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Unitary orbits of normal operators in von Neumann algebras
, 2005
"... Abstract. The starting points for this paper are simple descriptions of the norm and strong * closures of the unitary orbit of a normal operator in a von Neumann algebra. The statements are in terms of spectral data and do not depend on the type or cardinality of the algebra. We relate this to sever ..."
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Abstract. The starting points for this paper are simple descriptions of the norm and strong * closures of the unitary orbit of a normal operator in a von Neumann algebra. The statements are in terms of spectral data and do not depend on the type or cardinality of the algebra. We relate this to several known results and derive some consequences, of which we list a few here. Exactly when the ambient von Neumann algebra is a direct sum of σfinite algebras, any two normal operators have the same normclosed unitary orbit if and only if they have the same strong*closed unitary orbit if and only if they have the same strongclosed unitary orbit. But these three closures generally differ from each other and from the unclosed unitary orbit, and we characterize when equality holds between any two of these four sets. We also show that in a properly infinite von Neumann algebra, the strongclosed unitary orbit of any operator, not necessarily normal, meets the center in the (nonvoid) left essential central spectrum of Halpern. One corollary is a “type III Weylvon NeumannBerg theorem ” involving containment of essential central spectra. 1.
Finding decompositions of a class of separable states
 Linear Alg. Appl
"... Abstract. We consider the class of separable states which admit a decomposition i A i ⊗ B i with the B i 's having independent images. We give a simple intrinsic characterization of this class of states. Given a density matrix in this class, we construct such a decomposition, which can be chos ..."
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Abstract. We consider the class of separable states which admit a decomposition i A i ⊗ B i with the B i 's having independent images. We give a simple intrinsic characterization of this class of states. Given a density matrix in this class, we construct such a decomposition, which can be chosen so that the A i 's are distinct with unit trace, and then the decomposition is unique. We relate this to the facial structure of the set of separable states. The states investigated include a class that corresponds (under the ChoiJamio lkowski isomorphism) to the quantum channels called quantumclassical and classicalquantum by Holevo.
ACTIVE LATTICES DETERMINE AW*ALGEBRAS
, 2012
"... We prove that AW*algebras are determined by their projections, their symmetries, and the action of the latter on the former. We introduce active lattices, which are formed from these three ingredients. More generally, we prove that the category of AW*algebras is equivalent to a full subcategory ..."
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Cited by 6 (4 self)
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We prove that AW*algebras are determined by their projections, their symmetries, and the action of the latter on the former. We introduce active lattices, which are formed from these three ingredients. More generally, we prove that the category of AW*algebras is equivalent to a full subcategory of active lattices. Crucial ingredients are an equivalence between the category of piecewise AW*algebras and that of piecewise complete Boolean algebras, and a refinement of the piecewise algebra structure of an AW*algebra that enables recovering its total structure.
2014) Continuity of the maximumentropy inference
 Communications in Mathematical Physics 330(3) 1263–1292
"... We study the inverse problem of inferring the state of a finitelevel quantum system from expected values of a fixed set of observables, by maximizing a continuous ranking function. We have proved earlier that the maximumentropy inference can be a discontinuous map from the convex set of expected ..."
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We study the inverse problem of inferring the state of a finitelevel quantum system from expected values of a fixed set of observables, by maximizing a continuous ranking function. We have proved earlier that the maximumentropy inference can be a discontinuous map from the convex set of expected values to the convex set of states because the image contains states of reduced support, while this map restricts to a smooth parametrization of a Gibbsian family of fully supported states. Here we prove for arbitrary ranking functions that the inference is continuous up to boundary points. This follows from a continuity condition in terms of the openness of the restricted linear map from states to their expected values. The openness condition shows also that ranking functions with a discontinuous inference are typical. Moreover it shows that the inference is continuous in the restriction to any polytope which implies that a discontinuity belongs to the quantum domain of noncommutative observables and that a geodesic closure of a Gibbsian family equals the set of maximumentropy states. We discuss eight descriptions of the set of maximumentropy states with proofs of accuracy and an analysis of deviations. Index Terms – inference under constraints, continuous, open, maximumentropy inference, exponential family, information topology, information projection.
Complete positivity of the map from a basis to its dual basis
 Journal of Mathematical Physics
"... Abstract. The dual of a matrix ordered space has a natural matrix ordering that makes the dual space matrix ordered as well. The purpose of these notes is to give a condition that describes when the linear map taking a basis of Mn to its dual basis is a complete order isomorphism. We exhibit “natur ..."
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Abstract. The dual of a matrix ordered space has a natural matrix ordering that makes the dual space matrix ordered as well. The purpose of these notes is to give a condition that describes when the linear map taking a basis of Mn to its dual basis is a complete order isomorphism. We exhibit “natural ” orthonormal bases for Mn such that this map is an order isomorphism, but not a complete order isomorphism. Included among such bases is the Pauli basis. Our results generalize the Choi matrix by giving conditions under which the role of the standard basis {Eij} can be replaced by other bases. Given a vector space V there is no “natural ” linear isomorphism between V and the dual space V d, but each time we fix a basis B = {vi: i ∈ I} for V there is a dual basis B ̃ = {δi: i ∈ I} for V d satisfying δi(vj) = 0, i 6 = j 1, i = j and this allows us to define a (basis dependent) linear isomorphism between V and V d. Definition 1. If B is a basis of Mn, the linear map from Mn to Mdn taking each member of B to the corresponding member of the dual basis is denoted by DB, and is called the duality map. We let ΓB = D−1B: Mdn →Mn denote the inverse of this map. Note that if f ∈Mdn, and B is a basis ofMn, then ΓB(f) = b∈B f(b)b. In particular, when V = Mn (the space of n × n complex matrices), and we let E = {Ei,j: 1 ≤ i, j ≤ n} denote the standard matrix units, then the map ΓE: Mdn →Mn satisfies ΓE(f) = n∑ i,j=1 f(Ei,j)Ei,j.
unknown title
"... For the full state space K of B(C m ⊗ C n) each nonextreme point can be decomposed into extreme points in many different ways. But for the space S of separable states the situation is totally different. While nonextreme points with many different decompositions exist (and are easy to find) in S as ..."
