
Journal of Convex Analysis 28 (2021), No. 3, 847870 Copyright Heldermann Verlag 2021 The Face Generated by a Point, Generalized Affine Constraints, and Quantum Theory Stephan Weis 96231 Bad Staffelstein, Germany maths@weisstephan.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, purestate decomposition, minimal output entropy, operator Enorms. MSC: 52Axx, 47Axx, 81Qxx. [ Fulltextpdf (207 KB)] for subscribers only. 