models. We use a
+simple, NEATM-like model (Section 3) to model the observed
+NEA, and the consistency of the best-fit parameters is then
+checked by comparing the models to normalized flux data
+collected across multiple nights that represent a range of
+viewing geometries. We also compare the models to the
+absolute photometry collected by the NEOWISE spacecraft. By
+observing an object across multiple viewing geometries and
+combining normalized flux spectra with absolute photometry,
+we are able to place tight bounds on modeled NEA properties.
+These simple thermal model results are then compared to
+model results from SHERMAN (Magri et al. 2018), a complex
+thermophysical
+model;
+radar
+measurements;
+and
+other
+observations
+and
+analyses
+of
+the
+given
+object.
+These
+comparisons allow us to place constraints on the overall
+limitations of the simple thermal model and identify key factors
+that influence uncertainties in simple thermal model results.
+This analysis is performed on the well-studied NEA
+(285263) 1998 QE2 (hereafter referred to as QE2). QE2 is a
+spheroidal, binary NEA system, with an existing radar-derived
+shape model (Springmann et al. 2014). The secondary has a
+diameter ∼25% that of the primary (Springmann et al. 2014)
+and thus contributes only 6% of the total flux. Therefore, the
+primary object dominates the thermal emission from the
+system, and we neglect the secondary in our analysis. QE2 is
+an Xk-type asteroid in the Bus−DeMeo taxonomy, as derived
+from our SpeX prism spectra and a visible spectrum obtained
+by Hicks et al. (2013).
+As part of our investigation into the limitations of the
+NEATM-like model, we find a discrepancy in the currently
+accepted H-magnitude for QE2. We find that the current value
+is inconsistent with the size derived from the radar measure-
+ments of QE2. We investigate this discrepancy and discuss
+implications. As part of this investigation, we compare our
+results to previous studies to understand QE2ʼs composition
+and surface properties (Fieber-Beyer et al. 2020), as well as its
+spin state (Moskovitz et al. 2017). These comparisons allow us
+to further benchmark the uncertainties in the results of our
+method for placing tight constraints on NEA properties derived
+with simple thermal models.
+In Section 2 we discuss the data used for our analysis. In
+Section 3 we describe our simple, NEATM-like model, and in
+Section 4 we present the results for QE2 from this model. In
+Section 5 we describe our analysis of the uncertainties in these
+model results. We compare our simple, NEATM-like model
+results to model results from SHERMAN, radar data of QE2,
+and the results of other previous studies. We then discuss
+implications for the limitations of simple thermal models. We
+conclude with a summary of our results in Section 6.
+2. Spectral and Radar Data
+2.1. IRTF Observations
+The primary data used to constrain our models are normalized
+flux spectra obtained with the SpeX instrument at the NASA
+IRTF (Rayner et al. 2003). We use normalized flux, as it has
+smaller uncertainties relative to absolute photometry. These
+observations are carried out as part of our ongoing investigation
+into the physical properties of NEAs. We observed using both
+prism mode (0.8–2.5 μm) and Long-Wavelength Cross-Dispersed
+(LDX)1.9 mode (2.2–4.1 μm). Note that the observations of QE2
+presented here were done before the upgrade to SpeX that
+expanded the wavelength ranges of all settings.
+For QE2, observations were carried out over six nights, from
+2013 May 30 to 2013 July 10. Over this time, the solar phase
+angle of QE2 varied from 18°.0 to 39°.7, which let us observe
+different viewing geometries and illumination states. As a
+result, we see the thermal emission at different local times of
+day. This is important because it allows us to check the
+consistency of the fit parameters (Section 3). The various sub-
+Earth locations of QE2 that we observed are shown in Figure 1.
+A summary of the observational parameters for our six nights
+of SpeX data is shown in Table 1.
+2
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+All SpeX observations were done in pairs, nodding the
+telescope along a 15″ slit. We used exposure times of 15 s for
+our LXD data and 10–30 s for our prism data. The data were
+processed using the Spextool software package (Cushing et al.
+2004), and the spectra were extracted from summed images. In
+addition to the object, we observed solar-analog stars in a
+similar manner. At least one was a nearby G star within ∼5° of
+the object on the sky. All stars were compared to a well-
+characterized solar analog star on each night, and their spectra
+were corrected for slight spectral slope variations if necessary.
+Each asteroid–star pair was combined in a ratio after correcting
+each for atmospheric absorption lines. The spectra were then
+determined using a weighted average over all asteroid–star
+pairs and binned to form the final spectra. Bad data points were
+flagged and excluded from the fitting and averaging process.
+The detailed methods for this entire process are given in
+Howell et al. (2018).
+The data are broken up across each night into several
+independent sets of roughly 20–30 minutes each to sample
+different areas of the surface. QE2 has a rotation period of
+4.749 ± 0.002 hr (Springmann et al. 2014), meaning that each
+spectrum is separated by roughly 25°–40° of longitude at the
+equator. The sub-Earth latitudes and longitudes at the midtimes of
+the observations are shown in Figure 1. These sub-Earth
+coordinates are calculated using the shape model of Springmann
+et al. (2014). The LXD data for each of the six nights are shown in
+Figure 2. Each spectrum is normalized at 1.6 μm to give
+normalized flux. (Note that there is no significant thermal
+Figure 1. Sub-Earth locations on QE2 during observations as determined by a radar shape model (Springmann et al. 2014). (a) The pole solution with the “bumpy”
+topography in the northern hemisphere. (b) The pole solution with the topography partially in the southern hemisphere (Section 2.3). The range of sub-Earth locations
+observed indicates that QE2 was observed across multiple different viewing geometries. This range of observations is key for constraining QE2ʼs parameters using our
+NEATM-like model.
+3
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+Sub-EarthLocations
+17
+A)
+16
+Latitude (degree)
+Date
+15
+30May
+02 Jun
+4
+08 Jun
+十
+15Jun
+区
+18Jun
+13
+米
+10Jul
+12
+11
+B)
+米
+atitude(degree)
+5
+Date
+30May
+02 Jun
+区
+08 Jun
+15Jun
+18Jun
+-10
+米
+10Jul
+-15
+0
+60
+120
+180
+240
+300
+360
+Longitude(degree)contamination at this wavelength.) We use normalized flux
+because the relative uncertainties are much smaller than for
+absolutely calibrated photometry. We cover the range from
+completely reflected to thermally dominated to ensure that our
+simple thermal model is well constrained in both regimes. This
+technique has the advantage of being more flexible but the
+disadvantage that the data are highly correlated in wavelength.
+2.2. NEOWISE Observations
+In addition to our SpeX data, we fit our simple thermal
+model to data collected by NEOWISE. Unlike the SpeX data,
+which measure normalized flux, NEOWISE measures absolute
+photometry. Thus, fitting our simple thermal model to the
+NEOWISE data allows us to check that the best-fit parameters
+are consistent with both the spectrum shape and calibrated flux
+values. This provides an additional independent check on the
+consistency of the simple thermal model and allows us to
+identify any potential issues with the model not observed when
+fitting normalized flux data alone.
+We retrieve the NEOWISE data and associated uncertainties
+from the NASA/IPAC Infrared Science Archive (Mainzer
+et al. 2011a, 2014).7 We do not use the raw images, but instead
+retrieve processed data that list the magnitudes and uncertain-
+ties for channels W1 (effective wavelength 3.4 μm) and W2
+(effective wavelength 4.6 μm) for each time the object was
+observed. We remove data points that are flagged for potential
+contamination, such as by cosmic-ray hits, and average
+together all remaining observations. The uncertainty in the
+NEOWISE data is dominated by systematic errors and not
+statistical noise. All observations, except one, have similar
+uncertainties. We thus take a weighted average of the
+observations and adopt the variance of the overall data set,
+divided by the square root of the number of observations minus
+one, as our 1σ uncertainties. For QE2, all observations were
+taken over a short time interval such that the change in QE2ʼs
+orbital position was minimal. Therefore, we averaged together
+all available observations, resulting in one averaged set of data
+points from eight individual observations that span roughly 29
+hr and approximately six rotation periods. The individual
+observations are evenly distributed across the rotation phase. A
+summary of the observational parameters for the averaged
+observation is given in Table 1. A list of the individual
+observations is given in Table 2.
+After retrieval, the data are then converted from NEOWISE
+magnitudes to Fλ units following the procedures outlined in the
+WISE Data Processing Handbook (Wright et al. 2010; Cutri
+et al. 2012). For this process we apply a final blackbody color
+correction corresponding to a 221 K object. This blackbody
+temperature is determined by fitting ideal blackbody curves to
+the NEOWISE data in an iterative process until the corrected
+NEOWISE data and ideal blackbody curves converge. The
+blackbody temperature used for the initial correction is
+calculated using the theoretical blackbody temperature relation
+T
+L
+A
+r
+1
+16
+,
+1
+H
+sb
+4
+2
+(
+)
+( )
+
+s
+p
+=
+-
+where Le is the solar luminosity, A is the Bond albedo, rH is the
+object–Sun distance, and σsb is the Stefan–Boltzmann constant.
+Table 1
+Summary of Observations, Including Values Input Directly into the NEATM-like Model
+Date
+Set
+Midtime
+rH (au)
+Δ (au)
+α (deg)
+Instrument
+2013 May 30
+A
+06:46:50
+1.046 8
+0.040 3
+34.3
+SpeX
+2013 May 30
+B
+07:22:08
+1.046 8
+0.040 3
+34.2
+SpeX
+2013 May 30
+C
+08:36:57
+1.049 8
+0.040 2
+33.9
+SpeX
+2013 Jun 02
+A
+06:51:57
+1.052 2
+0.040 1
+18.3
+SpeX
+2013 Jun 02
+B
+07:08:19
+1.052 2
+0.040 1
+18.3
+SpeX
+2013 Jun 02
+C
+07:17:50
+1.052 2
+0.040 1
+18.3
+SpeX
+2013 Jun 02
+D
+07:34:17
+1.052 2
+0.040 1
+18.2
+SpeX
+2013 Jun 08
+A
+08:12:16
+1.067 1
+0.060 5
+30.0
+SpeX
+2013 Jun 08
+B
+09:25:01
+1.067 2
+0.060 8
+30.1
+SpeX
+2013 Jun 08
+C
+09:37:14
+1.067 2
+0.060 8
+30.1
+SpeX
+2013 Jun 08
+D
+10:38:10
+1.067 4
+0.061 0
+30.2
+SpeX
+2013 Jun 08
+E
+10:50:40
+1.067 4
+0.061 1
+30.2
+SpeX
+2013 Jun 15
+A
+11:06:28
+1.091 0
+0.098 8
+38.8
+SpeX
+2013 Jun 15
+B
+12:16:11
+1.091 2
+0.099 1
+38.8
+SpeX
+2013 Jun 18
+A
+13:07:51
+1.103 3
+0.116 9
+39.7
+SpeX
+2013 Jul 10
+A
+10:23:08
+1.218 8
+0.256 2
+34.0
+SpeX
+2013 Jul 10
+B
+10:29:19
+1.218 9
+0.256 2
+34.0
+SpeX
+2013 Jul 10
+C
+11:10:20
+1.219 0
+0.256 4
+34.0
+SpeX
+2013 Jul 10
+D
+11:49:53
+1.219 2
+0.256 6
+34.0
+SpeX
+2013 Jul 10
+E
+13:09:29
+1.219 6
+0.257 0
+34.0
+SpeX
+2017 Jul 01
+A
+10:51:35
+1.767 6
+1.445 3
+35.1
+NEOWISE
+Note. Set refers to different data sets on a given night. Midtime is the midtime of observation for the data set in UTC time. (Each SpeX observation spans roughly
+20–30 minutes, while the NEOWISE observation spans 29 hr. Thus, each SpeX spectrum is separated by roughly 25°–40° of longitude.) rH is the Sun–object distance,
+Δ is the Earth–object distance, and α is the solar phase angle. Note that the observations are carried out across a range of solar phase angles and viewing geometries.
+7
+https://www.ipac.caltech.edu/doi/irsa/10.26131/IRSA144
+4
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+Figure 2. Processed LXD data sets for each night of observations with SpeX and NEOWISE. (a–f) SpeX data for each of the six nights. The different letters within
+each panel indicate different data sets collected each night (Table 1). The y-axis is normalized flux, normalized to 1.6 μm. (Note that there is no significant thermal
+contamination at this wavelength.) (g) NEOWISE data in absolute flux density. Note that the NEOWISE data are plotted over a different wavelength range. We plot
+both the 1σ and 3σ uncertainties. (h) The “A“ data set for each night of SpeX data. These spectra highlight how different viewing geometries across the different nights
+produce a range of spectral slopes. We see that changes in viewing geometry produce changes in the spectra shape both within nights and across all nights of
+observations. Modeling these differences allows us to place tighter constraints on NEA properties.
+5
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+125
+A)
+B)
+100
+xnI
+2013 May 30
+2013 Jun 02
+Normalized 
+75
+ＡB
+50
+CD
+25
+0
+125
+C
+D)
+100
+ Flux
+Normalized
+75
+2013 Jun 15
+2013 Jun 08
+A
+A
+50
+BCDE
++
+25
+0
+125
+F)
+E)
+100
+Normalized
+75
+2013 Jul 10
+2013 Jun 18
+AB-
++A
++
+50
++
+CDE
++
+25
+3.2
+3.7
+4.2
+7
+3.2
+3.7
+4.2
+Wavelength (microns)
+Wavelength (microns)
+1e-20
+125
+G)
+H)
+_wn
+8e-21
+100
+SpeX, All Nights
+2017 Jul 01
+Normalized I
+2013 May 30
+6e-21
+75
+2013 Jun 02
+3g
+2013 Jun 08
++
+2013 Jun 15
+4e-21
+50
+2013 Jun 18
+2013 Jul 10
+25
+2e-21
+0e+00
+3.2
+2.5
+3.5
+4.5
+3.7
+4.2 
+3.0
+4.0
+5.0
+Wavelength (microns)
+Wavelength (microns)The Bond albedo is estimated according to the method
+described in Lebofsky & Spencer (1989):
+A
+G p
+0.29
+0.684
+,
+2
+(
+)
+( )
+=
++
+where G is the slope parameter in the HG magnitude system
+(Bowell et al. 1989) and p is the visual geometric albedo. The
+standard assumption of G = 0.15 is used, and p is taken from
+the model fits to the SpeX data. Note that since the fitting
+process is iterative, choices of the initial guess parameters do
+not strongly affect the final result. The end products of this
+conversion process are flux densities reported in units of W
+cm−2 μm−1, which match the units of our simple thermal
+model output. The final NEOWISE data for QE2 are shown in
+Figure 2. We show the data with both 1σ and 3σ uncertainties.
+2.3. Radar Shape Model
+As part of our investigation into the limitations of simple
+thermal models, we compare the results of our NEATM-like
+models to many other data sources and models, including radar
+images and a radar shape model. The radar image is a direct
+measurement of the size that only depends on the viewing
+geometry and the speed of light. A spheroidal object, such as
+QE2, shows a radius in radar range at nearly all aspects and is a
+robust size estimate. We compare the radar size to sizes derived
+from our NEATM-like model, based on the magnitude and
+albedo. We emphasize that this information is not used as an
+input of our NEATM-like model and is only used to compare
+with our NEATM-like model results.
+The radar shape model for QE2 is described by Springmann
+et al. (2014). The model is constructed using observations from
+the Arecibo Observatory and Goldstone. Data used were
+collected between 2013 May 31 and June 9, during QE2ʼs close
+approach to Earth. These radar images are used to derive a
+shape model as described in Magri et al. (2011). A nonlinear
+iterative process is used to adjust synthetic radar images to
+match the observations by minimizing the difference between
+them. This process is described in detail in several papers for
+other objects (Magri et al. 2011; Nolan et al. 2013). The shape
+model of QE2 is preliminary, and the complete analysis is
+beyond the scope of this paper. However, the derived diameter
+of the principal axes of QE2 is robust and reliable as a
+comparison to values obtained here. This analysis gives a
+diameter for QE2 of 3.2 ± 0.3 km and a diameter of the
+secondary of 800 ± 80 m. QE2 is spheroidal, with a few
+dominant surface features.
+Springmann
+et
+al.
+(2014)
+find
+a
+rotation
+rate
+of
+4.749 ± 0.002 hr for QE2 and two possible pole solutions,
+both of which are prograde. One of these solutions, which we
+refer to as the A solution, places most of the “bumpy”
+topography of QE2 in the northern hemisphere. This solution
+has a pole position of λ = 119° and β = 55°, where λ is the
+ecliptic pole longitude and β is the ecliptic pole latitude. The
+second solution, which we refer to as the B solution, places the
+“bumpy” topography partially in the southern hemisphere. This
+solution has a pole position of λ = 158° and β = 41°. Both
+solutions are shown in Figure 3.
+3. NEATM-like Model
+The simple thermal model we use to fit the data is based on
+the
+Standard
+Thermal
+Model
+(Lebofsky
+et
+al.
+1986;
+Lebofsky & Spencer 1989) and NEATM (Harris 1998). Our
+Figure 3. Sky views of QE2 on 2013 July 10 that show the radar shape model from Springmann et al. (2014). The arrows indicate the pole and spin direction. Left: the
+A solution with a pole position of λ = 119° and β = 55°. Right: the B solution with a pole position of λ = 158° and β = 41°.
+Table 2
+List of Individual NEOWISE Observations Used to Obtain the Single
+Averaged NEOWISE Data Set
+Date
+Midtime
+m1 (mag)
+σ1 (mag)
+m2 (mag)
+σ2 (mag)
+2017 Jun 30
+19:32:22
+16.768
+0.468
+13.609
+0.136
+2017 Jun 30
+22:41:03
+16.256
+0.253
+13.844
+0.158
+2017 Jul 01
+03:23:49
+16.625
+0.392
+13.944
+0.172
+2017 Jul 01
+06:32:19
+16.966
+0.535
+13.658
+0.131
+2017 Jul 01
+15:58:00
+16.449
+0.338
+14.193
+0.292
+2017 Jul 01
+19:06:30
+16.927
+0.522
+13.694
+0.135
+2017 Jul 01
+22:15:00
+16.557
+0.476
+13.801
+0.161
+2017 Jul 02
+01:23:41
+16.539
+0.375
+13.770
+0.124
+2017 Jul 01
+10:51:35
+16.528
+0.092
+13.758
+0.051
+Note. Midtime is the midtime of observation in UTC time. m1 and m2 are the
+NEOWISE reported magnitudes for W1 (effective wavelength 3.4 μm) and W2
+(effective wavelength 4.6 μm), respectively. σ1 and σ2 are the NEOWISE
+reported magnitude uncertainties for W1 and W2, respectively. The last row is
+the averaged observation.
+6
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+model is a variation of these models that we call our NEATM-
+like model (Howell et al. 2018). Like these models, for a given
+set of asteroid parameters, our NEATM-like model produces a
+theoretical thermal emission spectrum of the object that can be
+fit to any subset of the visible to near-IR spectra of an asteroid.
+However, our model also utilizes a simple incorporation of the
+rotation rate of the object that allows us to model the thermal
+inertia. The thermal inertia is a measurement of how well the
+object’s surface retains heat energy from the Sun and is
+measured in J m−2 s−1/2 K−1 (hereafter referred to as TIU for
+thermal inertia units). By determining the thermal inertia, in
+combination with the rotation rate, our NEATM-like model is
+able to account for differences across the day and night sides of
+an object. Thus, when incorporating many different observa-
+tions of a single object, taken at different viewing geometries,
+we are able to model how changes in thermal inertia affect the
+thermal emission of an object. Overall, this incorporation
+allows us to get a more robust picture of the properties of the
+object. We note that other than this addition, this model is
+functionally similar to the standard NEATM model.
+In addition to incorporating these parameters, our model also
+makes the typical assumption of a spherical shape for the
+asteroid. It also assumes subsolar and subobserver points on the
+asteroid’s equator and prograde rotation at a fixed rotation rate.
+(The NEATM-like model does not account for shape effects,
+and the radar-derived shape model of Springmann et al. 2014 is
+only used to compare to the NEATM-like model results to
+investigate the limitations of the NEATM-like model.) The
+model also incorporates a free-floating beaming parameter—a
+scaling factor between the observed and predicted flux from the
+asteroid. This factor accounts for additional effects not included
+in the model, such as surface roughness, deviations from a
+spherical shape, local shadowing, and nonzero obliquity. The
+beaming parameter generally ranges between ∼0.5 and 2.0,
+with higher values usually occurring at higher phase angles or
+for more irregularly shaped asteroids.
+Overall, our model includes three free-floating parameters:
+the visual geometric albedo, thermal inertia, and beaming
+parameter. The output of each run is a model spectrum of the
+asteroid, based on the input parameters, for each combination
+of the free-floating parameters. Thus, identifying best-fit
+parameters requires inspecting the model results and making
+direct comparisons to the data.
+For a given object, the consistency of these fit parameters
+can be checked by comparing the results to thermal infrared
+data collected across multiple nights that represent a range of
+viewing geometries. This is important because many combina-
+tions of albedo, thermal inertia, and beaming parameter can fit
+any individual observation. By comparing model results for a
+single object to data taken at multiple different viewing
+geometries of that object, we can thus identify consistent values
+of albedo and thermal inertia that fit every observation,
+breaking degeneracies in the solution. The beaming parameter
+is allowed to vary, as it is expected to change in value across
+each observation. Thus, across multiple different viewing
+geometries, only a tight range of albedo and thermal inertia
+values will fit every observation. This is true even when the
+beaming parameter is allowed to vary, as more extreme
+deviations in albedo or thermal inertia would require increas-
+ingly extreme values of the beaming parameter to fit the
+observations, and realistic beaming parameters are generally
+constrained to the range of ∼0.5–2.0 (Delbó et al. 2003). Note
+that these comparisons are done solely to constrain the
+parameter fits of the NEATM-like model and are separate
+from the comparisons done as part of our investigation into the
+limitations of the NEATM-like model (Section 5).
+The fixed model inputs for our NEATM-like model are the
+object's rotation period, a visible-to-near-IR reflectance ratio,
+Earth–object and Sun–object distances, solar phase angle,
+emissivity, and spherical equivalent diameter. For QE2, we use
+a rotation period of 4.749 ± 0.002 hr that was used by a
+previously derived radar shape model (Springmann et al. 2014).
+We also use a spherical equivalent diameter of 3.2 km from the
+same shape model. We note that since the shape of QE2 is very
+close to spherical, the assumption of spherical shape by the
+NEATM-like model is a very good assumption. The visible-to-
+near-IR reflectance ratio is estimated to be 1.127 using our
+SpeX prism spectra and a visible spectrum obtained by Hicks
+et al. (2013). This is a color correction factor used to relate the
+visible albedo to the near-infrared albedo at 1.6 μm, chosen as
+the normalization wavelength of the spectra. Earth–object and
+Sun–object distances, as well as solar phase angle, are
+calculated for each observation using JPL Horizons8 based
+on the midtime of observation for each data set. These values
+are listed in Table 1. The emissivity is set to 0.9.
+4. NEATM-like Model Results
+We generate NEATM-like models for each of our normal-
+ized flux SpeX data sets and our single absolute photometry
+NEOWISE data set. Models are generated across a wide range
+of albedos, thermal inertias, and beaming parameters. Models
+are then compared to the data using an objective function to
+constrain QE2ʼs properties. For any given data set, models of
+varying parameters change monotonically (Figure 4). These
+models are sorted by calculating a reduced χ2 between the
+model and the data. When performing this calculation, we only
+consider data points between 3.00 and 4.05 μm, as this is the
+region of strongest thermal emission without significant
+overlap with atmospheric water vapor lines. For the NEOWISE
+data set, both NEOWISE data points are used.
+It is important to note that the reduced χ2 value we calculate
+is not a formal χ2, as it does not reach a minimum at unity and
+does not go up by a value of 1 when the model is 1σ away from
+the data. This is because the uncertainties in the data are
+dominated by systematic effects, not statistical errors. The data
+points are not independent, as they are strongly correlated in
+wavelength and are affected by changing effects such as
+atmospheric conditions on different days, viewing geometry,
+and rotational changes of the asteroid. As a result, this
+calculation can be used to sort the goodness of fit of models for
+a given data set but cannot be used to compare models across
+data sets. Thus, for each data set, we use this method to identify
+the range of albedos and thermal inertias that produce models
+that lie within the 1σ uncertainties of the data. Figure 5 shows
+the variation in models that were accepted to fit the data for
+each data set. (Note that for the NEOWISE data we also
+examine the models that fit within the 3σ uncertainties. This
+range is also shown for the NEOWISE data.) Any models
+within the shown region are considered to fit the data. All other
+models for the given data set are discarded, as they are poor fits
+to the data.
+8
+https://ssd.jpl.nasa.gov/horizons/
+7
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+For each data set, we then have a range of albedos and
+thermal inertias that can be said to fit that given data set. These
+individual fit spaces are shown in Figure 6. (For the NEOWISE
+data, we show the models that fit the 1σ uncertainties.) Overall,
+we have 21 such data sets: 20 data sets spread across six nights
+of IRTF SpeX observations, and 1 set of NEOWISE data. To
+identify the range of albedos and thermal inertias that describe
+QE2 overall, we then search for the region of overlap between
+each of these 21 different model sets. These results are shown
+in Figure 7. There is a clear section in the parameter space of
+models that fit nearly every data set. We define this region as
+the best-fit space.
+All models within this space are consistent with the SpeX
+data, but do not fit the NEOWISE data with 1σ uncertainties.
+We then examine models that fit the NEOWISE data with 3σ
+uncertainties, and find that all models within the best-fit space
+are consistent with the NEOWISE data. This could be because
+the NEOWISE observations were taken at a much higher Sun–
+object distance than the SpeX data. As a result, QE2 was much
+colder at the time of these observations which may be
+introducing complexities to the thermal emission that our
+simple thermal model is not able to capture. Such effects may
+be better understood using a more complex thermophysical
+model, however a full investigation of this effect is beyond the
+scope of this work.
+Overall, our analysis gives best-fit ranges of 0.05–0.10 for
+the visual geometric albedo and 0–425 TIU for the thermal
+inertia. Note that there is a correlation such that higher thermal
+inertias require lower albedos. Results are summarized in
+Table 3.
+In general, we find a preference for lower beaming
+parameters of ∼0.55–0.80. Beaming parameter results are
+shown in Figure 8. We remind the reader that we expect the
+beaming parameter to change across observations, and so we
+do not attempt to fit for a single overall value of the beaming
+parameter. These values are calculated by taking the best-fit
+beaming parameter value for a fixed albedo of 0.07 and a fixed
+thermal inertia of 150 TIU. These values are chosen because
+they are near the center of the best-fit region. The NEOWISE
+beaming parameters are calculated using the 3σ uncertainties as
+they are the results consistent with the SpeX data. As expected,
+the beaming parameter is generally higher for higher phase
+angles. The exceptions to this trend are July 10 and the
+NEOWISE data, both of which have substantially greater rH
+and Δ values than the other nights. These larger distances also
+explain the noisier data observed on July 10.
+5. Limits of the NEATM-like Model
+In calculating our best-fit model ranges, we compared our
+model results across many data sets taken at different viewing
+geometries of QE2 (Figure 1). These comparisons have
+allowed us to place tighter constraints on our modeled albedo
+and thermal inertia than would be possible with single
+observations. These albedos and thermal inertias can then be
+compared to results from more complex thermophysical
+models, radar data, and other observations to identify how
+accurately the NEATM-like model was able to constrain the
+properties of QE2. Our model results also provide us with a
+range of beaming parameter values that change as a function of
+viewing geometry. Analyzing these changes in beaming
+parameter can allow us to identify the unmodeled factors
+limiting the accuracy of our NEATM-like model. Overall, by
+comparing our model results to previous studies of QE2, we
+can gain insight into the limitations of simple thermal models
+as applied to a single object. In the subsections below we walk
+through comparisons of our simple thermal model results to
+various other models and data sets. For each comparison, we
+discuss in what ways our simple thermal model results differ
+and discuss implications for the factors affecting the uncertain-
+ties of simple thermal model results.
+5.1. Albedo, Size, and H-magnitude
+Our modeled visual geometric albedo for QE2 of 0.05–0.10
+is higher than but overlaps with previously published values of
+0.03 0.02
+0.03
+-
++
+(Moskovitz et al. 2017) and 0.04 ± 0.01 (Fieber-
+Beyer et al. 2020). We can use our modeled albedo, in
+combination with a radar-derived size, to estimate QE2ʼs H-
+magnitude. This is given by the relationship
+⎛
+⎝
+⎞
+⎠
+H
+p
+D
+5 log
+1329 km ,
+3
+10
+( )
+= -
+where p is the albedo and D is the object diameter in kilometers
+(Pravec & Harris 2007, Equation (3)). Using the diameter of
+3.2 ± 0.3 km given by Springmann et al. (2014) and our
+modeled albedo range of 0.05–0.10, we get an H-magnitude of
+15.4–16.6. This value is lower than (but partially overlaps with)
+previously given H-magnitude values of 16.4 (Trilling et al.
+2010) and 17.3 (Moskovitz et al. 2017) for QE2.
+However, the radar shape model constrains the diameter with
+high accuracy. The radar-derived shape can be considered a
+true constraint on QE2ʼs size, as size can be measured directly
+from a radar image (Ostro 1985). Figure 9 shows a radar image
+of QE2 taken by the Arecibo Telescope on 2013 June 10. The
+vertical extent of the image shows distance from the observer to
+the terminator of the object. Thus, the resolution of the pixels,
+combined with knowledge of the speed of light, directly gives
+the object’s radius. In this image, QE2 covers 210 pixels in the
+vertical extent at 7.5 m pixel−1, giving an apparent radius of
+1575 m or a diameter of 3.15 km. However, using an H-
+magnitude of 17.3 and albedos of 0.05–0.10 gives a diameter
+Figure 4. A range of NEATM-like models compared to one of our SpeX data
+sets. As either the albedo or thermal inertia changes monotonically, the models
+correspondingly change monotonically across the data. This property allows us
+to identify a range of models that fit the data and is typical to all of our data
+sets. All models that fall within the 1σ error bars of the data would be
+considered good fits to the data. As such, in this case only the pV = 0.06 and
+Γ = 100 TIU model would be considered a good fit. Models shown here all
+have η = 0.86. Changes in beaming parameter can also monotonically affect
+how the models fit to the data.
+8
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+Variation in NEATM-like Models
+125
+十
+30 May A
+100
+pv =0.03,  =100
+Normalized Flux
+pv =0.06, I =100
+pv =0.09, I =100
+75
+pv =0.06, I =0
+王
+pv =0.06, I =200
+王
+王
+王
+正
+50
+25
+3.2
+3.7
+4.2
+Wavelength (microns)Figure 5. The variation in NEATM-like models that were accepted to fit the data for each data set. Any models within the shaded region are considered to fit the data.
+All other models for the given data set are discarded, as they are poor fits to the data. An objective function is used to identify which models fall within the shown
+region (Section 4). (a–f) SpeX data. The y-axis is normalized flux. The spectra are offset for clarity. (g) NEOWISE data in absolute flux density. Note that the
+NEOWISE data are plotted over a different wavelength range. For the NEOWISE data we examine models that fit within both the 1σ and 3σ uncertainties. Both
+regions are shown.
+9
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+250
+A)
+B)
+200
+xn
+2013 Jun 02
+2013 May 30
+ABCD
+ABC
+Normali
+100
+50
+0
+250
+C)
+D)
+2013 Jun 08
+2013 Jun 15
+A
+BCDE
+A
+Normal
+B
+100
+50
+0
+250
+E)
+F)
+200
+lux
+2013 Jul 10
+2013 Jun 18
+十+
+ABCDE
++ A
+100
+50
+0
+2.7
+3.2
+3.7
+4.2
+3.2
+3.7
+4.2
+Wavelength (microns)
+Wavelength (microns)
+1e-20
+G)
+wn
+9e-21
+8e-21
+2017 Jul 01
+7e-21
+6e-21
+1g
+3g
+5e-21
+4e-21
+3e-21
+2e-21
+1e-21
+0e+00
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+Wavelength (microns)Figure 6. Reduced χ2 maps for each of the spectra as fit by our simple, NEATM-like model. Warmer colors mean higher values (worse fits), and cooler colors mean
+lower values (better fits). Note that different max values are used for different spectra, as the reduced χ2 are not directly comparable across different spectra
+(Section 4). Each χ2 map is equivalent to showing the range of models that fit a given data set.The fit space of the NEOWISE data corresponds to the 1σ uncertainties
+10
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+0.18
+0.16
+X
+30 May A
+30 May B
+7.00
+30 May C
+6
+0.14
+5.25
+0.12
+3.50
+1.75
+0.00
+0.06
+0.04
+0.02
+0.18
+0.16
+X
+02 Jun A
+02 Jun B
+02 Jun C
+0.14
+2
+0.12
+0.06
+0.04
+0.02
+0.18
+0.16
+x
+02 Jun D
+08 Jun A
+08 Jun B
+0.14
+6
+4
+0.12
+2
+2
+0
+0.06
+0.04
+0.02
+0.18
+0.16
+X
+08 Jun C
+08 Jun D
+08 Jun E
+0.14
+3
+0.12
+2
+0 0.08
+0.06
+0.04
+0.02
+0.18
+0.16
+x
+15 Jun A
+15 Jun B
+18 Jun A
+6
+6
+12
+0.14
+4
+0.12
+2
+?
+0.06
+0.04
+0.02
+0.18
+0.16
+10 Jul A
+10 Jul B
+10 Jul C
+0.14
+3
+0.12
+2
+0.06
+0.04
+0.02
+0.18
+0.16
+10 Jul D
+10 Jul E
+3.00
+0.14
+5.25
+.25
+0.12
+3.50
+.50
+1.75
+0.75
+0.00
+1.2
+0.00
+0.9
+0.6
+0.06
+0.3
+0.04
+NEOWISE
+0.0
+0.02
+0 50 100 150 200 250 300 350 400 450 500 550 050 100 150 200 250 300 350 400 450 500 550 050 100 150 200 250 300 350 400 450 500 550
+Thermal Inertia (TIU)
+Thermal Inertia (TIU)
+Thermal Inertia (TIU)between 1.5 and 2.1 km, well outside of the 1σ errors of the
+radar measurement.
+We investigate this unusually large discrepancy in the H-
+magnitude by looking at existing observations. Using an H-
+magnitude value and an assumed G value, we can calculate
+predicted apparent magnitudes. These predicted apparent
+magnitudes can then be compared to observed apparent
+magnitudes reported to the Minor Planet Center (MPC).9
+Ephemeris values are calculated for QE2 using JPL Horizons10
+at 1-day intervals throughout 2013. We then calculate predicted
+apparent magnitudes for the H-magnitude consistent with the
+radar-determined
+size and our modeled albedo, the H-
+magnitude used by Moskovitz et al. (2017), and a range of G
+values from 0 to 0.15. This was done following the procedure
+in Bowell et al. (1989). These predicted apparent magnitudes
+are then compared to all the apparent magnitudes listed in the
+MPC. The results are shown in Figure 10. We see that H-
+magnitudes of neither 16.0 nor 17.3 perfectly match the data,
+but instead provide an upper and lower bound, respectively.
+However, we note that an H-magnitude of 16.0 appears to
+provide a more reasonable fit than an H-magnitude of 17.3.
+So what could be causing these H-magnitude differences?
+One possible explanation is related to the G parameter. The G
+parameter is often assumed to be 0.15 and is not fitted directly.
+Figure 10 shows that for H = 16.0 lower G values fit better,
+while for H = 17.3 higher G values fit better. For QE2, we
+would expect a lower G value, as lower G values are generally
+preferred
+for
+low-albedo
+objects
+owing
+to
+the
+smaller
+opposition effect. However, we note that the differences do
+not exceed ∼0.5 mag and thus cannot fully explain the
+discrepancy.
+Another possible explanation is related to color effects;
+however, the color of QE2 is very close to solar, and thus this is
+also unlikely to be a large factor in this case. The discrepancy
+could also be due to the secondary contributing to the
+magnitude. Using the radar shape model (Springmann et al.
+2014), we can calculate the effective diameter of the combined
+primary and secondary to be 3.3 ± 0.3 km. Using our modeled
+albedo range, this gives an H-magnitude difference of only
+∼0.1 and thus is an ignorable contribution to the uncertainty.
+Therefore, none of these effects by themselves can fully explain
+the observed differences. Overall, given the accuracy of the
+radar measurement of the diameter and our tightly constrained
+albedo range, the true H-magnitude cannot be as high as 17.3.
+It is therefore likely that a better estimate of the H-magnitude
+lies somewhere in between 16.0 and 17.3. Furthermore, our
+analysis shows that higher H-magnitudes and thus lower
+albedos are
+likely favored
+for QE2, potentially
+further
+constraining the results from our simple thermal model.
+5.2. Wavelength Range of Observations
+We can also analyze our results by leveraging the large
+wavelength range of our observations. Our observations span
+0.8–4.1 μm, and thus we are able to observe both the thermally
+dominated region of the spectra (3.0 μm) and the thermal tail
+(∼2.0–2.5 μm). We are therefore able to compare our model
+fits to both regions. This is notable because many studies (e.g.,
+Moskovitz et al. 2017) rely only on the tail region. We show
+this comparison for a selection of our data sets in Figure 11.
+We find that in nearly all cases the models that best fit the
+thermally dominated region also fit the tail region. However,
+for some dates (such as some data sets for 2013 July 10), an
+albedo increase of ∼0.02 relative to the model that fits the
+thermally dominated region is required to fit the tail region.
+This implies that QE2 may have an inhomogeneous surface and
+that we may be observinglocal thermal variations. Such
+variations could impart a wavelength-dependent change in the
+flux, thus creating the observed discrepancy. Another possibi-
+lity is that some other effect, such as surface roughness, that our
+NEATM-like model does not account for may be causing this
+mismatch. This result is important because it shows the dangers
+of relying on only a limited spectral region to derive surface
+properties such as albedo.
+5.3. Surface Topography
+The potential effects of a surface inhomogeneity can be
+investigated by comparing our NEATM-like model results to
+results from a more complex thermal model. In addition to our
+NEATM-like models, we generate models using SHERMAN.
+SHERMAN is a more complex thermophysical model that
+Figure 7. Final best-fit space for the visible geometric albedo and thermal
+inertia for QE2 using our simple, NEATM-like model. The color of the points
+represents the number of data sets that are fit by the associated parameter
+values. A cooler color means that the given parameters are consistent with more
+data sets. White indicates models consistent with £ 2 data sets. The black line
+outlines the region of best fit. This region corresponds to the region of overlap
+between all the individual model ranges found to fit each individual data set
+(Section 4). There is a correlation such that higher thermal inertias require
+lower albedos. All models were run with the same fixed model inputs listed in
+Section 3 and using ephemeris inputs listed in Table 1. This figure is generated
+using the results from the 1σ uncertainties on the NEOWISE data.
+Table 3
+Best-fit Model Ranges for the Three Free-floating Parameters of Our NEATM-
+like Model
+Parameter
+Range
+Albedo
+0.05–0.10
+Thermal inertia
+0–425 TIU
+Beaming parameter
+∼0.55–0.80
+Note. Albedo is visual geometric albedo. We expect the albedo and thermal
+inertia to be consistent across all data sets, and thus the ranges given represent
+the uncertainty in our model results. However, we expect the range of
+acceptable beaming parameters to change across observations, and thus the
+range given represents the range of values observed across all data sets.
+9
+https://minorplanetcenter.net/
+10 https://ssd.jpl.nasa.gov/horizons/
+11
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+Number of Data Sets Fit per Mode
+Geometric Albedo
+0.18
+0.16
+Number
+0.14
+20
+0.12
+12
+0.10
+3
+0.08
+0.06
+Visual
+0.04
+0.02
+0.00
+0
+50100150200250300350400450500550
+Thermal Inertia (TIU)takes account of the object’s shape and that can separate the
+effects of obliquity and self-shadowing. (For a full description
+of SHERMAN, see Magri et al. 2018.) We give SHERMAN
+the radar-derived shape model of QE2 (Springmann et al.
+2014), as well as our SpeX thermal infrared data. We also input
+a reflectance spectrum from our prism data, as well as a Hapke
+Figure 8. Plot of fitted beaming parameters as a function of solar phase angle adjusted so that 0° corresponds to QE2ʼs minimum phase angle during its close approach
+to Earth. We also compare our beaming parameters to those found by Moskovitz et al. (2017). Note the introduction of negative phase angles to differentiate between
+observations taken before (positive values) and after (negative values) opposition. The error bars represent the range of beaming parameters. The range is calculated by
+identifying models that fit the data with fixed albedo and thermal inertia (Section 4). Moskovitz et al. (2017) values are taken from their Figure 3. We see that our data
+exhibit roughly the same trend where the data overlap, but that our beaming values are significantly offset from the Moskovitz et al. (2017) values.
+Figure 9. Radar image of QE2 taken by the Arecibo Telescope on 2013 June 10. The vertical extent of the image shows distance from the observer to the terminator of
+the object. The horizontal extent shows Doppler shift, with blueshift to redshift going left to right. The resolution of the pixels, combined with knowledge of the speed
+of light, directly gives the object’s radius. In this image, QE2 covers 210 pixels in the vertical extent at 7.5 m pixel−1, giving an apparent radius of 1575 m or a
+diameter of 3.15 km.
+12
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+Beaming Parameter vs. Solar Phase Angle
+2.0
+1.9
+1.8
+1.7
+Moskovitz et al. (2017)
+1.6
+This Work
+May11
+1.5
+1.4
+1.3
+n
+1.2
+May 30巫
+Jun15
+巫 Jun 08
+1.1
+Jul5
+Jun 02
+1.0
+0.9
+0.8
+Jun18Jun15
+May 30 巫
+0.7
+Jul 10巫
+巫 Jun 08
+巫 Jun 02
+0.6
+NEOWISE 巫
+0.5
+-30
+-20
+-10
+0
+10
+20
+30
+40
+50
+60
+Adiusted αscattering function. SHERMAN has three free-floating para-
+meters: visual geometric albedo, thermal inertia, and crater
+fraction. The crater fraction is a proxy for surface roughness
+and describes the fraction of each model facet covered with
+hemispherical craters, following the method of Lagerros
+(1998). SHERMAN outputs a modeled thermal spectrum that
+we then compare with our thermal infrared data.
+SHERMAN is a forward model, so we generate many
+models across different values of the free-floating parameters to
+match to our data. Some preliminary model results are shown
+in Figure 12. We find that an albedo of 0.053, thermal inertia of
+200 TIU, and crater fraction of 70% can roughly match the
+data. These values are also consistent with the results of the
+NEATM-like model.
+The SHERMAN results also show that the topography of
+QE2 is affecting the thermal emission.Using SHERMAN, we
+run models using both possible pole solutions. The results
+show slight differences in the model fits to the data between
+these solutions, with a clear preference for the B solution,
+implying that these features are most likely located in QE2ʼs
+southern hemisphere (Figure 12). Thus, topography is likely
+playing a role for QE2 and is likely affecting the uncertainties
+in the simple thermal model results. Furthermore, topography
+may be one of the effects being captured by variations in our
+NEATM-like model’s beaming parameter.
+5.4. Beaming Parameter Trends
+The NEATM-like model’s beaming parameter is a scaling
+factor that accounts for additional effects not included in the
+model. As such, we can analyze the trend in our measured
+beaming parameters across each night of observation to
+understand the limitations of our NEATM-like model. We find
+beaming parameters that range from 0.54 to 0.78. These values
+therefore differ significantly from the value of η = 1.2 predicted
+by Harris (1998) for NEAs. Our modeled beaming parameters
+are instead much closer to the η ≈ 0.75 value predicted by
+Lebofsky et al. (1986) for main belt objects. Since the beaming
+parameter accounts for additional factors not incorporated into
+the NEATM-like model, we can use these differences to
+identify potential properties affecting QE2ʼs thermal emission.
+QE2 is a particularly good target for this analysis owing to its
+extremely spherical shape. Therefore, shape effects are likely a
+very small contributor to changes in the beaming parameter.
+One potential method for investigating beaming parameters
+is by looking for trends as a function of solar phase angle.
+Moskovitz et al. (2017) previously applied this method to QE2.
+Using beaming parameter as a proxy for thermal emission,
+Moskovitz et al. (2017) identified QE2 as a prograde rotator.
+We investigate this trend by showing the phase angle for QE2,
+which has a minimum value of 17°.1 on June 3, along with the
+fitted beaming parameters for the best-fit NEATM models for
+each night. We compare our results to those found by
+Moskovitz et al. (2017) in Figure 8.
+We find that our beaming parameter values do exhibit
+roughly the same trend as the Moskovitz et al. (2017) data but
+are significantly offset from the Moskovitz et al. (2017) data.
+We find much lower beaming parameter values than the
+Moskovitz et al. (2017) values of ∼1.1–1.4. We also find a
+range of thermal inertias that is overlapping with, but lower
+than their estimated range of ∼200–400 TIU done by
+comparing their NEATM results to more complex models.
+This is not unexpected, as our beaming parameter has been
+Figure 10. Plot of predicted apparent magnitudes for QE2 compared to all magnitudes reported to the MPC. All observations are from 2013 during QE2ʼs close
+approach to Earth. The predicted apparent magnitudes were calculated using ephemeris from JPL Horizons at 1-day intervals throughout 2013. We used H = 16.0 (a
+value from our modeled H-magnitude range) and H = 17.3 (the H-magnitude from Moskovitz et al. 2017), as well as a range of G values. We see that a lower H-
+magnitude, more consistent with our modeled range, agrees with the data for low G values.
+13
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+Predicted Apparent Magnitudes Compared to MPC
+Observations of QE2 in 2013
+10
+H,G
+MPc Observations
+12
+H=16, G=0
+H=16, G=0.05
+H=16, G=0.15
+H=17.3, G=0
+H=17.3, G=0.05
+14
+H=17.3, G=0.15
+18
+20
+Jan
+Mar
+Jun
+Sep
+Dec
+Timeseparated from the thermal inertia. The Moskovitz et al. (2017)
+beaming parameter must account for all the effects of thermal
+inertia, as they do not model thermal inertia explicitly, unlike
+our NEATM-like model, which does incorporate thermal
+inertia.
+Another possible explanation for why we observe different
+beaming parameters is because of our expanded wavelength
+range (Section 5.2). We incorporate data up to 4.05 μm in our
+NEATM-like model, while Moskovitz et al. (2017) only
+incorporate data up to 2.5 μm. As shown in Figure 11,
+Figure 11. Plot of NEATM-like models with varying visual geometric albedos across a selected range of dates. The data sets shown for May 30 and June 15 are the
+“A” data sets. All models shown have thermal inertia and beaming parameters that are within the best-fit ranges for the given date. Each row is a different data set. The
+left panels show the tail region, and the right panels show the thermally dominated region. We see that for July 10 A the models that fit the thermally dominated region
+do not fit the tail region and vice versa. An increase in albedo of ∼0.02 is required to fit the tail region for July 10 A. This is indicative of some kind of surface
+inhomogeneity.
+14
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+4
+125
+Normalized Flux
+100
+3
+75
+30 May
+30 May
+pv =0.07
+pv =0.07
+50
+25
+0
+0
+125
+4
+100
+3
+Normalized Flux
+L
+75
++ 15 Jun
+15 Jun
+Normali
+pv =0.08
+pv =0.08
+50
+25
+0
+125
+4
+100
+3
+Normalized Flux
+L
+10 Jul A
+ 10 Jul A
+pv =0.07
+75
+pv =0.07
+pv =0.08
+pv =0.08
+Normali
+pv =0.09
+pv =0.09
+50
+25
+0
+0
+125
+4
+100
+3
+FI
+Normalized 
+Normalized 
+75
+10 Jul D
+12
+10 Jul D
+pv =0.09
+pv =0.09
+50
+25
+0
+2.0
+2.2
+2.4
+2.6
+2.8
+3.0
+3.2
+3.7
+4.2
+Wavelength (microns)
+Wavelength (microns)mismatches in model fits between the thermally dominated
+region and tail region of the spectra are possible. We check this
+by comparing the Moskovitz et al. (2017) fits to our data at
+longer wavelengths (Figure 13). As expected, we see that
+although the Moskovitz et al. (2017) models fit the tail region,
+they do not fit the thermally dominated region.
+The differences in measured beaming parameters could also
+be related to the illumination geometry of QE2. The technique
+used by Moskovitz et al. (2017) relies on assuming that the
+observations of the asteroids were made with equatorial
+illumination and thus may not be as robust when viewing an
+object with a different illumination geometry. (Moskovitz et al.
+2017 also recognize this possibility.) Although the observations
+of QE2 are made at low sub-Earth latitudes, it is possible that
+the discrepancy in the beaming parameters could arise from
+high subsolar latitudes. For QE2 these can range from ∼30° to
+∼45° for the A pole solution or from ∼10° to ∼15° for the B
+pole solution. Thus, because QE2 is not being observed
+looking directly at its equator, this means that self-shadowing
+from topographical features on the asteroid’s surface is likely to
+be important. Even for the more equatorial illuminated B pole
+solution, self-shadowing could still be playing a significant
+role, as QE2 does not have an equatorial ridge and thus still has
+topographical variation at the equator. This agrees with our
+SHERMAN results that show the importance of topography on
+QE2, which may be contributing to observed temperature
+differences (Section 5.3). Thus, this may further explain why
+our beaming parameter results differ from those of Moskovitz
+et al. (2017).
+6. Summary and Conclusions
+We present simple thermal model fits using our NEATM-
+like model for the NEA (285263) 1998 QE2. Furthermore,
+we compare these model results to more complex thermo-
+physical models, radar data, and other existing analyses of
+QE2 to understand the key factors affecting the uncertainties
+in simple thermal model results. For our simple thermal
+model fits, QE2 was observed with the SpeX instrument on
+the NASA IRTF on six nights in 2013, representing a range
+of viewing and illumination geometries. Additional data were
+acquired by the NEOWISE spacecraft in 2017. A visual
+geometric albedo between 0.05 and 0.10 and thermal inertia
+between 0 and 425 TIU are found to be consistent with all six
+nights of SpeX data. These results are also consistent with the
+NEOWISE absolute photometry at the 3σ level. These
+constraints are more robust than they would be using
+NEOWISE observations alone, due to the larger uncertainties
+on absolute photometry. The general model agreement with
+both
+absolute
+flux
+and
+normalized
+flux
+measurements
+increases our confidence in our model results, while also
+allowing us to benefit from the smaller uncertainties on
+normalized flux data. This is possible because of our
+incorporation of data representing a range of viewing
+geometries. As a result, we are able to break degeneracies
+in model results based on a single night of observations.
+In order to constrain the limits of simple thermal models as
+applied to a single object, we compare our results to more
+complex thermophysical models and previous observations.
+We find that our modeled albedo values are higher than but
+overlap with previously published values (Moskovitz et al.
+2017; Fieber-Beyer et al. 2020) and are consistent with results
+from the complex thermophysical model SHERMAN. We also
+identify a discrepancy in the resulting H-magnitude value when
+using the radar-derived size measurement (Springmann et al.
+2014). Based on the tight constraints we place on QE2ʼs albedo
+and the tighter constraints Springmann et al. (2014) place on
+QE2ʼs diameter, we believe that the true H-magnitude value
+must be brighter than current measurements suggest. As a
+result, the true albedo is likely toward the lower end of the
+range we identify using our NEATM-like model.
+We also leverage the wide wavelength range of our data set
+to compare our best-fit model results to both the tail region and
+thermally dominated region of our spectra. We find that for
+some dates, although our models fit the thermally dominated
+region well, they require a higher albedo to fit the tail region.
+This highlights the need to incorporate data across a wide
+wavelength range when modeling asteroid surface properties.
+We posit that these differences may be due tolocal thermal
+variations, but a full investigation is beyond the scope of
+this work.
+In addition to these discrepancies, we also find differences
+between our modeled beaming parameters and existing models.
+The most likely source of these differences may be the
+orientation of QE2 and wavelength range of data used.
+Observing these differences has also allowed us to infer that
+topography may play a significant role in determining the
+thermal emission of QE2. Thus, in this case, the inability to
+model self-shadowing effects from topographical variations
+may be a key limiting aspect of the simple thermal models.
+Furthermore, this analysis again shows the importance of
+incorporating data from a wide wavelength range when
+working with simple thermal models.
+Overall, our work has demonstrated our ability to place
+tighter constraints on the results of simple thermal models by
+comparing
+data
+taken
+across
+multiple
+different
+viewing
+geometries. By combining normalized flux with absolute
+photometry, we are able to place tighter constraints than would
+be possible with absolute photometry alone. Finally, we are
+able to place some constraints on the limits of simple thermal
+models as applied to single objects, finding that topography,
+viewing geometry, and the wavelength range of data used can
+all affect simple thermal model results. This work is important
+Figure 12. SHERMAN model results for June 8 and July 10 using both the A
+and B pole solutions. All models have a visual geometric albedo of 0.053,
+thermal inertia of 200 TIU, and crater fraction of 70%. We see a clear
+preference for the B pole solution in the June 8 data and a slight preference for
+the B pole solution in the July 10 data. Thus, we see that QE2ʼs topography
+may be playing a role in shaping its thermal emission. We also note that the
+albedo and thermal inertia are consistent with our NEATM-like model.
+15
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+SHERMAN Models for o8 Jun
+and 10 Ju
+125
+Normalized Flux
+08 Jun
+100
+08 Jun - A
+08 Jun - B
+75
+ 10 Jul
+10 Jul- A
+10 Jul - B
+50
+25
+0
+2.7
+3.2
+3.7
+4.2
+Wavelength (microns)for diagnosing cases (such as QE2) where more detailed
+analysis of an object may be required to fully understand its
+properties.
+Being able to extract more information from simple thermal
+models, like our NEATM-like model, will be critical as we
+move into the future of large survey missions such as LSST and
+NEO Surveyor. The large data volumes produced by these
+missions will necessitate the use of simple models to make full
+use of the data. Using these data as efficiently as possible will
+require further insights into the limitations of simple thermal
+models. As this work shows, although these models are reliable
+for statistical measurements of large groups of objects, the
+results
+for
+individual
+objects
+may
+be
+subject
+to
+great
+uncertainties. Addressing these issues will therefore allow us
+to make full use of these models and gain even greater insights
+into fields such as planet formation, asteroid dynamics, and
+planetary defense.
+This work was partially funded by the NASA YORPD
+program (NASA grant 80NSSC21K0658) and NSF AST
+1856411. S.A.M. was supported by the University of Arizona,
+Lunar and Planetary Laboratory, Lieutenant Colonel Kenneth
+Rondo Carson and Virginia Bryan Carson Graduate Fellow-
+ship. This material is based on work supported by the National
+Science Foundation Graduate Research Fellowship Program
+under grant No. DGE-2137419. Any opinions, findings, and
+conclusions or recommendations expressed in this material are
+those of the author(s) and do not necessarily reflect the views of
+the National Science Foundation. S.E.M. was supported by
+NASA’s Near-Earth Object Observations Program through
+grant 80NSSC19K0523.
+ORCID iDs
+Samuel A. Myers
+https://orcid.org/0000-0001-8500-6601
+Ellen S. Howell
+https://orcid.org/0000-0002-7683-5843
+Figure 13. Plot of best-fit models from Moskovitz et al. (2017) compared to our longer-wavelength data. These models are generated using our simple, NEATM-like
+model. All data sets shown are the “A” data set for the given date. All models shown have zero thermal inertia and albedos of 0.03, as per the Moskovitz et al. (2017)
+fits. The shown η values correspond to the ranges reported for each date by Moskovitz et al. (2017). The left panels show the tail region, and the right panels show the
+thermally dominated region. We see that the models fit the tail region well, as expected. However, we note that these models do not fit the thermally dominated region.
+This discrepancy may explain why we find different modeled beaming parameters than Moskovitz et al. (2017).
+16
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+
+125
+4
+100
+3
+Normalized Flux
+30 May
+30 May
+n=1.10
+n =1.10
+Normalized |
+n =1.15
+75
+n =1.15
+n =1.20
+n
+=1.20
+2
+50
+25
+0
+0
+125
+4
+100
+Normalized Flux 
+Normalized 
+02 Jun
+75
+02 Jun
+n =1.05
+n =1.05
+2
+n =1.10
+n =1.10
+n =1.15
+50
+王全
+25
+0
+0
+125
+4
+100
+Normalized Flux 
+Normalized
+15 Jun
+15 Jun
+75
+n =1.10
+n =1.10
+乡乡乡乡多
+n =1.15
+n =1.15
+n =1.20
+n =1.20
+50
+25
+0
+2.0
+2.2
+2.4
+2.6
+2.8
+3.0
+3.2
+3.7
+4.2
+Wavelength (microns
+Wavelength (microns)Christopher Magri
+https://orcid.org/0000-0002-2200-4622
+Ronald J. Vervack, Jr.
+https://orcid.org/0000-0002-
+8227-9564
+Yanga R. Fernández
+https://orcid.org/0000-0003-1156-9721
+Sean E. Marshall
+https://orcid.org/0000-0002-8144-7570
+Patrick A. Taylor
+https://orcid.org/0000-0002-2493-943X
+References
+Arai, T., Yoshida, F., Hong, P., et al. 2020, LPSC, 51, 2924
+Belton, M., Veverka, J., Thomas, P., et al. 1992, Sci, 257, 1647
+Belton, M. J., Chapman, C. R., Klaasen, K. P., et al. 1996, Icar, 120, 1
+Bowell, E., Hapke, B., Domingue, D., et al. 1989, in Asteroids II, ed.
+R. P. Binzel, T. Gehrels, & M. S. Matthews (Tucson, AZ: Univ. Arizona
+Press), 524
+Cushing, M. C., Vacca, W. D., & Rayner, J. T. 2004, PASP, 116, 362
+Cutri, R., Wright, E., Conrow, T., et al. 2012, Explanatory Supplement to the
+WISE All-Sky Data Release Products, NASA/JPL
+Delbó, M., Harris, A. W., Binzel, R. P., Pravec, P., & Davies, J. K. 2003, Icar,
+166, 116
+Dollfus, A. 1971, IAU Colloq. Proc. 12, Physical Studies of Minor Planets
+(Cambridge: Cambridge Univ. Press), 25
+Ďurech, J., Vokrouhlicky`, D., Baransky, A., et al. 2012, A&A, 547, A10
+Fieber-Beyer, S. K., Kareta, T., Reddy, V., & Gaffey, M. J. 2020, Icar, 347,
+113807
+Harris, A. W. 1998, Icar, 131, 291
+Hicks, M., Lawrence, K., Chesley, S., et al. 2013, ATel, 5132, 1
+Hinkle, M., Howell, E., Fernández, Y., et al. 2022, Icar, 382, 114939
+Howell, E., Magri, C., Vervack, R., Jr., et al. 2018, Icar, 303, 220
+Howell, E., Nolan, M., Lejoly, C., et al. 2020, AAS/DPS Meeting, 52, 409.02
+Howell, E. S., Vervack, R., Jr., Nolan, M., et al. 2012, AAS/DPS Meeting, 44,
+110.07
+Hudson, R. S., & Ostro, S. J. 1994, Sci, 263, 940
+Jones, J. 2018, PhD thesis, Univ. Central Florida
+Jurić, M., Ivezić, Ž., Lupton, R. H., et al. 2002, AJ, 124, 1776
+Lagerros, J. S. V. 1998, A&A, 332, 1123
+Lauretta, D., DellaGiustina, D., Bennett, C., et al. 2019, Natur, 568, 55
+Lebofsky, L. A., & Spencer, J. R. 1989, in Asteroids II, ed. R. P. Binzel,
+T. Gehrels, & M. S. Matthews (Tucson, AZ: Univ. Arizona Press), 128
+Lebofsky, L. A., Sykes, M. V., Tedesco, E. F., et al. 1986, Icar, 68, 239
+Magri, C., Howell, E. S., Nolan, M. C., et al. 2011, Icar, 214, 210
+Magri, C., Howell, E. S., Vervack, R. J., Jr., et al. 2018, Icar, 303, 203
+Magri, C., Ostro, S. J., Scheeres, D. J., et al. 2007, Icar, 186, 152
+Mainzer, A., Bauer, J., Cutri, R., et al. 2014, ApJ, 792, 30
+Mainzer, A., Bauer, J., Grav, T., et al. 2011a, ApJ, 731, 53
+Mainzer, A., Grav, T., Bauer, J., et al. 2011b, ApJ, 743, 156
+Marchis, F., Kaasalainen, M., Hom, E. F. Y., et al. 2006, Icar, 185, 39
+Marchis, F., & Vega, D. 2014, AGUFM, 2014, P43F–08
+Marshall, S. E., Howell, E. S., Magri, C., et al. 2017, Icar, 292, 22
+Masiero, J. R., Mainzer, A., Bauer, J., et al. 2021, PSJ, 2, 162
+Masiero, J. R., Wright, E., & Mainzer, A. 2019, AJ, 158, 97
+Millis, R. L., & Dunham, D. W. 1989, in Asteroids II, ed. R. P. Binzel,
+T. Gehrels, & M. S. Matthews (Tucson, AZ: Univ. Arizona Press), 148
+Morrison, D., & Teller, E. 1995, in Hazards Due to Comets and Asteroids, ed.
+T. Gehrels, M. S. Matthews, & A. Schumann (Tucson, AZ: Univ. Arizona
+Press), 1135
+Moskovitz, N. A., Polishook, D., DeMeo, F. E., et al. 2017, Icar, 284, 97
+Nolan, M. C., Magri, C., Howell, E. S., et al. 2013, Icar, 226, 629
+Ostro, S. J. 1985, PASP, 97, 877
+Pravec, P., & Harris, A. W. 2007, Icar, 190, 250
+Rayner, J., Toomey, D., Onaka, P., et al. 2003, PASP, 115, 362
+Springmann, A., Taylor, P., Howell, E., et al. 2014, LPSC, 45, 1313
+Taylor, P. A., Howell, E., Nolan, M., et al. 2014, in Asteroids, Comets,
+Meteors, ed. K. Muinonen et al. (Helsinki) 524
+Taylor, P. A., Rivera-Valentín, E., & Aponte-Hernandez, B. 2019, EPSC-DPS
+Joint Meeting, 51, 689
+Trilling, D. E., Mueller, M., Hora, J., et al. 2010, AJ, 140, 770
+Vereš, P., Jedicke, R., Fitzsimmons, A., et al. 2015, Icar, 261, 34
+Veverka, J., Robinson, M., Thomas, P., et al. 2000, Sci, 289, 2088
+Wright, E. L., Eisenhardt, P. R., Mainzer, A. K., et al. 2010, AJ, 140, 1868
+17
+The Planetary Science Journal, 4:5 (17pp), 2023 January
+Myers et al.
+