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For the full state space K of B(C m ⊗ C n) each nonextreme point can be decomposed into extreme points in many different ways. But for the space S of separable states the situation is totally different. While nonextreme points with many different decompositions exist (and are easy to find) in S as well as in K, there are in S also plenty of points for which the decomposition is unique. A separable state is said to be of “decomposition length p ” (or just “of length p”) if it can be expressed as a convex combination of p pure product states but not of fewer, and we show in this article that the set of all separable states of length at most max (m, n) has an open dense subset of states with unique decomposition into pure product states. Actually, we exhibit such a dense open subset consisting of states with the property that they generate a face of S which is a simplex, from which the uniqueness follows. We also define a broader class of states that we show have a unique
unknown title
"... a convex combination of pure product states. It is natural to ask the extent to which this decomposition is unique. That is the main topic of this article. For the full state space K of B(C m ⊗ C n) each nonextreme point can be decomposed into extreme points in many different ways. But for the spac ..."
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a convex combination of pure product states. It is natural to ask the extent to which this decomposition is unique. That is the main topic of this article. For the full state space K of B(C m ⊗ C n) each nonextreme point can be decomposed into extreme points in many different ways. But for the space S of separable states the situation is totally different. While nonextreme points with many different decompositions exist (and are easy to find) in S as well as in K, there are in S also plenty of points for which the decomposition is unique. DiVincenzo, Terhal, and Thapliyal [4] defined the optimal ensemble cardinality of a separable state ρ to be k if k is the minimal number of pure product states required for a convex decomposition of ρ. Lockhart [11] used the term “optimal ensemble length ” for the same notion. For brevity, we will simply call this number the length of ρ, and we denote the set of separable states of length at most k by Sk. We show in
Symmetry, Compact Closure and Dagger Compactness for Categories of Convex Operational Models
"... Abstract In the categorical approach to the foundations of quantum theory, one begins with a symmetric monoidal category, the objects of which represent physical systems, and the morphisms of which represent physical processes. Usually, this category is taken to be at least compact closed, and more ..."
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Abstract In the categorical approach to the foundations of quantum theory, one begins with a symmetric monoidal category, the objects of which represent physical systems, and the morphisms of which represent physical processes. Usually, this category is taken to be at least compact closed, and more often, dagger compact, enforcing a certain selfduality, whereby preparation processes (roughly, states) are interconvertible with processes of registration (roughly, measurement outcomes). This is in contrast to the more concrete "operational" approach, in which the states and measurement outcomes associated with a physical system are represented in terms of what we here call a convex operational model: a certain dual pair of ordered linear spaces generally, not isomorphic to one another. On the other hand, state spaces for which there is such an isomorphism, which we term weakly selfdual, play an important role in reconstructions of various quantuminformation theoretic protocols, including teleportation and ensemble steering. In this paper, we characterize compact closure of symmetric monoidal categories of convex operational models in two ways: as a statement about the existence of teleportation protocols, and as the principle that every process allowed by that theory can be realized as an instance of a remote evaluation protocol hence, as a form of classical probabilistic conditioning. In a large class of cases, which includes both the classical and quantum cases, the relevant compact closed categories are degenerate, in the weak sense that every object is its own dual. We characterize the daggercompactness of such a category (with respect to the natural adjoint) in terms of the existence, for each system, of a symmetric bipartite state, the associated conditioning map of which is an isomorphism. * Perimeter Institute for Theoretical Physics hbarnum@perimeterinstitute.ca † Oxford University Computing Laboratory, ross.duncan@comlab.ox.ac.uk ‡ Department of Mathematics, Susquehanna University, wilce@susqu.edu 1 Categorical Semantics and Quantum Foundations One natural way to formalize a physical theory is as some kind of category, C, the objects of which are the systems, and the morphisms of which are the processes, contemplated by that theory. In order to provide some apparatus for representing compound systems, it is natural to assume further that C is a symmetric monoidal category. In the categorical semantics for quantum theory pioneered by Abramsky and Coecke There is an older tradition, stemming from Mackey's work on the foundations of quantum mechanics It is obviously of interest to see how far such convex operational theories can be treated formally, that is, as categories, and more especially, as symmetric monoidal categories; equally, one would like to know how much of the special structure assumed in the categor2 ical approach can be given an operational motivation. 1 Some first steps toward addressing these issues are taken in Organization and Notation Sections 2 and 3 provide quick reviews of the categorytheoretic and the convex frameworks, respectively, mainly following [1] for the former and We assume that the reader is familiar with basic categorytheoretic ideas and notation, as well as with the probabilistic machinery of quantum theory. We write C, D etc. for categories, A ∈ C, to indicate that A is an object of C, and C(A, B) for the set of morphisms between objects A, B ∈ C. Except as noted, all vector spaces considered here will be finitedimensional and real. We write Vec R for the category of finitedimensional real vector spaces and linear maps. The dual space of a vector space A is denoted by A * . An ordered vector space is a real vector space V equipped with a regular that is, closed, convex, pointed, generating cone V + , and ordered by the relation x ≤ y ⇔ y − x ∈ A + . A linear mapping φ : V → W between ordered linear spaces V and W is positive if φ(V + ) ⊆ W + . We write L +