Continuity of the maximum-entropy inference: convex geometry and numerical ranges approach
Leiba Rodman, Ilya M. Spitkovsky, Arleta Szkola, and Stephan Weis
Contact the author: Please use for correspondence this email.
Submission date: 26. Feb. 2015
published in: Journal of mathematical physics, 57 (2016), art-no. 1
DOI number (of the published article): 10.1063/1.4926965
MSC-Numbers: 81P16, 62F30, 52A20, 54C10, 62H20, 47A12, 52A10
Keywords and phrases: maximum-entropy inference, quantum inference, continuity, convex body, irreducible many-party correlation, Quantum correlation, numerical range
Download full preprint: PDF (447 kB)
We study the continuity of an abstract generalization of the maximum-entropy inference — a maximizer. It is defined as a right-inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations. Further, we use numerical range methods in the case of quantum inference which refers to two observables. One result is a complete characterization of points of discontinuity for 3 × 3 matrices.