Discontinuities in the Maximum-Entropy Inference

TL;DR

This paper investigates the discontinuities in maximum-entropy inference for quantum systems, revealing that such discontinuities occur with non-commuting observables and are characterized by the openness of a related linear map.

Contribution

It provides an example of discontinuity in quantum maximum-entropy inference and characterizes these discontinuities through the mathematical property of the associated linear map.

Findings

Discontinuities occur for non-commuting observables.

Maximum-entropy inference is continuous for mutually commuting observables.

Discontinuities are linked to the openness of a linear map.

Abstract

We revisit the maximum-entropy inference of the state of a finite-level quantum system under linear constraints. The constraints are specified by the expected values of a set of fixed observables. We point out the existence of discontinuities in this inference method. This is a pure quantum phenomenon since the maximum-entropy inference is continuous for mutually commuting observables. The question arises why some sets of observables are distinguished by a discontinuity in an inference method which is still discussed as a universal inference method. In this paper we make an example of a discontinuity and we explain a characterization of the discontinuities in terms of the openness of the (restricted) linear map that assigns expected values to states.