4tE2T4oBgHgl3EQf6ghP/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:aef272c1677ba1436c01820944549ba651aad355905d74920720e7ab5a904b01
+size 5767213

6dE4T4oBgHgl3EQf1w1z/content/2301.05293v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:546ad3e3679b77d1560012c50ba1916a28131baa28beb091990bf95cf991c97f
+size 1214683

6dE4T4oBgHgl3EQf1w1z/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d7e763d90b5db3fb88749f0f9577e58b42fe5148ea832e662ef3bfa110671bbf
+size 3014701

6dE4T4oBgHgl3EQf1w1z/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:dc6e2c6eebbc3ccf6b0e6cb88e28c8e0699f4590666a4d43c380460f257aea80
+size 110783

6tFJT4oBgHgl3EQflyzE/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e4fb770bfa9060b3da96b40ebdcab24c074b76086689896a010b112e2316c0de
+size 4784173

79E1T4oBgHgl3EQf7gUu/content/2301.03534v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:0431aee017607322495e3c3e40b2c880e0723b61a07ea001177fb45304031dc8
+size 1716904

79E1T4oBgHgl3EQf7gUu/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e34c881b3315283487b02d1443cefcf2636c7bdc34ae1e332062bd5cf3e2c71f
+size 165819

79E2T4oBgHgl3EQflQc0/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6d36b1fb15f89760121f141881dc1110a93a513e8f5a8941343726e1f275a395
+size 1638445

89E4T4oBgHgl3EQf3A3O/content/2301.05303v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:529667f0b58a410d71f60f811a8d41a521c9fa65a7ddc6ab5901154ba1ad27c4
+size 576700

89E4T4oBgHgl3EQf3A3O/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:75f7fdb5b560c9945a61d5f2b323e8161a87e0e7f4133a3670e07f43e7f4bc7b
+size 3670061

89E4T4oBgHgl3EQf3A3O/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:96c5050e58a642ac2e0a0ebc6ca12ea2219d626df2b1b5ed8a1a101befcf0a5c
+size 143484

8NFQT4oBgHgl3EQfHzW5/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:646920da3bcaacb31e70a098d459d16f0ecb08ac954638dbf3dffe5cea77ad6d
+size 7667757

