Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization

Fritz, Tobias; Netzer, Tim; Thom, Andreas

doi:10.1137/16m1100642

Cited by 45 publications

(64 citation statements)

References 29 publications

(67 reference statements)

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“…In particular, it determines the operator space S A up to unit preserving and completely isometric isomorphism (see [3,Theorem 2.4.2] or [9,Theorem 5.1]). The matrix range has been used and studied in recent works, in the contexts of the UCP interpolation problem [8] (following [17]), finite-dimensional/compact representability of operator systems [19,20] (following [16]), and extremal problems in matrix convex sets [12] (following [14]).…”

Section: Background and Statement Of Main Resultsmentioning

confidence: 99%

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Compressions of compact tuples

Passer

Shalit

2019

Linear Algebra and its Applications

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We study the matrix range of a tuple of compact operators on a Hilbert space and examine the notions of minimal, nonsingular, and fully compressed tuples. In this pursuit, we refine previous results by characterizing nonsingular compact tuples in terms of matrix extreme points of the matrix range. Further, we find that a compact tuple A is fully compressed if and only if it is multiplicity-free and the Shilov ideal is trivial, which occurs if and only if A is minimal and nonsingular. Fully compressed compact tuples are therefore uniquely determined up to unitary equivalence by their matrix ranges. We also produce a proof of this fact which does not depend on the concept of nonsingularity.This notion of minimality was used in [8, Section 6], and in the finite-dimensional case it corresponds precisely to the notion of minimal pencil used earlier in [17] (and to the notion of σ-minimal pencil used in [23]). Using matricial polar duality [11], one can show that 2010 Mathematics Subject Classification. 47A20, 47A13, 46L07, 47L25. Key words and phrases. Matrix convex set; matrix range; matrix extreme point; operator system; structure of compact tuples.

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Section: Background and Statement Of Main Resultsmentioning

confidence: 99%

“…K} is the largest matrix convex set whose first level is K. These two sets are equal precisely when K is a simplex by [20, Theorem 4.1] (see also [16,Theorem 4.7] for a similar result with the assumption that K is a polytope).…”

Section: Matrix Convexity and Extreme Pointsmentioning

confidence: 96%

Compressions of compact tuples

Passer

Shalit

2019

Linear Algebra and its Applications

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“…Roughly speaking, the C * -envelope of an operator system is the "smallest" C * -algebra containing that operator system [P02]. In this language, Theorem 1.1 shows that every operator system with a finite-dimensional realization (see [FNT17]) is completely normed by its finite dimensional boundary representations. For further material related to operator systems, completely positive maps, boundary representations, and the C * -envelope we direct the reader to [Ham79], [D96], [MS98], [F00], [F04], [FHL18], and [PSS18].…”

Section: Remarksmentioning

confidence: 99%

Arveson extreme points span free spectrahedra

Evert

Helton

2019

Math. Ann.

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Let SM n (R) g denote g-tuples of n × n real symmetric matrices. Given tuples X = (X 1 , . . . , X g ) ∈ SM n1 (R) g and Y = (Y 1 , . . . , Y g ) ∈ SM n2 (R) g , a matrix convex combination of X and Y is a sum of the formMatrix convex sets are sets which are closed under matrix convex combinations. A key feature of matrix convex combinations is that the g-tuples X, Y , and V * 1 XV 1 + V * 2 Y V 2 do not need to have the same size. As a result, matrix convex sets are a dimension free analog of convex sets.While in the classical setting there is only one notion of an extreme point, there are three main notions of extreme points for matrix convex sets: ordinary, matrix, and absolute extreme points. Absolute extreme points are closely related to the classical Arveson boundary. A central goal in the theory of matrix convex sets is to determine if one of these types of extreme points for a matrix convex set minimally recovers the set through matrix convex combinations.This article shows that every real compact matrix convex set which is defined by a linear matrix inequality is the matrix convex hull of its absolute extreme points, and that the absolute extreme points are the minimal set with this property. Furthermore, we give an algorithm which expresses a tuple as a matrix convex combination of absolute extreme points with optimal bounds. Similar results hold when working over the field of complex numbers rather than the reals.

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“…Finally we review the definition of minimal operator system (see, for example, Ref. [FNT16] for more details).…”

Section: Definition 8 a Spectrahedron Is A Set Of The Following Formmentioning

confidence: 99%

Separability for mixed states with operator Schmidt rank two

Cuevas

Drescher²,

Netzer³

2019

Quantum

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The operator Schmidt rank is the minimum number of terms required to express a state as a sum of elementary tensor factors. Here we provide a new proof of the fact that any bipartite mixed state with operator Schmidt rank two is separable, and can be written as a sum of two positive semidefinite matrices per site. Our proof uses results from the theory of free spectrahedra and operator systems, and illustrates the use of a connection between decompositions of quantum states and decompositions of nonnegative matrices. In the multipartite case, we prove that any Hermitian Matrix Product Density Operator (MPDO) of bond dimension two is separable, and can be written as a sum of at most four positive semidefinite matrices per site. This implies that these states can only contain classical correlations, and very few of them, as measured by the entanglement of purification. In contrast, MP-DOs of bond dimension three can contain an unbounded amount of classical correlations.

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Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization

Cited by 45 publications

References 29 publications

Compressions of compact tuples

Compressions of compact tuples

Arveson extreme points span free spectrahedra

Separability for mixed states with operator Schmidt rank two

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