Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach

TL;DR

This paper investigates the continuity properties of a generalized maximum-entropy inference using convex geometry and numerical ranges, revealing conditions for discontinuities and applying results to quantum correlations and $3\times 3$ matrices.

Contribution

It introduces a convex geometry approach to analyze the continuity of maximum-entropy inferences and characterizes discontinuities in quantum inference for small matrices.

Findings

Discontinuities occur at limits of extremal points that are not extremal.

Complete characterization of discontinuities for $3\times 3$ matrices.

Application to quantum correlations and inference with two observables.

Abstract

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 $3\times 3$ matrices.