Quantum convex support

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...

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Bibliographic Details
Published in:Linear algebra and its applications 2011-12, Vol.435 (12), p.3168-3188
Main Author: Weis, Stephan
Format: Article
Language:English
Subjects:
Algebra
Convex support
Duality
Exact sciences and technology
Face lattice
Functional analysis
Linear and multilinear algebra, matrix theory
Mathematical analysis
Mathematics
Mean value
Poonem
Projection lattice
Sciences and techniques of general use
Spectrahedron
State space
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Summary: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 1990s boom in semi-definite 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. Our main result 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.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2011.06.004