8tE1T4oBgHgl3EQf7wU5/content/tmp_files/2301.03537v1.pdf.txt

@@ -0,0 +1,2016 @@
+1
+TinyVers: A Tiny Versatile System-on-chip with
+State-Retentive eMRAM for ML Inference at the
+Extreme Edge
+Vikram Jain, Sebastian Giraldo, Jaro De Roose, Linyan Mei, Bert Boons, and Marian Verhelst
+Abstract—Extreme edge devices or Internet-of-thing nodes
+require both ultra-low power always-on processing as well as
+the ability to do on-demand sampling and processing. Moreover,
+support for IoT applications like voice recognition, machine
+monitoring, etc., requires the ability to execute a wide range
+of ML workloads. This brings challenges in hardware design to
+build flexible processors operating in ultra-low power regime.
+This paper presents TinyVers, a tiny versatile ultra-low power
+ML system-on-chip to enable enhanced intelligence at the Ex-
+treme Edge. TinyVers exploits dataflow reconfiguration to enable
+multi-modal support and aggressive on-chip power management
+for duty-cycling to enable smart sensing applications. The SoC
+combines a RISC-V host processor, a 17 TOPS/W dataflow
+reconfigurable ML accelerator, a 1.7 µW deep sleep wake-up
+controller, and an eMRAM for boot code and ML parameter
+retention. The SoC can perform up to 17.6 GOPS while achieving
+a power consumption range from 1.7 µW-20 mW. Multiple ML
+workloads aimed for diverse applications are mapped on the SoC
+to showcase its flexibility and efficiency. All the models achieve
+1-2 TOPS/W of energy efficiency with power consumption below
+230 µW in continuous operation. In a duty-cycling use case for
+machine monitoring, this power is reduced to below 10 µW.
+Index Terms—Extreme edge, tinyML, machine learning accel-
+erators, ultra-low power, system-on-chip.
+I. INTRODUCTION
+E
+Xtreme edge devices [1] or Internet-of-Things (IoT)
+nodes mostly perform non-vision tasks and can achieve
+good accuracy, even with small and lightweight neural network
+(NN) models [2]. This is in contrast to more traditional tasks
+designed for processing image data and contain millions to
+billions of parameters and operations with high hardware re-
+source demands. Consider the Google voice assistant as an ex-
+ample, which needs only 14 kilo bytes (kB) of NN parameters
+to run a keyword-spotting application on edge devices [3]. The
+insight that not all applications require maximum accuracy,
+large and complex NN models, has resulted in a new paradigm
+of ML application development, called tinyML or ML at the
+extreme edge [4]. This trend, at its core, has been driven by the
+V. Jain, L. Mei, and M. Verhelst are with the Department of Electrical
+Engineering - MICAS, KU Leuven, Belgium.
+S. Giraldo was with the Department of Electrical Engineering - MICAS,
+KU Leuven, Belgium. He is now with B12 Consulting, Belgium.
+J. De Roose and B. Boons were with the Department of Electrical
+Engineering - MICAS, KU Leuven, Belgium. They are now with Magics
+Technologies, Belgium.
+© 2023 IEEE. Personal use of this material is permitted. Permission from
+IEEE must be obtained for all other uses, in any current or future media,
+including reprinting/republishing this material for advertising or promotional
+purposes, creating new collective works, for resale or redistribution to servers
+or lists, or reuse of any copyrighted component of this work in other works.
+requirements imposed by battery-operated, performance- and
+power-constrained IoT nodes. Most IoT sensor nodes consist
+of a microcontroller unit (MCU) with a subset of sensors, a
+memory for storing acquired data, a CPU and a wireless data
+transceiver. The presence of these MCUs for data collection
+provides opportunities to process data very close to the sensor
+when the NN model is small, and avoids the high penalty of
+raw data transmission to more powerful edge or cloud units.
+Yet, this local ML processing, brings several new chal-
+lenges: 1.) As these nodes are battery-operated, the system is
+typically severely power or energy constrained requiring ultra-
+low power operation, with the ability to idle. 2.) the MCU,
+moreover, has limited compute power and memory space,
+resulting in a critical trade-off between model size, execution
+performance and hardware complexity; 3.) despite the need
+for efficiency, the system should also be flexible enough
+to support different classes of NN models across different
+applications, and 4.) it should have a small footprint. Several
+hardware for ML have been proposed in the recent literature
+and can be divided into three main categories: 1) extremely
+specialized edgeML accelerators designed for ultra-low power
+operation with little to no flexibility at low performance [5]–
+[8], 2) multi-modal edgeML accelerators providing medium
+level of flexibility with high performance at medium to high
+power consumption [9]–[13], and, 3) commercial-off-the-shelf
+(COTS) MCUs delivering higher flexibility but at low perfor-
+mance and medium power consumption [14]–[16]. Most of
+these hardware designs do not meet all the requirements of an
+extreme edge device. An exception is Vega [17] which presents
+a complete SoC, however, the specialized accelerator of Vega
+does not have the flexibility to handle all DNN workloads.
+Thus, a new class of flexible ultra-low power (ULP) platforms
+towards extreme edge deployment is needed.
+In this context, this work presents TinyVers [18], a highly
+adaptive SoC platform which significantly enhances the trade-
+off between energy efficiency and flexibility needed in extreme
+edge devices, through the use of: A.) a RISC-V proces-
+sor extended with a flexible ML accelerator (FlexML) with
+dataflow reconfiguration supporting diverse ML workloads and
+support for efficient zero-skipping in block structured sparsity
+and deconvolution; B.) an embedded magnetoresistive random
+access memory (eMRAM) for non-volatile storage enabling
+standalone operation with efficient power-down (or idling);
+C.) a programmable wake-up controller (WuC) supporting
+different power-on and idle modes to enable both always-on
+inference as well as on-demand and duty-cycled smart sensing
+arXiv:2301.03537v1  [cs.AR]  9 Jan 2023
+
+2
+and computation used in typical tinyML IoT applications. The
+SoC provides users flexibility not only in mapping diverse
+ML workloads for diverse tinyML applications, but also in
+supporting various use cases such as duty-cycling and smart
+sensing. We demonstrate TinyVers’ capabilities and improve-
+ments over state-of-the-art (SotA) on diverse applications in
+machine monitoring, anomaly detection, audio signal analysis,
+and image classification through the use of both deep learning
+as well as traditional ML workloads.
+The rest of the paper is organized as follows. The basics of
+ML compute kernels is introduced in Section II. Section III
+discusses the architecture overview of TinyVers, followed by
+Section IV providing further details of the FlexML accelerator.
+Section V provides details on how the software stack for
+ML deployment on TinyVers is undertaken. Subsequently,
+Section VI presents the experimental results of mapping
+different workloads and application use cases. Finally, Sec-
+tion VII compares TinyVers’ performance with related works
+and Section VIII concludes the paper.
+II. ALGORITHMIC BACKGROUND
+ML applications heavily exploit deep neural networks
+(DNN) with traditional convolutional (CNN) and fully con-
+nected (FC) layers. However, a plethora of new NN layer
+topologies are emerging. Some examples of these are the use
+of temporal convolutional networks (TCN) used in audio tasks
+like keyword spotting [19]–[21], or auto-encoders (AE) using
+convolution and deconvolution pairs in machine monitoring
+and anomaly detection tasks [22]–[24]. Morever, also machine
+learning models not relying on neural network layers are still
+used in extreme edge IoT nodes, such as support vector ma-
+chines (SVM) [25] used in novelty and anomaly detection ap-
+plications. The execution efficiency of all these workloads can
+can be improved with orders of magnitude when deployed on
+specialized accelerators. Yet, the wide variety in the compute
+kernels of interest complicates their efficient mapping on a
+single hardware platform. The following subsections deal with
+the different ML operation characteristics, their categorization
+into mathematical operations, and their hardware implications.
+A. Convolution and Dense Operation
+Convolutional and dense layers are the most common com-
+pute kernels used in DNNs and they can be decomposed
+into matrix-matrix multiplication (MMM) and matrix-vector
+multiplication (MVM) resp.. These two matrix operations can
+be represented mathematically as nested for loops as shown in
+Fig. 1. Most ML compute kernels can be categorized into one
+of these two mathematical operations, with some special layers
+requiring extra hardware changes. One such kernel is the TCN
+layer which can be represented as a 1D CNN and requires extra
+support for programmable dilation which is similar to strides
+in a convolution. Recurrent neural networks (RNN) like long
+short-term memory (LSTM) and gated recurrent unit (GRU)
+can be decomposed to MVM with need for extra hardware
+for activation functions. These hardware changes would be
+discussed further in Section IV.
+C
+C
+IY
+FY
+OY
+OX
+IX
+FX
+K
+*
+K
+Convolution Operation = MMM
+Input FMAP
+Weights
+Output FMAP
+C
+C
+*
+K
+Dense Operation = MVM
+Input FMAP
+Weights
+Output FMAP
+TCN
+CNN
+GAN
+AE
+LSTM
+FC
+SVM
+K
+for(y=0 to Y-1); for each output row
+ for(x=0 to X/N-1); for each output column
+  for(k=0 to K/N-1); for each output channel
+   for(c=0 to C-1); for each input channel
+    for(fy=0 to Fy-1); for each filter row
+     for(fx=0 to Fx-1); for each filter column
+      o[k][x][y] += i[c][x+fx][y+fy]*w[k][c][fx][fy]
+for(k=0 to K/N-1); for each output channel
+ for(c=0 to C/N-1); for each input channel
+  o[k] += i[c]*w[k][c]
+PE
+Spatial Unrolling X
+Temporal
+ Unrolling
+Spatial Unrolling Y
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+Fig. 1. Different ML models and their mathematical representation in terms
+of MMM and MVM. The nested for loop representation can be mapped onto
+specialized accelerators through spatial and temporal unrolling.
+When mapping MMMs and MVMs on specialized hardware
+accelerators, the nested for loops can be unrolled spatially
+and temporally, which is called dataflow in literature [26].
+On a 2D processing element (PE) array, two for loops can
+be spatially unrolled, i.e., the loops can be parallelized along
+the X and Y dimensions, as shown in Fig. 1. In the rest
+of the paper, this spatial unrolling is represented as (Spatial
+Unrolling X)|(Spatial Unrolling Y). The remaining for loops
+are temporally unrolled, i.e., sequential execution. Depending
+on the available parallelism and available re-usability, the
+spatial unrolling (X and Y) needs to be configurable, to be
+able to efficiently map all workloads, detailed in Section IV-B.
+B. Deconvolution
+Autoencoders used in many machine monitoring applica-
+tions consist of an encoder and a decoder pair, which tries
+to reconstruct the input data. After training on normal data,
+a reconstruction error signals an anomaly in the test data.
+Deconvolution or transposed convolution are used in these
+autoencoders and are built by combining the convolution and
+upsampling into a single operation. Deconvolution can be
+mapped as a convolution (MMM) but needs extra hardware
+to support zero-skipping of input for efficient mapping. Hard-
+ware modification can improve the mapping efficiency of this
+operation, and better exploit its inherent sparsity, as will be
+discussed in Section IV-C.
+C. Support Vector Machines (SVMs)
+SVMs are ML algorithms used for classification and re-
+gression tasks. When classification of input data between
+normal behavior and an anomaly is required, a binary classifier
+called a one-class support vector machine (OC-SVM) can be
+
+3
+used [27], [28]. The decision function of a OC-SVM using the
+radial basis function (RBF) kernel is given by the equation (1).
+For the Laplacian kernel, the L2 norm is replaced by L1 norm.
+f(x) =
+N
+�
+i=0
+αi · exp
+−∥x−svi∥2
+2σ2
+− b
+(1)
+where x is the input vector with length D, sv are the support
+vectors with length D, N is the number of support vectors,
+σ the standard deviation, α the Lagrange multiplier, and b the
+bias. The number of support vectors N, in combination with
+the vector length D, can become large in these workloads,
+making the L1 and L2 norm calculation complex, and their
+deployment can gain orders of magnitude in performance
+when deployed on specialized accelerators. The D and N
+dimensions of the norm operations can be treated similar to
+C and K dimensions of a dense layer (MVM) and can be
+spatially unrolled on the PE array. In addition to unrolling
+the norms, extra hardware to support squaring, subtraction,
+rounding and absolute operation needs to be added to each
+PE. The result of the norm calculation can then be used by a
+CPU core to compute the overall kernel.
+D. Structured Sparsity
+Exploiting sparsity in DNNs can help to reduce the com-
+putational complexity and memory requirements, by skipping
+zeros and compressing the NN parameters. However, random
+pruning or unstructured sparsity tends to be hard to efficiently
+map on hardware and requires special logic for zero-skipping
+and load balancing [29]–[31]. The structure of sparsity (gran-
+ularity of pruning) has high impact on hardware efficiency and
+prediction accuracy. Some works have found that unstructured
+sparsity achieves better prediction accuracy than structured
+sparsity but structured sparsity tends to be more hardware
+amenable and improves computational efficiency [30]. Thus, a
+structured sparse model could be trained with more iterations
+to revert back closer to the same prediction accuracy achieving
+similar overall efficiency/cost. Moreover, more coarse-grained
+sparsity can reduce the additional memory requirements im-
+posed for storing indices of non-sparse data.
+With all of these diverse ML workloads and their charac-
+teristics in mind, a platform which can efficiently map all of
+the above, needs to be designed.
+III. TINYVERS HARDWARE ARCHITECTURE
+TinyVers, as shown in Fig. 2, is a heterogeneous SoC
+consisting of a single core RISC-V processor, a flexible ML
+accelerator called FlexML, a 512 kB shared level-2 (L2)
+SRAM memory, a micro-DMA (uDMA) for data movement
+between peripherals/memory, a 512 kB eMRAM for non-
+volatile storage, and a WuC for power management. The SoC
+development is rooted in the PULPissimo platform [32]. It
+embeds a 2 kB read-only memory (ROM), which acts as the
+first stage boot loader (FSBL) and also controls boot from
+JTAG, external SPI flash or the eMRAM. Two communication
+busses are used: 1.) a logarithmic interconnect, which enables
+a tightly-coupled data memory (TCDM) providing single cycle
+eMRAM
+(512 KB)
+ROM
+Shared Memory L2 (512 kB)
+GPIO UART
+SPI
+I2C
+I2S
+CPI
+JTAG
+SCAN 
+CHAINS
+eMRAM 
+CNTL
+LP Data acq. Memory L2 
+(64 kB)
+TCDM interconnect
+uDMA
+DMA
+Source
+Source
+Sink
+RISC-V
+APB
+WuC (RTC
+& 
+Power FSM)
+2D SIMD
+Array
+8x8
+Weight L1 
+Memory
+Instruction 
+Memory
+Activation L1 
+Memory
+DMA
+Control 
+Registers
+Logic PD
+LP Data 
+Acq. Mem 
+Data Acq. 
+Mem PD
+L1 PD
+UDMA 
+PD
+AON PD
+MRAM 
+PD
+Power 
+Modes
+PD= Power Domain
+Boot
+OFF
+OFF
+OFF
+OFF
+OFF
+OFF
+OFF
+OFF
+OFF
+OFF
+OFF
+OFF
+ON/OFF
+OFF
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+ON
+Active
+Data Acq.
+LP Data Acq.
+Deep Sleep
+% VDD WAKE
+* VDD SCL
+** VDD SRAM
+^^ VDD MRAM
+# VCS MRAM
+ V+ bias
+V- bias
+^^ 
+ 
+# 
+% 
+% 
+Data 
+Mover 
+FSM
+FlexML Accelerator
+* 
+* 
+* 
+* 
+* 
+* 
+* 
+* 
+** 
+** 
+** 
+** 
+** 
+FlexML 
+Control 
+Unit
+Fig. 2. Overview of the complete TinyVers SoC showing the different power
+domains (PD) with their constituting modules and the power modes supported.
+access to the shared L2, and 2.) the APB standard bus, which
+is used for controlling different memory mapped modules.
+The interface between the SoC and FlexML accelerator is
+based on the HWPE framework presented in [33]. Using the
+streamers from [33], data is moved to-and-from the shared L2
+memory with the help of FlexML’s DMA engine which is a
+FSM controlling the data (un)loading of its private memories
+and double buffering operation. Several peripheral interface
+protocols are supported by the SoC including UART, SPI, I2C,
+I2S, and CPI, in addition to having 32 general purpose IOs
+(GPIO). Separate clocks are used for the main core logic,
+the peripheral interfaces, and the always-on domain which
+includes the WuC and the IO pads.
+A. Smart Sensing Modes for TinyML
+IoT tinyML applications typically operate by collecting data
+across a specified time window through an array of sensors,
+after which the collected data can be processed to make
+decisions. In many applications, the time window across which
+the data needs to be collected before processing can start,
+can vary from a few ms to sec. Moreover, during the sensor
+data collection, many modules of the MCU are not used since
+no heavy processing is done yet. This brings opportunities in
+improving power saving in many tinyML applications: during
+data collection, only the modules necessary for moving the
+windowed data from the sensor peripheral interfaces to the
+memory need to remain active, while e.g. the CPU can be
+put to sleep. Furthermore, in applications which work on
+time series data like audio, the memory requirement for the
+windowed data is small (< 64 kB), such that also a large part
+of the main memory of the MCU can be powered-down to
+avoid leakage power of the unused memory section.
+To this end, TinyVers introduces two tinyML optimized
+data acquisition power modes: 1.) ‘Data acq.’ and 2.) ‘LP
+data acq.’, as shown in Fig. 2. The data acq. mode, targeted
+towards applications with large sample data like vision, keeps
+the uDMA module and the complete shared L2 memory (512
+
+4
+Full Active
+Data Acq
+LP Data Acq
+0
+100
+200
+300
+31
+20
+8
+325
+77
+10
+356
+97
+18
+Power(µW)
+Dynamic
+Leakage
+Total
+Fig. 3.
+Power simulation of post-synthesis netlist undertaken in Cadence
+Genus tool for the three power modes. In all the three modes, I2S data is
+collected at a sampling frequency of 44.1 kHz for a window of 2 seconds.
+Full active power reported includes configuration of uDMA by RISC-V core
+and interrupt handling procedure, in addition to data collection.
+Power
+uDMA
+Power
+OFF
+Power
+ON
+Switch
+Power 1
+Switch
+Power 2
+Power
+OFF
+Reset
+Isolate
+Clk
+enable
+Power
+ON
+Top level FSM
+Bottom level FSM
+Power
+Logic & 
+L1
+Power
+MRAM
+Power
+L2 &
+L2 udma
+Fig. 4. Flow diagram showing the hierarchical FSM used in the WuC.
+kB) powered up. In contrast to that, the LP data acq. mode
+only keeps part of the shared L2 memory (64 kB) powered up,
+in addition to the uDMA. This mode is specifically targeted
+towards applications which needs time series and audio data
+like keyword spotting, machine monitoring, biosignal analysis,
+etc. Fig. 3 shows an estimation of the power saving that can
+be achieved when moving from a full active mode to the
+two tinyML sensing modes, with almost 3.5× improvement
+between the full active and data acq. modes and 5.5× between
+data acq. and LP data acq. modes.
+B. Power Management
+Aggressive power management is pursued in TinyVers on
+top of standard low power design. The SoC is divided into 6
+switchable power domains and 1 always-on domain (AON),
+as shown in Fig. 2. Each switchable power domain consists
+of multiple power gating switches, which isolate the VDD
+of the power domain from the global VDD supply. These
+power gating switches are controlled by control signals driven
+from the WuC of the AON domain. All interconnect crossings
+between the power domains are equipped with bidirectional
+level shifters and isolation cells, such that the individual supply
+voltages of the domains can be controlled independently.
+The smart WuC is in charge of this power management
+control, relying on a real-time counter (RTC). The counter
+can be programmed by the RISC-V core with millisecond
+granularity. The RISC-V core can instruct the WuC to bring
+the SoC into one of the five supported power modes shown in
+Fig. 2. To this end, the WuC encompasses hierarchical finite-
+state machines (FSM) driven by the RTC, as shown in Fig. 4,
+controlling the power-up and power-down of the complete SoC
+and the different power domains. The top level FSM controls
+the sequence of power-up/down of the different power domains
+and the bottom level FSMs control the fine-grain sequence to
+(de)activate the isolation cells and the power gating switches
+of the individual power domains.
+Emerging memories like ReRAM, MRAM, FeRAM, PCM,
+etc. [34], [35], have shown promise in building cost-effective
+embedded non-volatile memories (NVM) targeting applica-
+tions in edge computing for automotive or industry 4.0. NVM
+memories can be used as the storage space for boot code
+and other parameters that need to be stored. This enables
+two things: 1.) Duty-cycling can be used as a means of
+reducing power consumption in applications which do not
+require always-on operation; and 2.) the SoC does not need
+to go to a central cloud server in order to fetch its boot codes
+and NN parameters when it is power-cycled. Moreover, the
+availability of the NVM embedded on-chip, avoids the high
+energy cost of fetching data from off-chip.
+MRAM promoted as a universal memory, uses magnetic
+polarity to store data in its bitcells [36]. Being non-volatile
+and almost as dense as traditional SRAM, they are a good fit
+for tinyML applications using extreme edge SoCs. With this
+in mind, TinyVers integrates a 512 kB embedded MRAM on-
+chip, enabling extreme power management strategies for smart
+sensing and on-demand computation. In the SoC, the eMRAM
+acts as a non-volatile storage for the boot code that the RISC-
+V needs to wake-up and start processing, and can also store the
+NN parameters of the mapped ML workloads. The eMRAM
+can, finally, also be used as a non-volatile scratchpad space
+for storing windowed data in smart sensing applications. The
+interface between eMRAM and the shared L2 memory uses
+the uDMA unit and the design is based on the work of [17].
+IV. FLEXML ACCELERATOR
+This section firstly describes the architecture overview of the
+FlexML accelerator, followed by the dataflow reconfiguration
+used for flexible mapping, efficient zero-skipping used for
+deconvolution and structured sparsity, and finally the hardware
+for supporting SVM, as briefly discussed in Section II.
+A. FlexML Architecture Overview
+The FlexML accelerator is TinyVers’ specialized, versatile
+hardware accelerator. FlexML is designed to efficiently support
+the large diversity in ML workloads for tinyML applications,
+while exploiting the data reuse present in individual layer
+characteristics. This is achieved through a zero-latency runtime
+dataflow reconfiguration, discussed in Section IV-B. As shown
+in Fig. 5, FlexML encompasses an 8×8 single instruction
+multiple data (SIMD) array of processing elements (PE),
+wherein each processing element consists of a precision-
+scalable multiply-accumulate (MAC) unit with support for INT
+8/4/2 [37], shown in Fig. 6. As a result of the precision-
+scalability, the SIMD array can be reconfigured to be a
+8×8/16/32 array of INT8/4/2 MAC units, resp.. Each PE per-
+forms 1/2/4 MAC operations per cycle based on the selected
+precision (INT8/4/2) and the results are accumulated in a 32-
+bit register with full/partial output stationarity, reducing the
+movement cost of the large bit-width partial sums. The final
+output is passed through a ReLU function (if enabled), fol-
+lowed by re-quantization to the selected precision and written
+
+5
+DMA Engine
+Inst. 
+Mem
+Cntl
+FSM
+Sparsity 
+Mem
+2x2 kB
+Weight 
+Mem (L1)
+2x32 kB
+Adder Trees
+Act 
+Mem (L1)
+2x32 kB
+NLFG
+&
+Max
+Pool
+Input FIFO (L0)
+SIMD PE Array 8x8
+IX
+Layer Type
+ucode 
+Inst.
+K
+Fx
+Fy
+Input 
+pointer
+Weight
+Pointer
+.........
+IY
+C
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+Fig. 5. FlexML accelerator architecture overview with ucode instruction.
+*
+>>
+ABS
+REG
+REG
+Round
+Sub
+ReLU
+Overflow
+Control
+Input 
+Activation
+Output 
+Activation
+1
+0
+0
+0
+Input 
+Weight
+From Neighbor
+PE
+-
++
+8b
+20b
+8b
+16b
+…
+… …
+…
+… …
+unused
+4b
+4b
+Gated
+4b
+4b
+Gated
+12b
+9b
+2b 2b 2b
+…
+… …
+…
+… …
+Gated
+2b
+Gated
+2b
+2b
+2b
+2b
+8b
+6b
+Fig. 6.
+Block diagram of the processing elements used in the flexML
+accelerator, showing the precision-scalable MAC unit and the additional
+hardware to support SVM.
+back to the activation L1. Mixed precision quantization can
+help in improving performance of DNN models when moving
+below 8 bits precision. However, the hardware overhead of
+mixed precision can reduce the overall efficiency of PEs
+due to varying bandwidth and serialized dataflow [38]. Thus,
+FlexML only supports symmetric precision for its weights and
+activation. In addition, a simple shift and ReLU is used for
+normalization of output, which also keeps hardware overhead
+low. In order to maintain accuracy of the models, a hardware
+aware training framework, mentioned in Section V, is used.
+Supporting the SIMD PE array, are private level-1 (L1)
+SRAM based memories for storing both weights (64 kB)
+and activations (64 kB). Both the weight L1 and activation
+L1 are composed of two 32 kB banks operating in a ping-
+pong manner to overlap data writing and reading, improving
+the overall performance. An intermediate memory level L0 is
+provided between the activation L1 and the PE array. This
+L0 memory is a FIFO buffer of size 16×8 bits, used to
+improve data locality when doing shifting window operation
+in convolution. Furthermore, a separate non-linear function
+generator (NLFG) and a max pooling unit are provided. The
+for(y=0 to Y-1); for each output row
+ for(x=0 to X/8-1); for each output column
+  for(k=0 to K/8-1); for each output channel
+   for(c=0 to C-1); for each input channel
+    for(fy=0 to Fy-1); for each filter row
+     for(fx=0 to Fx-1); for each filter column
+     parfor(k=0 to 8-1); spatial unrolled output channel
+     parfor(x=0 to 8-1); spatial unrolled output column
+      o[k][x][y] += i[c][x+fx][y+fy]*w[k][c][fx][fy]
+for(k=0 to K/8-1); for each output channel
+ for(c=0 to C/8-1); for each input channel
+ parfor(k=0 to 8-1); spatial unrolled output channel
+ parfor(c=0 to 8-1); spatial unrolled input channel
+  o[k] += i[c]*w[k][c]
+FIFO
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+C
+Weight Memory
+FIFO
+MMM
+MVM
+Weight Memory 
+OX
+K
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+PE
+K
+Bank 0
+Bank 1
+Bank 3
+Bank 0
+Bank 1
+Bank 3
+Bank 7
+Bank 7
+Fig. 7. Diagram showing the dataflow reconfiguration used to switch from
+OX|K dataflow (left) for MMM to C|K dataflow for MVM. The nested for
+loops below show the addition of parfor loops for the spatial unrolling used.
+NLFG uses LUT-based linear approximation to generate the
+various activation functions (other than ReLU) used in NN
+models such as tanh, sigmoid, etc. To control the dataflow
+and control flow inside the accelerator, a control unit with
+FSMs fetches ucode instructions from the instruction memory,
+decodes the instruction and deploys the relevant layer on the
+PE array by updating the control signals and counters that
+track the workload. The ucode instructions are generated by a
+pseudo-compiler built in python (Section V), and consists of
+CISC-like layerwise long instructions with hyperparameters
+and shown in Fig. 5. The control unit is also extended to
+enable support for efficient zero-skipping of activations in the
+case of deconvolution and zero-skipping of pruned weights in
+conjunction with the sparsity index memories (Section IV-C).
+B. Dataflow Reconfiguration
+In order to efficiently map the diverse set of ML workloads,
+runtime dataflow reconfiguration is supported in the FlexML
+accelerator at no latency overhead. The configurability enables
+efficient mapping of both: 1.) MMMs used for CNN, decon-
+volution and TCN, exploiting both input and weight spatial
+data reuse under an OX|K dataflow with output stationarity,
+and 2.) MVMs used for FC, RNNs and norm calculation
+of SVMs with batch size 1, exploiting the available input
+spatial data reuse under a C|K dataflow with partial output
+stationarity. Multiple previous works have proposed dataflow
+reconfiguration in hardware to optimally map different work-
+loads [39]–[41]. However, these works suffer from large hard-
+ware overhead and latency for diverse dataflow support and
+are not suitable for extreme edge devices. This work limits the
+dataflows to two optimal mapping schemes, thereby, keeping
+hardware and power overhead low. Moreover, none of the prior
+works have looked into mapping of TCN, AE, and SVM on
+the same hardware accelerator. Fig. 7 shows the OX|K (left)
+and C|K dataflow (right) and their hardware implementation,
+resp.. In the OX|K dataflow, the spatial unrolling is applied to
+the OX and the K dimension of the nested for loop, allocating
+the unrolled OX dimension along the columns and the unrolled
+K along the rows of the SIMD PE array of dimension 8×8.
+
+6
+Input FIFO
+Input FIFO
+Input data from activation L1
+Normal operation
+Deconvolution operation
+Input data from activation L1
+0
+0
+0
+0
+0
+0
+0
+0
+Ctrl
+Control Unit
+Fetch instruction and enable 
+deconvolution
+Control the demux and muxes 
+for deconvolution
+Skip rows and columns with all zeros
+Cycle #0: Enable demux to push data to input FIFO
+Cycle #1: Set Ctrl to 0, data sent to PEs -> a 0 b 0 c 0 d 0
+Cycle #2: Set Ctrl to 1, data sent to PEs -> 0 b 0 c 0 d 0 e, 
+Shift 0 into input FIFO
+Cycle #3: Set Ctrl to 0, data sent to PEs -> b 0 c 0 d 0 e 0
+IX
+Input Activation
+Filter
+Deconvolution layer in software
+*Orange represents pruned pixels
+IY
+Fig. 8. Representation of deconvolution layer in software (top left), control
+unit running the zero-skip operation (bottom left), the architectural change
+required on the L0 FIFO to support deconvolution (top right), and cycle by
+cycle operation of the FIFO and PEs (bottom right).
+The rest of the for loops are temporally unrolled as shown
+in the nested for loops in Fig. 7, resulting in an output
+stationary dataflow. Under this dataflow regime, the activation
+L1 memory multicasts input activation data in the vertical
+dimension to the L0 FIFO memory, which fetches 8 words
+in the first cycle followed by single word during the shifting
+window operation, thereby, reducing the memory bandwidth
+and number of memory fetches by utilizing the reuse oppor-
+tunity. The weight L1 memory provides data in the horizontal
+dimension, providing 8 words using 2 internal banks, where
+each word is multi-cast along the row. Due to the output
+stationarity, accumulation continues till the final output is
+generated which are then systolically shifted out vertically to
+the activation L1, requiring 8 cycles to complete the output
+write-back. The input data shifting inside the L0 FIFO is made
+programmable to support the variable dilation used in TCNs
+or variable strides in general.
+The alternative C|K dataflow is used for MVM, as this
+workload cannot utilize the OX|K dataflow efficiently due
+to lack of re-usability of weights. Under this dataflow, the
+C dimension is spatially unrolled along the vertical column
+dimension and the K dimension along the horizontal row
+dimension. The activation L1 memory multicasts 8 words of
+input activation along the vertical dimension, bypassing the
+L0 FIFO memory. With a batch size of 1, no weight reuse
+is available and, thus, each PE needs a new weight every
+cycle. In order to meet this requirement, the weight memory
+utilizes all of its 8 banks to unicast 64 different weight words
+to the PEs. PE rows operate on different input channels (C) of
+the same output channel (K). Hence, once the required MAC
+operations per PE are done, the outputs of PEs of the same
+row are accumulated using an adder tree and one final output
+per row is shifted out to the activation memory.
+C. Efficient Zero-skipping for Deconvolution and Blockwise
+Structured Sparsity
+The FlexML accelerator supports efficient zero-skipping of
+deconvolution workloads. As shown in Fig. 8, the input FIFO
+Sparsity Index Mem 
+storing the sparse nature 
+Sparse 
+block
+1
+0101
+0000
+0000
+3
+C
+0
+1010
+2
+C
+C
+K
+*
+K
+x8
+K
+8
+8
+Blockwise Str. Sparsity for CNN
+Input
+Weight Matrix
+Weights
+Blockwise Str. Sparsity for FC/RNN
+Output
+Control Unit
+Fetch sparsity index for block 1 to 8 from sparsity memory
+Check bit wise, if one present then update counters of 
+control FSM to skip current C
+Fetch next blockwise index and repeat
+*Orange represents pruned pixels
+Fig. 9. Blockwise structured sparsity applied to CNN and dense layers (top),
+control unit operation in tandem with sparsity index memory to support zero-
+skipping (bottom).
+is designed such that when in deconvolution mode, it only
+fetches one set of words and shuffles it with zero padding. The
+control unit skips the rows and columns with zeros that would
+result in redundant computation, resulting in a performance
+gain of up to 2× compared to running deconvolution in
+convolution mode with upsampling.
+TinyVers also supports structured sparsity, more specifically,
+blockwise kernel-level sparsity (2D) for both convolutional
+and dense layers [29], [31]. In this scheme, shown in Fig. 9,
+complete input channels of the filter kernels are pruned with a
+constraint that a block size of 8 filter kernels (K = 8) should
+share the same pruning. The block size is decided by the
+dimension of the PE array and the spatial unrolling of K along
+the horizontal dimension of the 2D PE array. In our case, the
+selected block size makes controlling the dataflow and control
+flow easier. Applying the same channel pruning to all the 8
+filter kernels mapped in parallel on the PE array makes the
+mapping efficiency higher as all the rows can still operate with
+a common control logic, and enables not only energy savings,
+but also throughput benefits. For this, the FlexML accelerator
+consists of specialized sparsity index memories which store the
+bit encoded indices of the pruned channel groups. Fig. 9 shows
+the sparsity index memory and the control flow logic used in
+the control unit. Before every filter kernel block increment, the
+control unit fetches an index memory word and checks the data
+bit-by-bit for sparsity state, as the input channels increment.
+If a sparse channel is detected, the complete computation of
+the channel is skipped, thus, avoiding any zero computation.
+D. Support Vector Machine
+The L1 and L2 norm of OC-SVM requires modification
+of the PEs in order to use the same hardware for mapping
+the workload. As shown in Fig, 6, each PE is extended with
+a subtraction block, absolute unit, rounding unit, and the
+modification of the multiplier to also enable squaring for the
+norm calculation within the PE array. The input data vector x
+and the support vector svi are of dimension D and the number
+of support vectors is N. When used in the C|K dataflow, the
+
+7
+1.00E-01
+L1 MEMORY
+2.5 mm
+2.5 mm
+L2 
+MEM
+RISC-V
+&
+ACCEL
+uDMA
+eMRAM
+WuC
+Fig. 10. Measurement setup and chip microphotograph.
+D dimension of the input data vector of x is unrolled and
+multicasted vertically (C) along the PE array, while the N
+dimension of the support vector svi are unrolled and unicasted
+horizontally (K). The results of the N norm calculations,
+computed in the PEs, are then sent to the shared L2 memory
+where it is then post-processed by the RISC-V core with the
+GNU C in-built exponential function, multiplication with α
+and summation over N to generate the final output shown in
+equation (1).
+V. DEPLOYMENT OF NEURAL NETWORKS ON TINYVERS
+Hardware used for ML applications also requires a user
+programmable full stack that can translate ML algorithms
+directly from existing ML training and inference frameworks
+like Tensorflow, Keras, PyTorch, etc. This makes the quick and
+easy deployment of various ML workloads onto an existing
+hardware possible. A python based pseudo-compiler frame-
+work created for TinyVers taking into account its heterogeneity
+is created. An ML algorithm is first quantized to selected
+precision using the QKeras framework [42] for quantized-
+aware training. The quantization-aware training framework
+takes into consideration the hardware constraints such as
+symmetric quantization and the shift based scaling of output
+in the PEs of the accelerator. The quantized model is then
+passed to a python-based NN compilation which takes in the
+hardware description and provides a set of C-based header
+files for the RISC-V core, consisting of ucode instructions for
+the accelerator, NN parameters and also a golden model for
+verification of the mapped workload.
+VI. CHIP IMPLEMENTATION AND MEASUREMENT
+The TinyVers chip microphotograph shown in Fig. 10 was
+implemented and fabricated in GlobalFoundries 22FDXTM.
+The figure shows the different sub-modules used in the SoC
+and detailed in previous sections. Fig. 10 also shows the lab
+setup used for measurements and benchmarking. The follow-
+ing subsections details the measurements and benchmarking
+done on the SoC for power, energy efficiency and performance.
+A. Peak Performance Analysis
+First, a peak performance analysis is undertaken using a
+single CNN layer with 32 input channels, 32 output channels
+and a 3×3 filter kernel. Selection of the used layer for peak
+@Vdd Mem, Vdd Logic
+0.5
+1.0
+1.5
+2.0
+3.0
+2.5
+3
+6
+9
+12
+18
+15
+5
+Clock Frequency (MHz)
+Peak energy eff. (TOPS/W)
+Throughput (GOPS)
+10
+20
+30
+40
+50
+100
+120
+150
+@0.5, 0.4V
+@0.55, 0.5V
+@0.65, 0.5V
+@0.65, 0.6V
+@0.65, 0.6V
+@0.65, 0.6V
+@0.8, 0.8V
+@0.8, 0.8V
+@0.8, 0.8V
+0.586
+2.47
+2.0
+1.9
+1.85
+1.43
+1.44
+0.833
+0.838
+0.863
+1.17
+2.35
+4.69
+3.52
+5.86
+11.7
+14.1
+17.6
+Fig. 11. Peak performance analysis of CNN3×3 layer.
+5 MHz
+10 MHz
+20 MHz
+30 MHz
+40 MHz
+50 MHz
+100 MHz
+120 MHz
+150 MHz
+0
+5,000
+10,000
+15,000
+20,000
+Power(µW)
+WuC
+L2
+L2uDMA
+L1
+Logic
+DMA
+Mram(P)
+Mram(A)
+Fig. 12. Power breakdown of the peak perf. analysis with CNN3×3. MRAM
+power consumption is negligible as it is OFF in active mode. MRAM(A) and
+MRAM(P) represents MRAM array and MRAM periphery resp..
+performance is driven by the fact that convolutional layers
+with a 3×3 filter kernel are the most commonly used layer
+in modern DNN models. The hyperparameter selection of the
+CNN layer is driven by the constraint of maximum utilization
+of the PE array and the size of the private L1 memories
+of the accelerator. The 8 bit quantized activation and non-
+sparse (structured) weights of the CNN are generated using
+the compiler framework using the Google speech dataset for
+keyword spotting [43] and verified against the golden model
+for functional correctness.
+Fig. 11 plots the peak energy efficiency and the throughput
+with respect to the clock frequency while sweeping the voltage
+supply of the logic and memories for the benchmarked CNN
+layer. For fair comparison with other SotA chips, no body
+biasing is applied. Fig. 12 shows the power breakdown of
+individual modules when running the benchmarking layer. The
+SoC shows a large flexibility in delivered performance ranging
+from high energy efficiency/low throughput of 2.5 TOPS/W,
+586 MOPS when operating at a clock frequency of 5 MHz
+with 0.4 V logic, 0.5 V memories, to low energy efficiency
+/ high throughput of 0.8 TOPS/W, 17.6 GOPS operating at
+150 MHz with 0.8 V logic and memories. This provides a
+large range for extreme edge tinyML applications to operate,
+trading-off between speed and energy efficiency.
+B. Workload Benchmarks
+Using the peak energy efficiency operating point (5 MHz,
+0.4 V logic and 0.5 V memory) from Section VI-A, further
+performance analysis of different synthetic and actual real-
+time benchmarks are evaluated. Table I shows the SoCs
+flexibility through mapping of different ML layers and full
+
+MICAS
+naikraveraels
+ADIGILENT
+二
+店
+K2_VDD_L1K2_0D_L
+ZedBoardLens: E20:X80
+2022/02/038
+TABLE I
+WORKLOAD BENCHMARKS
+Workload
+Acc.
+Power
+(µW)
+Peak
+perf.
+(GOPS)
+Peak
+(effective NZ)
+energy eff.
+(TOPS/W)
+Synthetic
+CNN@8b
+-
+237
+0.586
+2.47(2.47)
+CNN@4b
+-
+197
+1.17
+5.94(5.94)
+CNN@2b
+-
+197
+2.35
+11.9(11.9)
+CNN@8b,
+-
+239
+1.03
+4.31(2.46)
+50% sparse
+CNN@8b,
+-
+212
+3.64
+17.1(2.76)
+87.5% sparse
+FC/RNN/SVM,
+-
+140
+0.116
+0.829(0.829)
+batch=16
+Deconv@8b
+-
+235
+1.36
+5.78(2.49)
+Real-time
+TCN (KWS)
+93.3%∗
+193
+0.204
+1.05(1.05)
+CAE
+-
+209
+0.442
+2.11(1.27)
+ResNet-8
+82%+
+228
+0.267
+1.17(1.17)
+OC-SVM
+-
+129
+0.126
+0.972(0.972)
+∗ 12-class task, baseline=93.46%, + baseline=85%
+2%
+ResNet8
+OC-SVM
+CAE
+TCN
+1%
+31%
+2%
+21%
+41%
+4%
+0%
+29%
+2%
+19%
+44%
+6%
+WuC
+L2
+L2uDMA
+L1
+Logic
+DMA
+Mram(P)
+Mram(A)
+0%
+31%
+18%
+44%
+5%
+1%
+47%
+3%
+15%
+25%
+9%
+Power: 129 μW, 
+Latency: 4.3ms
+Power: 228 μW, 
+Latency: 76ms
+Power: 193 μW, 
+Latency: 11ms
+Power: 209 μW, 
+Latency: 30ms
+Fig. 13. Energy breakdown showing the distribution of measured energy of the
+chip modules for running a single inference of the four real-time workloads
+on FlexML and RISC-V with input data already available in L2 memory.
+The power and latency measurements starts from setting up of accelerator
+parameters by RISC-V, data movement from L2 to L1, inference computations,
+and ends with post processing by RISC-V core. MRAM power consumption
+is negligible as it is OFF in active mode.
+workloads. The CNN layer from Section VI-A is extended and
+measured with different precision and blockwise structured
+sparsity (BSS) levels. When moving to lower precision of INT-
+4 and INT-2, the peak throughput improves by 2× and 4×
+while the peak energy efficiency improves by 2.4× and 4.8×
+resp., achieving a maximum of 11.9 TOPS/W at INT-2. As
+shown in Table I, at 8 bit precision with 50% BSS (16/32
+input channels pruned) the performance improves by around
+1.7× while at 87.5% BSS (28/32 input channels pruned)
+the performance increases by approximately 6.9×. Further
+performance improvement can be gained when moving to
+lower precision, however, low precision combined with high
+BSS levels can cause a large drop in accuracy and, thus, is not
+explored in this benchmarking. Other synthetic benchmarks
+such as FC, RNN, SVM and a deconvolutional layer similar
+to the CNN layer in terms of hyperparameters are explored
+and the results are shown in the table. For the dense layers,
+batching of 16 is used.
+Finally, 4 real-time application benchmarks are used to
+TABLE II
+MEASUREMENT RESULTS OF DIFFERENT LOW POWER MODES.
+Power Mode
+AON
+Freq.
+(kHz)
+Core
+Freq.
+(MHz)
+Power
+(µW)
+Wakeup
+Latency
+(µs)
+Deep Sleep
+33
+-
+1.7
+788
+LP Data acq.∗
+33
+5
+23.6
+788
+Data acq.∗
+33
+5
+67
+788
+∗ @Fs=44.1 kHz
+Wake-up latency (s)
+4
+8
+12
+16
+24
+20
+0.033 
+Clock Frequency (MHz)
+Power (μW)
+1 
+5 
+10 
+788 μs  
+26 μs  
+5.2 μs  
+2.6 μs  
+20 
+40 
+1.3 μs  650 ns  
+1.7 μW  2.1 μW  
+5.8 μW  
+7.8 μW  
+12.7 μW  
+22.8 μW  
+Fig. 14. Deep sleep power-latency-frequency tradeoff.
+show the capabilities of the SoC: 1.) keyword spotting (KWS)
+using TCN model [21], [44] on google speech dataset, 2.)
+continuous machine monitoring with a convolutional auto-
+encoder (CAE) [24] on MIMII dataset [45], 3.) ResNet-8
+image classification on CIFAR-10 used in MLPerfTM tiny
+benchmark [46], and 4.) Novelty detection with OC-SVM [47].
+Table I shows the peak performance characteristics of these
+benchmarks on the SoC, more specifically the RISC-V core
+and FlexML, with 8-bit precision, a single inference, and
+assuming all input data is available in the shared L2 memory.
+For TCN and ResNet-8, hardware-aware quantization was used
+and the energy and performance metrics were measured, while
+for the CAE and OC-SVM workloads, random inputs and
+weights were used. All the 4 workloads can be deployed with
+less than 230 µW of continuous real-time power at peak energy
+efficiency between 1-2 TOPS/W. This means that the SoC can
+provide high level of flexibility in workload mapping at sub-
+mW power to enable truly power efficient tinyML application
+on extreme edge devices. Fig. 13 shows the power breakdown
+of the 4 real-time workloads. For OC-SVM (dense operation),
+the power consumption of memory dominates, due to the lack
+of re-usability of weights leading to more data fetches. On the
+other hand, power breakdown of CNN based workloads (TCN,
+ResNet8 and CAE) shows equal distribution between memory
+and logic as the dataflow exploits maximum re-usability.
+C. Power Management
+Table II shows the measured real-time power of the different
+low power modes of the SoC detailed in Section III-B. In deep
+sleep mode the SoC operates with an AON clock frequency
+of 33 kHz. In this mode, only the AON domain consisting of
+the WuC and the logic controlling the IO pads stays powered
+
+9
+1.00E-06
+1.00E-05
+1.00E-04
+1.00E-03
+1.00E-02
+1.00E-01
+Power (W)
+Time (ms)
+0
+280
+140
+Erase followed by write output 
+to MRAM
+TCN processing (16 batch)
+Boot from MRAM
+I2S LP data acq. (2s window)
+Deep sleep
+Vdd SCL, Mem: 0.55 V
+Vdd AON: 0.7 V
+Core Freq. : 5 MHz
+2000 2140 2280
+Fig. 15.
+Instantaneous power trace showing the KWS application scenario
+with one full period of smart sensing and TCN processing followed by idling.
+ON. The resulting deep sleep power measured is 1.7 µW when
+operating at 0.7 V voltage supply. When compared to the peak
+power measured for the CNN layer, the deep sleep power is
+12,000× lower. The measure latency of waking up the SoC
+from deep sleep mode to active mode is 788 µs. This wake-
+up latency can be traded off to deep sleep power by sweeping
+the AON clock frequency. Fig. 14 plots the this trade-off for
+the measured power and wake-up latency when sweeping the
+AON clock frequency. Applications that need low latency can
+operate the AON clock at 40 MHz to attain a wake-up latency
+of 650 ns at a real-time power of 22.8 µW.
+Table II also shows the measured power for the two tinyML
+optimized power modes of data acq. and LP data acq. These
+power modes are measured with an I2S protocol based win-
+dowed test vector collection with the AON clock frequency at
+33 kHz and the core and peripheral clock frequency at 5 MHz.
+The SoC is programmed to collect I2S audio data through its
+uDMA at a sampling frequency of 44.1 kHz and a sampling
+window of 2 second. The sampling clock is generated by the
+SoC using the 5 MHz clock and lasts for the duration of
+sampling window. The data acq. or LP data acq. mode is then
+initiated and power is measured. The measured power for LP
+data acq. and data acq. is 23.6 µW and 67 µW resp. which is
+850× and 300× reduced power consumption compared to the
+peak power, when the core and peripheral frequencies can be
+dynamically lowered to 5 MHz.
+D. Instantaneous Power Trace
+In order to show the complete end-to-end application de-
+ployable on the SoC and to show the SoC’s full ML func-
+tionality, duty cycling and features of power management,
+two applications are mapped onto the heterogeneous SoC
+with windowed data collection done in the LP data acq.
+mode: keyword spotting with a TCN model operating in
+continuous mode [21]; and a machine monitoring use case with
+a Mel Frequency Energy Coefficient (MFEC) based feature
+extraction with a CAE in duty cycled mode [24].
+1) Keyword-spotting Application: The first application sce-
+nario is the keyword-spotting with TCN model. In this ap-
+plication scenario, audio data from a microphone of window
+1.00E-06
+1.00E-05
+1.00E-04
+1.00E-03
+1.00E-02
+1.00E-01
+Power (W)
+MFEC processing on RISC-V
+Autoencooder processing on FlexML
+Boot from MRAM
+I2S LP data acq. (1s window)
+Deep sleep
+Time (ms)
+Vdd SCL, Mem: 0.55 V
+Vdd AON: 0.7 V
+Core Freq. : 5 MHz
+Fig. 16. Instantaneous power trace showing the machine monitoring appli-
+cation scenario with one period of smart sensing, FE, and CAE processing
+followed by idling.
+size 2 seconds (16 batches) at a sampling frequency of 44.1
+kHz is collected using the I2S peripheral interface protocol,
+the collected data is simultaneously stored in the special L2
+uDMA memory using the SoC’s uDMA with the SoC being
+in the LP data acq. mode. After 2 seconds the SoC wake’s
+up into active mode and the collected data is processed using
+the TCN model from Section VI-B. The output of the TCN
+processing is then stored into the MRAM for future processing
+or transmission while the SoC can either go into deep sleep
+mode or collect new windowed sampling data. Fig. 15 shows
+the complete instantaneous power consumption trace of the
+KWS application scenario. When operating in this duty-cycled
+mode, the average power of the complete application is 173
+µW. The power can be further reduced to 10-20 µW by using
+the deep sleep power mode of the SoC during periods of no
+sensing or computation.
+2) Machine Monitoring Application: Machine monitoring
+used for predictive maintenance is the second application
+scenario selected. In this scenario, I2S peripheral interface
+protocol is used to collect audio data from a microphone with
+window size 1 second at a sampling frequency of 16 kHz.
+The collection of I2S audio data is operated in the LP data
+acq. mode of the SoC. Once the complete windowed data is
+collected, the SoC switches to the active mode in which the
+RISC-V core is used for the MFEC based feature extraction
+followed by running the CAE on the accelerator. Fig. 16
+plots the instantaneous power trace of running the machine
+monitoring application. Unlike the previous application which
+works on raw audio data, the CAE model need pre-processing
+MFEC data. As the MFEC algorithm is not supported on
+the accelerator, it is executed on the RISC-V core with
+INT16 precision instead of INT32 or FP32 to reduce power
+consumption [52]. The power trace plots show that running
+large feature extraction on RISC-V is not energy efficient
+taking large time to complete owing to single core operation.
+The average power for continuous operation remains below
+164 µW, but for this use case, 9.5 µW is consumed with a
+duty cycling of 0.05. The MFEC execution on the RISC-V
+
+10
+TABLE III
+PERFORMANCE COMPARISON WITH STATE-OF-THE-ART.
+[48]
+[17]
+[49]
+TinyVers
+[50]
+[51]
+Extreme Edge SoCs
+edgeML Accelerators
+Technology
+28nm FDSOI
+22FDX
+55nm
+22FDX
+28nm
+65nm
+Die Area (mm2)
+4.5
+12
+10
+6.25
+0.55
+16
+Applications
+IoT GP, DNN,
+IoT GP, DNN,
+IoT GP, DNN,
+IoT GP, DNN+,
+Always-on KWS
+DNN
+NSA
+NSA
+NSA
+Trad. ML, NSA
+Supported
+CNN,
+CNN,
+CNN,
+CNN,
+DSCNN
+CNN,
+ML layers
+FC/RNN
+FC/RNN
+FC/RNN
+FC/RNN, GAN,
+FC/RNN
+AE, TCN, SVM
+Architecture
+1×RI5CY+
+10×RI5CY+
+9×RI5CY
+1×RI5CY+
+DSCNN
+DNN
+ML accel.
+ML accel.
+FlexML accel.
+accel.
+accel.
+SRAM
+464 kB (40 kB)
+128 kB(L1)
+64 kB (L1)
+132 kB (L1)
+2 kB
+256 kB
+(State retentive)
+(16-1600 kB (L2))
+(512 kB (L2))
+(64/512 kB (L2))
+eNVM
+-
+4 MB MRAM
+-
+512 kB MRAM
+-
+-
+Deep sleep power (µW)
+-
+1.7
+3.6
+1.7
+-
+-
+SRAM ret.
+6.4
+2.8-123.7
+30
+23.6-67
+-
+-
+sleep power (µW)
+Int precision (bits)
+8, 16, 32
+8, 16, 32
+8, 16, 32
+2, 4, 8
+8
+1-16
+Supply voltage (V)
+0.45-0.9
+0.5-0.8
+1-1.2
+0.4-0.9
+0.41
+0.63-1.1
+Max frequency (MHz)
+350
+450
+250
+150
+0.04
+200
+Power range
+6.4µW-96mW
+1.7µW-49.4mW
+3.6µW-75mW
+1.7µW-20mW
+0.51µW
+3.2-297mW
+Best ML perf.
+36 GOPS
+32.2 GOPS
+12 GOPS
+17.6 GOPS
+2.3 MOPS+
+691.2 GOPS
+@8b∗
+@8b∗
+@8b∗
+@8b∗∗
+@8b∗∗
+@8b ∗∗
+Best ML eff.
+1.3 TOPS/W@
+1.3 TOPS/W@
+200 GOPS/W@
+2.47 TOPS/W@
+4.5 TOPS/W@
+5.57 TOPS/W,
+@Perf
+2.8 GOPS, 8b∗
+15.6 GOPS, 8b∗
+7 GOPS, 8b∗
+0.58 GOPS, 8b∗∗
+2.3 MOPS, 8b∗∗
+8b∗∗
+11.9 TOPS/W@
+11.6 TOPS/W,
+2.4 GOPS, 2b∗∗
+4b∗∗
++ estimated at 90% utilization of MACs, ∗ Matmul, ∗∗ CNN, 1 MAC = 2 Ops
+can be optimized using special DSP extensions available with
+the PULP libraries, which is left for future work.
+VII. COMPARISON WITH SOTA
+Table III shows the comparison of our SoC with SoTA on
+two fronts: on one hand, comparing with existing extreme edge
+SoCs (left), and on the other hand, with edge ML accelera-
+tors (right). Our SoC has similar or increased flexibility in
+application mapping compared to the extreme edge SoCs on
+the left, with much improved energy efficiency and power.
+TinyVers supports not only the IoT general processing (GP),
+DNNs and near-sensor analytics (NSA) like [17], [48], [49],
+but also DNN+ such as TCN and AE and traditional ML
+like SVM, all at better energy efficiency because of efficient
+mapping. This is evident from the best energy efficiency of
+2.47 TOPS/W for running a CNN layer on TinyVers. The
+energy efficiency is further enhanced to 11.9 TOPS/W when
+the CNN workload is quantized to INT2. Compared to the
+extreme edge SoCs, TinyVers provides support for precision
+scalability and, thus, can take advantage of improved perfor-
+mance using quantization. Furthermore, by utilizing support
+for block structured sparsity, TinyVers can reach a peak
+performance of 17 TOPS/W for an 8-bit CNN layer. This is
+much higher than the efficiencies reported by [17], [48], [49].
+Compared to the edge ML accelerators on the right,
+TinyVers shows much more flexibility at comparable per-
+formance metrics in terms of energy efficiency and power
+consumption. The edgeML accelerators only support a single
+or few models extremely efficiently, but this approach has
+drawbacks in deployment for extreme edge devices. For ex-
+ample, [50] can only perform KWS with depthwise separable
+CNN and its performance is much lower than TinyVers with
+comparable energy efficiency. UNPU [51], can only support
+CNN and FC/RNN layers and also does not have a complete
+standalone SoC, which effects efficiency at the system level.
+Moreover, these edgeML accelerators cannot support any kind
+of duty cycling as they lack power management and retention
+memory support. TinyVers supports the multi-modal require-
+ments of extreme edge devices at relatively similar energy
+efficiencies of the order of TOPS/W. Moreover, it adds the
+possibility of extreme low power idle states for duty-cycling
+use cases to enable < 10µW operation, shown empirically in
+Section VI-D. To summarize, TinyVers brings the best of both
+worlds of extreme edge processors and edgeML accelerators.
+VIII. CONCLUSION
+TinyML applications at the extreme edge needs not only
+heterogeneous SoCs with flexible accelerators to support di-
+verse workloads, but also adaptive power management for
+different duty-cycling operations. Moreover, to enable such
+adaptive power management, the need for embedded non-
+volatile memories arises. TinyVers extends a RISC-V core
+with a flexible ML accelerator supporting a diverse set of
+ML workload mapping in terms of diverse compute kernels,
+different precision and structured sparsity conditions. Further-
+more, the inclusion of a WuC and an eMRAM enables the
+adaptive power management required in many duty-cycling
+use cases. Measurement result shows that the chip can achieve
+an energy efficiency range of 0.8-17 TOPS/W at 0.58 GOPS
+to 17.6 GOPS of throughput. The different low power modes
+enable the chip to achieve power range from 1.7µW-20 mW.
+The application of machine monitoring takes advantage of the
+
+11
+deep sleep mode to consume only 9.5µW of power at a duty
+cycle of 0.05. Thus, TinyVers takes a step towards creating a
+new class of ultra-low power extreme edge SoCs.
+ACKNOWLEDGMENTS
+The authors would like to thank ETHZ for their support
+on PULP platform and GlobalFoundries for 22FDX tapeout
+support. The work has been supported under ISAAC project
+(FOD Economie Belgium Energietransitiefonds (oproep II)) in
+collaboration with Magics Technologies and received funding
+from the Flemish Government (AI Research Program).
+REFERENCES
+[1] J. Portilla, G. Mujica, J.-S. Lee, and T. Riesgo, “The Extreme Edge at
+the Bottom of the Internet of Things: A Review,” IEEE Sensors Journal,
+vol. 19, no. 9, pp. 3179–3190, 2019.
+[2] Y. Zhang, N. Suda, L. Lai, and V. Chandra, “Hello Edge: Keyword
+Spotting on Microcontrollers,” CoRR, vol. abs/1711.07128, 2017.
+[Online]. Available: http://arxiv.org/abs/1711.07128
+[3] P. Warden and D. Situnayake, Tinyml: Machine learning with tensorflow
+lite on arduino and ultra-low-power microcontrollers.
+O’Reilly Media,
+2019.
+[4] TinyML. [Online]. Available: https://www.tinyml.org/
+[5] P. Jokic, E. Azarkhish, R. Cattenoz, E. T¨uretken, L. Benini, and
+S. Emery, “A Sub-mW Dual-Engine ML Inference System-on-Chip for
+Complete End-to-End Face-Analysis at the Edge,” in 2021 Symposium
+on VLSI Circuits, 2021, pp. 1–2.
+[6] P. P. Bernardo, C. Gerum, A. Frischknecht, K. L¨ubeck, and O. Bring-
+mann, “UltraTrail: A Configurable Ultralow-Power TC-ResNet AI
+Accelerator for Efficient Keyword Spotting,” IEEE Transactions on
+Computer-Aided Design of Integrated Circuits and Systems, vol. 39,
+no. 11, pp. 4240–4251, 2020.
+[7] J. S. P. Giraldo, S. Lauwereins, K. Badami, and M. Verhelst, “Vocell: A
+65-nm Speech-Triggered Wake-Up SoC for 10-µW Keyword Spotting
+and Speaker Verification,” IEEE Journal of Solid-State Circuits, vol. 55,
+no. 4, pp. 868–878, 2020.
+[8] J. Giraldo and M. Verhelst, “Laika: A 5uW Programmable LSTM
+Accelerator for Always-on Keyword Spotting in 65nm CMOS,” in
+ESSCIRC 2018 - IEEE 44th European Solid State Circuits Conference
+(ESSCIRC), 2018, pp. 166–169.
+[9] D. Im, G. Park, Z. Li, J. Ryu, S. Kang, D. Han, J. Lee, and H.-J.
+Yoo, “DSPU: A 281.6mW Real-Time Depth Signal Processing Unit
+for Deep Learning-Based Dense RGB-D Data Acquisition with Depth
+Fusion and 3D Bounding Box Extraction in Mobile Platforms,” in 2022
+IEEE International Solid- State Circuits Conference (ISSCC), vol. 65,
+2022, pp. 510–512.
+[10] Y. Ju and J. Gu, “A 65nm Systolic Neural CPU Processor for Com-
+bined Deep Learning and General-Purpose Computing with 95% PE
+Utilization, High Data Locality and Enhanced End-to-End Performance,”
+in 2022 IEEE International Solid- State Circuits Conference (ISSCC),
+vol. 65, 2022, pp. 1–3.
+[11] H. Mo, W. Zhu, W. Hu, G. Wang, Q. Li, A. Li, S. Yin, S. Wei, and L. Liu,
+“9.2 A 28nm 12.1TOPS/W Dual-Mode CNN Processor Using Effective-
+Weight-Based Convolution and Error-Compensation-Based Prediction,”
+in 2021 IEEE International Solid- State Circuits Conference (ISSCC),
+vol. 64, 2021, pp. 146–148.
+[12] B. Moons, R. Uytterhoeven, W. Dehaene, and M. Verhelst, “14.5 en-
+vision: A 0.26-to-10tops/w subword-parallel dynamic-voltage-accuracy-
+frequency-scalable convolutional neural network processor in 28nm
+fdsoi,” in 2017 IEEE International Solid-State Circuits Conference
+(ISSCC).
+IEEE, 2017, pp. 246–247.
+[13] Y.-H. Chen, T. Krishna, J. S. Emer, and V. Sze, “Eyeriss: An energy-
+efficient reconfigurable accelerator for deep convolutional neural net-
+works,” IEEE journal of solid-state circuits, vol. 52, no. 1, pp. 127–138,
+2016.
+[14] GAPuino. [Online]. Available: https://greenwaves-technologies.com/
+product/gapuino/
+[15] Arduino Nano BLE 33. [Online]. Available: https://docs.arduino.cc/
+hardware/nano-33-ble/
+[16] Silicon Labs Thunderboard. [Online]. Available: https://www.silabs.
+com/development-tools/thunderboard/thunderboard-sense-two-kit
+[17] D. Rossi, F. Conti, M. Eggiman, A. D. Mauro, G. Tagliavini, S. Mach,
+M. Guermandi, A. Pullini, I. Loi, J. Chen, E. Flamand, and L. Benini,
+“Vega: A Ten-Core SoC for IoT Endnodes With DNN Acceleration and
+Cognitive Wake-Up From MRAM-Based State-Retentive Sleep Mode,”
+IEEE Journal of Solid-State Circuits, vol. 57, no. 1, pp. 127–139, 2022.
+[18] V. Jain, S. Giraldo, J. D. Roose, B. Boons, L. Mei, and M. Verhelst,
+“TinyVers: A 0.8-17 TOPS/W, 1.7 µW-20 mW, Tiny Versatile System-
+on-chip with State-Retentive eMRAM for Machine Learning Inference
+at the Extreme Edge,” in 2022 IEEE Symposium on VLSI Technology
+and Circuits (VLSI Technology and Circuits), 2022, pp. 20–21.
+[19] S. Bai, J. Z. Kolter, and V. Koltun, “An Empirical Evaluation
+of Generic Convolutional and Recurrent Networks for Sequence
+Modeling,” CoRR, vol. abs/1803.01271, 2018. [Online]. Available:
+http://arxiv.org/abs/1803.01271
+[20] S. Choi, S. Seo, B. Shin, H. Byun, M. Kersner, B. Kim, D. Kim,
+and S. Ha, “Temporal Convolution for Real-time Keyword Spotting
+on Mobile Devices,” CoRR, vol. abs/1904.03814, 2019. [Online].
+Available: http://arxiv.org/abs/1904.03814
+[21] E. A. Ibrahim, J. Huisken, H. Fatemi, and J. Pineda de Gyvez, “Keyword
+Spotting using Time-Domain Features in a Temporal Convolutional
+Network,” in 2019 22nd Euromicro Conference on Digital System
+Design (DSD), 2019, pp. 313–319.
+[22] J. K. Chow, Z. Su, J. Wu, P. S. Tan, X. Mao, and Y.-H. Wang, “Anomaly
+detection of defects on concrete structures with the convolutional autoen-
+coder,” Advanced Engineering Informatics, vol. 45, p. 101105, 2020.
+[23] T. B. Duman, B. Bayram, and G. ˙Ince, “Acoustic anomaly detection us-
+ing convolutional autoencoders in industrial processes,” in International
+Workshop on Soft Computing Models in Industrial and Environmental
+Applications.
+Springer, 2019, pp. 432–442.
+[24] G. Coelho, P. Pereira, L. Matos, A. Ribeiro, E. C. Nunes, A. Ferreira,
+P. Cortez, and A. Pilastri, “Deep Dense and Convolutional Autoencoders
+for Machine Acoustic Anomaly Detection,” in Artificial Intelligence
+Applications and Innovations, I. Maglogiannis, J. Macintyre, and L. Il-
+iadis, Eds.
+Springer International Publishing, 2021, pp. 337–348.
+[25] C. Cortes and V. Vapnik, “Support-vector networks,” Machine learning,
+vol. 20, no. 3, pp. 273–297, 1995.
+[26] L. Mei, P. Houshmand, V. Jain, S. Giraldo, and M. Verhelst, “ZigZag:
+Enlarging Joint Architecture-Mapping Design Space Exploration for
+DNN Accelerators,” IEEE Transactions on Computers, vol. 70, no. 8,
+pp. 1160–1174, 2021.
+[27] Z. Noumir, P. Honeine, and C. Richard, “On simple one-class classifica-
+tion methods,” in 2012 IEEE International Symposium on Information
+Theory Proceedings.
+IEEE, 2012, pp. 2022–2026.
+[28] P. Perera, P. Oza, and V. M. Patel, “One-Class Classification:
+A Survey,” CoRR, vol. abs/2101.03064, 2021. [Online]. Available:
+https://arxiv.org/abs/2101.03064
+[29] H. Mao, S. Han, J. Pool, W. Li, X. Liu, Y. Wang, and W. J. Dally,
+“Exploring the Regularity of Sparse Structure in Convolutional Neural
+Networks,” CoRR, vol. abs/1705.08922, 2017. [Online]. Available:
+http://arxiv.org/abs/1705.08922
+[30] T. Hoefler, D. Alistarh, T. Ben-Nun, N. Dryden, and A. Peste, “Sparsity
+in Deep Learning: Pruning and Growth for Efficient Inference and
+Training in Neural Networks,” J. Mach. Learn. Res., vol. 22, no. 1,
+jan 2021.
+[31] D. Kadetotad, S. Yin, V. Berisha, C. Chakrabarti, and J.-s. Seo, “An
+8.93 TOPS/W LSTM Recurrent Neural Network Accelerator Featuring
+Hierarchical Coarse-Grain Sparsity for On-Device Speech Recognition,”
+IEEE Journal of Solid-State Circuits, vol. 55, no. 7, pp. 1877–1887,
+2020.
+[32] P. D. Schiavone, D. Rossi, A. Pullini, A. Di Mauro, F. Conti, and
+L. Benini, “Quentin: an Ultra-Low-Power PULPissimo SoC in 22nm
+FDX,” in 2018 IEEE SOI-3D-Subthreshold Microelectronics Technology
+Unified Conference (S3S), 2018, pp. 1–3.
+[33] F. Conti, P. D. Schiavone, and L. Benini, “XNOR Neural Engine:
+A Hardware Accelerator IP for 21.6-fJ/op Binary Neural Network
+Inference,” IEEE Transactions on Computer-Aided Design of Integrated
+Circuits and Systems, vol. 37, no. 11, pp. 2940–2951, 2018.
+[34] J. Boukhobza, S. Rubini, R. Chen, and Z. Shao, “Emerging NVM: A
+Survey on Architectural Integration and Research Challenges,” ACM
+Trans. Des. Autom. Electron. Syst., vol. 23, no. 2, nov 2017. [Online].
+Available: https://doi.org/10.1145/3131848
+[35] S. Yu and P.-Y. Chen, “Emerging Memory Technologies: Recent Trends
+and Prospects,” IEEE Solid-State Circuits Magazine, vol. 8, no. 2, pp.
+43–56, 2016.
+[36] E. K¨ult¨ursay, M. Kandemir, A. Sivasubramaniam, and O. Mutlu, “Eval-
+uating STT-RAM as an energy-efficient main memory alternative,”
+
+12
+in 2013 IEEE International Symposium on Performance Analysis of
+Systems and Software (ISPASS), 2013, pp. 256–267.
+[37] L. Mei, M. Dandekar, D. Rodopoulos, J. Constantin, P. Debacker,
+R. Lauwereins, and M. Verhelst, “Sub-Word Parallel Precision-Scalable
+MAC Engines for Efficient Embedded DNN Inference,” in 2019 IEEE
+International Conference on Artificial Intelligence Circuits and Systems
+(AICAS), 2019, pp. 6–10.
+[38] V. Camus, L. Mei, C. Enz, and M. Verhelst, “Review and Benchmarking
+of Precision-Scalable Multiply-Accumulate Unit Architectures for Em-
+bedded Neural-Network Processing,” IEEE Journal on Emerging and
+Selected Topics in Circuits and Systems, vol. 9, no. 4, pp. 697–711,
+2019.
+[39] W. Lu, G. Yan, J. Li, S. Gong, Y. Han, and X. Li, “FlexFlow: A Flexible
+Dataflow Accelerator Architecture for Convolutional Neural Networks,”
+in 2017 IEEE International Symposium on High Performance Computer
+Architecture (HPCA), 2017, pp. 553–564.
+[40] H. Kwon, A. Samajdar, and T. Krishna, “Maeri: Enabling flexible
+dataflow mapping over dnn accelerators via reconfigurable intercon-
+nects,” ACM SIGPLAN Notices, vol. 53, no. 2, pp. 461–475, 2018.
+[41] Y.-H. Chen, T.-J. Yang, J. Emer, and V. Sze, “Eyeriss v2: A flexible
+accelerator for emerging deep neural networks on mobile devices,” IEEE
+Journal on Emerging and Selected Topics in Circuits and Systems, vol. 9,
+no. 2, pp. 292–308, 2019.
+[42] C. N. Coelho, A. Kuusela, S. Li, H. Zhuang, T. Aarrestad, V. Loncar,
+J. Ngadiuba, M. Pierini, A. A. Pol, and S. Summers, “Automatic
+heterogeneous quantization of deep neural networks for low-latency
+inference on the edge for particle detectors,” 2020. [Online]. Available:
+https://arxiv.org/abs/2006.10159
+[43] Speech
+Commands:
+A
+Public
+Dataset
+for
+Single-Word
+Speech
+Recognition. [Online]. Available: :http://download.tensorflow.org/data/
+speech commands v0/
+[44] J. S. P. Giraldo, V. Jain, and M. Verhelst, “Efficient Execution of
+Temporal Convolutional Networks for Embedded Keyword Spotting,”
+IEEE Transactions on Very Large Scale Integration (VLSI) Systems,
+vol. 29, no. 12, pp. 2220–2228, 2021.
+[45] H.
+Purohit,
+R.
+Tanabe,
+K.
+Ichige,
+T.
+Endo,
+Y.
+Nikaido,
+K.
+Suefusa,
+and
+Y.
+Kawaguchi,
+“MIMII
+Dataset:
+Sound
+Dataset
+for
+Malfunctioning
+Industrial
+Machine
+Investigation
+and
+Inspection,” CoRR, vol. abs/1909.09347, 2019. [Online]. Available:
+http://arxiv.org/abs/1909.09347
+[46] C. R. Banbury, V. J. Reddi, P. Torelli, J. Holleman, N. Jeffries, C. Kir´aly,
+P. Montino, D. Kanter, S. Ahmed, D. Pau, U. Thakker, A. Torrini,
+P. Warden, J. Cordaro, G. D. Guglielmo, J. M. Duarte, S. Gibellini,
+V. Parekh, H. Tran, N. Tran, W. Niu, and X. Xu, “MLPerf Tiny
+Benchmark,” CoRR, vol. abs/2106.07597, 2021. [Online]. Available:
+https://arxiv.org/abs/2106.07597
+[47] K. Yang, S. Kpotufe, and N. Feamster, “An Efficient One-Class
+SVM for Anomaly Detection in the Internet of Things,” CoRR, vol.
+abs/2104.11146, 2021. [Online]. Available: https://arxiv.org/abs/2104.
+11146
+[48] I. Miro-Panades, B. Tain, J.-F. Christmann, D. Coriat, R. Lemaire,
+C. Jany, B. Martineau, F. Chaix, A. Quelen, E. Pluchart, J.-P. Noel,
+R. Boumchedda, A. Makosiej, M. Montoya, S. Bacles-Min, D. Briand,
+J.-M. Philippe, A. Valentian, F. Heitzmann, E. Beigne, and F. Cler-
+midy, “SamurAI: A 1.7MOPS-36GOPS Adaptive Versatile IoT Node
+with 15,000× Peak-to-Idle Power Reduction, 207ns Wake-Up Time
+and 1.3TOPS/W ML Efficiency,” in 2020 IEEE Symposium on VLSI
+Circuits, 2020, pp. 1–2.
+[49] E. Flamand, D. Rossi, F. Conti, I. Loi, A. Pullini, F. Rotenberg, and
+L. Benini, “GAP-8: A RISC-V SoC for AI at the Edge of the IoT,”
+in 2018 IEEE 29th International Conference on Application-specific
+Systems, Architectures and Processors (ASAP), 2018, pp. 1–4.
+[50] W. Shan, M. Yang, T. Wang, Y. Lu, H. Cai, L. Zhu, J. Xu, C. Wu, L. Shi,
+and J. Yang, “A 510-nW Wake-Up Keyword-Spotting Chip Using Serial-
+FFT-Based MFCC and Binarized Depthwise Separable CNN in 28-nm
+CMOS,” IEEE Journal of Solid-State Circuits, vol. 56, no. 1, pp. 151–
+164, 2021.
+[51] J. Lee, C. Kim, S. Kang, D. Shin, S. Kim, and H.-J. Yoo, “UNPU: An
+Energy-Efficient Deep Neural Network Accelerator With Fully Variable
+Weight Bit Precision,” IEEE Journal of Solid-State Circuits, vol. 54,
+no. 1, pp. 173–185, 2019.
+[52] M. Fariselli, M. Rusci, J. Cambonie, and E. Flamand, “Integer-Only
+Approximated MFCC for Ultra-Low Power Audio NN Processing on
+Multi-Core MCUs,” in 2021 IEEE 3rd International Conference on
+Artificial Intelligence Circuits and Systems (AICAS), 2021, pp. 1–4.
+

