Quantum Convex Support

TL;DR

This paper explores the geometry and face structure of quantum convex support, linking it to spectrahedra and projections in matrix algebras, thus integrating convex geometry with quantum information theory.

Contribution

It introduces a novel analysis of the face lattice of quantum convex support via projections, connecting convex geometry with matrix algebra techniques.

Findings

Face lattice of convex support mapped to projections in A

Connection established between convex support and spectrahedra

Framework enables algebraic and geometric analysis of quantum states

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.