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Journal of Convex Analysis 28 (2021), No. 3, [final page numbers not yet available]
Copyright Heldermann Verlag 2021



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

Stephan Weis
96231 Bad Staffelstein, Germany
maths@weis-stephan.de

Maksim Shirokov
Steklov Mathematical Institute, Moscow, Russia
msh@mi.ras.ru



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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