AdE3T4oBgHgl3EQfsgtu/content/tmp_files/2301.04668v1.pdf.txt

@@ -0,0 +1,867 @@
+Trapped Ion Quantum Computing using Optical Tweezers and the Magnus Effect
+M. Mazzanti,1 R. Gerritsma,1, 2 R. J. C. Spreeuw,1, 2 and A. Safavi-Naini2, 3
+1Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, 1098 XH Amsterdam, Netherlands
+2QuSoft, Science Park 123, 1098 XG Amsterdam, the Netherlands
+3Institute for Theoretical Physics, Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands
+(Dated: January 13, 2023)
+We consider the implementation of quantum logic gates in trapped ions using tightly focused optical tweezers.
+Strong polarization gradients near the tweezer focus lead to qubit-state dependent forces on the ion. We show
+that these may be used to implement quantum logic gates on pairs of ion qubits in a crystal. The qubit-state
+dependent forces generated by this effect live on the plane perpendicular to the direction of propagation of
+the laser beams opening new ways of coupling to motional modes of an ion crystal. The proposed gate does
+not require ground state cooling of the ions and does not rely on the Lamb-Dicke approximation, although the
+waist of the tightly focused beam needs to be comparable with its wavelength in order to achieve the needed
+field curvature. Furthermore, the gate can be performed on both ground state and magnetic field insensitive
+clock state qubits without the need for counter-propagating laser fields. This simplifies the setup and eliminates
+errors due to phase instabilities between the gate laser beams. Finally, we show that imperfections in the gate
+execution, in particular pointing errors < 30 nm in the tweezers reduce the gate fidelity from F ≳ 0.99998 to
+≳ 0.999.
+Trapped ions are one of the most mature platforms for the
+implementation of quantum computing and quantum logic
+gates have been implemented with very high fidelity in these
+systems [1, 2]. Usually, the quantum logic gates in trapped
+ions rely on state-dependent forces applied to the ions by
+laser fields or magnetic fields.
+The exchange of motional
+quanta between the ions then leads to effective qubit-qubit in-
+teractions. Several recent works have explored how the use
+of state-of-the-art optical tweezer technology can benefit the
+trapped ion quantum computer. Optical tweezers can be used
+to confine atoms very strongly by inducing a dipole in them
+and find application in neutral atomic quantum simulators, in
+which tweezers are used to levitate individual atoms [3–7].
+In trapped ions, tweezers may be used to tune the soundwave
+spectrum in the ion crystal and thereby to program the inter-
+actions between the qubits [8–10]. Furthermore, in a recent
+work [11] we have proposed combining state-dependent opti-
+cal tweezers with oscillating electric fields to build a universal
+trapped ion quantum computer with extremely long-ranged in-
+teractions between the qubits.
+In this work, we consider another scenario, in which we
+make use of the strong polarization gradients that occur in op-
+tical tweezers. We note that strong gradients in optical po-
+tentials have been previously investigated to implement two-
+qubit gates without the need for ground-state cooling [12–
+14]. However, our approach utilizes the state-dependent dis-
+placement of the tweezer potential due to polarization gradi-
+ents [15–17]. We propose to use this optical analogue of the
+Magnus effect to implement quantum logic gates in trapped
+ions.
+Setup – We consider linearly x-polarized, Gaussian tweez-
+ers, pointing in the −y direction and tightly focused at two
+qubits between which we wish to implement a quantum logic
+gate. The quantum computing platform here considered is a
+linear crystal of N alkali-like trapped ions of mass m. In the
+focal plane the ions experience a strong polarization gradient
+along the x direction, such that the polarization is linear (x)
+in the center and circular (σ±)z in the wings of the Gaussian.
+b)
+a)
+⃗y
+⃗x
+⃗z
+mj
+P1/2
+−1/2
++1/2
+|0⟩
+|1⟩
+Ω−
+S1/2
+Ω+
+nX+
++
++
++
++
++
++
++
+−6 −4 −2
+0
+2
+4
+6
+x/λ
+−4
+−2
+0
+2
+4
+z/λ
+(σ−)z
+−6 −4 −2
+0
+2
+4
+6
+x/λ
+−4
+−2
+0
+2
+4 (σ+)z
+−6 −4 −2
+0
+2
+4
+6
+x/λ
+−4
+−2
+0
+2
+4
+z/λ
+(σ−)z
+−6 −4 −2
+0
+2
+4
+6
+x/λ
+−4
+−2
+0
+2
+4 (σ+)z
+Laguerre-Gaussian
+Gaussian
+FIG. 1. Schematic representation of the two-qubit gate. a) We apply
+tweezers propagating along the −y direction on the two ions forming
+the gate. The tweezer intensity can be decomposed into three polar-
+ization components. b) Simplified level scheme of an alkaline-earth
+like ion without nuclear spin showing the encoding of the qubit in
+its Zeeman ground states. The two polarization components of the
+tweezer couple to different states in the P1/2 manifold with detuning
+∆. This causes the minima of the tweezer potentials to be shifted by
+an amount ±λ depending on the qubit state. Bottom : main polar-
+ization components for a Gaussian and Laguerre-Gaussian (l = 1,
+n = 0) tightly focused tweezer.
+A direct calculation [18] decomposing the field in the focal
+plane into its circular components (σ±)z (and πz) shows that,
+to a good approximation, the circular components are near-
+Gaussian distributions, displaced in opposite directions along
+the x axis. We depict this setup in Fig. 1. Note that the circu-
+lar components rotate in the xy plane, i.e. a plane containing
+the k vector of the light. As shown in Fig. 1, the (σ±)z com-
+ponent is displaced by an amount ±λ ≡ ±λ/2π, with λ the
+arXiv:2301.04668v1  [quant-ph]  11 Jan 2023
+
+2
+tweezer wavelength. As the total field is the superposition
+of two displaced Gaussians, its intensity is slightly elongated
+along x. Hollow tweezers (Gaussian-Laguerre) can be used
+instead of Gaussian ones. This will provide the needed field
+curvature while keeping near-zero intensity at the center of
+the beam, drastically reducing the probability of off-resonant
+scattering that might limit the gate fidelity.
+For simplicity, we first consider ions without nuclear spin,
+such as 40Ca+, 88Sr+, 138Ba+ and 174Yb+. The qubits are
+encoded in the electronic ground states 2S1/2 and |0⟩ = |j =
+1/2, mj = 1/2⟩ and |1⟩ = |j = 1/2, mj = −1/2⟩ with
+j the total electronic angular momentum and mj its projec-
+tion on the quantization axis. The magnetic field lies along
+the z-direction and the tweezers are polarized along the x-
+direction, such that the ions experience linearly polarized laser
+light. The direction along the x-axis is the long direction of
+the ion trap, with trap frequency ωx. The motion of the ions
+along the x-direction can be described by collective modes of
+harmonic motion with frequencies ωm and mode vectors bi,m,
+with m labeling the mode and i the ion [19].
+We choose the detuning between the tweezers and the D1
+transition to be large enough to avoid photon scattering, but
+much smaller than the spin-orbit coupling splitting of the 2P
+state. In this way, we can neglect coupling to the P3/2 state.
+In what follows we will show that this requirement can be
+satisfied experimentally. Close to the center of the tweezer,
+strong polarization gradients appear and as a result, the two
+qubit states experience slightly different tweezer potentials. In
+particular, as we show in Fig. 1(a), the optical Magnus effect
+causes each qubit state to experience a tweezer potential that
+is offset from the apparent center of the tweezer by ∼ λ [16].
+Hence, we may approximate the tweezer potential as :
+ˆU(x) = −U0 exp
+�
+−2(ˆx + ˆσzλ)2/w2
+0
+�
+(1)
+≈ − ˜U0 + 1
+2mω2
+twˆx2 + gx ˆσz
+(2)
+with ωtw =
+�
+4 ˜U0(w2
+0 − 4λ2)/(mw4
+0), g = 4 ˜U0λ/w2
+0, and
+˜U0 = U0 exp(−2λ2/w2
+0) ≈ U0. Here U0 is the tweezer po-
+tential in the center and the beam waist is w0. Our approxima-
+tion replaces the tweezer potential with a harmonic potential
+and is valid for w0 ≫ lm, with lm =
+�
+ℏ/2mωm. The last
+term in U(x) is the result of the spin-dependent force g cou-
+pling the internal state of the qubit, ˆσz, to its motion ˆx. Thus,
+the optical Magnus effect allows us to straightforwardly im-
+plement a quantum gate.
+Tweezer Hamiltonian – In the interaction picture with re-
+spect to ˆH0 = ℏωmˆa†
+mˆam the tweezer Hamiltonian on ions i
+and j is:
+ˆH1 = A(t)
+�1
+2mω2
+tw
+�
+ˆx2
+i + ˆx2
+j
+�
++ g
+�
+ˆσ(i)
+z ˆxi + ˆσ(j)
+z ˆxj
+��
+.
+(3)
+Here, ˆxi = �
+m lmbim
+�
+ˆame−iωmt + ˆa†
+meiωmt�
+is the posi-
+tion operator of ion i in the interaction picture, with ˆa†
+m the
+creation operator for the mode m, and 0 ≤ A(t) ≤ 1 speci-
+fies the time-dependence of the tweezer intensity. The qubit-
+state independent terms in ˆH1 do not alter the dynamics of the
+quantum gate. We ignore these terms and arrive at:
+ˆH2 = A(t)g
+�
+ˆxiˆσ(i)
+z
++ ˆxjˆσ(j)
+z
+�
+,
+(4)
+which takes the form of a spin-phonon coupling Hamilto-
+nian reminiscent of the Mølmer-Sørenson scheme for phonon-
+mediated quantum gates in trapped ions [20]. However, at this
+stage we still have various choices available for A(t), depend-
+ing on which type of quantum gate we would like to imple-
+ment. For instance, pulsed A(t) could be used to perform fast
+gates. Here, we choose A(t) to obtain a geometric phase gate.
+For this, we set 2A(t) = 1−cos(νt+φ) where φ = 0 assures
+a smooth ramp of the tweezer intensity and ν = ωc + δ with
+the subscript c denoting the center-of-mass (c.o.m.) mode for
+which ωc = ωx and bi,c = 1/
+√
+N. We write the operators ˆxi
+and ˆxj in terms of ˆac and ˆa†
+c and perform the rotating wave
+approximation to arrive at:
+ˆH3 =
+glc
+4
+√
+N
+�
+ˆaceiδt + ˆa†
+ce−iδt� �
+ˆσ(i)
+z
++ ˆσ(j)
+z
+�
+.
+(5)
+To derive the qubit-qubit interactions forming the geomet-
+ric phase gate, we perform a unitary transformation ˆU1 =
+e−iδˆa†
+c ˆact to eliminate the time dependence, followed by a
+Lang-Firsov [21] transformation, ˆU2 = exp
+�
+ˆα
+�
+ˆa†
+c − ˆac
+���
+with ˆα = − ˜g
+δ
+�
+ˆσ(i)
+z
++ ˆσ(j)
+z
+�
+. Disregarding qubit-independent
+terms, we obtain
+Heff = 2˜g2
+ℏδ ˆσ(i)
+z ˆσ(j)
+z ,
+(6)
+with ˜g = glc/(4
+√
+N) = ˜η ˜U0, with the proportionality factor
+˜η = λlc/(
+√
+Nw2
+0). This Hamiltonian generates qubit-qubit
+interactions that can be used to implement a geometric phase
+gate by setting the gate time τ = 2π/δ and ˜g2τ
+ℏ2δ = π/4.
+Characterization of the gate – We analyse the gate dynam-
+ics by performing numerical simulation of the full dynamics
+generated by the Hamiltonian ˆHsim = ˆH0 + ˆU (xi) + ˆU (xj)
+for a two dimensional ion crystal where the tweezers po-
+tentials ˆU (xi;j) on ions i and j have been expanded up to
+fourth-order including spin-independent terms. We use real-
+istic experimental parameters: ∼ 156 µW of tweezer laser
+power focused to a waist of w0 ∼ 0.5 µm and tuned 15 THz
+to the red from the 2S1/2 →
+2P1/2 transition in 174Yb+
+(λ = 369.5 nm). This results in ˜U0/h ∼ 1.6 MHz, ˜g/h =
+2.1 kHz/
+√
+N, and setting δ = 2π × 12.2 kHz/
+√
+N the gate
+time for the geometric phase gate is τ = 170
+√
+Nµs. With a
+calculated qubit-state independent tweezer potential of ωtw ∼
+2π × 37 kHz, the center-of-mass mode frequency (ωc/2π ∼
+1 MHz) is shifted by ∼ 2ω2
+tw/ωcN ∼ 2π × 710/N Hz.
+This shift can easily be taken into account by correcting δ
+accordingly. In these estimates, we neglected the contribu-
+tion from other dipole allowed transitions, that are detuned by
+∼ 66 THz (the relatively weak 2S1/2 → 3[3/2]3/2 transition)
+and 115 THz (the strong D2 line) or more.
+We consider the gate unitary with a spin-echo sequence
+given by U(0, τ) = X⊗2U(τ/2, τ)X⊗2U(0, τ/2), where
+
+3
+FIG. 2. We calculate the gate fidelity for a ground state cooled ion
+¯nc, ¯ns = 0 (blue), sub-Doppler cooled thermal state with ¯nc =
+0.62, ¯ns = 0.23 (orange) and ¯nc = 15, ¯ns = 0.23 (red, using in
+this case a Fock cutoff nc ≤ 120, ns ≤ 10). (a) Process fidelity
+of the two-qubit Magnus gate for different gate times. (b) Effects of
+misalignment ϵ (orange) and intensity noise Λ1/τ (blue) on the gate
+fidelity. The size of each intensity noise data point represents the
+standard deviation of 20 simulation where we generated a random
+Gaussian noise with σ = Λ1/τ on each of the two pulses. This
+implies a noise on the laser intensity at frequency 1/τ that can not
+be removed by the spin-echo sequence.
+X⊗2 is a qubit flip on both qubits. This spin echo sequence is
+needed in order to remove local rotations on the qubits states
+and possible timing errors. We calculate the unitary time evo-
+lution operator U(0, τ) for a system of two ions with their
+motional c.o.m. and stretch modes and truncate their respec-
+tive Hilbert spaces to nc ≤ 18 and ns ≤ 10. In figure 2 we
+show the process fidelity of the gate assuming the ions are in
+their motional ground state (¯n = 0) as a function of gate time.
+The gate fidelity of F = 0.999988 with nc = ns = 0 rivals
+the current standard approaches. Moreover, the performance
+of our gate is robust to the thermal occupation of the motional
+modes. We characterize the gate performance in presence of
+thermal phonons using the average gate fidelity [18, 22] and
+find that it depends weakly on the motional state of the two
+ions. In fact, using ¯nc = 0.62, ¯ns = 0.23, the fidelity is
+almost unaltered at Fth = 0.999989.
+One of main experimental challenges is perfect tweezer
+alignment. We have studied the resilience of the gate against
+misalignment of the tweezer in the x-direction, which we
+denote by ϵ.
+In the presence of misalignment,
+˜U0
+→
+|0⟩
+|1⟩
+∆
+mF
+−1
+0
++1
+171Yb+
+FIG. 3. Relevant energy levels of 171Yb+ for implementing the gate
+on hyperfine qubit splitted by ωq. The coupling can be achieved
+using a pair of Raman beams detuned from the upper state 2P1/2 by
+∆. In the brackets are the angular contributions to the various dipole
+transition elements.
+TABLE I. Main sources of gate errors. We estimate γph as the prob-
+ability of a off-resonant scattering in for 174Yb+ during the gate time
+(τ = 240 µs) for a Gaussian and Laguerre-Gaussian beams. Other
+typical sources of errors are misalignment (ϵ), tweezer intensity noise
+(Λ1/τ) and timing (∆τ). The values here reported are for laser pa-
+rameters used in our numerical simulations.
+Error source
+γph
+Gaussian
+γph
+Laguerre-Gaussian
+ϵ
+30 nm
+Λ1/τ
+0.5%
+∆τ
+±5 µs
+1 − F
+2 × 10−3
+10−6
+1.3 × 10−3 9.3 × 10−5 2.7 × 10−4
+U0 exp−2(ϵ+ˆσzλ)2/ω2
+0.
+Thus, the misalignment has two ef-
+fects: (i) it changes the tweezer potential at the center of the
+tweezer and therefore the phase accumulation in the phase
+gate, and (ii) it shifts the potential in a qubit-state-dependent
+way. The second contribution is corrected to lowest order by
+a spin-echo sequence. Figure 2(b) shows the infidelity as
+a function of ϵ. Here we assume that the tweezers are mis-
+aligned on both ions in the same way which seems the experi-
+mentally most likely case. The unitary U(0, τ) leads to phase
+space trajectories for ⟨x(t)⟩ and ⟨px(t)⟩ associated with the
+c.o.m. motion[18]. As expected, we find approximately cir-
+cular phase-space orbits for the even parity states |00⟩, |11⟩,
+and very little motion for the odd parity ones. We see that ev-
+ery state combination leads to ion motion, but the difference
+in motion still leads to a high fidelity of ≳ 0.999 as shown in
+Figure 2(b).
+Clock state case – While the calculation was performed
+for the electron spin qubit states in 174Yb+, it should also
+be possible to use the hyperfine clock states |F = mF = 0⟩
+and |F = 1, mF = 0⟩ in 171Yb+. This qubit is insensitive to
+magnetic field noise and coherence times of up to an hour have
+been measured [23]. In this case, the tweezers are formed by a
+bichromatic co-propagating laser field detuned by ∆ from the
+D1 transition at 369.5 nm with overall detuning ∆ ≪ ωFS, the
+fine structure splitting. We set the frequency difference in the
+
+4
+bichromatic tweezer to 12.6 GHz, corresponding to the tran-
+sition between the qubit states [24]. The tweezer laser then
+causes Raman coupling between the qubit states via two dis-
+tinct paths. In the first path, the qubits are coupled via the state
+|P1/2, F = 1, mF = −1⟩ due to the σ− polarization compo-
+nent in the tweezer. In the other, the qubits are coupled via
+the state |P1/2, F = 1, mF = +1⟩ due to the σ+ component
+in the tweezer. We denote the Rabi frequencies of each path
+as Ω±
+1,2(x). The corresponding Raman couplings of each path
+interfere destructively in the center of the tweezer due to a rel-
+ative minus sign between Ω+
+1 (x) and Ω+
+2 (x) in their Clebsch-
+Gordan coefficient, ∝ (Ω−
+1 (0)Ω−
+2 (0)+Ω+
+1 (0)Ω+
+2 (0))/∆ = 0.
+However, the Magnus effect causes a strong position depen-
+dence of the relative strength of both paths of magnitude
+Ωeff(x) = Ω−
+1 (x)Ω−
+2 (x)
+∆
++ Ω+
+1 (x)Ω+
+2 (x)
+∆
+≈ Ω2
+∆
+4λx
+w2
+0
+,
+(7)
+where we assumed x ≪ λ ≪ w0 and |Ω±
+i (0)| = Ω/
+√
+2 with
+i = 1, 2, such that both laser frequencies have the same power.
+As a result, a qubit state-dependent force appears as in Eq.
+(4), except that we must now replace ˆσ(i,j)
+z
+→ ˆσ(i,j)
+x
+and the
+gate takes the form of the usual Mølmer-Sørensen interaction
+∝ ˆσ(i)
+x ˆσ(j)
+x
+[20]. Amplitude modulation via A(t) allows again
+for resonant enhancement of the gate.
+In addition to the Raman coupling, we obtain a tweezer po-
+tential (AC Stark shift) for each qubit state of magnitude
+δ|k⟩
+AC(x) =
+�
+i=1,2
+�
+j=+,−
+|Ωj
+i(x)|2
+∆i,|k⟩
+(8)
+with ∆1,|0⟩ = ∆ − ωq, ∆2,|0⟩ = ∆, ∆1,|1⟩ = ∆ and ∆2,|1⟩ =
+∆ + ωq. This causes an additional trapping potential Φ(x) ≈
+1
+2mω2
+twx2 that is independent of the qubit state as before, as
+well as a position-dependent differential Stark shift δAC(x) =
+δ|1⟩
+AC(x) − δ|0⟩
+AC(x). In the limit ωq ≪ |∆|,
+δAC(x) ≈ − ωq
+∆2
+�
+i=1,2
+�
+j=+,−
+|Ωj
+i(x)|2
+(9)
+= −ωq
+∆
+˜U0(x)
+(10)
+This differential Stark shift is estimated to be small, δAC/2π ≈
+2.7 kHz for the numbers used in the simulations, and can be
+compensated by a corresponding Raman detuning.
+Photon scattering on the D1 transition can be estimated as
+γph ∼ ˜U0Γ/(ℏ∆) ∼ 13 s−1 with Γ = 1.23 × 108 s−1 in
+Yb+. This adverse effect may be reduced significantly by em-
+ploying hollow tweezers [11, 25, 26] at the expense of added
+complexity. For a hollow beam with a waist w0 = 0.5 µm
+and ∼ 160 µW we obtain a reduction in scattering rate of
+∼ 10−6 s−1. As long as ωtw ≪ Ωrf, the drive frequency of the
+Paul trap, no parametric excitations can occur and micromo-
+tion of the ions is not a problem. Other errors, such as due to
+intensity noise of the laser, heating of the ions due to electric
+field noise and decoherence due to magnetic field noise have
+the same effect as in other gate implementations. Finally, we
+note that because the tweezers are far detuned from the closest
+transitions, the exact overall frequency of the tweezer laser is
+irrelevant.
+Conclusions In conclusion, we have described a novel
+type of quantum phase gate based on the optical Magnus ef-
+fect using optical tweezers in a linear chain of trapped ions.
+The main benefit is that the gate does not require counter-
+propagating laser fields, greatly simplifying the setup and
+eliminating errors due to phase instabilities between the gate
+laser beams. Furthermore, the state-dependent force gener-
+ated by the Magnus effect allows to perform the gate by cou-
+pling to motional modes on the plane perpendicular to the di-
+rection of propagation of the tweezers allowing novel experi-
+mental implementations. The proposed gate does not require
+ground state cooling and can perform a quantum logic gate
+on any pair of ion qubits by spatial addressing. The expected
+gate fidelity rivals the state of the art also for ions that are not
+cooled to the ground-state of motion.
+ACKNOWLEDGEMENTS
+This work was supported by the Netherlands Organiza-
+tion for Scientific Research (Grant Nos.
+680.91.120 and
+680.92.18.05, (R.G.). A.S.N is supported by the Dutch Re-
+search Council (NWO/OCW), as part of the Quantum Soft-
+ware Consortium programme (project number 024.003.037).
+[1] C. Ballance, T. Harty, N. Linke, M. Sepiol, and D. Lucas,
+Phys. Rev. Lett. 117, 060504 (2016).
+[2] J. Gaebler, T. Tan, Y. Lin, Y. Wan, R. Bowler, A. Keith,
+S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. Wineland,
+Phys. Rev. Lett. 117, 060505 (2016).
+[3] D. Barredo, S. d. Léséleuc, V. Lienhard, T. Lahaye, and
+A. Browaeys, Science 354, 1021 (2016).
+[4] M. Endres, H. Bernien, A. Keesling, H. Levine, E. R. Anschuetz,
+A. Krajenbrink, C. Senko, V. Vuletic, M. Greiner, and M. D.
+Lukin, Science 354, 1024 (2016).
+[5] M. Norcia, A. Young, and A. Kaufman, Phys. Rev. X 8, 041054
+(2018).
+[6] H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang,
+S. Ebadi, H. Bernien, M. Greiner, V. Vuleti´c, H. Pichler, and
+M. D. Lukin, Phys. Rev. Lett. 123, 170503 (2019).
+[7] A. Browaeys and T. Lahaye, Nature Physics 16, 132–142 (2020).
+[8] T. Olsacher, L. Postler, P. Schindler, T. Monz, P. Zoller, and L. M.
+Sieberer, PRX Quantum 1, 020316 (2020).
+[9] Y. H. Teoh, M. Sajjan, Z. Sun, F. Rajabi, and R. Islam,
+Phys. Rev. A 104, 022420 (2021).
+[10] J. D. Arias Espinoza,
+M. Mazzanti,
+K. Fouka,
+R. X.
+Schüssler, Z. Wu, P. Corboz, R. Gerritsma, and A. Safavi-Naini,
+Phys. Rev. A 104, 013302 (2021).
+[11] M. Mazzanti, R. X. Schüssler, J. D. Arias Espinoza, Z. Wu,
+R. Gerritsma, and A. Safavi-Naini, Phys. Rev. Lett. 127, 260502
+
+(2021).
+[12] J. I. Cirac and P. Zoller, Nature 404, 579 (2000).
+[13] M. Šašura and A. M. Steane, Phys. Rev. A 67, 062318 (2003).
+[14] T. Calarco, J. I. Cirac, and P. Zoller, Phys. Rev. A 63, 062304
+(2001).
+[15] K.-P. Wang, J. Zhuang, X.-D. He, R.-J. Guo, C. Sheng, P. Xu,
+M. Liu, J. Wang, and M.-S. Zhan, Chinese Phys. Lett. 37, 044209
+(2020).
+[16] R. J. Spreeuw, Phys. Rev. Lett. 125, 233201 (2020).
+[17] R. J. C. Spreeuw, Nanophotonics 11, 633 (2022).
+[18] See supplementary material at link to be inserted by editor for
+definitions, technical details and supporting calculations,.
+[19] D. F. V. James, Appl. Phys. B 66, 181 (1998).
+[20] K. Mølmer and A. Sørensen, Phys. Rev. Lett. 82, 1835 (1999).
+[21] I. G. Lang and Y. A. Firsov, J. Exp. Theor. Phys. 3, 27, 443
+(1968).
+[22] M. A. Nielsen, Physics Letters A 303, 249 (2002).
+[23] P. Wang, C.-Y. Luan, M. Qiao, M. Um, J. Zhang, Y. Wang,
+X. Yuan, M. Gu, J. Zhang, and K. Kim, Nat. Com. 12, 233
+(2021).
+[24] S. Olmschenk, K. C. Younge, D. L. Moehring, D. N. Matsuke-
+vich, P. Maunz, and C. Monroe, Phys. Rev. A 76, 052314 (2007).
+[25] C. T. Schmiegelow, J. Schulz, H. Kaufmann, T. Ruster, U. G.
+Poschinger, and F. Schmidt-Kaler, Nat. Com. 7, 12998 (2016).
+[26] M. Drechsler, S. Wolf, C. T. Schmiegelow, and F. Schmidt-
+Kaler, arXiv:2104.07095 (2021).
+
+Supplementary material for :
+Trapped Ion Quantum Computing using Optical Tweezers and the Magnus Effect
+M. Mazzanti,1 R. Gerritsma,1, 2 R. J. C. Spreeuw,1, 2 and A. Safavi-Naini2, 3
+1Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, 1098 XH Amsterdam, Netherlands
+2QuSoft, Science Park 123, 1098 XG Amsterdam, the Netherlands
+3Institute for Theoretical Physics, Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands
+(Dated: January 13, 2023)
+APPENDIX I : OPTICAL MAGNUS EFFECT
+A key characteristic of a tightly focused beam is the strong
+field curvature near the focus. This not only affects the local
+intensity but also its polarization structure. To calculate this,
+we take a superposition of plane waves labeled by their wave
+vector in spherical coordinates, k = (k, θ, φ). Taking k =
+ω/c as fixed we write
+E(r) ∝
+� 2π
+0
+dφ
+� π
+0
+dθ sin θ ux(θ, φ) a(θ, φ) eik·r
+with ux(θ, φ) a polarization vector obtained by co-rotating
+the x unit vector when k is rotated from z to (θ, φ), such
+that ux(θ, φ) · k = 0, see also Ref. [S1]. In the calcula-
+tion we center the beam around θ = 0, and the focal plane
+is given by r = (x, y, 0).
+The shape of the beam is de-
+termined by the amplitude function a(θ, φ). For a Gaussian
+beam we set a(θ, φ) = exp(−θ2/w2
+θ); for the lowest or-
+der (l = 1) Laguerre-Gaussian (LG) beam we set a(θ, φ) =
+θ exp(iφ − θ2/w2
+θ). After performing the above integral we
+rotate the results for tweezers propagating along the −y di-
+rection. Finally, the circular field components σ± shown in
+Fig. 1 of the main text are obtained as the projection onto unit
+vectors (x ± iy)/
+√
+2. In Figure S-1, all three polarization
+components for a Laguerre-Gaussian beam are shown. Note
+that the σ− and σ+ components have similar intensity while
+the π-polarization is suppressed by a factor ∼ 100.
+−6 −4 −2 0
+2
+4
+6
+x/λ
+−4
+−2
+0
+2
+4
+z/λ
+(σ−)z
+−6 −4 −2 0
+2
+4
+6
+x/λ
+−4
+−2
+0
+2
+4 (π)z
+−6 −4 −2 0
+2
+4
+6
+x/λ
+−4
+−2
+0
+2
+4 (σ+)z
+FIG. S-1. Intensity of the polarization components for a LG beam
+calculated at the focus. The π-polarization component has been en-
+hanced by a factor 100 to make it visible. Here we set wθ = 0.6
+APPENDIX II : PHASE-SPACE DYNAMICS
+We study the phase-space dynamics of the ions by simulat-
+ing the time dependent Hamiltonian using trotterization with
+time-steps of 10−4 τ. At each time-step we evaluate the ex-
+pectation value of the ⟨ˆx⟩ and ⟨ˆp⟩ for the center of mass mode.
+As expected, we find approximately circular phase-space or-
+bits for the even parity states |00⟩, |11⟩, and very little motion
+for the odd parity ones. In Fig. S-2 it is possible to see the evo-
+lution in phase-space for all the four spin states in case of per-
+fectly aligned and slightly misaligned tweezers. As described
+in the main text we simulate numerically the full Hamiltonian
+defined as ˆHsim = ˆH0 + ˆU (xi) + ˆU (xj) where in case of
+misalignment ϵ, ˆU (x) reads as :
+U(x) ≈ −U0 e−2((ˆx−ˆϵ)+ˆσzλ)2/w2
+0
+≈ − ˜U0 + 4 ˜U0
+ˆσzλ − ˆϵ
+w2
+0
+ˆx
++ 1
+2
+˜U0
+�
+4
+�
+w2
+0 − 4λ2�
+w4
+0
+�
+ˆx2 − 1
+2
+˜U0
+�
+16
+�
+ˆϵ2 − 2ˆσzˆϵλ
+�
+w4
+0
+�
+ˆx2
+−
+�
+8 ˜U0ˆσzλ3w2
+0 − 4
+�
+3ˆϵ2 + λ2�
+3w6
+0
+�
+ˆx3
++
+�
+8 ˜U0ˆϵ3w2
+0 − 4
+�
+ˆϵ2 + 3λ2�
+3w6
+0
+�
+ˆx3
+−
+�
+2 ˜U0ˆσzλˆϵ−48w2
+0 + 64
+�
+ˆϵ2 + λ2�
+3w8
+0
+�
+ˆx4
++
+�
+2 ˜U0
+3w4
+0 − 24w2
+0
+�
+ˆϵ2 + λ2�
++ 16
+�
+ˆϵ4 + 6ˆϵ2λ2 + λ4�
+3w8
+0
+�
+ˆx4.
+with
+˜U0 = U0e−2(ˆϵ+ˆσzλ)2/w2
+0
+A small tweezer misalignment ϵ gives rise to new spin-
+dependent terms in the Hamiltonian that shift the trapping po-
+tential in a state dependent way. In Fig.S-2 is shown how the
+dynamics is affected in the case where the tweezers are mis-
+aligned by 30 nm.
+APPENDIX III : GATE FIDELITY
+We characterize the gate by calculating the average process
+fidelity as follows : [S2]:
+¯F( ˆUid, ˆU ˆ
+Hsim) =
+�
+j tr
+�
+ˆUidˆσ†
+j ˆU †
+idˆσj( ˆU ˆ
+Hsim)
+�
++ d2
+d2 (d + 1)
+,
+
+2
+−1.4
+−0.7
+0.0
+0.7
+1.4
+⟨ˆx⟩
+−1.4
+−0.7
+0.0
+0.7
+1.4
+⟨ˆpx⟩
+ϵ = 0
+−2.1 −1.4 −0.7 0.0
+0.7
+1.4
+⟨ˆx⟩
+−1.4
+−0.7
+0.0
+0.7
+1.4
+⟨ˆpx⟩
+ϵ = 30 nm
+ψ↑↑
+ψ↓↑
+ψ↑↓
+ψ↓↓
+FIG. S-2. Center of mass mode phase-space dynamics for perfectly
+aligned tweezer (left) and for 30 nm misaligned ones (right). For the
+simulation we used the same parameters as for τ/2 = 120 µs point
+in Figure 1(a) of the main text.
+where ˆUid is the unitary of an ideal geometric phase gate and
+ˆσj( ˆU ˆ
+Hsim) ≡ trFS( ˆU ˆ
+Hsim [|n⟩⟨n| � ˆσj] ˆU †
+ˆ
+Hsim) projects the
+unitary matrix generated by the time evolution of the Hamil-
+tonian used for the simulations ˆU ˆ
+Hsim on the Fock state |n⟩
+and on a d-dimensional representation Pauli matrices.
+[S1] R. J. Spreeuw, Phys. Rev. Lett. 125, 233201 (2020).
+[S2] M. A. Nielsen, Physics Letters A 303, 249 (2002).
+

AdE3T4oBgHgl3EQfsgtu/content/tmp_files/load_file.txt

@@ -0,0 +1,488 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf,len=487
+page_content='Trapped Ion Quantum Computing using Optical Tweezers and the Magnus Effect  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Mazzanti,1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Gerritsma,1, 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Spreeuw,1, 2 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Safavi-Naini2, 3  1Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, 1098 XH Amsterdam, Netherlands  2QuSoft, Science Park 123, 1098 XG Amsterdam, the Netherlands  3Institute for Theoretical Physics, Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands  (Dated: January 13, 2023)  We consider the implementation of quantum logic gates in trapped ions using tightly focused optical tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Strong polarization gradients near the tweezer focus lead to qubit-state dependent forces on the ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We show  that these may be used to implement quantum logic gates on pairs of ion qubits in a crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The qubit-state  dependent forces generated by this effect live on the plane perpendicular to the direction of propagation of  the laser beams opening new ways of coupling to motional modes of an ion crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The proposed gate does  not require ground state cooling of the ions and does not rely on the Lamb-Dicke approximation, although the  waist of the tightly focused beam needs to be comparable with its wavelength in order to achieve the needed  field curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Furthermore, the gate can be performed on both ground state and magnetic field insensitive  clock state qubits without the need for counter-propagating laser fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This simplifies the setup and eliminates  errors due to phase instabilities between the gate laser beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Finally, we show that imperfections in the gate  execution, in particular pointing errors < 30 nm in the tweezers reduce the gate fidelity from F ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='99998 to  ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Trapped ions are one of the most mature platforms for the  implementation of quantum computing and quantum logic  gates have been implemented with very high fidelity in these  systems [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Usually, the quantum logic gates in trapped  ions rely on state-dependent forces applied to the ions by  laser fields or magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  The exchange of motional  quanta between the ions then leads to effective qubit-qubit in-  teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Several recent works have explored how the use  of state-of-the-art optical tweezer technology can benefit the  trapped ion quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Optical tweezers can be used  to confine atoms very strongly by inducing a dipole in them  and find application in neutral atomic quantum simulators, in  which tweezers are used to levitate individual atoms [3–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  In trapped ions, tweezers may be used to tune the soundwave  spectrum in the ion crystal and thereby to program the inter-  actions between the qubits [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Furthermore, in a recent  work [11] we have proposed combining state-dependent opti-  cal tweezers with oscillating electric fields to build a universal  trapped ion quantum computer with extremely long-ranged in-  teractions between the qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  In this work, we consider another scenario, in which we  make use of the strong polarization gradients that occur in op-  tical tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We note that strong gradients in optical po-  tentials have been previously investigated to implement two-  qubit gates without the need for ground-state cooling [12–  14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' However, our approach utilizes the state-dependent dis-  placement of the tweezer potential due to polarization gradi-  ents [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We propose to use this optical analogue of the  Magnus effect to implement quantum logic gates in trapped  ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Setup – We consider linearly x-polarized, Gaussian tweez-  ers, pointing in the −y direction and tightly focused at two  qubits between which we wish to implement a quantum logic  gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The quantum computing platform here considered is a  linear crystal of N alkali-like trapped ions of mass m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In the  focal plane the ions experience a strong polarization gradient  along the x direction, such that the polarization is linear (x)  in the center and circular (σ±)z in the wings of the Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  b)  a)  ⃗y  ⃗x  ⃗z  mj  P1/2  −1/2  +1/2  |0⟩  |1⟩  Ω−  S1/2  Ω+  nX+  +  +  +  +  +  +  +  −6 −4 −2  0  2  4  6  x/λ  −4  −2  0  2  4  z/λ  (σ−)z  −6 −4 −2  0  2  4  6  x/λ  −4  −2  0  2  4 (σ+)z  −6 −4 −2  0  2  4  6  x/λ  −4  −2  0  2  4  z/λ  (σ−)z  −6 −4 −2  0  2  4  6  x/λ  −4  −2  0  2  4 (σ+)z  Laguerre-Gaussian  Gaussian  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Schematic representation of the two-qubit gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' a) We apply  tweezers propagating along the −y direction on the two ions forming  the gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The tweezer intensity can be decomposed into three polar-  ization components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' b) Simplified level scheme of an alkaline-earth  like ion without nuclear spin showing the encoding of the qubit in  its Zeeman ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The two polarization components of the  tweezer couple to different states in the P1/2 manifold with detuning  ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This causes the minima of the tweezer potentials to be shifted by  an amount ±λ depending on the qubit state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Bottom : main polar-  ization components for a Gaussian and Laguerre-Gaussian (l = 1,  n = 0) tightly focused tweezer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  A direct calculation [18] decomposing the field in the focal  plane into its circular components (σ±)z (and πz) shows that,  to a good approximation, the circular components are near-  Gaussian distributions, displaced in opposite directions along  the x axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We depict this setup in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Note that the circu-  lar components rotate in the xy plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' a plane containing  the k vector of the light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 1, the (σ±)z com-  ponent is displaced by an amount ±λ ≡ ±λ/2π, with λ the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='04668v1  [quant-ph]  11 Jan 2023  2  tweezer wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' As the total field is the superposition  of two displaced Gaussians, its intensity is slightly elongated  along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Hollow tweezers (Gaussian-Laguerre) can be used  instead of Gaussian ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This will provide the needed field  curvature while keeping near-zero intensity at the center of  the beam, drastically reducing the probability of off-resonant  scattering that might limit the gate fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  For simplicity, we first consider ions without nuclear spin,  such as 40Ca+, 88Sr+, 138Ba+ and 174Yb+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The qubits are  encoded in the electronic ground states 2S1/2 and |0⟩ = |j =  1/2, mj = 1/2⟩ and |1⟩ = |j = 1/2, mj = −1/2⟩ with  j the total electronic angular momentum and mj its projec-  tion on the quantization axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The magnetic field lies along  the z-direction and the tweezers are polarized along the x-  direction, such that the ions experience linearly polarized laser  light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The direction along the x-axis is the long direction of  the ion trap, with trap frequency ωx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The motion of the ions  along the x-direction can be described by collective modes of  harmonic motion with frequencies ωm and mode vectors bi,m,  with m labeling the mode and i the ion [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  We choose the detuning between the tweezers and the D1  transition to be large enough to avoid photon scattering, but  much smaller than the spin-orbit coupling splitting of the 2P  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In this way, we can neglect coupling to the P3/2 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  In what follows we will show that this requirement can be  satisfied experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Close to the center of the tweezer,  strong polarization gradients appear and as a result, the two  qubit states experience slightly different tweezer potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In  particular, as we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 1(a), the optical Magnus effect  causes each qubit state to experience a tweezer potential that  is offset from the apparent center of the tweezer by ∼ λ [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Hence, we may approximate the tweezer potential as :  ˆU(x) = −U0 exp  �  −2(ˆx + ˆσzλ)2/w2  0  �  (1)  ≈ − ˜U0 + 1  2mω2  twˆx2 + gx ˆσz  (2)  with ωtw =  �  4 ˜U0(w2  0 − 4λ2)/(mw4  0), g = 4 ˜U0λ/w2  0, and  ˜U0 = U0 exp(−2λ2/w2  0) ≈ U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Here U0 is the tweezer po-  tential in the center and the beam waist is w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Our approxima-  tion replaces the tweezer potential with a harmonic potential  and is valid for w0 ≫ lm, with lm =  �  ℏ/2mωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The last  term in U(x) is the result of the spin-dependent force g cou-  pling the internal state of the qubit, ˆσz, to its motion ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Thus,  the optical Magnus effect allows us to straightforwardly im-  plement a quantum gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Tweezer Hamiltonian – In the interaction picture with re-  spect to ˆH0 = ℏωmˆa†  mˆam the tweezer Hamiltonian on ions i  and j is:  ˆH1 = A(t)  �1  2mω2  tw  �  ˆx2  i + ˆx2  j  �  + g  �  ˆσ(i)  z ˆxi + ˆσ(j)  z ˆxj  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  (3)  Here, ˆxi = �  m lmbim  �  ˆame−iωmt + ˆa†  meiωmt�  is the posi-  tion operator of ion i in the interaction picture, with ˆa†  m the  creation operator for the mode m, and 0 ≤ A(t) ≤ 1 speci-  fies the time-dependence of the tweezer intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The qubit-  state independent terms in ˆH1 do not alter the dynamics of the  quantum gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We ignore these terms and arrive at:  ˆH2 = A(t)g  �  ˆxiˆσ(i)  z  + ˆxjˆσ(j)  z  �  ,  (4)  which takes the form of a spin-phonon coupling Hamilto-  nian reminiscent of the Mølmer-Sørenson scheme for phonon-  mediated quantum gates in trapped ions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' However, at this  stage we still have various choices available for A(t), depend-  ing on which type of quantum gate we would like to imple-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' For instance, pulsed A(t) could be used to perform fast  gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Here, we choose A(t) to obtain a geometric phase gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  For this, we set 2A(t) = 1−cos(νt+φ) where φ = 0 assures  a smooth ramp of the tweezer intensity and ν = ωc + δ with  the subscript c denoting the center-of-mass (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=') mode for  which ωc = ωx and bi,c = 1/  √  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We write the operators ˆxi  and ˆxj in terms of ˆac and ˆa†  c and perform the rotating wave  approximation to arrive at:  ˆH3 =  glc  4  √  N  �  ˆaceiδt + ˆa†  ce−iδt� �  ˆσ(i)  z  + ˆσ(j)  z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  (5)  To derive the qubit-qubit interactions forming the geomet-  ric phase gate, we perform a unitary transformation ˆU1 =  e−iδˆa†  c ˆact to eliminate the time dependence, followed by a  Lang-Firsov [21] transformation, ˆU2 = exp  �  ˆα  �  ˆa†  c − ˆac  ��  with ˆα = − ˜g  δ  �  ˆσ(i)  z  + ˆσ(j)  z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Disregarding qubit-independent  terms, we obtain  Heff = 2˜g2  ℏδ ˆσ(i)  z ˆσ(j)  z ,  (6)  with ˜g = glc/(4  √  N) = ˜η ˜U0, with the proportionality factor  ˜η = λlc/(  √  Nw2  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This Hamiltonian generates qubit-qubit  interactions that can be used to implement a geometric phase  gate by setting the gate time τ = 2π/δ and ˜g2τ  ℏ2δ = π/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Characterization of the gate – We analyse the gate dynam-  ics by performing numerical simulation of the full dynamics  generated by the Hamiltonian ˆHsim = ˆH0 + ˆU (xi) + ˆU (xj)  for a two dimensional ion crystal where the tweezers po-  tentials ˆU (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='j) on ions i and j have been expanded up to  fourth-order including spin-independent terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We use real-  istic experimental parameters: ∼ 156 µW of tweezer laser  power focused to a waist of w0 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='5 µm and tuned 15 THz  to the red from the 2S1/2 →  2P1/2 transition in 174Yb+  (λ = 369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='5 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This results in ˜U0/h ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='6 MHz, ˜g/h =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='1 kHz/  √  N, and setting δ = 2π × 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='2 kHz/  √  N the gate  time for the geometric phase gate is τ = 170  √  Nµs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' With a  calculated qubit-state independent tweezer potential of ωtw ∼  2π × 37 kHz, the center-of-mass mode frequency (ωc/2π ∼  1 MHz) is shifted by ∼ 2ω2  tw/ωcN ∼ 2π × 710/N Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  This shift can easily be taken into account by correcting δ  accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In these estimates, we neglected the contribu-  tion from other dipole allowed transitions, that are detuned by  ∼ 66 THz (the relatively weak 2S1/2 → 3[3/2]3/2 transition)  and 115 THz (the strong D2 line) or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  We consider the gate unitary with a spin-echo sequence  given by U(0, τ) = X⊗2U(τ/2, τ)X⊗2U(0, τ/2), where  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We calculate the gate fidelity for a ground state cooled ion  ¯nc, ¯ns = 0 (blue), sub-Doppler cooled thermal state with ¯nc =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='62, ¯ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='23 (orange) and ¯nc = 15, ¯ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='23 (red, using in  this case a Fock cutoff nc ≤ 120, ns ≤ 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' (a) Process fidelity  of the two-qubit Magnus gate for different gate times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' (b) Effects of  misalignment ϵ (orange) and intensity noise Λ1/τ (blue) on the gate  fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The size of each intensity noise data point represents the  standard deviation of 20 simulation where we generated a random  Gaussian noise with σ = Λ1/τ on each of the two pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This  implies a noise on the laser intensity at frequency 1/τ that can not  be removed by the spin-echo sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  X⊗2 is a qubit flip on both qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This spin echo sequence is  needed in order to remove local rotations on the qubits states  and possible timing errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We calculate the unitary time evo-  lution operator U(0, τ) for a system of two ions with their  motional c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' and stretch modes and truncate their respec-  tive Hilbert spaces to nc ≤ 18 and ns ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In figure 2 we  show the process fidelity of the gate assuming the ions are in  their motional ground state (¯n = 0) as a function of gate time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  The gate fidelity of F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='999988 with nc = ns = 0 rivals  the current standard approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Moreover, the performance  of our gate is robust to the thermal occupation of the motional  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We characterize the gate performance in presence of  thermal phonons using the average gate fidelity [18, 22] and  find that it depends weakly on the motional state of the two  ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In fact, using ¯nc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='62, ¯ns = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='23, the fidelity is  almost unaltered at Fth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='999989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  One of main experimental challenges is perfect tweezer  alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We have studied the resilience of the gate against  misalignment of the tweezer in the x-direction, which we  denote by ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  In the presence of misalignment,  ˜U0  →  |0⟩  |1⟩  ∆  mF  −1  0  +1  171Yb+  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Relevant energy levels of 171Yb+ for implementing the gate  on hyperfine qubit splitted by ωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The coupling can be achieved  using a pair of Raman beams detuned from the upper state 2P1/2 by  ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In the brackets are the angular contributions to the various dipole  transition elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Main sources of gate errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We estimate γph as the prob-  ability of a off-resonant scattering in for 174Yb+ during the gate time  (τ = 240 µs) for a Gaussian and Laguerre-Gaussian beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Other  typical sources of errors are misalignment (ϵ), tweezer intensity noise  (Λ1/τ) and timing (∆τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The values here reported are for laser pa-  rameters used in our numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Error source  γph  Gaussian  γph  Laguerre-Gaussian  ϵ  30 nm  Λ1/τ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='5%  ∆τ  ±5 µs  1 − F  2 × 10−3  10−6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='3 × 10−3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='3 × 10−5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7 × 10−4  U0 exp−2(ϵ+ˆσzλ)2/ω2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Thus, the misalignment has two ef-  fects: (i) it changes the tweezer potential at the center of the  tweezer and therefore the phase accumulation in the phase  gate, and (ii) it shifts the potential in a qubit-state-dependent  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The second contribution is corrected to lowest order by  a spin-echo sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Figure 2(b) shows the infidelity as  a function of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Here we assume that the tweezers are mis-  aligned on both ions in the same way which seems the experi-  mentally most likely case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The unitary U(0, τ) leads to phase  space trajectories for ⟨x(t)⟩ and ⟨px(t)⟩ associated with the  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' motion[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' As expected, we find approximately cir-  cular phase-space orbits for the even parity states |00⟩, |11⟩,  and very little motion for the odd parity ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We see that ev-  ery state combination leads to ion motion, but the difference  in motion still leads to a high fidelity of ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='999 as shown in  Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Clock state case – While the calculation was performed  for the electron spin qubit states in 174Yb+, it should also  be possible to use the hyperfine clock states |F = mF = 0⟩  and |F = 1, mF = 0⟩ in 171Yb+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This qubit is insensitive to  magnetic field noise and coherence times of up to an hour have  been measured [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In this case, the tweezers are formed by a  bichromatic co-propagating laser field detuned by ∆ from the  D1 transition at 369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='5 nm with overall detuning ∆ ≪ ωFS, the  fine structure splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We set the frequency difference in the  4  bichromatic tweezer to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='6 GHz, corresponding to the tran-  sition between the qubit states [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The tweezer laser then  causes Raman coupling between the qubit states via two dis-  tinct paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In the first path, the qubits are coupled via the state  |P1/2, F = 1, mF = −1⟩ due to the σ− polarization compo-  nent in the tweezer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In the other, the qubits are coupled via  the state |P1/2, F = 1, mF = +1⟩ due to the σ+ component  in the tweezer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' We denote the Rabi frequencies of each path  as Ω±  1,2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The corresponding Raman couplings of each path  interfere destructively in the center of the tweezer due to a rel-  ative minus sign between Ω+  1 (x) and Ω+  2 (x) in their Clebsch-  Gordan coefficient, ∝ (Ω−  1 (0)Ω−  2 (0)+Ω+  1 (0)Ω+  2 (0))/∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  However, the Magnus effect causes a strong position depen-  dence of the relative strength of both paths of magnitude  Ωeff(x) = Ω−  1 (x)Ω−  2 (x)  ∆  + Ω+  1 (x)Ω+  2 (x)  ∆  ≈ Ω2  ∆  4λx  w2  0  ,  (7)  where we assumed x ≪ λ ≪ w0 and |Ω±  i (0)| = Ω/  √  2 with  i = 1, 2, such that both laser frequencies have the same power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  As a result, a qubit state-dependent force appears as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  (4), except that we must now replace ˆσ(i,j)  z  → ˆσ(i,j)  x  and the  gate takes the form of the usual Mølmer-Sørensen interaction  ∝ ˆσ(i)  x ˆσ(j)  x  [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Amplitude modulation via A(t) allows again  for resonant enhancement of the gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  In addition to the Raman coupling, we obtain a tweezer po-  tential (AC Stark shift) for each qubit state of magnitude  δ|k⟩  AC(x) =  �  i=1,2  �  j=+,−  |Ωj  i(x)|2  ∆i,|k⟩  (8)  with ∆1,|0⟩ = ∆ − ωq, ∆2,|0⟩ = ∆, ∆1,|1⟩ = ∆ and ∆2,|1⟩ =  ∆ + ωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This causes an additional trapping potential Φ(x) ≈  1  2mω2  twx2 that is independent of the qubit state as before, as  well as a position-dependent differential Stark shift δAC(x) =  δ|1⟩  AC(x) − δ|0⟩  AC(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In the limit ωq ≪ |∆|,  δAC(x) ≈ − ωq  ∆2  �  i=1,2  �  j=+,−  |Ωj  i(x)|2  (9)  = −ωq  ∆  ˜U0(x)  (10)  This differential Stark shift is estimated to be small, δAC/2π ≈  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7 kHz for the numbers used in the simulations, and can be  compensated by a corresponding Raman detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Photon scattering on the D1 transition can be estimated as  γph ∼ ˜U0Γ/(ℏ∆) ∼ 13 s−1 with Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='23 × 108 s−1 in  Yb+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This adverse effect may be reduced significantly by em-  ploying hollow tweezers [11, 25, 26] at the expense of added  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' For a hollow beam with a waist w0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='5 µm  and ∼ 160 µW we obtain a reduction in scattering rate of  ∼ 10−6 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' As long as ωtw ≪ Ωrf, the drive frequency of the  Paul trap, no parametric excitations can occur and micromo-  tion of the ions is not a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Other errors, such as due to  intensity noise of the laser, heating of the ions due to electric  field noise and decoherence due to magnetic field noise have  the same effect as in other gate implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Finally, we  note that because the tweezers are far detuned from the closest  transitions, the exact overall frequency of the tweezer laser is  irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Conclusions In conclusion, we have described a novel  type of quantum phase gate based on the optical Magnus ef-  fect using optical tweezers in a linear chain of trapped ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  The main benefit is that the gate does not require counter-  propagating laser fields, greatly simplifying the setup and  eliminating errors due to phase instabilities between the gate  laser beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Furthermore, the state-dependent force gener-  ated by the Magnus effect allows to perform the gate by cou-  pling to motional modes on the plane perpendicular to the di-  rection of propagation of the tweezers allowing novel experi-  mental implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The proposed gate does not require  ground state cooling and can perform a quantum logic gate  on any pair of ion qubits by spatial addressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The expected  gate fidelity rivals the state of the art also for ions that are not  cooled to the ground-state of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This work was supported by the Netherlands Organiza-  tion for Scientific Research (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='120 and  680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='05, (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='N is supported by the Dutch Re-  search Council (NWO/OCW), as part of the Quantum Soft-  ware Consortium programme (project number 024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='037).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Ballance, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Harty, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Linke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Sepiol, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lucas,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 117, 060504 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Gaebler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Tan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Wan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Bowler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Keith,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Glancy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Coakley, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Knill, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Leibfried, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Wineland,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 117, 060505 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Barredo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Léséleuc, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lienhard, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lahaye, and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Browaeys, Science 354, 1021 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Endres, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Bernien, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Keesling, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Levine, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Anschuetz,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Krajenbrink, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Senko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Vuletic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Greiner, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Lukin, Science 354, 1024 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Norcia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Young, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Kaufman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' X 8, 041054  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Levine, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Keesling, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Semeghini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Omran, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Wang,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Ebadi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Bernien, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Greiner, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Vuleti´c, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Pichler, and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lukin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 123, 170503 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Browaeys and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lahaye, Nature Physics 16, 132–142 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [8] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Olsacher, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Postler, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Schindler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Monz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Zoller, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Sieberer, PRX Quantum 1, 020316 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [9] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Teoh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Sajjan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Sun, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rajabi, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Islam,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' A 104, 022420 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Arias Espinoza,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Mazzanti,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Fouka,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Schüssler, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Wu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Corboz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Gerritsma, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Safavi-Naini,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' A 104, 013302 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Mazzanti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Schüssler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Arias Espinoza, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Wu,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Gerritsma, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Safavi-Naini, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 127, 260502  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Cirac and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Zoller, Nature 404, 579 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Šašura and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Steane, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' A 67, 062318 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [14] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Calarco, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Cirac, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Zoller, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' A 63, 062304  (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [15] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Zhuang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' He, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Sheng, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Xu,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Wang, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Zhan, Chinese Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 37, 044209  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [16] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Spreeuw, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 125, 233201 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [17] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Spreeuw, Nanophotonics 11, 633 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [18] See supplementary material at link to be inserted by editor for  definitions, technical details and supporting calculations,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [19] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' James, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' B 66, 181 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [20] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Mølmer and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Sørensen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 82, 1835 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [21] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Firsov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 3, 27, 443  (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Nielsen, Physics Letters A 303, 249 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [23] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Luan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Qiao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Um, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Wang,  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Yuan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Gu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Zhang, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Kim, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 12, 233  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Olmschenk, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Younge, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Moehring, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Matsuke-  vich, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Maunz, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Monroe, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' A 76, 052314 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [25] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Schmiegelow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Schulz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Kaufmann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Ruster, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Poschinger, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Schmidt-Kaler, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 7, 12998 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Drechsler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Wolf, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Schmiegelow, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Schmidt-  Kaler, arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='07095 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  Supplementary material for :  Trapped Ion Quantum Computing using Optical Tweezers and the Magnus Effect  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Mazzanti,1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Gerritsma,1, 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Spreeuw,1, 2 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Safavi-Naini2, 3  1Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, 1098 XH Amsterdam, Netherlands  2QuSoft, Science Park 123, 1098 XG Amsterdam, the Netherlands  3Institute for Theoretical Physics, Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands  (Dated: January 13, 2023)  APPENDIX I : OPTICAL MAGNUS EFFECT  A key characteristic of a tightly focused beam is the strong  field curvature near the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' This not only affects the local  intensity but also its polarization structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' To calculate this,  we take a superposition of plane waves labeled by their wave  vector in spherical coordinates, k = (k, θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Taking k =  ω/c as fixed we write  E(r) ∝  � 2π  0  dφ  � π  0  dθ sin θ ux(θ, φ) a(θ, φ) eik·r  with ux(θ, φ) a polarization vector obtained by co-rotating  the x unit vector when k is rotated from z to (θ, φ), such  that ux(θ, φ) �� k = 0, see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' [S1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In the calcula-  tion we center the beam around θ = 0, and the focal plane  is given by r = (x, y, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  The shape of the beam is de-  termined by the amplitude function a(θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' For a Gaussian  beam we set a(θ, φ) = exp(−θ2/w2  θ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' for the lowest or-  der (l = 1) Laguerre-Gaussian (LG) beam we set a(θ, φ) =  θ exp(iφ − θ2/w2  θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' After performing the above integral we  rotate the results for tweezers propagating along the −y di-  rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Finally, the circular field components σ± shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 1 of the main text are obtained as the projection onto unit  vectors (x ± iy)/  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In Figure S-1, all three polarization  components for a Laguerre-Gaussian beam are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Note  that the σ− and σ+ components have similar intensity while  the π-polarization is suppressed by a factor ∼ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  −6 −4 −2 0  2  4  6  x/λ  −4  −2  0  2  4  z/λ  (σ−)z  −6 −4 −2 0  2  4  6  x/λ  −4  −2  0  2  4 (π)z  −6 −4 −2 0  2  4  6  x/λ  −4  −2  0  2  4 (σ+)z  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' S-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Intensity of the polarization components for a LG beam  calculated at the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' The π-polarization component has been en-  hanced by a factor 100 to make it visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Here we set wθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='6  APPENDIX II : PHASE-SPACE DYNAMICS  We study the phase-space dynamics of the ions by simulat-  ing the time dependent Hamiltonian using trotterization with  time-steps of 10−4 τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' At each time-step we evaluate the ex-  pectation value of the ⟨ˆx⟩ and ⟨ˆp⟩ for the center of mass mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  As expected, we find approximately circular phase-space or-  bits for the even parity states |00⟩, |11⟩, and very little motion  for the odd parity ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' S-2 it is possible to see the evo-  lution in phase-space for all the four spin states in case of per-  fectly aligned and slightly misaligned tweezers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' As described  in the main text we simulate numerically the full Hamiltonian  defined as ˆHsim = ˆH0 + ˆU (xi) + ˆU (xj) where in case of  misalignment ϵ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' ˆU (x) reads as :  U(x) ≈ −U0 e−2((ˆx−ˆϵ)+ˆσzλ)2/w2  0  ≈ − ˜U0 + 4 ˜U0  ˆσzλ − ˆϵ  w2  0  ˆx  + 1  2  ˜U0  �  4  �  w2  0 − 4λ2�  w4  0  �  ˆx2 − 1  2  ˜U0  �  16  �  ˆϵ2 − 2ˆσzˆϵλ  �  w4  0  �  ˆx2  −  �  8 ˜U0ˆσzλ3w2  0 − 4  �  3ˆϵ2 + λ2�  3w6  0  �  ˆx3  +  �  8 ˜U0ˆϵ3w2  0 − 4  �  ˆϵ2 + 3λ2�  3w6  0  �  ˆx3  −  �  2 ˜U0ˆσzλˆϵ−48w2  0 + 64  �  ˆϵ2 + λ2�  3w8  0  �  ˆx4  +  �  2 ˜U0  3w4  0 − 24w2  0  �  ˆϵ2 + λ2�  + 16  �  ˆϵ4 + 6ˆϵ2λ2 + λ4�  3w8  0  �  ˆx4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  with  ˜U0 = U0e−2(ˆϵ+ˆσzλ)2/w2  0  A small tweezer misalignment ϵ gives rise to new spin-  dependent terms in the Hamiltonian that shift the trapping po-  tential in a state dependent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='S-2 is shown how the  dynamics is affected in the case where the tweezers are mis-  aligned by 30 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  APPENDIX III : GATE FIDELITY  We characterize the gate by calculating the average process  fidelity as follows : [S2]:  ¯F( ˆUid, ˆU ˆ  Hsim) =  �  j tr  �  ˆUidˆσ†  j ˆU †  idˆσj( ˆU ˆ  Hsim)  �  + d2  d2 (d + 1)  ,  2  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4  ⟨ˆx⟩  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4  ⟨ˆpx⟩  ϵ = 0  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4  ⟨ˆx⟩  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='4  ⟨ˆpx⟩  ϵ = 30 nm  ψ↑↑  ψ↓↑  ψ↑↓  ψ↓↓  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' S-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Center of mass mode phase-space dynamics for perfectly  aligned tweezer (left) and for 30 nm misaligned ones (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' For the  simulation we used the same parameters as for τ/2 = 120 µs point  in Figure 1(a) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  where ˆUid is the unitary of an ideal geometric phase gate and  ˆσj( ˆU ˆ  Hsim) ≡ trFS( ˆU ˆ  Hsim [|n⟩⟨n| � ˆσj] ˆU †  ˆ  Hsim) projects the  unitary matrix generated by the time evolution of the Hamil-  tonian used for the simulations ˆU ˆ  Hsim on the Fock state |n⟩  and on a d-dimensional representation Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [S1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Spreeuw, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' 125, 233201 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content='  [S2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}
+page_content=' Nielsen, Physics Letters A 303, 249 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE3T4oBgHgl3EQfsgtu/content/2301.04668v1.pdf'}

