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Journal of Convex Analysis 28 (2021), No. 3, 847--870
Copyright Heldermann Verlag 2021

The Face Generated by a Point, Generalized Affine Constraints, and Quantum Theory

Stephan Weis
96231 Bad Staffelstein, Germany

Maksim Shirokov
Steklov Mathematical Institute, Moscow, Russia

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.

Keywords: Face generated by a point, extreme set, relative algebraic interior, generalized affine constraint, extreme point, generalized compactness, quantum state, pure-state decomposition, minimal output entropy, operator E-norms.

MSC: 52Axx, 47Axx, 81Qxx.

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