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arXiv · 2003.14302

The face generated by a point, generalized affine constraints, and quantum theory

Stephan Weis·Maksim Shirokov
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Abstract

We analyze faces generated by points in an arbitrary convex set and their relative algebraic interiors, which are nonempty as we shall prove. We show that by intersecting a convex set with a sublevel or level set of a generalized affine functional, the dimension of the face generated by a point may decrease by at most one. We apply the results to the set of quantum states on a separable Hilbert space. Among others, we show that every state having finite expected values of any two (not necessarily bounded) positive operators admits a decomposition into pure states with the same expected values. We discuss applications in quantum information theory.

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Stephan Weis, Maksim Shirokov. 2020-03-31. The face generated by a point, generalized affine constraints, and quantum theory. https://arxiv.org/abs/2003.14302

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