C9E1T4oBgHgl3EQfpwXL/content/2301.03336v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:80a62531597a567d8e6f4291accd276bee4b130b7cdeed012b2a0420cf5a90ea
+size 240950

C9E1T4oBgHgl3EQfpwXL/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:22604c0eb45f3bd9de0023003b8d84f56c71ce8650cb2c81d5f593f7420d1207
+size 2752557

C9E1T4oBgHgl3EQfpwXL/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d272bb2c8ccf7a533241a3417651b6e40bd9f1467b24bb2391eef4c7c9853f56
+size 113015

CtE1T4oBgHgl3EQfWAR_/content/tmp_files/2301.03109v1.pdf.txt

@@ -0,0 +1,1356 @@
+Cinematic Techniques in Narrative Visualization
+Matthew Conlen
+Our World in Data
+[email protected]
+Jeffrey Heer
+University of Washington
+[email protected]
+Hillary Mushkin
+California Institute of Technology
+[email protected]
+Scott Davidoff
+Jet Propulsion Laboratory
+California Institute of Technology
+[email protected]
+ABSTRACT
+The many genres of narrative visualization (e.g. data comics, data
+videos) each offer a unique set of affordances and constraints. To
+better understand a genre that we call cinematic visualizations—3D
+visualizations that make highly deliberate use of a camera to convey
+a narrative—we gathered 50 examples and analyzed their traditional
+cinematic aspects to identify the benefits and limitations of the form.
+While the cinematic visualization approach can violate traditional
+rules of visualization, we find that through careful control of the
+camera, cinematic visualizations enable immersion in data-driven,
+anthropocentric environments, and can naturally incorporate in-
+situ narrators, concrete scales, and visual analogies. Our analysis
+guides our design of a series of cinematic visualizations, created for
+NASA’s Earth Science Communications team. We present one as a
+case study to convey design guidelines covering cinematography,
+lighting, set design, and sound, and discuss challenges in creating
+cinematic visualizations.
+1
+INTRODUCTION
+Within narrative visualization [57], researchers have identified gen-
+res (such as data comics [3] and data videos [1]) that help better
+unpack and situate their specific application and the features that
+they employ. cinematic visualizations embed data into a three-
+dimensional, time varying scene, utilizing one or more cameras to
+direct the relationship between a viewer and the scene to tell a dra-
+matic data-driven story. This cinematic approach is different from
+the one typically used in information visualization, where graph-
+ics are reduced to a minimal form, incorporating only essential
+elements like axes and data-driven marks [64]. Cinematic visual-
+izations are more maximal: non-data marks are not compressed or
+reduced, instead entire digital worlds are built up around data points
+and included in the visible frame. This technique allows viewers
+to feel present in locations augmented with data-bound objects,
+known as data visceralizations [41]. Narrative documentary visual-
+izations [10] can be produced through the careful editorial direction
+of the cinematography, editing, mise-en-scène, and sound [5].
+Through an analysis of 50 existing cinematic visualizations, we
+identified four salient techniques (in-situ narrators, resolution of
+scale, anthropocentric perspective, and story-driven cameras) that
+cinematic visualizations employ to dramatically engage their au-
+dience through emotionally resonant data-stories. We show how
+these techniques are used throughout the examples analyzed, dis-
+cuss constraints associated with them, and reason about why cine-
+matic visualizations may be effective despite the known pitfalls of
+3D visualization.
+Using the lessons learned from this formal analysis, we produced
+a web-based article containing a series of cinematic visualizations
+relating to climate change, which was published by NASA’s Earth
+Science Communications team 1. We contribute the design process
+for one of these visualizations as a case study, presenting design
+artifacts that were created during our process (both successful
+and unsuccessful), and provide concrete guidelines for designers
+of cinematic visualizations. Our analysis and design artifacts are
+available at https://cinematic-visualization.github.io/.
+2
+RELATED WORK
+Narrative visualizations are used to improve memorability [7, 8],
+to instill empathy or emotion [9], to frame a message [33], and to
+improve engagement [19, 28]. Segel & Heer [57] provided an ini-
+tial characterization of the design space of narrative visualizations,
+which was later elaborated to include additional techniques [60].
+Hullman et al. [34] focused on the role of sequence in narrative
+visualization, characterizing a set of transition types and other high
+level strategies for sequencing visualizations. Tools have been cre-
+ated to support narrative visualization authoring [2, 11, 18, 56], and
+a small number of empirical evaluations of narrative visualizations
+have been conducted [9, 19, 46, 69]. Further work has investigated
+specific genres of narrative visualization such as data comics [3],
+and new genres have emerged beyond Segel & Heer’s initial set,
+such as “scrollytelling.” Here we add to the ongoing conversation
+around narrative visualization by identifying another such genre:
+cinematic visualization. Kosara & McKinley [39] identified the op-
+portunity for narrative visualization researchers to learn from other
+disciplines that engaged heavily with storytelling and multimedia,
+this paper draws on film art scholarship, incorporates a formal
+system of cinematic style into our analysis, discussion, and design
+of cinematic visualizations.
+2.1
+Data Videos & VR
+Data videos were included in the initial set of genres put forth by
+Segel & Heer [57] and first studied closely by Amini et al. [1]. Not
+all data videos are cinematic visualizations (for example, we do
+not consider a video consisting of a sequence of two-dimensional
+infographics to be cinematic), and not all cinematic visualizations
+1https://climate.nasa.gov/news/2933/visualizing-the-quantities-of-climate-change/
+arXiv:2301.03109v1  [cs.HC]  8 Jan 2023
+
+, ,
+Conlen, et al.
+Figure 1: The Dangers of Storm Surge (CV42) is a mixed reality video produced by the Weather Channel. The video opens with
+a close up shot of a news anchor wearing a rain jacket, standing in front of a house (1A). There are audible sounds of rain
+under the anchor’s voice and water dripping down the windows of the house. The camera pulls back revealing that the live
+anchor is being composited into a 3D scene of a suburban neighborhood during a storm surge (1B). There are very few data
+points actually encoded as visual elements. The piece simply shows water rising from zero, to three, to six, to nine feet (1C-D)
+as the anchor narrates with details in reference to the danger of storm surge associated with hurricanes.
+are data videos (for example one in which the visualization is deeply
+tied to the text of an interactive news article). While Amini et al.
+were primarily concerned with the narrative structure and attention
+cues of data videos, we additionally consider the visual and auditory
+style of cinematic visualizations in detail. Under our formal style
+system, our analysis of editing is most closely related to Amini’s
+work, however that is only one of four dimensions we consider.
+Bradbury & Guadagno [10] studied viewer preferences in docu-
+mentary narrative visualization (a subgenre of data videos in which
+data is presented using the techniques of documentary film), and
+found that audiences may prefer when documentary data videos
+include voice-over narration and on-screen narrators. We build on
+their analysis of the use of narrators and narration, in particular
+during our discussions of in-situ narrators that interact with data-
+bound objects digitally rendered into the space around them, and
+of the use of sound in cinematic visualizations. Video producers
+have extended the traditional documentary visualization format to
+enable interactivity such as user selected paths through the content
+and manipulable graphics [29, 63].
+Immersive data stories [35] have been discussed within the
+emerging field of immersive analytics [44] and have been shown
+to allow viewers to examine data at multiple scales, support im-
+mersive exploration, and create affective personal experiences with
+data [36]. Lee et al. [41] introduced data visceralizations, where
+physical quantities are visualized in 3D virtual reality scenes. This
+paper helps to bridge the gap between data visceralization and nar-
+rative visualization by showing how cinematic techniques can be
+used to create author-guided narrative visualizations using data vis-
+ceralizations. Cinematic visualizations similarly attempt to immerse
+viewers and create emotionally resonant experiences, although in
+contrast to immersive visualizations they are typically viewed on a
+standard 2D screen with limited (or no) user control of the camera.
+There are several toolkits for creating immersive data visualizations
+and data stories on augmented reality devices [21, 55, 59].
+2.2
+3D Computer Graphics
+Animation [62] has been a partner discipline with visualization for
+some time. Classic principles of animation [61] have been adapted
+for digital usage [40] and subsequently for information visualiza-
+tion [30]. With realistic camera models [38] and improving render-
+ing capabilities [20] digital animation became a tool to create Holly-
+wood films [31]. While 3D graphics have been used in visualization
+to limited success, e.g., to display hierarchical information [17], the
+use of 3D graphics in information visualization is often avoided.
+A broad body of research documents potential pitfalls, including
+that volume is not a perceptually effective encoding channel [16],
+and that 3D projections introduce distortion and occlusion [67].
+We find that designers of cinematic visualizations may intention-
+ally use suboptimal encodings in support of more visceral [41] and
+emotionally resonant [28] graphics.
+The use of 3D does find more regular application in scientific visu-
+alization [4, 65], including its use in storytelling [22, 43]. Borkiewicz
+used the term cinematic scientific visualization [6] to refer to a class
+of narrative data videos that focus on scientific data. Here we use
+cinematic visualization in a similar way but do not restrict the data
+to be strictly scientific or inherently spatial. Unlike Borkiewicz,
+our description encapsulates visualizations which are not embed-
+ded in films, but may be, for example, displayed as an animation
+accompanying a news article.
+2.3
+Film Art
+Bordwell and Thompson [5] define narrative and style as the two
+major formal systems of film. While prior work has examined se-
+quence [34] and narrative structure & attention cues [1] in data
+videos, we observe that cinematic style has far less visibility in the
+critical vocabulary of data visualization. Style plays a crucial roll
+in filmmaking, enabling directors to “confirm our expectations, or
+modify them, or cheat, or challenge them. [...] A director directs
+not only the cast and crew. A director also directs us, directs our
+attention, shapes our reaction.” [5] This paper brings Bordwell and
+Thompson’s formal system of cinematic style into the world of data
+visualization, and uses it to examine how narrative visualizations
+borrow techniques from cinema while departing from many of the
+traditional practices advocated by visualization research.
+Style consists of four features, which together make up a film’s
+style, each now briefly described. Mise-en-scène refers to every-
+thing that is seen in the frame, including lighting, actors, objects,
+backdrops, and so on [27]. Cinematography refers to the use of the
+
+Cinematic Techniques in Narrative Visualization
+, ,
+Figure 2: (A) In [REALISTIC] Elephant rocket fuel - Saturn V (CV29), a model Saturn V rocket takes off, however, instead of
+flames exiting the bottom of the spacecraft, elephants are expelled, the number of elephants represents the corresponding
+mass of fuel. This video may not make for a particularly effective visualization in terms of conveying precise quantities, but
+the style successfully uses humor in order to call attention to the fact that rocket launches use a quantity of fuel so great
+it is appropriate to measure it in terms of dozens of elephants. (B) In Here are 120 million Monopoly pieces, roughly one for
+every household in the United States (CV6) by the New York Times the pile of Monopoly pieces is first seen from a far, before
+the reader scrolls down the page to trigger the camera zooming in to the very top of the pile, dramatically revealing what a
+disproportionately small portion of families provide most political funding.
+camera, how shots are composed and framed [26]. By placing ele-
+ments at specific locations within the frame, they can be perceived
+either as the subject or the background of the image [25]. Editing
+is the composition of multiple pieces of footage in time or space,
+creating transitions between perspectives and scenes [54]. Sound
+is the audio used, whether it be music, voice over, or sounds from
+characters or objects on screen [32]. Our analysis of cinematic visu-
+alization identified techniques along these dimensions of style that
+designers can use to enhance their presentation of data narratives.
+3
+CINEMATIC VISUALIZATION SURVEY
+We collected cinematic visualization to analyze by surveying liter-
+ature on narrative visualization [1, 6, 34, 35, 43, 57, 60], browsing
+the information visualization awards website Information is Beau-
+tiful [45] and the PacificVIS storytelling contest [50], and searching
+for news articles, blog posts, conference talks, and videos which
+were described using combinations of the keywords cinematic, data,
+data video, dataviz, datavis, visualization, news, newsgames, immer-
+sive, mixed reality, 3d, and video. We searched the portfolios of the
+creators of the visualizations found initially and their collaborators.
+A full list of the cinematic visualizations can be seen in Figure 7 in
+the appendix of this paper; we refer to these studies by identifiers
+throughout the paper (e.g., CV4 refers to the fourth example in the
+table). Our analysis considered 50 cinematic visualizations. While
+the corpus is not exhaustive, the examples expose the variety of
+media (interactive news articles, YouTube videos, and TV segments)
+which cinematic visualizations occupy and the messages that they
+deliver. The examples visualized a broad range of data types, in-
+cluding datasets both with and without physical and geographic
+dimensions.
+Rather than empirically evaluate specific design patterns utilized
+in the visualizations, we turn to the means of understanding plot
+devices [57], sequencing [1], and film style [5]. We analyzed the
+style of each example along the dimensions of mise-en-scène, cine-
+matography, editing, and sound using the 4-step analysis process
+described by Bordwell and Thompson [5], a canonical method of
+film analysis. For each example we first identified the main com-
+municative goals of the visualizations, and then studied the salient
+techniques applied within the mise-en-scène, cinematography, edit-
+ing, and sound which supported these narrative goals. We then used
+iterative coding to categorize the salient techniques used across
+the examples. Usage of these techniques are shown in Figure 7, for
+example we recorded many ways in which a viewer’s attention
+is guided (through color, light, annotations, and narrators in the
+mise-en-scène) and use of cinematographic techniques like point-
+of-view perspective and user-controlled cameras. The table shows
+that the medium of the cinematic visualization has some impact on
+the techniques used, for example cinematic visualizations embed-
+ded in online articles rarely use sound, but often utilize user-paced
+segments, while those presented as videos make heavy use of sound.
+
+A
+B
+Here are 120 million
+ Monopoly pieces, roughly
+ one for every household
+in the United States.
+Just 158 families have
+provided nearly half of the
+early money for efforts to
+capture the White House., ,
+Conlen, et al.
+Figure 3: VFX Artist Reveals the True Scale of the Universe fea-
+tures a live-action narrator alongside scaled-down 3D mod-
+els of celestial bodies.
+3.1
+Design Techniques
+Through this analysis we identified salient recurring techniques that
+were frequently applied to support the communicative goals of the
+visualization, including the use of in-situ narrators, anthropocentric
+perspective, resolution of scale, and story-driven cameras.
+In-situ narrators mediate interactions with diegetic data.
+Perhaps the most novel technique that we identified in cinematic
+visualizations is the use of in-situ narrators, in which the mise-en-
+scène contains a character that interacts directly with on-screen,
+diegetic data.2 In contrast to traditional documentary visualization
+narrators who might participate from off-screen (“voice of god”) or
+refer to data visualizations rendered as two-dimensional holograms
+or composited over top the video [10], in-situ narrators are under-
+stood by the viewer to be able to see and interact with the diegetic
+data either through the use of superimposed data visceralizations
+2Something which is diegetic exists in the same universe as the characters on screen; we
+use the phrase diegetic data to refer to data-driven elements which are part of—rather
+than composited over—the scene shown in the frame.
+(CV35, 40, 42, 43) or, in one case, data physicalization [37] (CV41).
+This (typically) mixed reality environment serves an important
+role for narrative visualization, allowing the on-screen narrator to
+mediate interactions between the audience and the graphics, letting
+them provide additional context and push the storyline forward.
+These narrators, essential components of the mise-en-scène, can
+also help concretize a visualization’s anthropocentric perspective,
+reinforcing the idea that data is being displayed at a human scale.
+In The Dangers of Storm Surge (CV42), one exemplar of this
+technique (Figure 1) produced by the Weather Channel, a news
+anchor wearing a blue jacket explains the dangers associated with
+flooding due to storm surge. The graphics are coordinated with
+the narrator’s script and appear to respond to his dialogue, the
+composition of the frame inviting comparison between the man and
+the height of the water. The narrator is the primary subject from the
+start of the clip, positioned centrally in frame and maintaining focus
+due to visual cues like his bright blue coat, the circular platform
+upon which he stands, and the shot composition. To call attention
+to the water’s height at certain key moments, a brightly colored
+annotation is projected onto the crest of the surge.
+An anthropocentric perspective transports viewers and
+enables drama. One notable aspect of cinema is how the camera is
+able to transport the audience into the scene: people watching sus-
+pend disbelief [24] to allow themselves to wholeheartedly imagine,
+or “believe”, that they are in the scene, seeing things through the
+camera lens. That is, the camera’s perspective becomes the viewer’s
+point of view, they are one and the same. The height, angle, and
+distance of a camera in relation to objects in the scene all play a role
+in how a viewer will interpret and respond to the frame that they
+ultimately see [5]. When a camera is placed high above a setting,
+the viewer feels like they are also high above it. When a camera
+is placed at eye level, a viewer feels as if they are standing there
+watching the subject. For example, both CV1 and CV26 utilize unit
+visualizations and concrete scales to visualize quantities in relation
+to the size of Manhattan, but each uses perspective to impact the
+viewer’s experience in a different way. In CV1 the data being dis-
+played (plastic bottle usage) is not directly related to the locations
+being used as concrete scale referents, and an overview shot is
+used, letting the viewer absorb the scale of the data rather than
+the details and textures of the city itself. In contrast, CV26 begins
+with a shot from a camera placed at eye-level, looking at several
+of the city’s ubiquitous yellow taxis, transporting viewers to the
+city at street level, and forcing them to reckon with the data being
+displayed (New York City’s annual green house gas emissions) in a
+much more visceral way [41].
+Some cinematic visualizations place the camera perspective
+somewhere that is humanly impossible. However, if the audience
+suspends disbelief, the camera can carry the viewer through these
+otherwise inaccessible spaces, for example, CV12 shows an anima-
+tion of the Cassini spacecraft as it orbited and eventually crashed
+into Saturn. Choice and Chance (CV11), visualizes the events of the
+2016 Pulse night club shooting in Tampa Bay, positions a camera
+looking “through” the roof of a nightclub. Because the scene is
+shot using a digital model instead of a real location, the roof of the
+club can simply be removed and problems of occlusion go away.
+Changing perspectives can also shift the subject of the scene or
+add emotional content, for example, when the camera moves to
+
+A
+SUBSCRIBE
+B
+c
+Rige!Cinematic Techniques in Narrative Visualization
+, ,
+Figure 4: New York City’s greenhouse gas emissions as one-ton spheres of carbon dioxide gas, a cinematic visualization produced
+by Carbon Visuals and released online. The cinematic visualization uses a variety of different camera views, along with stark
+colors to guide viewers through an explanation of the scale of the city’s greenhouse gas emissions. The number of instances
+of the blue sphere is driven by the rate of emissions. As this number grows the city buildings serve as a concrete scale.
+reveal something that wasn’t already in the frame, the audience
+experiences seeing it for the first time. In Choice and Chance the
+camera moves to different vantage points throughout the model as
+the story progresses. The camera remains in an overview shot for
+the majority of the article, but moves to ground level at the climax,
+elevating the intensity of the shot by placing the viewer into the
+perspective of a bystander.
+Author-defined camera trajectories can be played, paused,
+and (lightly) modified by viewers. The cinematic visualizations
+that we analyzed tended to use author-driven narrative structures [57],
+with most user interactions consisting of the user clicking or scrolling
+to trigger the visualization to continue to the next stage (e.g., CV2,
+5-17, 21-22). Operationally, this requires animating the position
+and orientation of a digital camera model along a track specified
+by the author, and has been used heavily by cinematic visualiza-
+tions embedded in articles (16 out of 22). The other way in which
+(constrained) interactivity was employed was allowing the manipu-
+lation of 3D models. In most cases this means the user can position
+the camera at a particular location around the model (see CV17 for a
+stereotypical example). These models might be scientific (CV13,17)
+or cultural (CV5) objects that would be otherwise inaccessible to
+the audience viewing the visualization. It is common for orbital
+cameras to be used, constraining the camera’s focus to remain on
+a particular object of interest while allowing the user to exercise
+control over viewing angle and zoom level (Fig. 7D). Cinematic
+visualizations that support these interactions must be rendered in
+real-time, limiting the fidelity at which the models may be rendered.
+Visualization techniques are combined toward resolution
+of scale. While we traditionally think of 3D graphics as ineffective
+for encoding quantities [16], a recurring theme in our examples is
+the use of 3D graphics to visualize and communicate quantities of
+a massive scale (e.g., CV1, 6, 8, 26-28). Quantities at a scale beyond
+what we experience in daily life (i.e. hyperobjects [47]), like amount
+of carbon dioxide emitted from NYC annually (CV26), may be es-
+pecially difficult for people to picture because we rarely, if ever,
+interact with quantities of such a size. Cinematic visualizations can
+convey a quantity of scale in a concrete and affecting way by using
+cinematography to establish the viewer’s point of view from the
+ground, a position which often serves as the implicit zero point
+of a y-axis. We observed that several visualization techniques are
+naturally expressed in cinematic visualizations, including data vis-
+ceralizations [41], unit visualization [51] and concrete scales [14].
+For example, in CV27 the viewer sees a city park, including trees,
+people standing in a grassy field, and a ten meter tall blue sphere
+representing the actual size of one metric ton of CO2 (data vis-
+ceralization). As the scene progresses, many more spheres appear,
+each representing one metric ton of CO2 (unit visualization), until
+so many appear that the camera must zoom out, above the park,
+observing the growing pile of spheres in comparison to the city
+buildings (concrete scale).
+Objects which are used as backdrops—for example a city skyline
+(CV11) or parked car (CV42, Fig. 1)—may serve double duty as
+concrete scale referents and contextual elements. The use of 3D
+graphics affords designers the ability to use concrete scales (CV1,
+26) and visual analogies (CV29, 36) to (re-)contextualize the size of
+objects, and digital sets are constructed to facilitate comparisons
+that are impossible to make directly in the physical world (CV1,
+27) and use point-of-view perspective to impart a visceral sense of
+magnitude. The visual medium is rich with possibilities for analogy.
+For example, in [REALISTIC] Elephant rocket fuel - Saturn V (CV29,
+Fig. 2), designer Maxim Sachs renders the launch of the Saturn V
+rocket, except that the rocket expels elephants behind it as it travels,
+rather than exhaust. The elephants represent the mass of fuel that
+is being expended. By juxtaposing these images, Sachs is able to re-
+frame an abstract quantity of rocket fuel in terms that people may
+have more familiarity with, and do it with a sense of humor that
+may make the visualization overall more memorable or engaging
+for its audience [8]. In a more typical case, the narrator of CV40
+asks the audience to imagine if Earth were the size of a tennis ball,
+and then, using this new scale, shows the relative size of different
+planets, moons, and stars. These planets are compared against one
+another, rendered into real-world footage including a narrator who
+provides guidance and relevant facts about the celestial objects.
+
+, ,
+Conlen, et al.
+Figure 5: How Much is a Gigatonne? shows one gigatonne of ice in Central Park, New York. A digital set (A) is designed including
+multiple cameras, lighting, and data-driven and contextual elements. Footage from the various cameras is composed to create
+the final sequence (B-E). This was one of several videos that we developed for an article published on NASA’s climate website.
+View the full videos at https://cinematic-visualization.github.io/.
+They are shown embedded into several settings, for example an
+office, a Los Angeles street, and the New York City skyline.
+3.2
+Constraints
+The time-based format does not support a high data density.
+Traditional information graphics often present a data-dense display
+with minimal “non-data ink” [64] to remove possible distractions
+and optimize the display for tasks such as value look-up and com-
+parison. In some cases, designers may choose to add additional
+illustrative features to increase the memorability of the visualiza-
+tion [7]. In contrast, cinematic visualizations utilize diegetic data,
+embedded in a three dimensional scene with other elements which
+contextualize the scene (see CV35 for a striking example). In cine-
+matic visualizations (e.g. CV40,42) the elements surrounding the
+data fulfill a dual role as both data and non-data ink; they add
+spatial presence to the visualization [12], supporting a sense of
+transportation to the virtual world for viewers, while simultane-
+ously serving as guides and axes, points of reference for concrete
+scales [14]. Rather than densely packing data, we see that cinematic
+visualizations often only show one or a few data points in the frame,
+favoring to include additional contextual elements that help add
+emotional resonance to the data-story being told.
+Designers trade-off between perceptual effectiveness and
+dramatic narrative. Visualizations that employ 3D graphics are
+often ineffective perceptually. These graphics may use sub-optimal
+encoding channels like volume and can further bias judgement
+through distortion and occlusion. Cinematic visualizations are not
+appropriate when the task is centered around value judgements.
+Instead, we see cinematic visualizations effectively used when a
+rough estimate of values is sufficient and the precise value is not
+of central importance (e.g. CV29). Many of the cinematic visual-
+izations that we analyzed use a volume encoding to display data
+(CV1,6,26,27,35). Volume is a less effective encoding channel com-
+pared to position and may cause the audience to misestimate the
+true quantity. This trade-off may be acceptable depending on the
+data being presented and the precision with which the author hopes
+it will be apprehended.
+4
+CASE STUDY: HOW MUCH IS A
+GIGATONNE?
+We collected and studied the aforementioned cinematic visualiza-
+tions while exploring designs to support the communication ob-
+jectives of NASA’s Earth Science Communications Team. Climate
+change is a complex, multi-faceted issue of global importance [49]
+and the team is tasked with maintaining climate.nasa.gov, a website
+that tracks vital statistics about Earth’s climate, and delivers up-
+dates about global warming to a diverse global audience of millions
+of readers. The team uses traditional information graphics [48], as
+well as narrative visualizations (e.g., [53]), to highlight how scien-
+tists know that anthropogenic global warming is truly happening,
+what changes have taken place in Earth’s climate so far, and why it
+is an important topic for readers to understand even if it does not
+seem to be affecting them. However, the team sought data-driven
+stories that more viscerally engaged their audience and connect
+
+Digital set design
+Cam1 (God's eye view)
+D
+Cam2 (bird's eye view)
+Lighting: Global Illumination
+Data-driven element
+Geographic elements
+Cam3 (point-of-view)
+Texture from satellite images
+A
+Rendered output
+B
+Central Park
+D
+C
+ewYorkCit
+God's eye view (Establishing shot)
+Point-of-view (Establishing)
+Point-of-view (Initial action)
+Medium-long shot (Peak)Cinematic Techniques in Narrative Visualization
+, ,
+Figure 6: We explored many different designs, these were left on the cutting room floor. The designs were dropped for reasons
+including poor perceptual effectiveness (A-C), locations too small for the scale of the data (D-F), and designs too illustrative
+and not physically accurate enough (G-H). It was particularly difficult to identify locations that were broadly recognizable
+from a 3D reconstruction but also suitable to server as a concrete scale referent.
+the planetary scale data of climate change to a human scale that
+readers can readily understand.
+Within the domain of climate change communication is a range
+of research investigating how to effectively communicate the latest
+science to a broad audience. High level principles of climate change
+communication have been synthesized by the Center for Research
+on Environmental Decisions [58]. We think cinematic visualizations
+are well suited to satisfy principles “Get Your Audience’s Attention“
+and “Translate Scientific Data Into Concrete Experience.” Here
+we describe how our work creates connections between ongoing
+investigations in narrative visualization, computer graphics, and
+film art to achieve this.
+Guided by editorial priorities set by NASA’s Earth Science Com-
+munication team, we produced an article consisting of a several
+cinematic visualizations to communicate massive quantities related
+to climate change. We endeavoured to make them interpretable and
+meaningful to a broad public audience. These visualizations were
+eventually published to an audience of millions. Here we describe
+our design process to create cinematic visualizations, identifying a
+general workflow of use to practitioners who wish to create this
+type of visualization themselves, and to tool-builders who wish to
+provide better support for authoring cinematic visualizations in
+the future. As with visualization production in general, these steps
+are not necessarily linear; rather, the process is iterative and error
+prone, and may require going back to earlier steps if it becomes
+apparent that a design is not working. We experienced many failed
+attempts (see Figure 6) before arriving at our final designs.
+4.1
+Pre-Production
+Narrative. Quantities of ice loss are measured in gigatonnes, a
+unit of mass corresponding to one million metric tons. Statistics
+about ice loss are often reported using this unit, for example Earth’s
+polar ice caps are losing about 426 gigatonnes of ice per year, at
+the time of writing. The scale of the unit here hides the fact that
+426 gigatonnes is a massive amount of ice. Our goal was to provide
+a visualization that would allow our audience to better interpret
+these statistics going forward. We collected statistics on ice loss in
+Greenland and Antarctica (the two ice sheets) over the course of
+significant periods, such as the amount of ice lost between 2002-
+2017 when NASA’s Grace satellite was actively observing the polar
+ice caps, or since the start of the 20th century (5,000 and 49,000
+gigatonnes, respectively).
+We settled on cinematic visualization because it is a natural fit
+for the use of concrete scales, we wanted to draw people’s attention,
+there is a relatively small amount of data that we are showing, and
+we wanted to display the data in a context that conveyed corporeal
+urgency. Given the affordances identified in Section 3, a cinematic
+visualization was an appropriate choice for our task of visualizing
+quantities related to climate change in a way that would capture
+the attention of our audience and allow them to comprehend the
+data in a concrete way. We ultimately chose the form factor for our
+visualization to be an interactive article containing a series of short
+cinematic visualizations. The visualizations were embedded as pre-
+rendered videos, which could be loaded dynamically, allowing for
+a certain amount of interactivity. Depending on the use case, one
+must determine whether real-time rendering is needed or not. Using
+real-time rendering limits the level of photorealism [52], but enables
+another level of interactivity, letting the user control the camera
+and interact with elements in the scene (Fig. 7D). We intended
+the narrative structure of our visualization to be largely author-
+driven [57], and decided that real-time rendering was not required.
+After determining that a cinematic visualization was appropri-
+ate, we began outlining possible scripts and creating storyboards
+in which we sketched ideas for locations, cinematography, and se-
+quencing of shots. We first sought to identify locations that would
+serve as effective backdrops, allowing people to gain a concrete
+understanding of the size of data in familiar locations. We consid-
+ered natural locations like the Grand Canyon, Monument Valley,
+Mt. Everest, and Uluru, urban environments like Houston, New
+York City, San Francisco, and St. Louis, and other man-made sites
+like football stadiums and the Hoover Dam. Within each of these
+environments we created sketches to help determine the camera
+placement, mise-en-scène, data, and annotations that the visualiza-
+tions would require, and wrote rough scripts to define the narrative
+structure.
+While we wanted to place data in a variety of different envi-
+ronments so that our diverse audience would be able to connect,
+
+2000
+1979
+2009
+Carbon
+Emissions
+7021
+M, ,
+Conlen, et al.
+ultimately many of these locations were not used. See Figure 6 for
+examples of some of the locations that were not able to support
+both focus and context at an anthropocentric perspective. The final
+article consisted of videos visualizing one, then 5,000, then 49,000
+gigatonnes of ice. The videos were embedded throughout the text
+of an article which provided context. In the first and last videos the
+user could click to choose to play videos displaying the relevant
+quantity of ice in different locations. Here we look closely at the
+design process for the first video, showing one gigatonne of ice in
+Central Park, New York City.
+4.2
+Principal Photography
+With the storyboards and scripts ready, the source footage that
+would make up the final video needed to be created. We chose to
+use Blender for this process, which provides both an interactive
+GUI-based interface as well as a Python API that allowed us to
+load, transform, and bind data to objects in a 3D scene. We created
+renders for many different scenes, although ultimately ended up
+using a small number of them in our published pieces.
+Mise-en-scène. The elements that constitute the mise-en-scène
+of a cinematic visualization need to be created and arranged. Be-
+cause many of our scenes take place in real-world locations, we
+were able to utilize existing open data sets to import geographic
+data, including 3D models of buildings and terrain data. In addition
+to elements derived from real-world locations, we added elements
+which would be parameterized by data, for example the large block
+of ice placed in Central Park (Figure 5). After the models have been
+created, they need to be assigned a material, which (along with
+lighting) will determine how they appear in final renders. We chose
+to use a flat shading for the buildings and other environmental
+elements. This gave these elements less visual weight while still al-
+lowing them to be easily identifiable. We considered using a similar
+flat style for the data elements, but ultimately decided to add a more
+photorealistic ice material which would allow the data to stand out
+against the buildings and reinforce the idea that we were showing
+a concrete amount of ice. While many of the examples that we saw
+utilize a studio lighting setup to control shadows and reflection, we
+opted to use simple global illumination to emulate the sun shining
+in our outdoor scene. This meant our lighting was realistic for the
+location and the setup was quite simple, but we were limited in our
+ability to use lighting as a tool to guide attention, as we saw it used
+(for example) in CV15.
+With the scene constructed, the next step was to bind the data.
+This was the point at which we realized that many of the set lo-
+cations were not going to work with the data we were hoping to
+visualize (“data changes everything” [66]). For example, a gigatonne
+of ice placed in a football stadium (Fig. 6D) would extend over 200
+kilometers into the sky, making it difficult to view both the diegetic
+data and the stadium itself simultaneously. For our visualizations
+we were simply assigning the dimensions of a primitive 3D object
+based on calculations related to the mass of ice melt over specific
+periods, along with the density of ice, in order to create blocks of
+ice which were physically representative of the quantity lost.
+Cinematography. After we incorporated our data into the scene
+it was time to add animation and cinematography. Blender supports
+a keyframe-based animation system which made it simple to add
+basic animations to the size and locations of elements in the scene,
+as well as the position and perspective of cameras. Working off of
+the storyboards that we had created, we placed cameras (shown
+in Figure 5) that would be physically realistic and familiar: we use
+three cameras, one a human point-of-view, one a bird’s eye view
+(as if it were taken from a helicopter circling the city), and one a
+"god’s eye view" taken from the perspective of a satellite overhead.
+The satellite camera allowed us to create an initial establishing shot,
+while the other cameras provided views that supported a ground-
+level view as well as an overview. When sequenced together, these
+camera perspectives allow us to present focus plus context [13] to
+the viewer, and support our narrative goals [1].
+4.3
+Post-Production
+Once the source material was created, we needed to edit it to form
+a coherent narrative, for example by combining multiple videos in
+sequence, adding annotations on top of the video to add context,
+and adding sound to add presence, guide attention, and provide
+details. Any visual effects must be added at this stage. For example,
+in the case of embedding digital data objects into physical footage
+of a narrator, a “match moving” process to align the digital and
+physical scenes would need to be performed [23].
+Editing. We combined footage from multiple cameras, compos-
+ing shots into a narrative structure, starting with establishing shots,
+then initial action, peak, and finally release [1]. The sequence of
+images is important to advance the role of narrative, pacing, and
+mood. Narrative visualizations often include annotations to provide
+additional context and explain to viewers what it is they are seeing.
+In the case of cinematic visualizations these annotations can be
+composited over the source footage using standard video editing
+software. Some examples that we saw embed annotations directly
+into the 3D scene itself, which requires them to be embedded in the
+source footage directly. We chose to composite annotations rather
+than include them “in-situ” as it facilitated more rapid iteration dur-
+ing the editing process, allowing us to change the timing, location,
+and content of annotations, without needing to re-render any of
+the source footage — a potentially time-consuming process.
+Sound. In our work we ultimately did not use audio, instead
+opting to embed the videos in a larger text article, which would
+serve to provide viewers with context for the visualization. This is
+a limitation and something to be explored more in future work, as
+audio can be a useful tool in cinematic visualization to set tone and
+drive narrative.
+4.4
+Publication
+Once the article was completed and approved for publication, it
+was posted to NASA’s climate website. We did not collect detailed
+metrics on how readers interacted with the videos on the article
+itself, but can see how users responded to posts on the NASA
+Climate Facebook, Instagram, and Twitter pages. These posts—
+which contained a link to the article and (in some cases) directly
+embedded the video set in New York City—were collectively viewed
+tens of thousands of times, received thousands of engagements
+(likes, comments, shares), and the article was subsequently shared
+by other organizations such as the United States Department of
+
+Cinematic Techniques in Narrative Visualization
+, ,
+Agriculture and the World Meteorological Organization, as well as
+by individual scientists and meteorologists.
+Across all of the social platforms users left 94 direct comments,
+with topics ranging from positive (for example, some explicitly
+expressing that they like this type of visualization “We need more
+of these types of comparisons in the media”, “This is an amazing
+visualization. Thanks NASA!”, or asking for similar visualizations
+of different quantities “It would be very interesting to see this illus-
+tration but with the predicted sea level after all the ice in Greenland
+and Antarctica melt. Can you show that?”) to concern about the
+data being visualized (“Oh my God. Come to our aid.”, “Thanks for
+helping us comprehend the enormity of this sad news!”, a GIF of
+a cartoon rodent crying) to climate change denial (“Where’s your
+proof”, “Wow, as much as 2 millimetres. Measured by satellite too”).
+The comments were distributed roughly uniformly across the three
+types (positive attitude toward visualization, concern about climate
+change, and climate change denial), but varied heavily across plat-
+forms, with users on Facebook expressing concern or denying that
+there is a climate problem, users on Instagram leaving both positive
+and concerned comments, and users on Twitter expressing a range
+of concern, denial, and a positive attitude toward the graphic.
+5
+DISCUSSION
+Cinematic visualizations can engage viewers with dramatic and
+visceral presentations of data, highlighting particularly important
+data points, and presenting an author-guided tour through data
+embedded in a relevant context. On the other hand, they may be
+poor choices for communicating large amounts of data and are
+not optimal in terms of perceptual effectiveness. If a cinematic
+visualization is appropriate, it will require a broad range of skills —
+such as cinematography, narrative, 3D modeling, video editing, and
+possibly acting — and a time-consuming iterative design process.
+5.1
+Challenges of Creating Cinematic
+Visualizations
+While cinematic visualizations can capture the attention of their
+audience and help viewers relate to the data in a concrete way,
+they can be challenging and time-consuming to produce. Here we
+discuss some of the challenges inherent in creating an effective
+cinematic visualization.
+One of the most apparent difficulties of cinematic visualization
+is the potentially overwhelming size of the design space. Works
+in this genre typically use three visual dimensions, plus time and
+sound. The methods that allow us to analyze and critique cinematic
+visualizations (e.g., [5]) do not necessarily help us to create them.
+That is, they are difficult to use generatively. While information
+designers are familiar with the attention to detail that is required
+when placing objects in a frame in order to achieve an effective
+visual hierarchy, in cinematic visualizations there are also objects
+outside of the frame that affect the style and tone of the visualization.
+For example, the placement of the camera in relation to the subjects,
+the focal length of the camera, and the placement and strength of
+light sources are all instrumental in creating a shot which can easily
+be decoded by viewers.
+There is a diversity of tasks that need to be completed in order
+to create a cinematic visualization, each requiring a separate set of
+skills. For example, in addition to skills required for traditional visu-
+alization (data analysis, transformation, and visualization) and nar-
+rative visualization (understanding audience, storytelling, graphic
+design), cinematic visualization will often make use of animation,
+cinematography, lighting, motion graphics, 3D modeling, sound
+design, video editing, and (sometimes) acting. The skills that make
+one a good 3D modeler are not necessarily the same skills that make
+one a good storyteller, and so graphics of this type often require
+a diverse team to create. Furthermore, for ray-tracing renderers,
+there is a large gap between prototypes and final rendered output,
+challenging the iterative design process.
+5.2
+Considerations for Cinematic Visualization
+Creators
+While cinematic visualizations share many of the same design goals
+of more traditional narrative visualization (e.g., guide the viewers’
+attention), the way in which these goals are operationalized differ.
+Here we highlight ways that these design goals were operational-
+ized across the four dimensions of style, both in our own work and
+in the examples analyzed. For a full breakdown of the techniques
+used in each example, see Figure 7.
+Mise-en-scène. Objects’ sizes, colors, shapes, textures, and place-
+ment in relation to one another can all be used create an effective
+visual hierarchy. For example, to guide a user’s attention in a cin-
+ematic visualization, a designer might choose to use lighting to
+cast a glow around an object (CV11), or change the object’s color
+(CV2, CV13) so that it stands out. In How Much is a Gigatonne,
+the ice’s large size, color, and shine draw a viewers attention to
+it in contrast with the surrounding buildings, which are smaller,
+grayscale, and matte. The mise-en-scène is designed both to com-
+municate information—including using narrators (CV42), diegetic
+data (CV35), and visual analogies (CV6)—and to add dramatic affect
+(e.g. CV11, CV40).
+Cinematography. Perspective can be used both to drive narra-
+tive and to set tone, as well as to provide focus plus context. The
+position (CV26), angle (CV28), or focus (CV2) of a camera can be
+modified so that the object becomes the focal point of the frame.
+To help narrow the large space of possible cinematic visualizations,
+and make effective use of the frame, designers of cinematic visual-
+ization may study how shots are composed and sequenced in films.
+In How Much is a Gigatonne?, we rendered footage from multiple
+cameras in order to create close-up, medium, and wide shots. Some
+cinematic visualizations enable limited user-control of the camera,
+for example letting the user trigger the next stage of animation
+(CV9) or rotate their perspective (CV13). Often the camera needs
+to track a particular object in the scene (CV12). If this object is in
+motion you may need to set your camera to track it. Planning the
+path of the camera so that the object of interest is not occluded by
+other objects and so that motion is smooth and visually pleasing
+can be difficult. This may be done algorithmically [15, 68] or by
+hand.
+Editing. Putting the footage into a particular order progressively
+reveals information to convey the authors’ intended message. Edi-
+tors may use footage from one camera at one location (CV29), or
+multiple cameras at multiple locations (CV40). The editing tech-
+niques used in data videos—particularly the use of establishing,
+
+, ,
+Conlen, et al.
+initial, peak, and release shots—has been studied in more depth by
+Amini et al. [1]. Similar to movie makers, creators of cinematic visu-
+alizations may use the technique of storyboarding to prototype and
+communicate their scenes in a lo-fidelity form before endeavouring
+on the time intensive task of 3D modeling and rendering. In How
+Much is a Gigatonne we use establishing shots to situate the viewer
+before initiating action from the perspective of the ground level (an
+anthropocentric perspective), before cutting to the vantage point
+of a helicopter, using the city skyline as a concrete scale.
+Sound. Audio can set tone (CV25), cue attention (CV28), and
+impart additional details through narration on (CV40) or off-screen
+(CV45). Music (CV29) and ambient sound (CV26) can affect the tone
+of the visualization and add presence to the scene, for example
+CV29 uses combines techno music and a visual analogy of of the
+weight of rocket fuel (measured in elephants) to create a humorous
+juxtaposition which may make the visualization more approachable
+and less dry. CV26 uses diegetic sound (taxi cabs honking) to rein-
+force the anthropocentric perspective. In How Much is a Gigatonne
+we did not use sound (neither did most of the other visualizations
+that we analyzed which used an “article” format), but effective use
+of both the visual and auditory channels has been shown to lead to
+improved outcomes in multimedia learning contexts [42].
+5.3
+Implications for Authoring Tools
+As cinematic visualization is a newly emerging genre, there is rel-
+atively little tool support to facilitate authoring of this type of
+visualization. Instead, creators turn to general purpose 3D software
+that was designed to support a breadth of use cases such as architec-
+tural design, modeling, and narrative animation. These tools, while
+powerful and expressive, may overwhelm users with complexity
+that is incidental to the task of creating a cinematic visualization.
+For example, objects are assigned materials which are powered by
+low-level shader code. One can not choose, e.g., between “realistic”
+or “cartoon” aesthetics but instead must compose low level shader
+components to achieve the desired look.
+These tools do not support the basic building blocks of visualiza-
+tion, such as easily ingesting data and binding data values to objects
+in a scene. Instead, users must write custom scripts to handle any
+such task. The interfaces in general are multi-modal: most 3D mod-
+eling work is done directly through a GUI, but data-driven work
+needs to be done in code; shaders are described using a directed
+graph. Authors are forced to context switch between drastically
+different environments, arguably making it harder to iterate.
+The task of 3D rendering can be computationally intensive. De-
+pending on the output resolution, complexity of the scene, and
+computing power available, a short (30 seconds) animation could
+take several hours to render. There is a large gap between the
+fidelity of the final renders and what a designer sees while con-
+structing the scene. This setup makes it important to create test
+renders frequently, but makes it hard to have a rapid feedback loop.
+5.4
+Limitations of our Work
+Our survey was limited to 50 examples, taken from a limited set of
+sources. While not exhaustive, the examples implement a range of
+design techniques across a variety of applications. We do not pro-
+vide an empirical evaluation of the work surveyed, instead choosing
+to use techniques of film criticism in order to analyze patterns used
+and identify the communication intentions of their producers. We
+similarly did not empirically evaluate our own work, and instead
+provide an account of our design process and detail our reasoning
+for important decisions that were made along the way. Our work
+does not fully utilize the design space of cinematic visualizations
+that we identified; for example, we did not use sound at all, and all
+narration was done through written text with a few small overlays
+in the video. The experience might be improved by incorporating
+narration either on-screen or off [10].
+6
+CONCLUSION
+We presented cinematic visualization, a genre of narrative visu-
+alization that uses techniques from cinema in order to enhance the
+presentation of data-driven stories. A central contribution of this
+work is to identify a new genre of narrative visualization that we
+then analyze in depth. The importance of genre is clear in other art
+forms like literature and cinema; however, it is invoked less often
+in the context of visualization research. We believe that this type of
+work is crucial for understanding the design of narrative visualiza-
+tions, and thinking rigorously about how they can be constructed
+and deployed. While past work on narrative visualization looked
+specifically at the narrative structure, here we look at both narra-
+tive and style as formal systems that contribute to the dramatic
+experience of watching a cinematic visualization. To do this, we
+turned to theory from another form of art, film, in order to provide
+grounding in the features of style, and used analysis techniques
+established in that domain to deconstruct our case studies.
+We analyzed a variety of examples of cinematic visualization and
+the techniques that they employ towards certain narrative applica-
+tions. Many of these visualizations show a relatively small amount
+of data (e.g., focusing on a single rate or quantity) as opposed to
+being data-dense. The non-data elements of the scene play an im-
+portant role: they are used to set the location in which the shot is
+taking place and provide cues to viewers about where they are, what
+they are looking at, and why it is relevant. This approach is quite
+different from typical information visualizations, where data may
+be reduced to a minimal form, such as a line or a bar. Cinematic
+visualization instead tends to be more maximal in its approach,
+such that the non-data ink is not reduced or omitted, but rather
+used to build up entire digital worlds around data points. This style
+encourages viewers to feel present in locations augmented with
+data objects, or to viscerally experience events that happened in
+the past, or are happening far away in the universe.
+Rendering data in 3D is a fraught endeavor, as the values being
+rendered can be obscured by humans’ relatively poor ability to
+estimate and compare volume, and because the 3D projection can
+introduce distortion when trying to read values. Why would the cre-
+ators choose to follow a cinematic path over one that more clearly
+and directly communicates the underlying data with precision?
+We argue that in choosing to treat a visualization as a cinematic
+experience, its authors might be looking beyond the immediate
+data, in order to viscerally ground that data in meaningful context.
+In other words, analytic precision is only one of several objectives
+that a visualization might help accomplish. In choosing 3D, we
+might diminish precision in service of other objectives.
+
+Cinematic Techniques in Narrative Visualization
+, ,
+ACKNOWLEDGEMENTS
+We would like to thank Susan Callery, Holly Shaftel, Randal Jackson,
+Daniel Bailey, Michael Gunson, Josh Willis, Joe Witte, and the
+Earth Science Communications Team at NASA’s Jet Propulsion
+Laboratory for their support of this work. A portion of this research
+was carried out at the Jet Propulsion Laboratory, California Institute
+of Technology, under a contract with the National Aeronautics and
+Space Administration (80NM0018D0004).
+REFERENCES
+[1] Fereshteh Amini, Nathalie Henry Riche, Bongshin Lee, Christophe Hurter, and
+Pourang Irani. 2015. Understanding data videos: Looking at narrative visualiza-
+tion through the cinematography lens. In Proceedings of the 33rd Annual ACM
+Conference on Human Factors in Computing Systems. ACM, 1459–1468.
+[2] Fereshteh Amini, Nathalie Henry Riche, Bongshin Lee, Andres Monroy-
+Hernandez, and Pourang Irani. 2016. Authoring data-driven videos with Dat-
+aClips. IEEE transactions on visualization and computer graphics 23, 1 (2016),
+501–510.
+[3] Benjamin Bach, Zezhong Wang, Matteo Farinella, Dave Murray-Rust, and
+Nathalie Henry Riche. 2018. Design patterns for data comics. In Proceedings
+of the 2018 CHI Conference on Human Factors in Computing Systems. ACM, 38.
+[4] M Pauline Baker and Colleen Bushell. 1995. After the storm: Considerations for
+information visualization. IEEE Computer Graphics and Applications 15, 3 (1995),
+12–15.
+[5] David Bordwell, Kristin Thompson, and Jeff Smith. 1997. Film art: An introduction.
+Vol. 7. McGraw-Hill New York.
+[6] Kalina Borkiewicz, AJ Christensen, Helen-Nicole Kostis, Greg Shirah, and Ryan
+Wyatt. 2019. Cinematic Scientific Visualization: The Art of Communicating
+Science. In ACM SIGGRAPH 2019 Courses (Los Angeles, California) (SIGGRAPH
+’19). ACM, New York, NY, USA, Article 5, 273 pages.
+https://doi.org/10.1145/
+3305366.3328056
+[7] Michelle A Borkin, Zoya Bylinskii, Nam Wook Kim, Constance May Bainbridge,
+Chelsea S Yeh, Daniel Borkin, Hanspeter Pfister, and Aude Oliva. 2015. Be-
+yond memorability: Visualization recognition and recall. IEEE transactions on
+visualization and computer graphics 22, 1 (2015), 519–528.
+[8] Michelle A Borkin, Azalea A Vo, Zoya Bylinskii, Phillip Isola, Shashank Sunkavalli,
+Aude Oliva, and Hanspeter Pfister. 2013. What makes a visualization memorable?
+IEEE Transactions on Visualization and Computer Graphics 19, 12 (2013), 2306–
+2315.
+[9] Jeremy Boy, Anshul Vikram Pandey, John Emerson, Margaret Satterthwaite,
+Oded Nov, and Enrico Bertini. 2017. Showing people behind data: Does anthro-
+pomorphizing visualizations elicit more empathy for human rights data?. In
+Proceedings of the 2017 CHI Conference on Human Factors in Computing Systems.
+ACM, 5462–5474.
+[10] Judd D Bradbury and Rosanna E Guadagno. 2020. Documentary narrative vi-
+sualization: Features and modes of documentary film in narrative visualization.
+Information Visualization 19, 4 (2020), 339–352.
+[11] Matthew Brehmer, Bongshin Lee, Nathalie Henry Riche, Darren Edge, Christo-
+pher White, Kate Lytvynets, and David Tittsworth. 2017. Microsoft Timeline
+Storyteller. https://timelinestoryteller.com.
+[12] Paul Cairns, Anna Cox, and A Imran Nordin. 2014. Immersion in digital games:
+review of gaming experience research. Handbook of digital games 1 (2014), 767.
+[13] S Card, J Mackinlay, and B Shneiderman. 1999. Readings in information visualiza-
+tion: using vision to think. Morgan Kaufmann.
+[14] Fanny Chevalier, Romain Vuillemot, and Guia Gali. 2013. Using concrete scales:
+A practical framework for effective visual depiction of complex measures. IEEE
+transactions on visualization and computer graphics 19, 12 (2013), 2426–2435.
+[15] Marc Christie and Eric Languénou. 2003. A constraint-based approach to camera
+path planning. In International Symposium on Smart Graphics. Springer, 172–181.
+[16] William S Cleveland and Robert McGill. 1984. Graphical perception: Theory, ex-
+perimentation, and application to the development of graphical methods. Journal
+of the American statistical association 79, 387 (1984), 531–554.
+[17] Andy Cockburn and Bruce McKenzie. 2000. An evaluation of cone trees. In
+People and Computers XIV—Usability or Else! Springer, 425–436.
+[18] Matthew Conlen and Jeffrey Heer. 2018. Idyll: A Markup Language for Authoring
+and Publishing Interactive Articles on the Web. In ACM User Interface Software &
+Technology (UIST). http://idl.cs.washington.edu/papers/idyll
+[19] Matthew Conlen, Alex Kale, and Jeffrey Heer. 2019. Capture & Analysis of Active
+Reading Behaviors for Interactive Articles on the Web. Computer Graphics Forum
+(Proc. EuroVis) (2019). http://idl.cs.washington.edu/papers/idyll-analytics
+[20] Robert L Cook, Loren Carpenter, and Edwin Catmull. 1987. The Reyes image
+rendering architecture. ACM SIGGRAPH Computer Graphics 21, 4 (1987), 95–102.
+[21] Maxime Cordeil, Andrew Cunningham, Benjamin Bach, Christophe Hurter,
+Bruce H Thomas, Kim Marriott, and Tim Dwyer. 2019. IATK: An Immersive
+Analytics Toolkit. In 2019 IEEE Conference on Virtual Reality and 3D User Interfaces
+(VR). IEEE, 200–209.
+[22] Donna Cox and PA Fishwick. 2006. Metaphoric mappings: The art of visualization.
+Aesthetic computing (2006), 89–114.
+[23] Tim Dobbert. 2006. Matchmoving: the invisible art of camera tracking. John Wiley
+& Sons.
+[24] Anthony J Ferri. 2007. Willing suspension of disbelief: Poetic faith in film. Lexington
+Books.
+[25] Michael Freeman. 2007. The photographer’s eye: composition and design for better
+digital photos. CRC Press.
+[26] Gustav Freytag. 1904. Technique of the drama: An exposition of dramatic composi-
+tion and art. Scott, Foresman.
+[27] John Gibbs. 2002. Mise-en-scène: film style and interpretation. Vol. 10. Wallflower
+Press.
+[28] Esther Greussing and Hajo G Boomgaarden. 2019. Simply bells and whistles?
+Cognitive effects of visual aesthetics in digital longforms. Digital Journalism 7, 2
+(2019), 273–293.
+[29] Neil Halloran. 2015. The Fallen of World War II. http://www.fallen.io/ww2/
+[30] Jeffrey Heer and George Robertson. 2007. Animated Transitions in Statistical Data
+Graphics. IEEE Trans. Visualization & Comp. Graphics 13, 6 (2007), 1240–1247.
+[31] Mark Henne, Hal Hickel, Ewan Johnson, and Sonoko Konishi. 1996. The making of
+toy story [computer animation]. In COMPCON’96. Technologies for the Information
+Superhighway Digest of Papers. IEEE, 463–468.
+[32] Tomlinson Holman. 2012. Sound for film and television. Taylor & Francis.
+[33] Jessica Hullman and Nick Diakopoulos. 2011. Visualization rhetoric: Framing
+effects in narrative visualization. IEEE transactions on visualization and computer
+graphics 17, 12 (2011), 2231–2240.
+[34] Jessica Hullman, Steven Drucker, Nathalie Henry Riche, Bongshin Lee, Danyel
+Fisher, and Eytan Adar. 2013. A deeper understanding of sequence in narrative
+visualization. IEEE Transactions on visualization and computer graphics 19, 12
+(2013), 2406–2415.
+[35] Petra Isenberg, Bongshin Lee, Huamin Qu, and Maxime Cordeil. 2018. Immersive
+Visual Data Stories. In Immersive Analytics. Springer, 165–184.
+[36] Alexander Ivanov, Kurtis Danyluk, Christian Jacob, and Wesley Willett. 2019.
+A Walk Among the Data. IEEE computer graphics and applications 39, 3 (2019),
+19–28.
+[37] Yvonne Jansen, Pierre Dragicevic, Petra Isenberg, Jason Alexander, Abhijit Karnik,
+Johan Kildal, Sriram Subramanian, and Kasper Hornbæk. 2015. Opportunities
+and challenges for data physicalization. In Proceedings of the 33rd Annual ACM
+Conference on Human Factors in Computing Systems. 3227–3236.
+[38] Craig Kolb, Don Mitchell, Pat Hanrahan, et al. 1995. A realistic camera model for
+computer graphics. In SIGGRAPH, Vol. 95. 317–324.
+[39] Robert Kosara and Jock Mackinlay. 2013. Storytelling: The next step for visual-
+ization. Computer 46, 5 (2013), 44–50.
+[40] John Lasseter. 1987. Principles of traditional animation applied to 3D computer
+animation. In ACM Siggraph Computer Graphics, Vol. 21. ACM, 35–44.
+[41] Benjamin Lee, Dave Brown, Bongshin Lee, Christophe Hurter, Steven Drucker,
+and Tim Dwyer. 2020. Data Visceralization: Enabling Deeper Understanding of
+Data Using Virtual Reality. arXiv preprint arXiv:2009.00059 (2020).
+[42] Renae Low and John Sweller. 2005. The modality principle in multimedia learning.
+The Cambridge handbook of multimedia learning 147 (2005), 158.
+[43] Kwan-Liu Ma, Isaac Liao, Jennifer Frazier, Helwig Hauser, and Helen-Nicole
+Kostis. 2011. Scientific storytelling using visualization. IEEE Computer Graphics
+and Applications 32, 1 (2011), 12–19.
+[44] Kim Marriott, Falk Schreiber, Tim Dwyer, Karsten Klein, Nathalie Henry Riche,
+Takayuki Itoh, Wolfgang Stuerzlinger, and Bruce H Thomas. 2018. Immersive
+Analytics. Vol. 11190. Springer.
+[45] David McCandless. 2012. Information is beautiful. Collins London.
+[46] Sean McKenna, N Henry Riche, Bongshin Lee, Jeremy Boy, and Miriah Meyer.
+2017. Visual narrative flow: Exploring factors shaping data visualization story
+reading experiences. In Computer Graphics Forum, Vol. 36. Wiley Online Library,
+377–387.
+[47] Timothy Morton. 2013. Hyperobjects: Philosophy and Ecology after the End of the
+World. U of Minnesota Press.
+[48] NASA. 2020. Vital signs of the planet. https://climate.nasa.gov/vital-signs/.
+[49] Rajendra K Pachauri, Myles R Allen, Vicente R Barros, John Broome, Wolfgang
+Cramer, Renate Christ, John A Church, Leon Clarke, Qin Dahe, Purnamita Das-
+gupta, et al. 2014. Climate change 2014: synthesis report. Contribution of Working
+Groups I, II and III to the fifth assessment report of the Intergovernmental Panel on
+Climate Change. Ipcc.
+[50] PacificVis. 2020. Visual Data Storytelling Contest. https://vimeo.com/pviscontest.
+[51] Deokgun Park, Steven M Drucker, Roland Fernandez, and Niklas Elmqvist. 2017.
+Atom: A grammar for unit visualizations. IEEE transactions on visualization and
+computer graphics 24, 12 (2017), 3032–3043.
+[52] Matt Pharr, Wenzel Jakob, and Greg Humphreys. 2016. Physically based rendering:
+From theory to implementation. Morgan Kaufmann.
+[53] Carol
+Rasmussen.
+2019.
+A
+Partnership
+Forged
+by
+Fire.
+https://climate.nasa.gov/news/2899/a-partnership-forged-by-fire/.
+
+, ,
+Conlen, et al.
+[54] Karel Reisz and Gavin Millar. 1971. The technique of film editing. (1971).
+[55] Donghao Ren. 2019. Visualization Authoring for Data-driven Storytelling. Ph.D.
+Dissertation. UC Santa Barbara.
+[56] Arvind Satyanarayan and Jeffrey Heer. 2014. Authoring narrative visualizations
+with ellipsis. In Computer Graphics Forum, Vol. 33. Wiley Online Library, 361–370.
+[57] Edward Segel and Jeffrey Heer. 2010. Narrative visualization: Telling stories
+with data. IEEE transactions on visualization and computer graphics 16, 6 (2010),
+1139–1148.
+[58] Debika Shome and Sabine Marx. 2009. The psychology of climate change com-
+munication: a guide for scientists, journalists, educators, political aides, and the
+interested public. Center for Research on Environmental Decisions, New York.
+[59] Ronell Sicat, Jiabao Li, JunYoung Choi, Maxime Cordeil, Won-Ki Jeong, Benjamin
+Bach, and Hanspeter Pfister. 2018. Dxr: A toolkit for building immersive data
+visualizations. IEEE transactions on visualization and computer graphics 25, 1
+(2018), 715–725.
+[60] Charles D Stolper, Bongshin Lee, N Henry Riche, and John Stasko. 2016. Emerging
+and recurring data-driven storytelling techniques: Analysis of a curated collection
+of recent stories. (2016).
+[61] Frank Thomas, Ollie Johnston, and Frank Thomas. 1995. The illusion of life:
+Disney animation. Hyperion New York.
+[62] J Thompson, Z Liu, W Li, and J Stasko. 2020. Understanding the Design Space
+and Authoring Paradigms for Animated Data Graphics. In Computer Graphics
+Forum, Vol. 39. Wiley Online Library, 207–218.
+[63] Hamish Todd. 2017. Virus, the Beauty of the Beast. http://viruspatterns.com/
+[64] Edward R Tufte. 2001. The visual display of quantitative information. Vol. 2.
+Graphics press Cheshire, CT.
+[65] Craig Upson, TA Faulhaber, David Kamins, David Laidlaw, David Schlegel, Jefrey
+Vroom, Robert Gurwitz, and Andries Van Dam. 1989. The application visual-
+ization system: A computational environment for scientific visualization. IEEE
+Computer Graphics and Applications 9, 4 (1989), 30–42.
+[66] Jagoda Walny, Christian Frisson, Mieka West, Doris Kosminsky, Søren Knudsen,
+Sheelagh Carpendale, and Wesley Willett. 2019. Data Changes Everything: Chal-
+lenges and Opportunities in Data Visualization Design Handoff. IEEE transactions
+on visualization and computer graphics 26, 1 (2019), 12–22.
+[67] Colin Ware. 2012. Information visualization: perception for design. Elsevier.
+[68] I-Cheng Yeh, Chao-Hung Lin, Hung-Jen Chien, and Tong-Yee Lee. 2011. Effi-
+cient camera path planning algorithm for human motion overview. Computer
+Animation and Virtual Worlds 22, 2-3 (2011), 239–250.
+[69] Qiyu Zhi, Alvitta Ottley, and Ronald Metoyer. 2019. Linking and Layout: Ex-
+ploring the Integration of Text and Visualization in Storytelling. In Computer
+Graphics Forum, Vol. 38. Wiley Online Library, 675–685.
+
+Cinematic Techniques in Narrative Visualization
+, ,
+Figure 7: We analyzed the style of 50 cinematic visualizations using the features of mise-en-scène, cinematography, editing, and
+sound. An HTML version of this table including URLs for each row can be found at https://cinematic-visualization.github.io/.
+
+Cinematography
+Mise-en-scene
+Editing
+Audio
+rative Text
+Composited Annotations
+Jser-controlled camera
+Realistic Background
+ - Opacity
+mera
+er-triggered steps
+3617 - L
+n-Situ Annotations
+Composited Narra
+Point-of-view can
+nera
+Visual Analogy
+rator
+it Visualizatiol
+Scale
+punos oeba
+Reality
+n-situ Narra
+notation
+Annotation
+Annotation
+dded
+ncrete
+erview
+pax
+ISIC
+0
+Author
+Publishel
+Titte
+cV1
+Drowning in plastic
+Simon Scarr, et al.
+Reuters
+CV2
+Is This the Neighborhood New York Deserves?
+Michael Kimmelman 
+NYT
+CV3
+Krigsskipet som krasjet og sank
+B
+Stangvik, et al.
+VG
+CV4
+How China Turned a City Into a Prison
+Chris Buckley, et al.
+NYT
+CV5
+The Forbidden City's unique architecture
+Marco Hernandez
+South China Morning Post
+CV6
+Here are 120 million Monopoly pieces, roughly one.
+Confessore, et al.
+NYT
+CV7
+Building Katie's New Face
+Jason Treat
+NatGeo
+CV8
+Mass Exodus: The scale of the Rohingya crisis
+Christian Inton
+Reuters
+This 3-D Simulation Shows Why Social Distancing.
+CV9
+Parshina-Kottas, et al.
+NYT
+CV10 Tracking China's Muslim Gulag
+Simon Scarr
+Reuters 
+CV11
+Choice and Chance
+Staff
+Tampa Bay Times
+cV12 Cassini's Grand Tour
+Nadia Drake, et al.
+NatGeo
+CV13 Resurecting a Dragon
+Brian T Jacobs
+NatGeo
+CV14Want to fireproof your house? Here's where to start
+Kyle Kim, et al.
+LATimes 
+CV15Apoll 11 -As They Shot It
+Jonathan Corum, et al. NYT
+CV16 The Atlas of Moons
+NatGeo Staff
+NatGeo
+CV17 Explore a Toad's Digital Clone
+Brian T. Jacobs
+NatGeo
+CV18 THE THOMAS FIRE: 40 DAYS OF DEVASTATION
+Joe Fox
+LATimes
+D.
+CV19 Is the Nasdag in Another Bubble?
+Roger Kenny, et al.
+Wall Street Journal
+CV20 A 3-D View of a Chart That Predicts The Econom...
+Gregor Aisch, et al.
+NYT
+CV21
+Inside the Taser
+Simon Scarr
+Reuters
+CV22 Seeing Earth from Outer Space
+Matthew Conlen
+The Pudding
+C.
+CV23 Of Catastrophes and Rescues: Making the Invisib..
+Peter Mindek, et al.
+PacificVis
+CV24A Visualization of Two-stage Autoignition of n-dod..
+Yucong Ye, et al.
+PacificVis
+CV25 How Mariano Rivera Dominates Hitters
+Graham Roberts, et al. NYTimes
+Real World Visuals
+Real World Visuals
+CV27 CCS:A2 degree solution
+Real World Visuals
+Real world Visuals
+CV28CARS
+Real World Visuals
+Real world Visuals
+CV29[REALISTIC] Elephant rocket fuel - Saturn V
+Maxim Sachs
+YOUTUBE
+CV30 University of Exeter greenhouse gas emissions in R..
+Real World Visuals
+Real world Visuals
+CV31if The World Were 100 People
+Gabriel Reilich, et al.
+Good Magazine
+CV32 Up - and down - from Ground Zero
+Graham Roberts, et al. NYT
+CV33 The Birth of a Virtual Cell
+Peter Mindek, et al.
+PacificVis
+CV34
+The Nuclear Threat - The Shadow Peace
+Neil Halloran
+Youtube
+CV35What f Carbon Left Your Tailpipe as Solid Chunks?
+Sukee Bennett
+PBS Nova
+CV36Stay Home, Flatten the Curve
+keta
+Youtube
+CV37 Chart Party: We decided to erase the three-pointer
+Jon Bois 
+Youtube
+cV38 200 Countries, 200 Years, 4 Minutes
+Hans Rosling
+Youtube
+CV39 The best stats you've ever seen
+Hans Rosling 
+TED 
+CV40 VFX Artist Reveals the True Scale of the Universe
+Wren Weichman
+Corridor Crew
+CV41
+Helge Ingstad
+Hallvard Sandberg
+NRKbeta
+CV42The dangers of storm surge
+The Weather Channel
+The Weather Channel
+CV43 Survive the Tornado
+The Weather Channel
+The Weather Channel
+CV44
+Television Elections Coverage
+KING5 TV
+KING5
+CV45 Powers of Ten
+Eames & Eames
+IBM
+CV46 Strange things happen when you rotate in 4 dimen...
+Hamish Todd
+Youtube
+CV47 Discovering Gale Crater
+Armand Emamdjomeh LATimes
+A
+CV48 Taiwan earthquake: Survivors found in rubble of Ta..
+ Malachy Brown,
+SketchFab / Australian
+cV49 Four of the Best Olympians, as You've Never Seen...
+John Branch
+NYT
+CV50 How We Created a Virtual Crime Scene to Investig.
+Malachy Brown,
+NYT
+CV13
+CV40
+CV26
+CV35
+D. Author-Guided
+A. In-situ Narrator
+B. Anthropocentric Perspective
+C. Resolving Scale
+Interactive Camera

