Quantum Convex Support

Abstract

Convex support, the mean values of a set of random variables, is central in information theory and statistics. Equally central in quantum information theory are mean values of a set of observables in a finite-dimensional C*-algebra A, which we call (quantum) convex support. The convex support can be viewed as a projection of the state space of A and it is a projection of a spectrahedron. Spectrahedra are increasingly investigated at least since the 1990's boom in semidefinite programming. We recall the geometry of the positive semi-definite cone and of the state space. We write a convex duality for general self-dual convex cones. This restricts to projections of state spaces and connects them to results on spectrahedra. Really new in this article is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections, using natural isomorphisms. The result encodes the face lattice of the convex support into a set of projections in A and enables the integration of convex geometry with matrix calculus or algebraic techniques.

27 pages, 5 figures, minor errors corrected, font changed from small to normal

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Cited by in corpus (17)

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• Entropy Distance: New Quantum Phenomena
• Non-Asymptotic Classical Data Compression with Quantum Side Information
• Discontinuity of Maximum Entropy Inference and Quantum Phase Transitions
• Maximizing the divergence from a hierarchical model of quantum states
• Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach
• Continuity of the Maximum-Entropy Inference
• Dissecting the qutrit
• Computable entanglement conversion witness that is better than the negativity
• Pre-images of extreme points of the numerical range, and applications
• Optimal arrangements of classical and quantum states with limited purity
• Morphophoric POVMs, generalised qplexes, and 2-designs
• Quantum marginals, faces, and coatoms
• A variation principle for ground spaces
• A Semidefinite Hierarchy for Containment of Spectrahedra