Continuity of the Maximum-Entropy Inference

TL;DR

This paper investigates the continuity properties of maximum-entropy inference in quantum systems, showing that it is generally continuous up to boundary points and characterizing when discontinuities occur.

Contribution

It proves that maximum-entropy inference is continuous at boundary points for arbitrary ranking functions and identifies conditions under which discontinuities are typical in quantum systems.

Findings

Inference is continuous up to boundary points in quantum systems.

Discontinuities in inference are typical and related to non-commutative observables.

A geodesic closure of Gibbsian families equals the set of maximum-entropy states.

Abstract

We study the inverse problem of inferring the state of a finite-level quantum system from expected values of a fixed set of observables, by maximizing a continuous ranking function. We have proved earlier that the maximum-entropy inference can be a discontinuous map from the convex set of expected values to the convex set of states because the image contains states of reduced support, while this map restricts to a smooth parametrization of a Gibbsian family of fully supported states. Here we prove for arbitrary ranking functions that the inference is continuous up to boundary points. This follows from a continuity condition in terms of the openness of the restricted linear map from states to their expected values. The openness condition shows also that ranking functions with a discontinuous inference are typical. Moreover it shows that the inference is continuous in the restriction to any polytope which implies that a discontinuity belongs to the quantum domain of non-commutative observables and that a geodesic closure of a Gibbsian family equals the set of maximum-entropy states. We discuss eight descriptions of the set of maximum-entropy states with proofs of accuracy and an analysis of deviations.