DNAyT4oBgHgl3EQfSPdR/content/2301.00081v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c33030482438553760e3f705a90dffd0a24af3a239aee3220492bb4bf778bc36
+size 919484

DNAyT4oBgHgl3EQfSPdR/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:abb77915eccc509eb6beeef1e880735c80ce68e35c6c6e9ef43156adeb44cc88
+size 499904

DdE0T4oBgHgl3EQfQQD8/content/2301.02192v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:8c36c010c7d02ce449c83540b4d4e369f3a2957026f688edde2be5eb68d9ea58
+size 990837

DdE0T4oBgHgl3EQfQQD8/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c1e6ab58362c0c64d4233399961389cfecc94f1e99911814b27ebe362098d548
+size 4259885

E9E4T4oBgHgl3EQfGwxn/content/2301.04897v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:79ee4158a221286f282a7c182aaa9c2e7dfb4cf53c90ddae49bbfb5039ded0cc
+size 313817

E9E4T4oBgHgl3EQfGwxn/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d5e52f50293bd42a9aa0547078eddce18fd1717e2aebe63983d9c8103432384b
+size 6225965

E9E4T4oBgHgl3EQfGwxn/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ce66d1867f54cb83ea83a10fd5d727c8b135c08b5ea96d00d5a21c606cdabf05
+size 203160

