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

arXiv eprints
 Pub Date:
 March 2020
 arXiv:
 arXiv:2003.14302
 Bibcode:
 2020arXiv200314302W
 Keywords:

 Mathematics  Functional Analysis;
 Mathematical Physics;
 Mathematics  Metric Geometry;
 Quantum Physics;
 52Axx;
 47Axx;
 81Qxx
 EPrint:
 Any comments are welcome, v2: labels have changed