Dilations, inclusions of matrix convex sets, and completely positive maps

Abstract

A matrix convex set is a set of the form $\mathcal{S}={\cup }_{n\ge 1}{\mathcal{S}}_n$ (where each ${\mathcal{S}}_n$ is a set of $d$-tuples of $n\times n$ matrices) that is invariant under UCP maps from $M_n$ to $M_k$ and under formation of direct sums. We study the geometry of matrix convex sets and their relationship to completely positive maps and dilation theory. Key ingredients in our approach are polar duality in the sense of Effros and Winkler, matrix ranges in the sense of Arveson, and concrete constructions of scaled commuting normal dilation for tuples of self-adjoint operators, in the sense of Helton, Klep, McCullough and Schweighofer. Given two matrix convex sets $\mathcal{S}={\cup }_{n\ge 1}{\mathcal{S}}_n,$ and $\mathcal{T}={\cup }_{n\ge 1}{\mathcal{T}}_n$, we find geometric conditions on $\mathcal{S}$ or on $\mathcal{T}$, such that ${\mathcal{S}}_1\subseteq {\mathcal{T}}_1$ implies that $\mathcal{S}\subseteq C\mathcal{S}$ for some constant $C$. For instance, under various symmetry conditions on $\mathcal{S}$, we can show that $C$ above can be chosen to equal $d$, the number of variables, and in some cases this is sharp. We also find an essentially unique self-dual matrix convex set $\mathcal{D}$, the self-dual matrix ball, for which corresponding inclusion and dilation results hold with constant $C=\sqrt{d}$. Our results have immediate implications to spectrahedral inclusion problems studied recently by Helton, Klep, McCullough and Schweighofer. Our constants do not depend on the ranks of the pencils determining the free spectrahedra in question, but rather on the "number of variables" $d$. There are also implications to the problem of existence of (unital) completely positive maps with prescribed values on a set of operators.

Proposition 6.3, Corollary 6.4 and Theorem 6.5 in the previous version do not hold in the generality claimed. In this version we added "non-singularity" assumption (Definition 6.3) under which these results hold; in the new version they appear as Proposition 6.7, Corollary 6.8 and Theorem 6.9. 52 pages

References in corpus (4)

• Extending quantum operations
• Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets
• The noncommutative Choquet boundary III: Operator systems in matrix algebras
• Dilations of matricies

Cited by in corpus (24)

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• Dilation theory in finite dimensions and matrix convexity
• Strongly peaking representations and compressions of operator systems
• Non-commutative manifolds, the free square root and symmetric functions in two non-commuting variables
• Compressions of compact tuples
• Dilations of $q$-commuting unitaries
• Bianalytic free maps between spectrahedra and spectraballs
• Noncommutative partial convexity via $\Gamma$-convexity
• On fixed points of self maps of the free ball
• Noncommutative polynomials describing convex sets
• A tensor norm approach to quantum compatibility
• Facial structure of matrix convex sets
• Dilations of unitary tuples
• Maximal violation of steering inequalities and the matrix cube
• Factorization of noncommutative polynomials and Nullstellensätze for the free algebra
• Factorizations of Quantum Channels
• On the matrix range of random matrices
• Noncommutative partially convex rational functions
• Matrix Extreme Points and Free extreme points of Free spectrahedra
• The Arveson boundary of a Free Quadrilateral is given by a noncommutative variety