E9FQT4oBgHgl3EQfRDbD/content/tmp_files/2301.13285v1.pdf.txt

@@ -0,0 +1,1386 @@
+Do entangled states correspond to entangled measurements under local transformations?
+Florian Pimpel,1, ∗ Martin J. Renner,2, 3, ∗ and Armin Tavakoli4
+1Atominstitut, Technische Universität Wien, Stadionallee 2, 1020 Vienna, Austria
+2University of Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology (VCQ), Boltzmanngasse 5, 1090 Vienna, Austria
+3Institute for Quantum Optics and Quantum Information - IQOQI Vienna,
+Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
+4Physics Department, Lund University, Box 118, 22100 Lund, Sweden
+We investigate whether pure entangled states can be associated to a measurement basis in which all vectors are
+local unitary transformations of the original state. We prove that for bipartite states with a local dimension that is
+either 2, 4 or 8, every state corresponds to a basis. Via numerics we strongly evidence the same conclusion also
+for two qutrits and three qubits. However, for some states of four qubits we are unable to find a basis, leading
+us to conjecture that not all quantum states admit a corresponding measurement. Furthermore, we investigate
+whether there can exist a set of local unitaries that transform any state into a basis. While we show that such a
+state-independent construction cannot exist for general quantum states, we prove that it does exist for real-valued
+n-qubit states if and only if n = 2, 3, and that such constructions are impossible for any multipartite system
+of an odd local dimension. Our results suggest a rich relationship between entangled states and iso-entangled
+measurements with a strong dependence on both particle numbers and dimension.
+Entanglement is a fundamental, broadly useful and an in-
+tensely studied feature of quantum mechanics.
+However,
+in spite being of arguably similar foundational significance,
+much less is known about the entanglement of joint quan-
+tum measurements than the entanglement of quantum states.
+Entangled measurements are crucial for seminal quantum in-
+formation protocols such as teleportation [1], dense coding
+[2] and entanglement swapping [3], which are instrumen-
+tal for various quantum technologies.
+Typically, they are
+based on the paradigmatic Bell basis, which is composed of
+the four maximally entangled states (|00⟩ ± |11⟩)/
+√
+2 and
+(|01⟩ ± |10⟩)/
+√
+2. In the same way that the Bell basis may
+be thought of as the measurement corresponding to the max-
+imally entangled state, it is natural to ask whether entangled
+states in general can be associated with a corresponding en-
+tangled measurement. Studying the relationship between en-
+tangled states and entangled measurements is not only inter-
+esting for understanding quantum mechanics. It is also an
+invitation to explore, in the context of quantum information
+applications, the largely uncharted terrain of entangled mea-
+surements beyond the Bell basis and its immediate general-
+isations. Most notably, entangled measurements beyond the
+Bell basis are also increasingly interesting for topics such as
+network nonlocality [4] and entanglement-assisted quantum
+communication [5, 6].
+Consider that we are given a pure quantum state |ψ⟩ com-
+prised of n subsystems, each of dimension d. Is it possible
+to find a measurement, namely an orthonormal basis of the
+global dn-dimensional Hilbert space, in which all basis states
+have the same degree of entanglement as |ψ⟩? Specifically, we
+want to decide the existence of dn strings, {Vj}dn
+j=1, of local
+unitary transformations,
+Vj =
+n
+�
+k=1
+U (j)
+k
+(1)
+∗ These authors contributed equally.
+where U (j)
+k
+is a d-dimensional unitary operator, such that the
+set of states |ψj⟩ ≡ Vj |ψ⟩ form a basis, i.e. | ⟨ψj|ψj′⟩ | =
+δjj′. If affirmative, we say that |ψ⟩ admits a basis and we call
+the set of basis vectors {|ψj⟩}dn
+j=1 a |ψ⟩-basis.
+Known examples of entangled measurements can be ac-
+commodated in this picture. For example, the Bell basis can
+be obtained from operating on |ψ⟩ = (|00⟩ + |11⟩)/
+√
+2 with
+the four strings of local unitaries {Vj}4
+j=1 = {11 ⊗ 11, 11 ⊗
+X, Z ⊗ 11, Z ⊗ X}, where X and Z are bit-flip and phase-
+flip Pauli operators. A well-known generalisation of the Bell
+basis to n systems of dimension d can be thought of as a
+|GHZn,d⟩-measurement where the relevant state is the higher-
+dimensional GHZ state |GHZn,d⟩ =
+1
+√
+d
+�d−1
+k=0 |k⟩⊗n. The
+corresponding strings of local unitaries are Vj = Zj1
+d ⊗Xj2
+d ⊗
+. . . ⊗ Xjn
+d |GHZn,d⟩ where j = j1 . . . jn ∈ {0, . . . , d − 1}n
+and where Zd = �d−1
+l=0 e
+2πi
+d l |l⟩⟨l| and Xd = �d−1
+l=0 |l + 1⟩⟨l|
+are generalised Pauli operators. More generally, any state that
+is locally maximally entanglable (for example graph states)
+is known to admit a basis via suitable unitaries of the form
+Vj = U j1
+1 ⊗ . . . ⊗ U jn
+n
+[7]. These states are characterised
+by the property that if each qubit is supplemented with a
+qubit ancilla and controlled unitary gates are performed on
+the state-ancilla pairs, then a maximally entangled bipartite
+state can be constructed between the collection of state-qubits
+and the collection of ancilla-qubits.
+However, this is far
+from a complete characterisation of the states that admit a
+basis, which is seen already in the restrictive form of the
+strings of unitaries. For example, the three-qubit W-state,
+|W3⟩ = (|001⟩+|010⟩+|100⟩)/
+√
+3, is not locally maximally
+entanglable but is neverthelss known to admit a basis [8]. In
+what follows, we set out to systematically explore whether
+entangled states admit a corresponding basis and then, as we
+will introduce later, whether such bases can be constructed
+even without prior knowledge of the state.
+Let us begin with considering the simplest situation, namely
+when |ψ⟩ is a state of two qubits. We constructively show that
+every such state admits a basis. To this end, we first apply
+the state-dependent local unitaries W A
+ψ ⊗ W B
+ψ that map |ψ⟩,
+arXiv:2301.13285v1  [quant-ph]  30 Jan 2023
+
+2
+via a Schmidt decomposition, into the computational basis,
+|ψS⟩ = λ |00⟩+
+√
+1 − λ2 |11⟩ for some coefficient 0 ≤ λ ≤ 1.
+Then, we consider the action of the following four strings of
+local unitaries
+�
+�
+�
+�
+�
+11 ⊗ 11
+11 ⊗ XZ
+XZ ⊗ Z
+XZ ⊗ X
+�
+�
+�
+�
+�
+.
+(2)
+One can verify that this transforms |ψS⟩ into a |ψ⟩-basis. No-
+tice that once the state has been rotated into the Schmidt form
+|ψS⟩, the subsequent unitaries (2) do not depend on λ. This
+construction can be extended to bipartite (n = 2) states of
+local dimension d = 4 and d = 8. Again via Schmidt de-
+composition, we can find state-dependent local unitaries that
+transform |ψ⟩ into |ψS⟩ = �d−1
+l=0 λl |ll⟩ for some Schmidt co-
+efficients �
+l λ2
+l = 1. In Appendix A, we show that there is a
+set of local unitaries that indeed leads to a |ψ⟩-basis indepen-
+dently of the specific Schmidt coefficients.
+It is natural to consider also the simplest case that is not
+of the above convenient form, namely that of two qutrits,
+(n, d) = (2, 3). This appears to be considerably different be-
+cause we fail to find strings of local unitaries that bring the
+Schmidt decomposition |ψS⟩ into a basis without explicit de-
+pendence on the Schmidt coefficients. Nevertheless, a basis
+might still be possible to construct by taking the Schmidt co-
+efficients into account when choosing the local unitaries. Ac-
+tually, this seems to always be possible. To arrive at this, we
+have used a numerical method. Let {|φj⟩}m
+j=1 be a set of states
+in a given Hilbert space. These states are pairwise orthogonal
+if and only if they realise the global minimum (zero) of the
+following objective function
+f({φj}) ≡
+�
+j̸=j′
+| ⟨φj|φj′⟩ |2.
+(3)
+For a given state |ψ⟩, we numerically minimise f({ψj}) over
+all possible strings {Vj}dn
+j=1 of local unitaries. To this end,
+we parameterise the local unitaries U (j)
+k
+using the scheme of
+Ref. [9]. For the two-qutrit case, we have randomly chosen
+1000 pairs of Schmidt coefficients (λ1, λ2) which (up to local
+unitaries) fully specifies the state. In each case we numerically
+minimise f({ψj}). Without exception, we find strings of lo-
+cal unitaries that yield a result below our selected precision
+threshold of f ≤ 10−6.
+Furthermore, we have also numerically investigated the
+case of three qubits, (n, d) = (3, 2). This scenario requires
+a different approach than the previous cases since multipar-
+tite states have no Schmidt decomposition. Instead, for any
+given three-qubit state |ψ⟩, there exists local unitary transfor-
+mations that map it onto the canonical form a |000⟩+b |011⟩+
+c |101⟩ + d |110⟩ + e |111⟩ where (b, c, d, e) are real num-
+bers and a is a complex number [10, 11]. Hence, up to lo-
+cal unitaries, the state space (after normalisation) is charac-
+terised by five real numbers. Later, we will provide an analyt-
+ical construction of a |ψ⟩-basis for the four-parameter family
+corresponding to restricting a to be real. However, we have
+not found an analytical basis construction for general three-
+qubit states, but we nevertheless conjecture that it exists. To
+evidence this, we have employed the previously introduced
+numerical search method. Again, we have randomly chosen
+1000 normalised sets of coefficients (a, b, c, d, e) and searched
+for the minimal value of f over all the strings of local qubit
+unitaries. In all cases, we find that f vanishes up to our se-
+lected precision of f ≤ 10−6.
+Given the above case studies, one might suspect that ev-
+ery pure quantum state admits a basis.
+Interestingly, this
+seems not to be true.
+While some states of four qubits,
+(n, d) = (4, 2), are found to admit a basis, for example
+a W state and doubly-excited Dicke state [23], it appears
+that most four-qubit states do not admit a basis. We have
+sampled many different four-qubit states and repeatingly at-
+tempted to numerically find a basis via the minimisation of
+(3), also using several different search algorithms. It was reg-
+ularly found that the estimated minimum is multiple orders
+of magnitude above our given precision threshold for a basis.
+For example, we searched for the minimum of f for the state
+2
+√
+6 |W⟩ +
+√
+2
+√
+6 |GHZ4,2⟩, with 100 randomised initial points,
+and never reached below f = 10−1, five orders of magnitude
+above our precision threshold. We have attempted to prove
+that no basis exists by employing semidefinite outer relax-
+ations of f over the set of dimensionally-restricted quantum
+correlations [12] combined with a modified sampling of the
+state and measurement space [13] and symmetrisation tech-
+niques [14] to efficiently treat the large number of single-qubit
+unitaries featured in this problem. However, the conjecture
+has resisted our efforts. A guiding intuition for the impossi-
+bility of a basis is to note that the number of free parameters is
+3n(2n − 1) whereas the number of orthogonality constraints
+(counting both the real and imaginary part) is 22n − 2n, and
+the latter is larger than the former only when n ≥ 4.
+Furthermore, if an n-qubit state |ψ⟩ does not admit a ba-
+sis, then the (n + 1)-qubit state |ψ′⟩ = |ψ⟩ ⊗ |0⟩ also does
+not admit a basis. By contradiction, suppose there are 2n+1
+unitaries V ′
+j = Vj ⊗ U (j)
+n+1 such that |⟨ψ′|(V ′
+j )†V ′
+k|ψ′⟩| =
+δjk ∀j, k ∈ {1, ..., 2n+1}. Divide the 2n+1 states U (j)
+n+1 |0⟩
+into two sets such that two orthogonal vectors are not in
+the same set (e.g. the northern and southern hemisphere
+of the Bloch ball).
+Consider the set that contains at least
+as many elements as the other one, hence, at least 2n el-
+ements.
+By construction, these states cannot be distin-
+guished on the last qubit, |⟨0|U (j)†
+n+1U (k)
+n+1|0⟩| ̸= 0.
+Since
+|⟨ψ′|(V ′
+j )†V ′
+k|ψ′⟩| = |⟨ψ|V †
+j Vk|ψ⟩| · |⟨0|U (j)†
+n+1U (k)
+n+1|0⟩|, we
+must have |⟨ψ|V †
+j Vk|ψ⟩| = δjk for all of those pairs, which
+contradicts that |ψ⟩ does not admit a basis. By induction, this
+argument shows that if our above conjecture holds, namely
+that some four-qubit states do not admit a basis, then the same
+holds for any number of qubits.
+Since not all pure quantum states admit a basis, and this
+seems to be typical rather than exceptional for four qubits, it
+is interesting to ask whether some distinguished families of
+n-qubit states can nevertheless admit a basis. This is well-
+known to be the case for n-qubit GHZ-states and graph-states
+
+3
+since they are locally maximally entanglable. More interest-
+ingly, a positive answer is also possible for states that are not
+of this kind: we construct a basis for the n-qubit W-state,
+|Wn⟩ =
+1
+√n
+�
+σ σ(|0⟩⊗n−1 |1⟩) where σ runs over all permu-
+tations of the position of “1”. Note that |W1⟩ = |1⟩ and that a
+|W1⟩-basis is obtained from the unitaries {11, X}. Now we ap-
+ply induction. Consider that the strings {V (n)
+j
+}2n
+j=1 generate a
+|Wn⟩-basis. One can then construct a basis for n+1 qubits as
+follows. For half of the basis elements, namely j = 1, . . . , 2n,
+define V (n+1)
+j
+= V (n)
+j
+⊗ 11 and for the other half, namely j =
+2n +1, . . . , 2n+1, define V (n+1)
+j
+= �n
+k=1 U (j)
+k Z ⊗X. As we
+detail in Appendix B, one can verify that {V (n+1)
+j
+|Wn+1⟩}j
+is a W-basis. We note that for the purpose of entanglement
+distillation, a different construction of a W-basis was given in
+Ref. [8].
+So far, we have considered whether a specific state can be
+associated to a specific measurement. In other words, the uni-
+tary constructions have been state-dependent. We now go fur-
+ther and introduce a complementary concept, namely whether
+there exist strings of local unitaries {Vj} that can transform
+any state in a space of states S into a basis, i.e. strings of local
+unitaries that satisfy
+∀ψ ∈ S,
+|⟨ψ|V †
+j Vj′|ψ⟩| = δjj′.
+(4)
+Naturally, this state-independent notion of basis construc-
+tion is much stronger than the previously considered state-
+dependent notion.
+In the most ambitious case, when we
+choose the space S to be the entire Hilbert space of n sub-
+systems of dimension d, i.e. S ≃ (Cd)⊗n, then a state-
+independent construction cannot exist. In fact, not even two
+orthogonal vectors can be state-independently constructed for
+the full quantum state space. To show this, we can w. l. g. set
+V1 = 11 and assume that there exists local unitaries {Uk}
+such that |ψ1⟩ = |ψ⟩ and |ψ2⟩ = �n
+k=1 Uk |ψ⟩ are orthog-
+onal for all |ψ⟩. Focus now on the particular state |ψ⟩ =
+�n
+k=1 |µk⟩ where |µk⟩ is some eigenvector of the unitary Uk.
+Since the eigenvalues of a unitary are complex phases, writ-
+ten eiϕk for Uk and |µk⟩, we obtain |ψ1⟩ = �n
+k=1 |µk⟩ and
+|ψ2⟩ = ei �n
+k=1 ϕk �n
+k=1 |µk⟩. These two states are evidently
+not orthogonal and hence we have a contradiction.
+Interestingly, the situation changes radically if we limit our
+state-independent investigation to all quantum states in a real-
+valued Hilbert space. That is, S ≃ (Rd)⊗n. Such real quan-
+tum systems have also been contrasted in the literature with
+their complex counterparts [15–17]. Let us momentarily ig-
+nore the n-partition structure of our Hilbert space and sim-
+ply consider two real states |ψ1⟩ = |ψ⟩ and |ψ2⟩ = U |ψ⟩
+obtained from a given real target state |ψ⟩ and a fixed (ψ-
+independent) unitary U.
+It holds that ψ1 and ψ2 are or-
+thogonal if and only if U is skew-symmetric.
+To prove
+this, assume first the skew-symmetry property U = −U T .
+Since for real states ⟨ψ1|ψ2⟩ = ⟨ψ2|ψ1⟩∗ is equivalent to
+⟨ψ|U|ψ⟩ = ⟨ψ|U †|ψ⟩∗ = ⟨ψ|U T |ψ⟩, skew-symmetry im-
+plies that ⟨ψ1|ψ2⟩ = 0. Conversely, assume that ⟨ψ|U|ψ⟩ = 0
+for all real-valued ψ. Choosing in particular |ψ⟩ = |k⟩ for
+k = 0, . . . , d − 1, it follows that all diagonal elements of U
+must vanish. Then, choose |ψ⟩ =
+1
+√
+2(|i⟩ + |j⟩) for any pair
+i ̸= j. This yield Uii+Ujj+Uij+Uji = 0, but since we know
+that the diagonals vanish we are left with just Uij = −Uji
+which defines a skew-symmetric operator.
+Returning to our n-partitioned real Hilbert space, and still
+w. l. g. taking V1 = 11, the above result demands that we find
+local unitaries such that
+U1 ⊗ . . . ⊗ Un = −U T
+1 ⊗ . . . ⊗ U T
+n .
+(5)
+This is only possible if U T
+k = ±Uk. Hence, all local unitaries
+must be either symmetric or skew-symmetric, and the number
+of the latter must be odd. When extended from two orthogonal
+states to a whole basis, we require that this property holds for
+every pair of distinct labels (j, j′) in the basis. In other words,
+we require that every string (Vj)†Vj′ with j ̸= j′ is skew-
+symmetric.
+The question becomes whether the above condition can
+be satisfied for a given scenario. Consider it first for qubit
+systems (d = 2). In Appendix C we show that the set of
+complex qubit unitaries that are either symmetric or skew-
+symmetric and whose products are again either symmetric
+or skew-symmetric, must obey a simple structure; they are
+essentially equivalent to the four Pauli-type operators P ≡
+{11, X, Z, XZ}. Thus, if a state-independent construction ex-
+ists, we can restrict to selecting one of these four operators for
+each of our local unitaries U (j)
+k . Interestingly, for the case of
+two qubits, (n, d) = (2, 2), a state-independent construction
+is possible. It is in fact given by Eq. (2). One can straightfor-
+wardly verify that the above criterion is satisfied, i.e. all local
+unitaries are selected from P and all pairs of products of uni-
+tary strings in (2) are skew-symmetric. Alternatively, one can
+easily verify that (2) maps every state �
+i,j=0,1 αij |ij⟩ into a
+basis, for any real coefficients αij. Furthermore, by the same
+token, a state-independent basis is also possible for every real
+state of three qubits, (n, d) = (3, 2). One explicit construc-
+tion that satisfies our necessary and sufficient criterion is the
+following set of eight strings of local unitaries
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+11 ⊗ 11 ⊗ 11
+Z ⊗ Z ⊗ XZ
+Z ⊗ XZ ⊗ 11
+XZ ⊗ 11 ⊗ 11
+Z ⊗ X ⊗ XZ
+X ⊗ 11 ⊗ XZ
+X ⊗ XZ ⊗ Z
+X ⊗ XZ ⊗ X
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+Again,
+one
+may
+easily
+verify
+that
+every
+real
+state
+�
+i,j,k=0,1 αijk |ijk⟩ is mapped into a basis.
+Two- and three-qubits are interesting cases because they
+are exceptional.
+As we now show, there exists no state-
+independent construction for real states of four or more qubits.
+We first prove this for n = 4 and then show that this im-
+plies impossibility also for n > 4. The four-qubit case con-
+tains 16 strings of unitaries and we know that each local uni-
+tary can w. l. g. be selected from P. Since we seek a state-
+independent construction, we can momentarily consider only
+the state |0000⟩. In order for it to be mapped into a basis, we
+
+4
+(2,2,R) (2,2,C) (3,2,R) (3,2,C) (4,2,R) (2,3,C) (2,4 or 8,C) (n, 2m + 1,R)
+State-dependent
+construction
+
+
+
+()
+()
+()
+
+− − −
+State-independent
+construction
+
+
+
+
+
+
+
+
+TABLE I: Overview of results. The first row indicates the scenario: (n, d, S) gives particle number, dimension and the type of state space
+respectively. The symbol indicates the existence of a basis under local unitaries. The symbol indicates that there in general can be no basis
+under local unitaries, i.e. at least one state admits no basis. Paranthesis indicates that the result is obtained from numerical search. The
+symbol − − − indicates that no investigation was made.
+see that Z acts trivially on every register and therefore each
+one of the 16 combinations of bit-flip or identity operators,
+{Xc1 ⊗ Xc2 ⊗ Xc3 ⊗ Xc4} for c1, c2, c3, c4 ∈ {0, 1}, must
+be featured in exactly one of the 16 unitary strings {Vj}16
+j=1.
+Let us now look only at six of these strings, namely those
+corresponding to having zero bit-flips (1 case), one bit-flip (4
+cases) and four bit-flips (1 case). W. l. g. fixing V1 = 11 (zero
+bit-flips), the strings take the form
+V1
+11
+⊗
+11
+⊗
+11
+⊗
+11
+V2
+XZr11 ⊗
+Zr12
+⊗
+Zr13
+⊗
+Zr14
+V3
+Zr21
+⊗ XZr22 ⊗
+Zr23
+⊗
+Zr24
+V4
+Zr31
+⊗
+Zr32
+⊗ XZr33 ⊗
+Zr34
+V5
+Zr41
+⊗
+Zr42
+⊗
+Zr43
+⊗ XZr44
+V6
+XZr51 ⊗ XZr52 ⊗ XZr53 ⊗ XZr54
+,
+(6)
+where rij
+∈ {0, 1} represent our freedom to insert a Z
+operator and thus realise the two relevant elements of P.
+Since every row must be skew-symmetric and the only skew-
+symmetric element in P is XZ, we must have r11 = r22 =
+r33 = r44 = 1 and r51 + r52 + r53 + r54 = 1 where ad-
+dition is modulo two. Moreover, every product of two rows
+must be skew-symmetric, i.e. the product must have an odd
+number of XZ operations. For the four middle rows, this im-
+plies rij + rji = 1 for distinct indices i, j ∈ {1, 2, 3, 4}. For
+the products V †
+6 Vj for j = 2, 3, 4, 5, the conditions for skew-
+symmetry respectively become
+r12 + r13 + r14 + r52 + r53 + r54 = 1
+r21 + r23 + r24 + r51 + r53 + r54 = 1
+r31 + r32 + r34 + r51 + r52 + r54 = 1
+r41 + r42 + r43 + r51 + r52 + r53 = 1.
+(7)
+Summing these four equations and using the previously es-
+tablished skew-symmetry conditions, one can cancel out all
+degrees of freedom rij and arrive at the contradiction 1 = 0.
+Hence, we conclude that the state-independent basis construc-
+tion for four qubits is impossible.
+For the case of five qubits, we can again assume w. l. g. that
+the 32 combinations of bit-flip or identity operators, {Xc1 ⊗
+Xc2⊗Xc3⊗Xc4⊗Xc5} for c1, c2, c3, c4, c5 ∈ {0, 1} must be
+featured in exactly one of the 32 unitary strings since the state
+|00000⟩ has to be mapped into an orthonormal basis. Suppose
+there is a state-independent construction that maps every real-
+valued five-qubit state into a basis, in especially any state of
+the form |ψ⟩ ⊗ |0⟩, where |ψ⟩ is an arbitrary real-valued four
+qubit state. Now consider the 16 strings where c5 = 0. Since
+the fifth qubit is always mapped to itself, it has to hold that the
+first four qubits are pairwise distinguishable. However, this
+implies a state-independent construction for four qubits which
+is in contradiction to the above. By induction, this implies that
+no state-independent construction can exist whenever n ≥ 4.
+The possibility of state-independent constructions for real-
+valued bi- and tri-partite systems draws heavily on the sim-
+ple structure of skew-symmetric qubit unitaries. If we con-
+sider real-valued systems of dimension d > 2, the situa-
+tion changes considerably.
+Using our necessary and suffi-
+cient condition, it follows immediately that state-independent
+constructions are impossible in all odd dimensions, i.e. when
+(n, d) = (n, 2m+1). This stems from the fact that there exists
+no skew-symmetric unitary matrix in odd dimensions. To see
+that, simply note that if A is skew-symmetric then det(A) =
+det
+�
+AT �
+= det(−A) = (−1)2m+1 det(A) = − det(A) and
+hence det(A) = 0, but that contradicts unitarity because the
+determinant of a unitary has unit modulus.
+In summary, we have investigated the correspondence be-
+tween entangled states and entangled measurements under lo-
+cal unitary transformations, both when the local transforma-
+tion can and cannot explicitly depend on the target state. Per-
+haps surprisingly, we have found that this problem is not so
+straightforward and has a strong dependence on both the num-
+ber of subsystems involved and their dimension. Our analyt-
+ical and numerical results and conjectures are summarised in
+Table I.
+The conspicuous open problem left by our work is to prove
+our conjecture that there exists states that do not admit a ba-
+sis under local unitaries. An interesting related question is if
+one can bound the relative volume of four-qubit states that do
+not admit a basis. Our numerical investigations suggest that
+nearly all four-qubit states should belong to this class. Fur-
+thermore, it would be useful to find analytical solutions for
+the three-qubit and two-qutrit state-dependent cases. More-
+over, for the state-independent considerations, we focused on
+real Hilbert spaces. A natural question is whether there ex-
+ists state-independent basis constructions for other interest-
+ing spaces. For example, if one restricts to bipartite states
+of a known entanglement entropy, can one construct a state-
+independent basis? The answer is clearly positive for the lim-
+iting cases of product states and maximally entangled states.
+Another interesting space to consider is the symmetric sub-
+space of n-qubit Hilbert space.
+
+5
+Our results may also have prospects in quantum informa-
+tion as one may now construct entangled measurements asso-
+ciated to entangled states. Recently there has been proposals
+of two-qubit entangled projections; the so-called Elegant Joint
+Measurements [18, 19] which have also been realised in vari-
+ous experiments [20? , 21]. The Elegant Joint Measurements
+can be seen as a particular type of |ψ⟩-basis where |ψ⟩ is a
+partially entangled two-qubit state. However, the basis addi-
+tionally has the feature that the collections of reduced states
+form a tetrahedron. This requirement goes beyond our prob-
+lem formulation, as we do not impose any structure on the
+reduced states of our bases. However, it suggests an avenue
+to identifying interesting and highly symmetric measurements
+by finding the particular |ψ⟩-basis that maximises the Hilbert
+space volume spanned its collection of reduced states.
+Finally, one of the notable shortcommings of traditional,
+GHZ based, multiqubit entanglement swapping protocols is
+that the loss of one particle renders the measurement separa-
+ble. However, some other states that are inequivalent to GHZ
+under LOCC can preserve their entanglement under reduc-
+tions. The existence of an iso-entangled basis composed of
+such states may constitute an avenue to more noise-resiliant
+entanglement swapping protocols which have natural quan-
+tum information applications.
+Note added.— During the late stage of our work, we be-
+came aware of the previous work [23] where i. a. bases are
+found for some Dicke states.
+ACKNOWLEDGMENTS
+We thank Hayata Yamasaki,
+Marcus Huber,
+Jakub
+Czartowski and Karol ˙Zyczkowski for discussions. A. T. ac-
+knowledges support from the Wenner-Gren Foundation and
+from the Wallenberg Centre for Quantum Technology. M. J.
+R. acknowledges financial support from the Austrian Science
+Fund (FWF) through BeyondC (F7103-N38), the Project No.
+I-2906, as well as support by the John Templeton Founda-
+tion through Grant 61466, The Quantum Information Struc-
+ture of Spacetime (qiss.fr), the Foundational Questions Insti-
+tute (FQXi) and the research platform TURIS. The opinions
+expressed in this publication are those of the authors and do
+not necessarily reflect the views of the John Templeton Foun-
+dation.
+[1] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and
+W. K. Wootters, Teleporting an unknown quantum state via dual
+classical and einstein-podolsky-rosen channels, Phys. Rev. Lett.
+70, 1895 (1993).
+[2] C. H. Bennett and S. J. Wiesner, Communication via one- and
+two-particle operators on einstein-podolsky-rosen states, Phys.
+Rev. Lett. 69, 2881 (1992).
+[3] M. ˙Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ek-
+ert, “event-ready-detectors” bell experiment via entanglement
+swapping, Phys. Rev. Lett. 71, 4287 (1993).
+[4] A. Tavakoli, A. Pozas-Kerstjens, M.-X. Luo, and M.-O. Renou,
+Bell nonlocality in networks, Reports on Progress in Physics
+85, 056001 (2022).
+[5] A. Tavakoli, J. Pauwels, E. Woodhead, and S. Pironio, Correla-
+tions in entanglement-assisted prepare-and-measure scenarios,
+PRX Quantum 2, 040357 (2021).
+[6] J. Pauwels, A. Tavakoli, E. Woodhead, and S. Pironio, Entan-
+glement in prepare-and-measure scenarios: many questions, a
+few answers, New Journal of Physics 24, 063015 (2022).
+[7] C. Kruszynska and B. Kraus, Local entanglability and multipar-
+tite entanglement, Phys. Rev. A 79, 052304 (2009).
+[8] A. Miyake and H. J. Briegel, Distillation of multipartite entan-
+glement by complementary stabilizer measurements, Phys. Rev.
+Lett. 95, 220501 (2005).
+[9] C. Spengler, M. Huber, and B. C. Hiesmayr, Composite param-
+eterization and haar measure for all unitary and special unitary
+groups, Journal of Mathematical Physics 53, 013501 (2012),
+https://doi.org/10.1063/1.3672064.
+[10] A. Acín, A. Andrianov, L. Costa, E. Jané, J. I. Latorre, and
+R. Tarrach, Generalized schmidt decomposition and classifi-
+cation of three-quantum-bit states, Phys. Rev. Lett. 85, 1560
+(2000).
+[11] H.
+A.
+Carteret,
+A.
+Higuchi,
+and
+A.
+Sudbery,
+Multi-
+partite
+generalization
+of
+the
+schmidt
+decomposition,
+Journal
+of
+Mathematical
+Physics
+41,
+7932
+(2000),
+https://doi.org/10.1063/1.1319516.
+[12] M. Navascués and T. Vértesi, Bounding the set of finite di-
+mensional quantum correlations, Phys. Rev. Lett. 115, 020501
+(2015).
+[13] A. Tavakoli, Semi-device-independent framework based on
+restricted distrust in prepare-and-measure experiments, Phys.
+Rev. Lett. 126, 210503 (2021).
+[14] A. Tavakoli, D. Rosset, and M.-O. Renou, Enabling computa-
+tion of correlation bounds for finite-dimensional quantum sys-
+tems via symmetrization, Phys. Rev. Lett. 122, 070501 (2019).
+[15] M.-O. Renou, D. Trillo, M. Weilenmann, T. P. Le, A. Tavakoli,
+N. Gisin, A. Acín, and M. Navascués, Quantum theory based on
+real numbers can be experimentally falsified, Nature 600, 625
+(2021).
+[16] W. K. Wootters, Entanglement sharing in real-vector-space
+quantum theory, Foundations of Physics 42, 19 (2012).
+[17] K.-D. Wu, T. V. Kondra, S. Rana, C. M. Scandolo, G.-Y. Xiang,
+C.-F. Li, G.-C. Guo, and A. Streltsov, Operational resource the-
+ory of imaginarity, Phys. Rev. Lett. 126, 090401 (2021).
+[18] N. Gisin, Entanglement 25 years after quantum teleportation:
+Testing joint measurements in quantum networks, Entropy 21,
+10.3390/e21030325 (2019).
+[19] A. Tavakoli, N. Gisin, and C. Branciard, Bilocal bell inequali-
+ties violated by the quantum elegant joint measurement, Phys.
+Rev. Lett. 126, 220401 (2021).
+[20] J.-F. Tang, Z. Hou, J. Shang, H. Zhu, G.-Y. Xiang, C.-F. Li, and
+G.-C. Guo, Experimental optimal orienteering via parallel and
+antiparallel spins, Phys. Rev. Lett. 124, 060502 (2020).
+[21] C.-X. Huang, X.-M. Hu, Y. Guo, C. Zhang, B.-H. Liu, Y.-
+F. Huang, C.-F. Li, G.-C. Guo, N. Gisin, C. Branciard, and
+A. Tavakoli, Entanglement swapping and quantum correla-
+tions via symmetric joint measurements, Phys. Rev. Lett. 129,
+030502 (2022).
+
+6
+[22] E. Bäumer, N. Gisin, and A. Tavakoli, Demonstrating the power
+of quantum computers, certification of highly entangled mea-
+surements and scalable quantum nonlocality, npj Quantum In-
+formation 7, 117 (2021).
+[23] Y.
+Tanaka,
+D.
+Markham,
+and
+M.
+Murao,
+Local
+encoding
+of
+classical
+information
+onto
+quantum
+states,
+Journal
+of
+Modern
+Optics
+54,
+2259
+(2007),
+https://doi.org/10.1080/09500340701403301.
+Appendix A: Basis construction for every bipartite state of local dimension d = 4 and d = 8
+Let the local dimension be a power of two, d = 2m, and index the d2 basis elements as (˜j, j) where ˜j = 0, 1, . . . , d − 1
+and j = 1, 2, . . . , d. Let W A
+ψ ⊗ W B
+ψ be the state-dependent local unitaries that transform the general state |ψ⟩ into the Schmidt
+basis, i.e. |ψS⟩ ≡ W A
+ψ ⊗ W B
+ψ |ψ⟩ = �d−1
+l=0 λl |l, l⟩, with the Schmidt coefficients λl ∈ R satisfying �
+l λ2
+l = 1. We now
+further decompose the individual d-dimensional registers as a string of m qubits, writing |l⟩ = |l1 . . . lm⟩. Thus, the Schmidt
+decomposed state reads
+|ψS⟩ =
+�
+l1,...,lm=0,1
+λl |l1 . . . lm, l1 . . . lm⟩ .
+(A1)
+Once the state has been put in the form (A1), we apply a set of local unitaries that is independent of the Schmidt coefficients.
+For d = 4 and ˜j = 0, the two sets of unitaries read as follows:
+˜j j
+U (˜j,j)
+1
+U (˜j,j)
+2
+U (˜j,j)
+1
+⊗ U (˜j,j)
+2
+|ψS⟩
+0 1
+11 ⊗ 11
+11 ⊗ 11
+λ00 |00, 00⟩ + λ01 |01, 01⟩ + λ10 |10, 10⟩ + λ11 |11, 11⟩
+0 2
+11 ⊗ X
+11 ⊗ XZ
+λ00 |01, 01⟩ − λ01 |00, 00⟩ + λ10 |11, 11⟩ − λ11 |10, 10⟩
+0 3
+X ⊗ 11
+XZ ⊗ Z
+λ00 |10, 10⟩ − λ01 |11, 11⟩ − λ10 |00, 00⟩ + λ11 |01, 01⟩
+0 4
+X ⊗ X XZ ⊗ X
+λ00 |11, 11⟩ + λ01 |10, 10⟩ − λ10 |01, 01⟩ − λ11 |00, 00⟩
+(A2)
+In addition, we define U (˜j,j)
+1
+:= X
+˜j
+4 U (˜j=0,j)
+1
+and U (˜j,j)
+2
+:= U (˜j=0,j)
+2
+, where Xd is the d-dimensional shift-operator Xd =
+�d−1
+l=0 |l + 1⟩⟨l|. Note that, the unitaries U (˜j,j)
+2
+coincide with the state-independent set for two qubits given in Eq. (2) and
+do not depend on ˜j. At the same time, U (˜j=0,j)
+1
+are the same as U (˜j,j)
+2
+where the Z gates are left out. We now show that
+{U (˜j,j)
+1
+⊗ U (˜j,j)
+2
+|ψS⟩}˜j,j is a basis of the bipartite Hilbert space. One can check directly that the four states with ˜j = 0 stated in
+Eq. (A2) above are pairwise orthogonal. We want to mention that we are exploiting the fact that U (˜j=0,j)
+2
+are the elements of a
+state-independent construction. To see the connection, note that the calculation for the state-independent two-qubit construction
+reads as follows:
+(11 ⊗ 11)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩ ,
+(A3)
+(11 ⊗ XZ)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |01⟩ − λ01 |00⟩ + λ10 |11⟩ − λ11 |10⟩ ,
+(A4)
+(XZ ⊗ Z)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |10⟩ − λ01 |11⟩ − λ10 |00⟩ + λ11 |01⟩ ,
+(A5)
+(XZ ⊗ X)(λ00 |00⟩ + λ01 |01⟩ + λ10 |10⟩ + λ11 |11⟩) = λ00 |11⟩ + λ01 |10⟩ − λ10 |01⟩ − λ11 |00⟩ .
+(A6)
+Since these states are pairwise orthogonal for arbitrary real coefficients λl1l2, the same holds true for the states in Eq. (A2).
+In addition, all of the states where ˜j = 0 are elements of the subspace spanned by |00, 00⟩, |01, 01⟩, |10, 10⟩ and |11, 11⟩.
+Hence, they form a basis of this four-dimensional subspace.
+By shifting now the first system we obtain a basis for the
+remaining orthogonal subspaces.
+More precisely, since we defined U (˜j,j)
+1
+= X
+˜j
+4 U (˜j=0,j)
+1
+the states where ˜j = 1 are
+esentially the same states as the ones in Eq. (A2) but with the first system shifted by one l → l ⊕ 1 (mod 4). For example,
+λ00 |11, 10⟩ − λ01 |00, 11⟩ − λ10 |01, 00⟩ + λ11 |10, 01⟩ is the state that corresponds to ˜j = 1 and j = 3. In this way, the
+four states where ˜j = 1 form a basis of the subspace spanned by |01, 00⟩, |10, 01⟩, |11, 10⟩ and |00, 11⟩ (or all states where
+|l + 1, l⟩). Analogously, the four states where ˜j = 2 (˜j = 3) form a basis of the subspaces spanned by the vectors with |l + 2, l⟩
+(|l + 3, l⟩). Altogether, the sixteen states {U (˜j,j)
+1
+⊗ U (˜j,j)
+2
+|ψS⟩}˜j,j form a basis of the entire sixteen dimensional Hilbert space.
+A similar construction can be found for d = 8 by using the state-independent construction of three qubits. Similar as above,
+
+7
+the set for ˜j = 0 reads as follows:
+˜j j
+U (˜j,j)
+1
+U (˜j,j)
+2
+U (˜j,j)
+1
+⊗ U (˜j,j)
+2
+|ψS⟩
+0 1
+11 ⊗ 11 ⊗ 11
+11 ⊗ 11 ⊗ 11
++λ000 |000, 000⟩ + λ001 |001, 001⟩ + λ010 |010, 010⟩ + λ011 |011, 011⟩
++λ100 |100, 100⟩ + λ101 |101, 101⟩ + λ110 |110, 110⟩ + λ111 |111, 111⟩
+0 2
+11 ⊗ 11 ⊗ X
+Z ⊗ Z ⊗ XZ
++λ000 |001, 001⟩ − λ001 |000, 000⟩ − λ010 |011, 011⟩ + λ011 |010, 010⟩
+−λ100 |101, 101⟩ + λ101 |100, 100⟩ + λ110 |111, 111⟩ − λ111 |110, 110⟩
+0 3
+11 ⊗ X ⊗ 11
+Z ⊗ XZ ⊗ 11
++λ000 |010, 010⟩ + λ001 |011, 011⟩ − λ010 |000, 000⟩ − λ011 |001, 001⟩
+−λ100 |110, 110⟩ − λ101 |111, 111⟩ + λ110 |100, 100⟩ + λ111 |101, 101⟩
+0 4
+X ⊗ 11 ⊗ 11
+XZ ⊗ 11 ⊗ 11
+(...)
+0 5
+11 ⊗ X ⊗ X
+Z ⊗ X ⊗ XZ
+(...)
+0 6
+X ⊗ 11 ⊗ X
+X ⊗ 11 ⊗ XZ
+(...)
+0 7
+X ⊗ X ⊗ 11
+X ⊗ XZ ⊗ Z
+(...)
+0 8
+X ⊗ X ⊗ X X ⊗ XZ ⊗ X
+(...)
+(A7)
+Again, we define U (˜j,j)
+1
+= X
+˜j
+8 U (˜j=0,j)
+1
+and U (˜j,j)
+2
+= U (˜j=0,j)
+2
+. The proof that this forms a basis of the 64-dimension Hilbert
+space is completely analogous to the case of d = 4 before. The eight states for ˜j = 0 form a basis of the eight-dimensional
+subspace spanned by |l1l2l3, l1l2l3⟩ (for li = 0, 1). Applying the shift operator X8 to the first system, one obtains bases of the
+other eight-dimensional orthogonal subspaces spanned by the vectors with
+��l + ˜j, l
+�
+. This approach cannot (immediately) be
+generalized to higher dimensions d = 2n, due to the lack of state-independent constructions for n ≥ 4 qubits. However, there is
+in principle no reason to restrict the unitaries on the second system to tensor products of single qubit Pauli gates as we do here.
+In principle, we could also consider general permutations with suitably chosen signs such that all terms cancel in this pairwise
+sense as above. Even when considering this larger class of possibilities, we made an exhaustive search and could not find any
+additional construction. Due to this, it seems unlikely that a construction exists in which the unitaries do not depend on the
+Schmidt coefficients.
+Appendix B: An n-qubit basis of W-states
+We define the n-qubit W-state as
+|W1⟩ ≡ |1⟩
+|W2⟩ ≡
+1
+√
+2 (|01⟩ + |10⟩)
+|W3⟩ ≡
+1
+√
+3 (|001⟩ + |010⟩ + |100⟩)
+|W4⟩ ≡ 1
+2 (|0001⟩ + |0010⟩ + |0100⟩ + |1000⟩)
+...
+(B1)
+Note that for one and two qubits, the definition is only introduced for sake of convenience. In general, we write
+|Wn⟩ ≡
+1
+√n
+�
+σ
+σ(|0⟩⊗n−1 |1⟩),
+(B2)
+where σ runs over all permutations of the position of “1”. It is also useful to write the state recursively as
+|Wn+1⟩ =
+�
+n
+n + 1 |Wn⟩ ⊗ |0⟩ +
+1
+√n + 1 |0⟩n ⊗ |1⟩
+(B3)
+Clearly, if we apply the local unitaries U (1)
+1
+= 11 and U (2)
+1
+= X to |W1⟩ we generate the trivial one-qubit W-basis {|0⟩ , |1⟩}.
+Assume now that the local unitaries {U (j)
+k } for k = 1, . . . n and j = 1, . . . , 2n yield a |Wn⟩-basis. We will now show that under
+this assumption we can construct a basis for |Wn+1⟩ and hence it follows from induction that a W-basis exists for any number
+of qubits.
+
+8
+We illustrate the induction step as follows,
+U (1)
+1
+⊗
+U (1)
+2
+⊗ . . . ⊗
+U (1)
+n
+⊗
+11
+U (2)
+1
+⊗
+U (2)
+2
+⊗ . . . ⊗
+U (2)
+n
+⊗
+11
+...
+...
+...
+U (2n)
+1
+⊗
+U (2n)
+2
+⊗ . . . ⊗
+U (2n)
+n
+⊗
+11
+U (1)
+1 Z
+⊗
+U (1)
+2 Z
+⊗ . . . ⊗
+U (1)
+n Z
+⊗ X
+U (2)
+1 Z
+⊗
+U (2)
+2 Z
+⊗ . . . ⊗
+U (2)
+n Z
+⊗ X
+...
+...
+...
+U (2n)
+1
+Z ⊗ U (2n)
+2
+Z ⊗ . . . ⊗ U (2n)
+n
+Z ⊗ X
+.
+(B4)
+We see that for the first 2n basis elements, we extend the unitaries for n qubits by tensoring with 11 for qubit number n + 1.
+For the latter 2n basis elements, we extend the unitaries for n qubits by multiplying all of them from the right by Z and finally
+tensoring with X for qubit number n + 1. As usual, we now write the string of unitaries associated to each row as V (n+1)
+j
+for
+n = 1, . . . , 2n+1. We similarly use V (n)
+j
+for the unitary strings for the case of n qubits.
+To see that this yields a basis, we first show that the first 2n basis elements (upper block of table, j = 1, . . . , 2n) are orthogonal.
+For this purpose, we use the recursion formula (B3) to write for j ̸= j′
+⟨Wn+1|(V (n+1)
+j′
+)†V (n+1)
+j
+|Wn+1⟩ =
+n
+n + 1⟨Wn0|(V (n)
+j′
+)†V (n)
+j
+⊗ 11|Wn0⟩ +
+1
+n + 1⟨0 . . . 01|(V (n)
+j′
+)†V (n)
+j
+⊗ 11|0 . . . 01⟩
++
+√n
+n + 1⟨Wn0|(V (n)
+j′
+)†V (n)
+j
+⊗ 11|0 . . . 01⟩ +
+√n
+n + 1⟨0 . . . 01|(V (n)
+j′
+)†V (n)
+j
+⊗ 11|Wn0⟩ = 0
+The first term is zero for all j′ ̸= j due to the induction hypothesis. The third and fourth terms are zero due to orthogonality in
+the last qubit register. The second term is zero for every j′ ̸= j there exists at least one qubit register k for which U (j′)
+k
+and U (j)
+k
+are composed of different numbers of bit-flips (X). The latter follows from the initial condition of using {11, X} to construct the
+|W1⟩-basis.
+The same procedure will analogously show that the latter 2n basis elements (lower block of the table, j = 2n + 1, . . . , 2n+1)
+are orthogonal. We are left with showing that every overlap between the upper and lower block, i.e. with any j′ = 1, . . . , 2n and
+any j = 2n + 1, . . . , 2n+1, also vanishes. For this we have
+⟨Wn+1|(V (n+1)
+j′
+)†V (n+1)
+j
+|Wn+1⟩ =
+n
+n + 1⟨Wn0|
+�
+(V (n)
+j′
+)†V (n)
+j
+⊗ X
+�
+n
+�
+k=1
+Z ⊗ 11|Wn0⟩
++
+1
+n + 1⟨0 . . . 01|
+�
+(V (n)
+j′
+)†V (n)
+j
+⊗ X
+�
+n
+�
+k=1
+Z ⊗ 11|0 . . . 01⟩
++
+√n
+n + 1⟨Wn0|
+�
+(V (n)
+j′
+)†V (n)
+j
+⊗ X
+�
+n
+�
+k=1
+Z ⊗ 11|0 . . . 01⟩
++
+√n
+n + 1⟨0 . . . 01|
+�
+(V (n)
+j′
+)†V (n)
+j
+⊗ X
+�
+n
+�
+k=1
+Z ⊗ 11|Wn0⟩
+Note that �n
+k=1 Z ⊗ 11 |Wn0⟩ = − |Wn0⟩ and �n
+k=1 Z ⊗ 11 |0 . . . 01⟩ = |0 . . . 01⟩. The first and second terms are both zero due
+to orthogonality in the final qubit register. We thus have
+⟨Wn+1|(V (n+1)
+j′
+)†V (n+1)
+j
+|Wn+1⟩ =
+√n
+n + 1⟨Wn|(V (n)
+j′
+)†V (n)
+j
+|0 . . . 0⟩ −
+√n
+n + 1⟨0 . . . 0|(V (n)
+j′
+)†V (n)
+j
+|Wn⟩
+=
+√n
+n + 1⟨Wn|(V (n)
+j′
+)†V (n)
+j
+− (V (n)
+j
+)†V (n)
+j′
+|0 . . . 0⟩ = 0.
+(B5)
+The last equality follows from the fact that it is sufficient, for given (j, j′), that there exist some register index k such that
+(U (j′))†
+kU (j)
+k
+− (U (j))†
+kU (j′)
+k
+= 0 in order for the overlap to vanish. This is always the case because due to our construction (see
+initial condition and the table), for every two unitaries there is at least one register k where the single-qubit unitaries differ by
+X, meaning that either (U (j)
+k , U (j′)
+k
+) = (11, X)/(Z, XZ), or the same with j ↔ j′ is true. The condition above is satisfied by
+all of these combinations. Hence we conclude that the proposed construction satisfies
+⟨Wn+1|(V (n+1)
+j
+)†V (n+1)
+j′
+|Wn+1⟩ = δjj′
+(B6)
+
+9
+and therefore yields a W-state basis for any number of qubits.
+Appendix C: The Pauli structure for state-independent qubit unitary constructions
+We consider the set of local unitaries P that are applied to the i-th qubit in the state-independent construction and show that
+without loss of generality, the set can be chosen to be the Pauli-type gates P ≡ {11, X, Z, XZ}. First, note that the set is finite
+since there are exactly 2n basis states. Next, we observe that the identity 11 has to be within the set P since we demand that
+V1 = 11. Furthermore, we can argue that the gate
+XZ =
+�
+0 −1
+1
+0
+�
+has to be within the set as well, since it is the only gate that maps every real qubit state to its orthogonal state. More precisely,
+if it is not used on the i-th qubit at least once, one can choose a real qubit state |φi⟩ such that none of the gates in P map
+|φi⟩ to its orthogonal vector. Hence if we apply the state-independent construction to the real-valued product state |φ⟩ =
+|0⟩1 ⊗ . . . |0⟩i−1 ⊗ |φi⟩ ⊗ |0⟩i+1 ⊗ . . . ⊗ |0⟩n none of the resulting 2n states are distinguishable on the i-th qubit, which is
+impossible if these states should form a basis of product states. Therefore, the gate XZ has to be within the set P. Apart from
+the gates 11 and XZ we can constrain which other qubit unitaries can be in the set P. We know that if we demand V1 = 11, every
+string of local unitaries (Vj) and their products (Vj)†Vj′ with j ̸= j′ have to be skew-symmetric. As a result, the local unitaries
+on each subsystem (hence, the unitaries in the set P) and also all their products have to be either symmetric or skew-symmetric.
+By neglecting a global phase, the general form of a unitary operator can be written as:
+U =
+�
+cos (θ)eiα
+sin (θ)eiβ
+− sin (θ)e−iβ cos (θ)e−iα
+�
+.
+(C1)
+The only skew-symmetric 2×2 unitary is, up to an irrelevant global phase, the Pauli-type operator XZ, which we already found
+to be necessarily in the set P. All the symmetric matrices of this form can be written as:
+U =
+�
+cos (θ)eiα
+i sin (θ)
+i sin (θ) cos (θ)e−iα
+�
+.
+(C2)
+If the gate U is in P, it is at some point multiplied with the gate XZ since the operator XZ is used at least once on the i-th qubit.
+Since we know that the result of this product has to be again either symmetric or skew-symmetric, we obtain that α = π/2, 3π/2
+due to:
+(XZ)†U =
+�
+0 1
+−1 0
+� �
+cos (θ)eiα
+i sin (θ)
+i sin (θ) cos (θ)e−iα
+�
+=
+�
+i sin (θ) cos (θ)e−iα
+− cos (θ)eiα
+−i sin (θ)
+�
+.
+(C3)
+The two possibilities for α = π/2, 3π/2 correspond to the two solutions
+U1 =
+�
+cos (θ)
+sin (θ)
+sin (θ) − cos (θ)
+�
+,
+U2 =
+�
+sin (θ) − cos (θ)
+− cos (θ) − sin (θ)
+�
+.
+(C4)
+We left the irrelevant global factor i for simplicity. Considering the additional degree of freedom of θ, we can restrict to the first
+class of solutions U1 since the second class U2 can be obtained by shifting θ by π/2. Hence, if we add a gate U to the set P, it
+has to be of the form given by U1 above. Now if we add two such gates to the set P, the product of U1 with another valid matrix
+U ′
+1 is
+U †
+1U ′
+1 =
+�
+cos (θ)
+sin (θ)
+sin (θ) − cos (θ)
+� �
+cos (θ′)
+sin (θ′)
+sin (θ′) − cos (θ′)
+�
+=
+=
+�
+cos (θ) cos (θ′) + sin (θ) sin (θ′)
+cos (θ) sin (θ′) − sin (θ) cos (θ′)
+sin (θ) cos (θ′) − cos (θ) sin (θ′)
+cos (θ) cos (θ′) + sin (θ) sin (θ′)
+�
+=
+�
+cos (θ − θ′)
+− sin (θ − θ′)
+sin (θ − θ′)
+cos (θ − θ′)
+�
+If both, U1 and U ′
+1, are in P, this product has to be again either symmetric, which is true if θ = θ′ or skew-symmetric, which is
+true if θ = θ′ + π/2. (Note that, also θ = θ′ + π and θ = θ′ + 3π/2 are possible solutions but we do not have to consider them
+
+10
+since they just differ by an irrelevant global factor of (−1) in one of the two unitaries.) Hence, U ′
+1 is either U1 or the unitary U2
+stated above. Hence, for each single-qubit subsystem, we can only use a set of operators P ≡ {11, U1, U2, XZ} for our basis
+construction.
+In a final step, we can show that we can restrict also θ. To see this, suppose a state-independent construction exists where we
+use the gates from the set P ≡ {11, U1, U2, XZ}. Now consider the construction where each gate U1 is replaced with W †U1W,
+each gate U2 with W †U2W, each gate XZ with W †XZW and each gate 11 with W †11W, where:
+W =
+�
+cos (α) − sin (α)
+sin (α)
+cos (α)
+�
+(C5)
+for some freely chosen parameter α. This also has to be a state-independent construction for any state with real coefficients,
+since W is a map from real states to real states, and all inner products between the basis states remain the same under this
+local transformation. Hence, if a state-independent construction exists with the gate set P ≡ {11, U1, U2, XZ}, another state-
+independent construction with the gate set P′ ≡ {W †11W, W †U1W, W †U2W, W †XZW} has to exist as well. Choosing
+α = θ/2, the set P′ ≡ {W †11W, W †U1W, W †U2W, W †XZW} becomes exactly P′ ≡ {11, X, Z, XZ}, which concludes the
+proof.
+

F9AzT4oBgHgl3EQfUfyk/content/tmp_files/2301.01268v1.pdf.txt

@@ -0,0 +1,1049 @@
+arXiv:2301.01268v1  [math.CV]  3 Jan 2023
+Proper holomorphic maps in Euclidean spaces
+avoiding unbounded convex sets
+Barbara Drinovec Drnovˇsek and Franc Forstneriˇc
+Abstract We show that if E is a closed convex set in Cn (n > 1) contained in a closed halfspace H such
+that E ∩ bH is nonempty and bounded, then the concave domain Ω = Cn \ E contains images of proper
+holomorphic maps f : X → Cn from any Stein manifold X of dimension < n, with approximation of
+a given map on closed compact subsets of X. If in addition 2 dim X + 1 ≤ n then f can be chosen an
+embedding, and if 2 dim X = n then it can be chosen an immersion. Under a stronger condition on E
+we also obtain the interpolation property for such maps on closed complex subvarieties.
+Keywords Stein manifold, holomorphic embedding, Oka manifold, minimal surface, convexity
+MSC (2010): 32H02, 32Q56; 52A20, 53A10
+Date: 3 January 2023
+In memoriam Nessim Sibony
+1. Introduction
+Let X be a Stein manifold. Denote by O(X, Cn) the Frechet space of holomorphic maps
+X → Cn endowed with the compact-open topology and write O(X, C) = O(X). A theorem of
+Remmert [36] (1956), Narasimhan [35] (1960), and Bishop [7] (1961) states that almost proper
+maps are residual in O(X, Cn) if dim X = n, proper maps are dense if dim X < n, proper
+immersions are dense if 2 dim X ≤ n, and proper embeddings are dense if 2 dim X < n. A
+proof is also given in the monograph [29] by Gunning and Rossi.
+It is natural to ask how much space proper maps need. We pose the following question.
+Problem 1.1. For which domains Ω ⊂ Cn are proper holomorphic maps (immersions,
+embeddings) X → Cn as above, with images contained in Ω, dense in O(X, Ω)?
+It is evident that Ω cannot be contained in a halfspace of Cn since every holomorphic map
+from C to a halfspace lies in a complex hyperplane. In this paper we give an affirmative answer
+for concave domains whose complement E = Cn \ Ω satisfies the following condition.
+Definition 1.2. A closed convex set E in a real or complex Euclidean space V has bounded
+convex exhaustion hulls (BCEH) if for every compact convex set K in V
+(1.1)
+the set h(E, K) = Conv(E ∪ K) \ E is bounded.
+Here, Conv denotes the convex hull. The following is our first main result.
+Theorem 1.3. Let E be an unbounded closed convex set in Cn (n > 1) with bounded convex
+exhaustion hulls. Given a Stein manifold X with dim X < n, a compact O(X)-convex set K in
+X, and a holomorphic map f0 : K → Cn with f0(bK) ⊂ Ω = Cn \ E, we can approximate f0
+uniformly on K by proper holomorphic maps f : X → Cn satisfying f(X \ ˚
+K) ⊂ Ω. The map
+f can be chosen an embedding if 2 dim X < n and an immersion if 2 dim X ≤ n.
+
+2
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+In this paper, a map f : K → Cn from a compact set K is said to be holomorphic if it is the
+restriction to K of a holomorphic map on an open neighbourhood of K.
+In particular, if f0(K) ⊂ Ω then the theorem gives uniform approximation of f0 by proper
+holomorphic maps f : X → Cn with f(X) ⊂ Ω. If bE is of class C 1 and strictly convex
+near infinity, we obtain an analogue of Theorem 1.3 with additional interpolation on a closed
+complex subvariety X′ of X such that f0 : X′ → Cn is proper holomorphic; see Theorem 4.2.
+Without the condition on the range, interpolation of proper holomorphic embeddings X ֒→ Cn
+on a closed complex subvariety was obtained by Acquistapace et al. [1] in 1975.
+The analogue of the BCEH condition for unbounded closed sets in Stein manifolds, with the
+convex hull replaced by the holomorphically convex hull, is used in holomorphic approximation
+theory of Arakelyan and Carleman type; see the survey in [18].
+It is evident that a closed convex set E ⊂ Rn has BCEH if and only if there is an increasing
+sequence K1 ⊂ K2 ⊂ · · · of compact convex sets exhausting Rn such that the set h(E, Kj) (see
+(1.1)) is bounded for every j = 1, 2, . . .. In particular, BCEH is a condition at infinity which is
+invariant under perturbations supported on a compact subset. For compact convex sets E ⊂ Cn,
+Theorem 1.3 was proved in [24]; in this case BCEH trivially holds.
+We show in Section 3 that a closed convex set E in Rn has BCEH if and only if E is
+continuous in the sense of Gale and Klee [26]; see Proposition 3.3. If E has BCEH then
+Conv(E ∪ K) is closed for any compact convex set K ⊂ Rn (see [26, Theorem 1.5]). If such E
+is unbounded, which is the main case of interest, there are affine coordinates (x, y) ∈ Rn−1 × R
+such that E = Eφ = {(x, y) ∈ Rn : y ≥ φ(x)} is the epigraph of a convex function
+φ : Rn−1 → R+ = [0, +∞) growing at least linearly near infinity (see Proposition 3.4). In
+particular, an unbounded closed convex set E ⊂ Cn with BCEH is of the form
+(1.2)
+E = Eφ = {z = (z′, zn) ∈ Cn : ℑzn ≥ φ(z′, ℜzn)}
+in some affine complex coordinates z = (z′, zn) on Cn, with φ as above. (Here, ℜ and ℑ denote
+the real and the imaginary part.) For a convex function φ of class C 1 we give a differential
+characterization of the BCEH condition on its epigraph Eφ; see Proposition 3.8. The BCEH
+property holds if the radial derivative of φ tends to infinity; see Corollary 3.9. On the other hand,
+there are convex functions of linear growth whose epigraphs have BCEH; see Example 3.10. By
+Proposition 3.11, a convex function φ with at least linear growth at infinity can be approximated
+uniformly on compacts by functions ψ ≤ φ of the same kind whose epigraphs Eψ have BCEH.
+This allows us to extend Theorem 1.3 as follows; see Section 4 for the proof.
+Corollary 1.4. The conclusion of Theorem 1.3 holds for any convex epigraph Eφ of the form
+(1.2) such that φ ≥ 0 and the set {φ = 0} is nonempty and compact.
+A closed convex set E ⊂ Cn with BCEH does not contain any affine real line (see Proposition
+3.4), and for n > 1 its complement Ω = Cn \ E is an Oka domain according to Wold and the
+second named author; see [25, Theorem 1.8]. This fact plays an important role in our proof
+of Theorem 1.3, given in Section 4. (The precise result from Oka theory which we shall use
+is stated as Theorem 4.1.) Among closed convex epigraphs (1.2), the class of sets with Oka
+complement is strictly bigger than the class of sets with BCEH. In particular, the former class
+contains many sets containing boundary lines, which is impossible for a set with BCEH.
+Problem 1.5. Is there a (not necessarily convex) set Eφ ⊂ Cn of the form (1.2) with φ ≥ 0 of
+sublinear growth for which Theorem 1.3 holds? Is there a set of this kind in C2 such that C2\Eφ
+contains the image of a proper holomorphic disc D = {z ∈ C : |z| < 1} → C2?
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+3
+Theorem 1.3 is the first general result in the literature providing proper holomorphic maps
+X → Cn from any Stein manifold of dimension < n whose images avoid large convex sets in
+Cn close to a halfspace, and with approximation of a given map on a compact holomorphically
+convex set in X. Without the approximation condition and assuming that dim X ≤ n − 2, there
+are proper holomorphic maps of X into a complex hyperplane in Cn \ E.
+On the other hand, there are many known results concerning proper holomorphic maps in
+Euclidean spaces and in more general Stein manifolds whose images avoid certain small closed
+subsets, such as compact or complete pluripolar ones, and results in which the source manifold
+is the disc D = {z ∈ C : |z| < 1}. Proper holomorphic discs in C2 avoiding closed complete
+pluripolar sets of the form E = E′ × C, with E′ ⊂ C, were constructed by Alexander [5] in
+1977. The first named author showed in [13] (2004) that for every closed complete pluripolar
+set E in a Stein manifold Y with dim Y > 1 and point p ∈ Y \ E there is a proper holomorphic
+disc f : D → Y with p ∈ f(D) ⊂ Y \ E. If Y = C2 there also exist embedded holomorphic
+discs with this property according to Borell et al. [8] (2008), and for dim Y ≥ 3 this holds
+by the general position argument. Proper holomorphic discs in C2 with images contained in
+certain concave cones were constructed by Globevnik and the second named author [23] in
+2001. They also constructed proper holomorphic discs in C2 with images in (C \ {0})2, and
+hence proper harmonic discs D → R2, disproving a conjecture by Schoen and Yau [37, p. 18].
+(Another construction of such maps was given by Boˇzin [9].) More generally, it was shown
+by Alarc´on and L´opez [4, Corollary 1.1] in 2012 that every open Riemann surface X admits a
+proper harmonic map to R2 which is the projection of a conformal minimal immersion X → R3.
+The aforementioned result from [23] was used by the first named author in [12] (2002) to classify
+closed convex sets in C2 whose complement is filled by images of holomorphic discs which are
+proper in C2. More recently, Forstneriˇc and Ritter [24] (2014) proved Theorem 1.3 in the case
+when E ⊂ Cn is a compact polynomially convex set and 2 dim X ≤ n (for immersions) or
+2 dim X < n (for embeddings), and for proper holomorphic maps X → Cn when dim X < n
+and E is a compact convex set. A further development in this direction is the analogue of
+Theorem 1.3 when Cn is replaced by a Stein manifold Y with the density property and E ⊂ Y
+is a compact O(Y )-convex set; see [22, Remark 4.5] and the references therein. However, in all
+mentioned results except those in [23, 12], the avoided sets are thin or compact.
+Without insisting on approximation, the theorem of Remmert, Bishop, and Narasimhan is not
+optimal with respect to the dimension of the target space. Indeed, it was shown by Eliashberg and
+Gromov [17] in 1992, with an improvement for odd dimensional Stein manifolds by Sch¨urmann
+[38] in 1997, that a Stein manifold X of dimension m ≥ 2 embeds properly holomorphically
+in Cn with n =
+�3m
+2
+�
++ 1, and for m ≥ 1 it immerses properly holomorphically in Cn with
+n =
+� 3m+1
+2
+�
+. (See also [20, Sect. 9.3].) However, the construction method in these papers,
+which relies on the Oka principle for sections of certain stratified holomorphic fibre bundles,
+does not give the density statement, and we do not know whether Theorem 1.3 holds for maps to
+these lower dimensional spaces. It is also an open problem whether every open Riemann surface
+embeds properly holomorphically in C2; see [20, Secs. 9.10-9.11] and the survey [21].
+Theorem 1.3 is proved in Section 4. The proof relies on two main ingredients. One is the
+result of Wold and the second named author [25, Theorem 1.8] which shows in particular that the
+complement Ω = Cn \E of a closed convex set E having BCEH is an Oka domain. The second
+main technique comes from the work of Dor [10, 11] (1993-95), following earlier papers by
+Stensønes [39] (1989) and Hakim [30] (1990). Dor constructed proper holomorphic immersions
+and embeddings of any smoothly bounded, relatively compact, strongly pseudoconvex domain
+D in a Stein manifold X into any pseudoconvex domain Ω in Cn under the dimension conditions
+
+4
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+in Theorem 1.3.
+Previously, Hakim [30] constructed proper holomorphic maps to balls in
+codimension one. The main idea is to inductively lift the image of bD under a holomorphic
+map f : ¯D → Ω to a given higher superlevel set of a strongly plurisubharmonic exhaustion
+function ρ : Ω → R+ in a controlled way, taking care not to decrease the value of ρ ◦ f very
+much anywhere on D during the process. When D is a finite bordered Riemann surface, this can
+be achieved by using approximate solutions of a Riemann-Hilbert boundary value problem (see
+[14]). In higher dimensions the proof is more subtle and uses carefully controlled holomorphic
+peak functions on ¯D to push a given map f : ¯D → Ω locally at a point z ∈ f(bD) in the direction
+of the zero set Sz of the holomorphic (quadratic) Levi polynomial of the exhaustion function
+ρ : Ω → R. At a noncritical point z ∈ Ω of ρ, Sz is a smooth local complex hypersurface and
+the restricted function ρ|Sz increases quadratically as we move away from z. If ρ is a strictly
+convex function, this can be achieved by pushing the image of f(bD) in the direction of suitably
+chosen affine complex hyperplanes. Dor’s construction was extended by the authors to maps
+from strongly pseudoconvex domains in Stein manifolds to an arbitrary Stein manifold Ω, and
+also to q-convex complex manifolds for suitable values of q ∈ N; see the papers [14, 15] from
+2007 and 2010, respectively. In those papers we introduced the technique of gluing holomorphic
+sprays of manifold-valued maps on a strongly pseudoconvex Cartan pair with control up to the
+boundary (a nonlinear version of the Cousin-I problem) and a systematic approach for avoiding
+critical points of a q-convex Morse exhaustion function on Ω.
+Earlier constructions of this type, using simpler holomorphic peak functions and higher
+codimension, were given in 1985 by Løw [34] and Forstneriˇc [19] who showed that every
+relatively compact strongly pseudoconvex domain D in a Stein manifold embeds properly
+holomorphically in a high dimensional Euclidean ball. A related result with interpolation on
+a suitable subset of the boundary of D is due to Globevnik [27] (1987). This peak function
+technique was inspired by the construction of inner functions on the ball of Cn by Løw [33] in
+1982, based on the work of Hakim and Sibony [31].
+We apply this technique to push the boundary f0(bD) ⊂ Ω = Cn \ E of a holomorphic map
+f0 : ¯D → Cn in Theorem 1.3 out of a certain compact convex cap C attached to E along a part
+of bC contained in bE and such that the set E1 = E ∪ C is convex and has bounded convex
+exhaustion hulls. At the same time, we ensure that the new map g : ¯D → Cn still sends D \ K
+to Ω. For a precise result, see Proposition 2.1. In the next step, we use that Ω1 = Cn \ E1 is
+an Oka domain (see Corollary 3.6). Since g(bD) ⊂ Ω1, we can apply the Oka principle (see
+Theorem 4.1) to approximate g by a holomorphic map f1 : X → Cn with f1(X \ D) ⊂ Ω1.
+Continuing inductively, we obtain a sequence of holomorphic maps X → Cn converging to a
+proper map satisfying Theorem 1.3. The details are given in Section 4.
+The analogues of Theorem 1.3 and Corollary 1.4 also hold for minimal surfaces in Rn.
+Theorem 1.6. Let n ≥ 3, and let φ : Rn−1 → R+ be a convex function such that the set {φ = 0}
+is nonempty and compact. Given an open Riemann surface X, a compact O(X)-convex set K
+in X, and a conformal minimal immersion f0 : U → Rn from a neighbourhood of K with
+f0(bK) ⊂ Ω = {y < φ(x)}, we can approximate f0 uniformly on K by proper conformal
+minimal immersions f : X → Rn (embeddings if n ≥ 5) satisfying f(X \ ˚
+K) ⊂ Ω.
+If in addition φ is of class C 1, strictly convex at infinity, and the epigraph Eφ = {y ≥ φ(x)}
+has BCEH then one can add to this statement the interpolation of the map on discrete sets, in
+analogy to Theorem 4.2.
+Theorem 1.6 is obtained by following the proof of Theorem 1.3, replacing Proposition 2.1
+by the analogous result obtained by the Riemann–Hilbert deformation method for conformal
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+5
+minimal surfaces (see [2] or [3, Chapter 6]). Furthermore, it has recently been shown by the
+authors [16, Corollary 1.5] that the complement of a closed convex set E ⊂ Rn (n ≥ 3) is
+flexible for minimal surfaces (an analogue of the Oka property in complex geometry) if and only
+if E is not a halfspace or a slab; clearly this includes all sets with BCEH.
+Another method for constructing proper minimal surfaces, which yields the same result in
+some examples not covered by Theorem 1.6, was developed by Alarc´on and L´opez [4] in 2012.
+They showed that Theorem 1.6 holds for any wedge domain Γ × R ⊂ R3, where Γ ⊂ R2 is an
+open cone with angle > π; see [4, Theorem 5.6]. The complement of this set is convex but it
+fails to satisfy the hypotheses of Theorem 1.6 due to the presence of lines in the boundary. An
+important difference between these two fields, which affects the possible construction methods,
+is that every open Riemann surface admits a proper harmonic map to the plane R2 (see [4,
+Theorem I]), while only few such surfaces admit proper holomorphic maps to C.
+The analogue of Problem 1.5 for minimal surfaces asks whether there is a domain in R3 of
+the form {x3 < φ(x1, x2)}, where φ : R2 → R+ is a function with sublinear growth, which
+contains minimal surfaces of hyperbolic type that are proper in R3, or just a proper hyperbolic
+end of a minimal surface. In particular, it would be interesting to know whether the domain
+below the upper half of a vertical catenoid has this property. On the other hand, the strong
+halfspace theorem of Hoffman and Meeks [32] says that the only proper minimal surfaces in R3
+contained in a halfspace are planes.
+2. Pushing a strongly pseudoconvex boundary out of a strictly convex cap
+Let O be a convex domain in Cn for some n > 1. Recall that a continuous function ρ : O → R
+is said to be strictly convex if for any pair of points a, b ∈ O we have that
+ρ(ta + (1 − t)b) < tρ(a) + (1 − t)ρ(b) for all 0 < t < 1.
+Assume now that ρt : O → R (t ∈ [0, 1]) is a continuous family of C 1 functions satisfying
+the following conditions:
+(a) For every t ∈ [0, 1] the function ρt is strictly convex. Note that dρt ̸= 0 on Mt := {ρt = 0}.
+(b) If 0 ≤ s < t ≤ 1 then ρt ≤ 0 on Ms.
+(c) There is an open relatively compact subset ω0 of M0 such that for every pair of numbers
+0 ≤ s < t ≤ 1 we have that Mt ∩ M0 = Mt ∩ Ms = M0 \ ω0.
+This means that the hypersurfaces Mt coincide on the subset M0 \ ω0, and as t ∈ [0, 1]
+increases the domains ωt = Mt \ M0 ⊂ Mt are pairwise disjoint and move into the convex
+direction. Each compact set of the form
+(2.1)
+Ct =
+�
+s∈[0,t]
+ωs for t ∈ [0, 1]
+is called a strictly convex cap with the base ω0. Note that bCt = ω0 ∪ ωt, Ct is strictly convex
+along ωt, strictly concave along ω0, and it has corners along ω0 ∩ ωt. As t ∈ [0, 1] increases to
+1, the caps Ct monotonically increase to C1 and they share the same base ω0. Likewise, for any
+0 ≤ s < t ≤ 1 the set Cs,t = �
+u∈[s,t] ωu is a strictly convex cap with the base ωs. The sets
+(2.2)
+Et = {z ∈ O : ρt(z) ≤ 0} for t ∈ [0, 1]
+are strictly convex along bEt = {ρt = 0}, they form a continuously increasing family in t, and
+Et = E0 ∪ Ct for every t ∈ [0, 1].
+
+6
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+Under these assumptions, we have the following result.
+Proposition 2.1. Let D be a smoothly bounded, relatively compact, strongly pseudoconvex
+domain in a Stein manifold X with dim X < n. Let the sets Et ⊂ O ⊂ Cn (t ∈ [0, 1]) be
+given by (2.2), and let f0 : ¯D → O be a map of class A ( ¯D) such that f0(bD) ∩ E0 = ∅. Given
+a compact set K ⊂ D such that f0(D \ K) ∩ E0 = ∅ and a number ǫ > 0, there is a map
+f : ¯D → O of class A ( ¯D) satisfying the following conditions:
+(i) f(bD) ∩ E1 = ∅,
+(ii) f(D \ K) ∩ E0 = ∅, and
+(iii) maxx∈K |f(x) − f0(x)| < ǫ.
+Recall that a map f : ¯D → O is said to be of class A ( ¯D) if it is continuous on ¯D and
+holomorphic on D. In our application of Proposition 2.1 in the proof of Theorem 1.3, the set O
+will be a ball (or the entire Euclidean space) and the hypersurfaces Mt = {ρt = 0} = bEt will
+be convex graphs over the coordinate hyperplane Cn−1 × R ⊂ Cn.
+In the proof of Proposition 2.1 we shall need the following lemma.
+Lemma 2.2. Assume that O is a convex open subset of Cn for n > 1, L is a compact subset of
+O, and ρ : O → R is a C 1 smooth strictly convex function. Then there is a number δ > 0 with
+the following property. If D is a smoothly bounded strongly pseudoconvex domain in a Stein
+manifold X of dimension dim X = m < n, K is a compact subset of D, and f : ¯D → O is a
+map of class A ( ¯D) such that
+(2.3)
+ρ(f(z)) > −δ for all z ∈ bD
+and
+ρ(f(z)) > 0 if z ∈ bD and f(z) /∈ L,
+then given η > 0 there is a map g : ¯D → O of class A ( ¯D) satisfying the following conditions:
+(i) ρ(g(z)) > 0 for z ∈ bD,
+(ii) ρ(g(z)) > δ for those z ∈ bD for which g(z) ∈ L,
+(iii) ρ(g(z)) > ρ(f(z)) − η for z ∈ D \ K, and
+(iv) |f(z) − g(z)| < η for z ∈ K.
+For m = 1, i.e., when D is a finite bordered Riemann surface, this is a simplified version of
+[14, Lemmas 6.2 and 6.3], which is proved by using approximate solutions of a Riemann–Hilbert
+boundary value problem. This method was employed in several earlier papers mentioned in [14].
+When ρ is strictly convex, C 1 smoothness suffices since in the proof we may take a continuous
+family of tangential linear discs to the sublevel set of ρ.
+For m ≥ 2, Lemma 2.2 is a simplified and slightly modified version of [15, Lemma 5.3].
+Besides the fact that we are considering single maps ¯D → O instead of sprays of maps, the only
+difference is that the assumption in [15, Lemma 5.3] that the set {ρ = 0} is compact is replaced
+by the assumption (2.3) saying that ρ(f(z)) for z ∈ bD may be negative only if f(z) lies in the
+compact set L ⊂ O. This hypothesis ensures that the lifting for a relatively big amount (the role
+of the constant δ) only needs to be made on a compact subset of O, while elsewhere it suffices to
+pay attention not to decrease ρ ◦ f by more than a given amount and to approximate sufficiently
+closely on K (the role of the constant η). The proof requires only a minor adaptation of [15,
+proof of Lemma 5.3], using its local version [15, Lemma 5.2] in a finite induction with respect
+to a covering of bD by small open sets on which there are good systems of local holomorphic
+peak functions. In fact, Lemma 2.2 corresponds to a simplified version of [15, Sublemma 5.4],
+which explains how to lift the image of bD with respect to ρ for a sufficiently large amount at
+those points in bD which the map f sends to a certain coordinate chart Ui in the target manifold.
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+7
+In our situation, the role of Ui is played by an open relatively compact neighbourhood of the set
+L ∩ {ρ = 0} in O, and there is no need to use the rest of the proof of [15, Lemma 5.3].
+Proof of Proposition 2.1. For t ∈ [0, 1] let δt > 0 be a number for which the conclusion of
+Lemma 2.2 holds for the function ρt and the compact set L = C1 (see (2.1)). The open sets
+Ut = {z ∈ O : −δt < ρt(z) < δt} for t ∈ [0, 1]
+form an open covering of C1, so there exists a finite subcovering {Utj} for 0 ≤ t1 < t2 < . . . <
+tk ≤ 1. Applying Lemma 2.2 we inductively find maps f1, . . . , fk ∈ A ( ¯D) such that for every
+j = 1, . . . , k we have that
+(a) fj(bD) ∩ Etj = ∅ (where Et is given by (2.2)),
+(b) fj(D \ K) ∩ E0 = ∅, and
+(c) |fj − fj−1| < ǫ/k on K.
+Note that conditions (a) and (b) hold for f0 and (c) is void. Assume inductively that for some
+j ∈ {1, . . . , k} we have maps f0, . . . , fj−1 satisfying these conditions. Applying Lemma 2.2
+with f = fj−1 and taking fj = g, condition (a) follows from part (i) in Lemma 2.2, (b) follows
+from (ii) provided that the number η > 0 in Lemma 2.2 is chosen small enough, and (c) follows
+from (iii) in Lemma 2.2 provided that η ≤ ǫ/k . This gives the map fj satisfying conditions
+(a)–(c) and the induction may continue. The map f = fk then satisfies the proposition.
+□
+Remark 2.3. Proposition 2.1 also holds, with the same proof, if ρt (t ∈ [0, 1]) are strongly
+plurisubharmonic functions of class C 2 satisfying dρt ̸= 0 on Mt = {ρt = 0}. Indeed, the
+results from [15], which are used in the proof, pertain to this case. In the present paper we shall
+only use the convex case under C 1 smoothness, which comes naturally in the construction.
+3. Closed convex sets with BCEH
+In the context of convex analysis, closed unbounded convex sets that share several important
+properties with compact convex sets were studied by Gale and Klee [26] in 1959.
+They
+introduced the class of continuous sets, and we show that this class coincides with the class
+of sets having BCEH, introduced in Definition 1.2; see Proposition 3.3. We then develop further
+properties of these sets which are relevant to the proof of our main theorems.
+By a ray in Rn, we shall mean a closed affine halfline. Let E be a closed convex subset of
+Rn. A boundary ray of E is a ray contained in the boundary of E. An asymptote of E is a ray
+L ⊂ Rn \ E such that dist(L, E) = inf{|x − y| : x ∈ L, y ∈ E} = 0. The function
+σ : {u ∈ Rn : |u| = 1} → R ∪ {+∞},
+σ(u) = sup{x · u : x ∈ E}
+is called the the support function of E. (Here, x · u denotes the Euclidean inner product.) A
+closed convex set E is said to be continuous in the sense of Gale and Klee [26] if the support
+function of E is continuous. Note that every compact convex set is continuous.
+The following result is a part of [26, Theorem 1.3] due to Gale and Klee; we only list those
+conditions that will be used. The last item (iv) uses also [26, Theorem 1.5].
+Theorem 3.1. For a closed convex subset E in Rn the following conditions are equivalent:
+(i) E is continuous.
+(ii) E has no boundary ray nor asymptote.
+(iii) For each point p ∈ Rn the convex hull Conv(E ∪ {p}) is closed.
+(iv) For every compact convex set K ⊂ Rn the set Conv(E ∪ K) is closed.
+
+8
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+Condition (iii) implies that the closed convex hull Conv(E ∪ {p}) is the union of the line
+segments connecting p to the points in E. It also shows that an unbounded continuous closed
+convex subset E of Rn is not contained in any affine hyperplane.
+Let us record the following observation which will be used in the sequel.
+Lemma 3.2. Let E ⊂ Rn be a closed convex set, p ∈ Rn\E, and L ⊂ Rn be an affine subspace
+containing p. Then, Conv(E ∪ {p}) ∩ L = Conv((E ∩ L) ∪ {p}).
+Proof. Set E′ = E ∩ L. It is obvious that Conv(E′ ∪ {p}) ⊂ Conv(E ∪ {p}) ∩ L. Conversely,
+since E is convex, every point q ∈ Conv(E ∪ {p}) belongs to a line segment from p to a point
+q′ ∈ E. If in addition q ∈ L and q ̸= p then q′ ∈ E′, and hence q ∈ Conv(E′ ∪ {p}).
+□
+Proposition 3.3. A closed convex set E ⊂ Rn has BCEH if and only if it is continuous in the
+sense of Gale and Klee [26].
+Proof. Since all closed bounded convex sets have BCEH and are continuous, it suffices to
+consider the case when the set E is unbounded.
+If E is not continuous then by Theorem 3.1 it has a boundary ray or an asymptote. Denote it
+by L, and let ℓ be the affine line containing L. Pick any affine 2-plane H ⊂ Rn containing ℓ.
+There is a point p ∈ H \(ℓ∪E). By considering rays from p to points q ∈ E approaching L and
+going to infinity (if L is a boundary ray, we can choose points q ∈ L), we see that the closure
+of the set h(E, p) = Conv(E ∪ {p}) \ E contains the parallel translate L′ ⊂ H+ of L passing
+through p, so h(E, p) is unbounded and hence E does not have BCEH.
+Assume now that E is a continuous and let us prove that it has BCEH. We need to show that
+for any closed ball B ⊂ Rn the set h(E, B) = Conv(E ∪ B) \ E is bounded. Assume to the
+contrary that there is a sequence xm ∈ h(E, B) with |xm| → ∞ as m → ∞. Since the sets E
+and B are convex, we have that
+xm = tmbm + (1 − tm)em for tm ∈ [0, 1], bm ∈ B, em ∈ E, and m ∈ N.
+Note that (1 − tm)|em| → ∞ as m → ∞. By compactness of the respective sets we may
+assume, passing to a subsequence, that em ̸= 0 for all m and the sequences tm, bm, and
+1
+|em|em
+are convergent. Denote their respective limits by t, b, and f. We have that
+xm = tmbm + (1 − tm)em = bm + (1 − tm)|em|
+� em
+|em| − bm
+|em|
+�
+= bm + (1 − tm)|em|fm
+where fm =
+� em
+|em|− bm
+|em|
+�
+. Note that limm→∞ fm = f. Pick a number α ≥ 0 and set p = b+αf.
+If m is large enough then (1−tm)|em| > α, so the point ym = bm+αfm lies on the line segment
+connecting bm and xm. Since xm ∈ Conv(E ∪ {bm}), it follows that ym ∈ Conv(E ∪ {bm}).
+Note that the sequence ym converges to p. Since E is continuous, Conv(E ∪ {b}) is closed by
+Theorem 3.1, so p = limm→∞ ym ∈ Conv(E ∪ {b}). Since this holds for every α ≥ 0, the ray
+L = {b + αf : α ∈ [0, ∞)} lies in Conv(E ∪ {b}). By Lemma 3.2 there is α0 ∈ [0, ∞) such
+that the ray L′ = {b + αf : α ≥ α0} lies in E. Since E is continuous, L is not a boundary ray
+of E by Theorem 3.1, thus L contains a point q = b+α1f ∈ E \bE for some α1 ≥ α0. Choose
+a neighbourhood Uq ⊂ E of q. For any large enough m we then have pm := bm + α1fm ∈ Uq.
+Let Lm = {bm + αfm : α ≥ 0}. Note that Lm ∩ Conv(E ∪ {bm}) = Conv((Lm ∩ E) ∪ {bm})
+by Lemma 3.2. However, for m large enough the point xm ∈ Lm lies on the opposite side of pm
+than bm, so xm belongs to Lm ∩ Conv(E ∪ {bm}) but not to Conv((Lm ∩ E) ∪ {bm}). This
+contradiction proves that E has BCEH.
+□
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+9
+Given a function φ : Rn−1 → R, the epigraph of φ is the set
+(3.1)
+E = Eφ = {(x, y) ∈ Rn−1 × R : y ≥ φ(x)}.
+Note that a function is convex if and only if its epigraph is convex.
+Proposition 3.4. If E ⊊ Rn is a closed unbounded convex set with BCEH then
+(i) E does not contain any affine real line, and
+(ii) for every affine line ℓ intersecting E in a ray and any hyperplane H transverse to ℓ, E is
+the epigraph of a convex function on H. In particular, there are affine coordinates (x, y)
+on Rn in which E is of the form (3.1) for a convex function φ : Rn−1 → R+ satisfying
+(3.2)
+lim inf
+|x|→+∞
+φ(x)
+|x|
+> 0.
+The condition (3.2) says that φ grows at least linearly at infinity. We show in Example 3.10
+that linear growth is possible.
+Proof. (i) Assume that ℓ ⊂ E is an affine line and let us prove that E does not have BCEH.
+Since E is a proper subset of Rn, there is a parallel translate ℓ′ of ℓ which is not contained in E,
+and hence ℓ′ \E contains a ray L. Let p be the endpoint of L, and let p′ ∈ L be an arbitrary other
+point. Since E ∩L = ∅, there is a ball B around p′ such that Conv(B ∪{p})∩E = ∅. Clearly,
+there is a point q ∈ B such that the ray Lq with the endpoint p and containing q intersects the
+line ℓ, so the line segment from p to q belongs to Conv(E ∪ {p}) \ E = h(E, p). By moving
+p′ ∈ L to infinity we see that h(E, p) is unbounded, so E does not have BCEH.
+(ii) Since E is unbounded, it contains a ray L. Denote by ℓ the affine line containing L. Let
+ℓ′ be any parallel translate of ℓ. Since E contains no affine lines by part (i), there is a point
+p ∈ ℓ′ \E. The closed convex hull of the union of L and p contains the parallel translate L′ ⊂ ℓ′
+of L passing through p. Since E has BCEH, we conclude that L′ ⊂ Conv(E ∪ {p}) and L′ \ E
+is bounded. Since E ∩ L′ is convex, L′ ∩ E is a closed ray with the endpoint on bE. This shows
+that E is a union of closed rays contained in parallel translates of the line ℓ, so it is an epigraph
+of a convex function defined on any hyperplane H ⊂ Rn transverse to ℓ. Choosing H such that
+H ∩ E = ∅ there are affine coordinates (x, y) on Rn with H = {y = 0} and ℓ = {x = 0}. In
+these coordinates, E is of the form (3.1) for a positive convex function φ.
+Finally, if condition (3.2) fails then there is a sequence (xk, yk) ∈ E with |xk| → +∞ and
+yk/|xk| → 0 as k → ∞. The union of the line segments Lk connecting p = (0, −1) ∈ Rn−1×R
+to (xk, yk), intersected with the lower halfspace y ≤ 0, is then an unbounded subset of
+h(E, p) = Conv(E ∪ {p}) \ E, contradicting the assumption that E has BCEH.
+□
+Remark 3.5. The growth condition (3.2) for an epigraph can always be achieved in suitable
+linear coordinates (even without the BCEH property) if there is a supporting hyperplane H ⊂ Rn
+for E such that the set E∩H is nonempty and compact. Indeed, we may then choose coordinates
+(x, y) on Rn such that H = {y = 0}, E ⊂ {y ≥ 0}, and 0 ∈ E. If the condition (3.2) fails,
+there is a sequence (xk, yk) ∈ E with |xk| → +∞ and yk/|xk| → 0 as k → ∞. After passing
+to a subsequence, a ray in E ∩ H lies in the closure of the union of the line segments Lk ⊂ E
+connecting the origin to (xk, yk), contradicting the assumption that the latter set is compact.
+Corollary 3.6. If E is a closed convex set in Cn (n > 1) having BCEH then Cn \ E is Oka.
+Proof. By Proposition 3.4 the set E does not contain any affine real line, and hence Cn \ E is
+Oka by [25, Theorem 1.8].
+□
+
+10
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+The following lemma shows that the BCEH condition is stable under uniform approximation.
+Lemma 3.7. Assume that φ : Rn−1 → R is a convex function whose epigraph Eφ (3.1) has
+BCEH. Then for any ǫ > 0 and convex function ψ : Rn−1 → R satisfying |φ − ψ| < ǫ the
+epigraph Eψ also has BCEH.
+Proof. If Eψ fails to have BCEH then by Theorem 3.1 and Proposition 3.3 it has a boundary
+ray or an asymptote, L. Since dist(L, Eψ) = 0 and Eψ is convex, dist(x, Eψ) converges
+to zero as x ∈ L goes to infinity.
+Thus, by making L shorter if necessary, we have that
+L ⊂ Eφ−2ǫ \ Eφ+2ǫ. Hence, L lies out of Eφ+2ǫ but the vertical translation of L for 4ǫ pushes it
+in Eφ+2ǫ. Since Eφ+2ǫ, being a translate of Eφ, has BCEH, this contradicts Proposition 3.4 (ii).
+The contradiction shows that Eψ has BCEH as claimed.
+□
+We now give a differential characterization of the BCEH property of an epigraph (3.1).
+Proposition 3.8. If φ : Rn−1 → R is a convex function of class C 1 satisfying condition (3.2),
+then the epigraph E = {(x, y) ∈ Rn : y ≥ φ(x)} has BCEH if and only if
+(3.3)
+lim
+|x|→∞ |x|
+�
+1 −
+φ(x)
+x · ∇φ(x)
+�
+= +∞.
+Proof. We first consider the case n = 2. Then, x is a single variable and (3.3) is equivalent to
+(3.4)
+lim
+x→+∞
+�
+x − φ(x)
+φ′(x)
+�
+= +∞
+and
+lim
+x→−∞
+�
+x − φ(x)
+φ′(x)
+�
+= −∞.
+For every x ∈ R such that φ′(x) ̸= 0 the number
+(3.5)
+ξ(x) = x − φ(x)
+φ′(x)
+is the first coordinate of the intersection of the tangent line to the graph of φ at the point (x, φ(x))
+with the first coordinate axis y = 0. By (3.2) and convexity of φ we have that |φ′(x)| is bounded
+away from zero for all sufficiently big |x|. This shows that conditions (3.4) are invariant under
+translations, so we may assume that φ ≥ 0 and φ(0) = 0. It is easily seen that the function ξ is
+increasing. If φ is of class C 2, we have that ξ′(x) = φ(x)φ′′(x)/φ′(x)2 ≥ 0.
+Assume now that conditions (3.4) hold.
+Pick a pair of sequences aj < bj in R with
+limj→∞ aj = −∞ and limj→∞ bj = +∞. The intervals Ij = [ξ(aj), ξ(bj)] then increase
+to R as j → ∞. We identify Ij with Ij × {0} ⊂ R2. Since φ is convex, its epigraph lies above
+the tangent line at any point. It follows that the set h(E, Ij) (see (1.1)) is the bounded region in
+R×R+ whose boundary consists of Ij, the two line segments Lj and L′
+j connecting the endpoints
+(ξ(aj), 0) and (ξ(bj), 0) of Ij to the respective points Aj = (aj, φ(aj)) and Bj = (bj, φ(bj))
+on bE, and the graph of φ over [aj, bj]. The supporting lines of Lj and L′
+j intersect at a point
+Cj in the lower halfspace y < 0, and we obtain a closed triangle ∆j with the endpoints Aj, Bj,
+and Cj. Note that ∆j ∩ (R × {0}) = Ij. Since φ grows at least linearly (see (3.2)), the triangles
+∆j ⊂ R2 exhaust R2 as j → ∞, and the set h(E, ∆j) (1.1) is bounded for every j. Hence,
+E has BCEH. This argument furthermore shows that for any point p = (0, −c) /∈ E there is
+a unique pair of tangent lines to bE passing through p such that, denoting by q1, q2 ∈ bE the
+respective points where these lines intersect bE, the convex hull Conv(E ∪ {p}) is the union of
+E and the triangle with vertices p, q1, q2.
+Conversely, if (3.3) fails then it is easily seen that E has a boundary ray or an asymptote, so
+it does not have BCEH. We leave the details to the reader.
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+11
+The case with n ≥ 3 now follows easily. Pick a unit vector v ∈ Rn−1, |v| = 1, and let Lv
+denote the 2-plane in Rn passing through the origin and spanned by v and en = (0, . . . , 0, 1).
+Then, Ev := E ∩ Lv = {(t, y) ∈ R2 : y ≥ φ(tv)} and the first condition in (3.4) reads
+(3.6)
+lim
+t→+∞
+�
+t −
+φ(tv)
+�n−1
+j=1 vj
+∂φ
+∂xj (tv)
+�
+= +∞.
+Writing x = tv with t ≥ 0 and v = x/|x|, this is clearly equivalent to (3.3). As before, let
+p = (0, . . . , 0, −c) /∈ E. If (3.3) holds then Conv(Ev ∪ {p}) ⊂ Lv is obtained by adding to
+Ev the triangle in Lv obtained by the two tangent lines to bEv passing through p as described in
+the case n = 2. The sizes of these triangles are uniformly bounded with respect to the direction
+vector |v| = 1, and condition (3.2) implies that these triangles increase to Lv as c → +∞,
+uniformly with respect to v. Since �
+|v|=1 Lv = Rn, Lemma 3.2 shows that
+Conv(E ∪ {p}) =
+�
+|v|=1
+Conv(Ev ∪ {p}),
+and hence E has BCEH. The converse is seen as in the special case n = 2.
+□
+Corollary 3.9. If φ : Rn−1 → R+ is a convex function of class C 1 such that
+lim
+|x|→+∞
+x · ∇φ(x)
+|x|
+= +∞,
+then the epigraph E = {(x, y) ∈ Rn : y ≥ φ(x)} has BCEH.
+Proof. By restricting to planes as in the above proof, it suffices to consider the case n = 2. We
+may assume that φ ≥ 0 and φ(0) = 0. Since φ is convex, g(x) = φ′(x) is an increasing function
+and the above condition reads limx→±∞ g(x) = ±∞. For any x0 > 0 and x ≥ x0 we have that
+ξ(x) := x −
+1
+g(x)
+� x
+0
+g(t)dt =
+� x
+0
+�
+1 − g(t)
+g(x)
+�
+dt ≥
+� x0
+0
+�
+1 − g(t)
+g(x)
+�
+dt.
+Letting x → +∞ we have that g(t)
+g(x) → 0 uniformly on t ∈ [0, x0], and hence the last integral
+converges to x0. Letting x0 → ∞ we see that limx→+∞ ξ(x) = +∞. The analogous argument
+applies when x → −∞. Hence, conditions (3.3) hold and therefore E has BCEH.
+□
+Example 3.10. There exist convex epigraphs (3.1) having BCEH where the function φ grows
+linearly, although it cannot be too close to linear near infinity in the absence of boundary rays
+and asymptotes. We give such an example in R2. Let g : R → (−1, 1) be an odd, continuous,
+increasing function with limx→+∞ g(x) = 1 and
+� ∞
+0 (1−g(x))dx = +∞. (An explicit example
+is g(x) = 2
+πArctan x.) Its integral φ(x) =
+� x
+0 g(t)dt for x ∈ R then clearly satisfies φ(x) ≥ 0,
+φ′(x) = g(x) ∈ (−1, +1) (hence φ grows linearly), and φ is convex. We now show that (3.3)
+holds. Let x > 0 be large enough so that g(x) > 0. We have that
+ξ(x) = x −
+1
+g(x)
+� x
+0
+g(t)dt =
+� x
+0
+�
+1 − g(t)
+g(x)
+�
+dt.
+Fix x0 > 0 and let x ≥ x0. Then, ξ(x) ≥
+� x0
+0 (1 − g(t)/g(x))dt. Since limx→+∞ g(x) = 1
+and ξ is increasing for large enough |x|, it follows that limx→+∞ ξ(x) ≥
+� x0
+0 (1 − g(t))dt.
+Sending x0 → +∞ gives limx→+∞ ξ(x) ≥
+� ∞
+0 (1 − g(t))dt = +∞. Similarly we see that
+limx→−∞ ξ(x) = −∞. Thus, (3.3) holds, and hence the epigraph of φ has BCEH.
+By using the idea in the above example we now prove the following approximation result,
+which extends Theorem 1.3 to a much bigger class of convex epigraphs (see Corollary 1.4).
+
+12
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+Proposition 3.11. Assume that φ : Rn−1 → R+ is a convex function such that the set {φ = 0}
+is nonempty and compact. Given numbers ǫ > 0 (small) and R > 0 (big) there is a smooth
+convex function ψ : Rn−1 → R such that ψ < φ on Rn−1, φ(x) − ψ(x) < ǫ for all |x| ≤ R,
+and the epigraph Eψ = {y ≥ ψ} has BCEH.
+Proof. By Remark 3.5 the function φ grows at least linearly near infinity (see (3.2)). Set
+(3.7)
+A = lim inf
+|x|→∞
+φ(x)
+|x| > 0.
+Since the set φ = 0 does not contain any affine line, Azagra’s result [6, Theorem 1 and
+Proposition 1] implies that for every ǫ > 0 there is a smooth strictly convex function ψ on
+Rn−1 satisfying φ − ǫ < ψ < φ. Replacing φ by ψ − minx ψ(x) ≥ 0 we may therefore assume
+that φ is smooth. By increasing the number R > 0 if necessary, we may assume that
+(3.8)
+φ(x)
+|x|
+≥ A
+2
+for all |x| ≥ R.
+Pick a number r ∈ (0, 1) close to 1 such that
+(3.9)
+(1 − r) sup
+|x|≤R
+φ(x) < ǫ.
+Choose a smooth increasing function h : R → R+ such that
+h(t) = 0 for t ≤ R,
+lim
+t→+∞ h(t) = 1,
+and
+� ∞
+0
+(1 − h(t))dt = +∞.
+(We can take a smoothing of the Arctan function used in Example 3.10.) Set
+H(x) =
+� |x|
+0
+h(s)ds
+for x ∈ Rn−1.
+Clearly, H ≥ 0 is a radially symmetric smooth convex function that vanishes on |x| ≤ R and
+satisfies H(x) ≤ |x| for all x ∈ Rn−1. With A and r as in (3.7) and (3.9) we set
+δ = A(1 − r)
+2
+.
+We claim that the function
+ψ(x) = rφ(x) + δH(x)
+for x ∈ Rn−1
+satisfies the conditions in the theorem. Clearly, ψ ≥ rφ is a smooth convex function. For
+|x| ≤ R we have H(x) = 0, so ψ(x) = rφ(x) ≤ φ(x) and φ(x) − ψ(x) = (1 − r)φ(x) < ǫ by
+(3.9). If |x| > R then φ(x)/|x| ≥ A/2 by (3.8) and H(x) < |x|, which implies
+ψ(x)
+|x|
+≤ rφ(x)
+|x| + δ ≤ φ(x)
+|x| .
+Indeed, we have that φ(x)
+|x| − r φ(x)
+|x| = (1 − r)φ(x)
+|x| ≥ A(1−r)
+2
+= δ. Hence, ψ ≤ φ on Rn−1.
+It remains to show that the epigraph Eψ satisfies BCEH. We shall verify (3.3), which is
+equivalent to (3.6) with uniform convergence with respect to the vector v = x/|x|. Write
+gv(t) = r∂φ(tv)
+∂t
+,
+k(t) = δh(t),
+˜gv(t) = ∂ψ(tv)
+∂t
+= gv(t) + k(t).
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+13
+The quantity in (3.6) associated to the function ψ is given by
+ξv(t)
+=
+t − ψ(tv)
+˜gv(t) =
+� t
+0
+�
+1 − gv(s) + k(s)
+gv(t) + k(t)
+�
+ds
+=
+� t
+0
+gv(t) − gv(s)
+gv(t) + k(t) ds +
+� t
+0
+k(t) − k(s)
+gv(t) + k(t)ds
+≥
+� t
+0
+gv(t) − gv(s)
+gv(t) + δ
+ds +
+� t
+0
+k(t) − k(s)
+gv(t) + δ ds,
+where the last inequality holds since the functions gv and k are nonnegative and increasing and
+k < δ. Pick c > 0. We will show that for large enough t > 0 and any unit vector v ∈ Rn−1 the
+above expression is bigger than or equal to c. Choose positive numbers t0, a, t1 as follows:
+t0 = 3c,
+a = max{3 max
+|v|=1 gv(t0), 3δ},
+� t1
+0
+(k(t1) − k(s))ds > ac.
+Such t1 exists since limt→+∞
+� t
+0(k(t) − k(s))ds = δ
+� ∞
+0 (1 − h(s))ds = +∞. Since the
+integrands in the bound for ξv(t) are nonnegative, we have for t ≥ max{t0, t1} and |v| = 1 that
+(3.10)
+ξv(t) ≥
+� t0
+0
+gv(t) − gv(s)
+gv(t) + δ
+ds +
+� t1
+0
+k(t) − k(s)
+gv(t) + δ ds.
+Assume that for some such (t, v) we have that gv(t) + δ ≥ a. Since a ≥ 3δ, it follows that
+gv(t) ≥ 2δ and hence
+gv(t)
+gv(t) + δ ≥ 2
+3.
+Furthermore, from a ≥ 3 max|v|=1 gv(t0) we get for 0 ≤ s ≤ t0 that
+gv(s)
+gv(t) + δ ≤ gv(t0)
+a
+≤ 1
+3.
+These two inequalities imply that the first integral in (3.10) is bounded below by t0/3 ≥ c. If
+on the other hand gv(t) + δ < a then the denominator of the second integral in (3.10) is at most
+a, so the integral is ≥ c by the choice of t1. This shows that ξv(t) ≥ c for all |v| = 1 and
+t ≥ max{t0, t1}. Since c was arbitrary, condition (3.3) holds and hence Eψ has BCEH.
+□
+The following observation will be used in the proof of Theorem 1.3.
+Proposition 3.12. Denote by B the open unit ball in Rn. Let Eφ ⊂ Rn be a closed convex
+set of the form (3.1) with C 1 boundary having BCEH, where the function φ : Rn−1 → R is
+bounded from below and strictly convex near infinity. Then there is an r0 > 0 such that for every
+r ≥ r0 the convex hull Conv(Eφ ∪ rB) = {y ≥ ψ(x)} is a closed convex set with BCEH, and
+ψ : Rn−1 → R is a convex function of class C 1 such that ψ ≤ φ and these functions agree near
+infinity. Furthermore, if r ≥ r0 is large enough then the function φt : Rn−1 → R defined by
+(3.11)
+φt(x) = (1 − t)φ(x) + tψ(x),
+x ∈ Rn−1
+is strictly convex for every t ∈ (0, 1), and for any 0 < t0 < t1 < 1 the closure of the set
+{(x, y) ∈ Rn : φt1(x) < y < φt0(x)}
+is a strictly convex cap with the base in the strictly convex hypersurface {y = φt0(x)}.
+
+14
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+Proof. Consider the function on Rn−1 given by
+˜φr(x) =
+�
+min{φ(x), −
+�
+r2 − |x|2},
+|x| < r,
+φ(x),
+|x| ≥ r.
+(Note that ˜φr may be discontinuous at the points of the sphere |x| = r.) The convex hull of its
+epigraph E˜φr equals Conv(E∪rB), which is closed by Theorem 3.1 (iv), and the set h(E, rB) =
+Conv(E ∪ rB) \ E is bounded since E has BCEH. By smoothing ˜φr we get a function ˜ψr of
+class C 1 which agrees with φ near infinity such that Conv(E ˜ψr) = Conv(E ∪ rB). By [28,
+Theorem 3.2] we conclude that Conv(E ∪ rB) has C 1 boundary, so it is the epigraph Eψr of a
+convex function ψr : Rn−1 → R of class C 1 which agrees with φ near infinity.
+Since φ grows at least linearly, there is a function τ(r) defined for r ∈ R+ large enough such
+that ψr(x) = −
+�
+r2 − |x|2 for |x| ≤ τ(r) and τ(r) → +∞ as r → +∞. By choosing r large
+enough, the compact set of points where the function φ fails to be strictly convex is contained
+in the ball |x| < τ(r). Since on this ball we have that ψr(x) = −
+�
+r2 − |x|2 which is strictly
+convex, the convex combinations φt in (3.11) of φ and ψ = ψr are strictly convex on Rn−1 for
+all 0 < t < 1. For such r, the last statement in the proposition is evident. (Note that the strictly
+convex functions ρt(x, y) = exp(ψt(x) − y) − 1 for t ∈ (0, 1) correspond to those used in
+Section 2.)
+□
+4. Proof of Theorem 1.3
+For the definition and the main theorem on Oka manifolds, see [20, Definition 5.4.1 and
+Theorem 5.4.4]. We shall use the following version of the Oka principle; see [22, Theorem 1.3].
+Theorem 4.1. Assume that X is a Stein manifold, K is a compact O(X)-convex set in X, X′
+is a closed complex subvariety of X, Ω is an Oka domain in a complex manifold Y , f : X → Y
+is a continuous map which is holomorphic on a neighbourhood of K, f|X′ : X′ → Y is
+holomorphic, and f(X \ ˚
+K) ⊂ Ω. Then there is a homotopy {ft}t∈[0,1] of continuous maps
+ft : X → Y connecting f = f0 to a holomorphic map f1 : X → Y such that for every t ∈ [0, 1]
+the map ft is holomorphic on a neighbourhood of K, it agrees with f on X′, it approximates f
+uniformly on K and uniformly in t ∈ [0, 1] as closely as desired, and ft(X \ ˚
+K) ⊂ Ω.
+Proof of Theorem 1.3. By Proposition 3.4 there are complex coordinates z = (z′, zn) on Cn
+such that the given set E is an epigraph of the form (1.2). We shall write z = (x, y) where
+x = (z′, ℜzn) ∈ Cn−1 × R ∼= R2n−1 and y = ℑzn ∈ R, so E = Eφ = {y ≥ φ(x)} where
+φ ≥ 0 is a convex function as in Proposition 3.4. Let the set K ⊂ X and the map f0 : K → Cn
+be as in the theorem; in particular, f0(bK) ⊂ Cn \ E. Thus, there are an open neighbourhood
+U ⊂ X of K and ǫ > 0 such that f0 is holomorphic in U and f0(U \ ˚
+K) ⊂ Cn \ Eφ−ǫ. By
+Azagra [6, Theorem 1.8] there is a a real analytic strictly convex function φ0 : R2n−1 → R such
+that φ − ǫ < φ0 < φ. Its epigraph E0 = {(x, y) ∈ Cn : y ≥ φ0(x)} is a closed strictly convex
+set with real analytic boundary which has BCEH by Lemma 3.7, and f0(U \ ˚
+K) ⊂ Cn \ E0.
+Let B denote the open unit ball in Cn centred at 0. Recall the notation h(E, K) in (1.1). Pick
+a number r0 > 0. We can find an increasing sequence rk > 0 diverging to infinity such that
+(4.1)
+h(E0, rkB) ⊂ rk+1B
+for k = 0, 1, 2, . . . .
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+15
+Indeed, since E0 has BCEH, the set h(E0, rkB) is bounded for each k, and hence (4.1) holds if
+the number rk+1 is chosen large enough. Set
+Ek+1 = Conv(E0 ∪ rkB) = E0 ∪ h(E0, rkB)
+for k = 0, 1, 2, . . ..
+We clearly have that E0 ⊂ E1 ⊂ · · · ⊂ �∞
+k=0 Ek = Cn. Furthermore, (4.1) shows that for
+j = 0, 1, . . . , k + 1 we have that E0 ⊂ Ej ⊂ E0 ∪ rk+1B and hence
+(4.2)
+Ek+2 = Conv(Ej ∪ rk+1B) for j = 0, 1, . . . , k + 1.
+Proposition 3.12 shows that for each k = 1, 2, . . . we have Ek = {y ≥ φk(x)} where φk a
+convex function of class C 1 which agrees with φ0 near infinity, and Ek has BCEH. Hence,
+Ωk = Cn \ Ek = {(x, y) ∈ Cn : y < φk(x)}
+is an Oka domain for every k = 0, 1, . . . by Corollary 3.6. In view of Ek+2 = Conv(Ek∪rk+1B)
+(see (4.2)), Proposition 3.12 also shows that if rk+1 is chosen large enough then the function
+(4.3)
+ψt = (1 − t)φk + tφk+2 : Cn−1 × R → R
+is strictly convex for every t ∈ (0, 1), and for each 0 < t0 < t1 < 1 the closure of the set
+(4.4)
+C = {(x, y) : ψt1 < y < ψt0}
+is a strictly convex cap as described in Section 2. (Note that the strictly convex functions
+ρt(x, y) = exp(ψt(x) − y) − 1 for t ∈ (0, 1) correspond to those used in Section 2.)
+Choose an exhaustion D0 ⊂ D1 ⊂ · · · ⊂ �∞
+k=0 Dk = X by smoothly bounded, relatively
+compact, strongly pseudoconvex domains with O(X)-convex closures such that K ⊂ D0 ⊂
+¯D0 ⊂ U. For consistency of notation we set D−1 = K. We now construct a sequence of
+holomorphic maps fk : ¯Dk → Cn satisfying the following conditions for k = 0, 1, 2, . . .:
+(a) fk(Dk \ Dk−1) ⊂ Ωk = Cn \ Ek,
+(b) fk+1(Dk \ Dk−1) ⊂ Ωk, and
+(c) fk+1 approximates fk uniformly on ¯Dk−1 as closely as desired.
+For k = 0 the initial map f0 in Theorem 1.3 satisfies condition (a) while conditions (b) and (c)
+are void. Assuming inductively that we found maps f0, . . . , fk satisfying these conditions, the
+construction of the next map fk+1 is made in two steps as follows.
+By compactness of the set fk(bDk) ⊂ Ωk = {y < φk(x)} we can choose t0 ∈ (0, 1) small
+enough such that f(bDk) ⊂ {y < ψt0(x)}, where the function ψt (t ∈ [0, 1]) is given by (4.3).
+By (4.1) we can also choose t1 ∈ (t0, 1) sufficiently close to 1 such that
+Ek+1 ⊂ {(x, y) : y ≥ ψt1(x)}.
+Proposition 2.1 applied to the map fk : ¯Dk → Cn, the set Ek, and the strictly convex cap
+C (4.4) (which corresponds to C1 in Proposition 2.1) gives holomorphic map gk : ¯Dk → Cn
+approximating fk on Dk−1 and satisfying
+(4.5) gk(bDk) ⊂ {(x, y) : y < ψt1(x)} ⊂ Cn \ Ek+1 = Ωk+1 and gk(Dk \ Dk−1) ⊂ Ωk.
+In the second step, we use that Ωk+1 is an Oka domain. Since Ωk+1 is contractible and
+gk(bDk) ⊂ Ωk+1 by (4.5), gk extends from ¯Dk to a continuous map X → Cn sending
+X \ Dk to Ωk+1. Theorem 4.1 applied to gk gives a holomorphic map fk+1 : ¯Dk+1 → Cn
+approximating gk on ¯Dk and satisfying fk+1(Dk+1 \ Dk) ⊂ Ωk+1 (which is condition (a) for
+k + 1) and fk+1(Dk \ Dk−1) ⊂ Ωk (condition (b)). Since fk+1 approximates gk on ¯Dk and gk
+approximates fk on ¯Dk−1, fk+1 also satisfies condition (c). This completes the induction step.
+
+16
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+If the approximations are close enough then the sequence fk converges uniformly on
+compacts in X to a holomorphic f : X → Cn. Conditions (a)–(c) and the fact that the sets
+Ek exhaust Cn imply that f is a proper holomorphic map satisfying f(X \ ˚
+K) ⊂ Ω0 = Cn \E0.
+To construct proper holomorphic immersions and embeddings in suitable dimensions given in
+the theorem, we use the general position argument at every step to ensure that every map fk in the
+sequence is an immersion or an embedding. (See e.g. [20, Corollary 8.9.3].) If the convergence
+is fast enough then the same holds for the limit map f by a standard argument.
+□
+Proof of Corollary 1.4. Given a holomorphic map f0 : K → Cn with f0(bK) ⊂ Cn \ Eφ as
+in Theorem 1.3, Proposition 3.11 furnishes a closed convex set Eψ ⊃ Eφ with BCEH such that
+f0(bK) ⊂ Cn \ Eψ. Applying Theorem 1.3 with Eψ gives the desired conclusion.
+□
+We have the following analogue of Theorem 1.3 with interpolation on a closed complex
+subvariety of X. Unlike in the above corollary, approximation of E from the outside by convex
+sets enjoying BCEH cannot be used since the subvariety f(X′) may have zero distance to bE.
+This results extends the case of [24, Theorem 15] when E is a compact convex set.
+Theorem 4.2. Let E be a closed convex set in Cn (n > 1) with C 1 boundary which is strictly
+convex near infinity and has bounded convex exhaustion hulls. Let X be a Stein manifold,
+K ⊂ X be a compact O(X)-convex set, U ⊂ X be an open set containing K, X′ be a
+closed complex subvariety of X, and f0 : U ∪ X′ → Cn be a holomorphic map such that
+f0|X′ : X′ → Cn is proper holomorphic and f0(bK ∪ (X′ \ K)) ∩ E = ∅. Given ǫ > 0 there
+exists a proper holomorphic map f : X → Cn satisfying the following conditions:
+(a) f(X \ ˚
+K) ⊂ Cn \ E,
+(b) ∥f − f0∥K < ǫ,
+(c) f|X′ = f0|X′.
+If 2 dim X ≤ n then f can be chosen an immersion (and an embedding if 2 dim X + 1 ≤ n)
+provided that f0|X′ is one.
+Proof. This is proved by a small modification of the proof of Theorem 1.3, similar to the one
+in [24, proof of Theorem 15]. The initial step in the proof, approximating E from the outside
+by a strictly convex set, is unnecessary since bE is strictly convex near infinity. The main (and
+essentially the only) change comes in the choice of the exhaustion Dk of the Stein manifold
+X. In the inductive step when constructing the map fk+1, we must assume in addition that
+fk(bDk ∩ X′) ⊂ Ωk+1 = Cn \Ek+1. Then, we push the image of bDk out of Ek+1 by the same
+method as before, using Proposition 2.1 but ensuring that the modifications are kept fixed on X′
+and small near bDk ∩ X′. This is possible since the method from [15] is applied locally near
+bDk (away from bDk ∩ X′), and these local modifications are glued together by preserving the
+value of the map on X′. We refer to [24, proof of Theorem 15] for a more precise description.
+This gives the next holomorphic map fk+1 : X → Cn satisfying fk+1(X \ Dk) ⊂ Ωk+1,
+fk+1|X′ = fk|X′, and conditions (b) and (c) in the proof of Theorem 1.3. We then choose the
+next domain Dk+1 ⊂ X big enough such that fk+1(bDk+1 ∩X′) ⊂ Ωk+2 = Cn \Ek+2. This is
+possible since the map fk+1|X′ = f0|X′ : X′ → Cn is proper, f0(X′ \ ˚
+K) ⊂ Ω = Cn \ E, and
+the domain Ωk+2 agrees with Ω near infinity by the construction. Clearly the induction step is
+now complete. Assuming that the approximations are close enough, the sequence fk converges
+to a limit holomorphic map f : X → Cn satisfying the stated conditions.
+□
+Acknowledgements. The first named author is supported by grants P1-0291, J1-3005, and N1-
+0137 from ARRS, Republic of Slovenia. The second named author is supported by the European
+Union (ERC Advanced grant HPDR, 101053085) and grants P1-0291, J1-3005, and N1-0237
+
+Proper holomorphic maps in Euclidean spaces avoiding unbounded convex sets
+17
+from ARRS, Republic of Slovenia. The authors wish to thank Antonio Alarc´on for helpful
+discussions and information concerning the case pertaining to minimal surfaces.
+References
+[1] F. Acquistapace, F. Broglia, and A. Tognoli. A relative embedding theorem for Stein spaces. Ann. Scuola Norm.
+Sup. Pisa Cl. Sci. (4), 2(4):507–522, 1975.
+[2] A. Alarc´on, B. Drinovec Drnovˇsek, F. Forstneriˇc, and F. J. L´opez. Every bordered Riemann surface is a complete
+conformal minimal surface bounded by Jordan curves. Proc. Lond. Math. Soc. (3), 111(4):851–886, 2015.
+[3] A. Alarc´on, F. Forstneriˇc, and F. J. L´opez. Minimal surfaces from a complex analytic viewpoint. Springer
+Monographs in Mathematics. Springer, Cham, 2021.
+[4] A. Alarc´on and F. J. L´opez. Minimal surfaces in R3 properly projecting into R2. J. Differential Geom.,
+90(3):351–381, 2012.
+[5] H. Alexander. Explicit imbedding of the (punctured) disc into C2. Comment. Math. Helv., 52(4):539–544, 1977.
+[6] D. Azagra. Global and fine approximation of convex functions. Proc. Lond. Math. Soc. (3), 107(4):799–824,
+2013.
+[7] E. Bishop. Mappings of partially analytic spaces. Amer. J. Math., 83:209–242, 1961.
+[8] S. Borell, F. Kutzschebauch, and E. F. Wold. Proper holomorphic disks in the complement of varieties in C2.
+Math. Res. Lett., 15(4):821–826, 2008.
+[9] V. Boˇzin. Note on harmonic maps. Internat. Math. Res. Notices, 1999(19):1081–1085, 1999.
+[10] A. Dor. Approximation by proper holomorphic maps into convex domains. Ann. Scuola Norm. Sup. Pisa Cl.
+Sci. (4), 20(1):147–162, 1993.
+[11] A. Dor. Immersions and embeddings in domains of holomorphy. Trans. Amer. Math. Soc., 347(8):2813–2849,
+1995.
+[12] B. Drinovec Drnovˇsek. Proper holomorphic discs avoiding closed convex sets. Math. Z., 241(3):593–596, 2002.
+[13] B. Drinovec Drnovˇsek. Proper discs in Stein manifolds avoiding complete pluripolar sets. Math. Res. Lett.,
+11(5-6):575–581, 2004.
+[14] B. Drinovec Drnovˇsek and F. Forstneriˇc. Holomorphic curves in complex spaces. Duke Math. J., 139(2):203–
+253, 2007.
+[15] B. Drinovec Drnovˇsek and F. Forstneriˇc. Strongly pseudoconvex domains as subvarieties of complex manifolds.
+Amer. J. Math., 132(2):331–360, 2010.
+[16] B. Drinovec Drnovˇsek and F. Forstneriˇc. Flexible domains for minimal surfaces in Euclidean spaces. J. Math.
+Anal. Appl., 517:126653, 2023.
+[17] Y. Eliashberg and M. Gromov. Embeddings of Stein manifolds of dimension n into the affine space of dimension
+3n/2 + 1. Ann. of Math. (2), 136(1):123–135, 1992.
+[18] J. E. Fornæss, F. Forstneriˇc, and E. Wold. Holomorphic approximation: the legacy of Weierstrass, Runge, Oka-
+Weil, and Mergelyan. In Advancements in complex analysis. From theory to practice, pages 133–192. Cham:
+Springer, 2020.
+[19] F. Forstneriˇc. Embedding strictly pseudoconvex domains into balls. Trans. Amer. Math. Soc., 295(1):347–368,
+1986.
+[20] F. Forstneriˇc. Stein manifolds and holomorphic mappings (The homotopy principle in complex analysis),
+volume 56 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer, Cham, second edition,
+2017.
+[21] F. Forstneriˇc. Holomorphic embeddings and immersions of Stein manifolds: a survey. In Geometric complex
+analysis, volume 246 of Springer Proc. Math. Stat., pages 145–169. Springer, Singapore, 2018.
+[22] F.
+Forstneriˇc.
+Recent
+developments
+on
+Oka
+manifolds.
+arXiv
+e-prints,
+2022.
+https://arxiv.org/abs/2006.07888.
+[23] F. Forstneriˇc and J. Globevnik. Proper holomorphic discs in C2. Math. Res. Lett., 8(3):257–274, 2001.
+[24] F. Forstneriˇc and T. Ritter. Oka properties of ball complements. Math. Z., 277(1-2):325–338, 2014.
+[25] F. Forstneriˇc and E. F. Wold. Oka domains in Euclidean spaces. Int. Math. Res. Not., to appear.
+https://arxiv.org/abs/2203.12883.
+[26] D. Gale and V. Klee. Continuous convex sets. Math. Scand., 7:379–391, 1959.
+[27] J. Globevnik. Boundary interpolation by proper holomorphic maps. Math. Z., 194(3):365–373, 1987.
+[28] A. Griewank and P. J. Rabier. On the smoothness of convex envelopes. Trans. Amer. Math. Soc., 322(2):691–
+709, 1990.
+[29] R. C. Gunning and H. Rossi. Analytic functions of several complex variables. AMS Chelsea Publishing,
+Providence, RI, 2009. Reprint of the 1965 original.
+
+18
+B. Drinovec Drnovˇsek and F. Forstneriˇc
+[30] M. Hakim. Applications holomorphes propres continues de domaines strictement pseudoconvexes de Cn dans
+la boule unit´e de Cn+1. (On the extension of proper holomorphic mappings from strictly pseudoconvex domains
+in Cn into the unit ball of Cn+1). Duke Math. J., 60(1):115–133, 1990.
+[31] M. Hakim and N. Sibony. Fonctions holomorphes born´ees sur la boule unite de Cn. Invent. Math., 67:213–222,
+1982.
+[32] D. Hoffman and W. H. Meeks, III. The strong halfspace theorem for minimal surfaces. Invent. Math.,
+101(2):373–377, 1990.
+[33] E. Løw. A construction of inner functions on the unit ball in Cp. Invent. Math., 67:223–229, 1982.
+[34] E. Løw. Embeddings and proper holomorphic maps of strictly pseudoconvex domains into polydiscs and balls.
+Math. Z., 190(3):401–410, 1985.
+[35] R. Narasimhan. Imbedding of holomorphically complete complex spaces. Amer. J. Math., 82:917–934, 1960.
+[36] R. Remmert. Sur les espaces analytiques holomorphiquement s´eparables et holomorphiquement convexes. C.
+R. Acad. Sci. Paris, 243:118–121, 1956.
+[37] R. Schoen and S. T. Yau. Lectures on harmonic maps. Conference Proceedings and Lecture Notes in Geometry
+and Topology, II. International Press, Cambridge, MA, 1997.
+[38] J. Sch¨urmann. Embeddings of Stein spaces into affine spaces of minimal dimension. Math. Ann., 307(3):381–
+399, 1997.
+[39] B. Stensønes. Proper holomorphic mappings from strongly pseudoconvex domains in C2 to the unit polydisc in
+C3. Math. Scand., 65(1):129–139, 1989.
+Barbara Drinovec Drnovˇsek
+Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubljana,
+Slovenia
+Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia.
+e-mail: [email protected]
+Franc Forstneriˇc
+Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubljana,
+Slovenia
+Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia
+e-mail: [email protected]
+

FdE1T4oBgHgl3EQf-wYy/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:04ea547d79178378b2eca882a49a8cb394d332d05c20c69e5c1540ed80abe7f9
+size 181453