Geometric approach to quantum statistical inference

TL;DR

This paper explores the geometric framework of quantum statistical inference, analyzing measures of distance between quantum states, and applies this approach to quantum estimation, speed limits, and thermodynamics.

Contribution

It provides a detailed geometric analysis of quantum statistical measures and their implications for quantum inference and related applications.

Findings

Identifies quantum divergence measures as geometric distances.

Establishes relations between quantum metrics and divergences.

Applies geometric methods to quantum estimation and thermodynamics.

Abstract

We study quantum statistical inference tasks of hypothesis testing and their canonical variations, in order to review relations between their corresponding figures of merit---measures of statistical distance---and demonstrate the crucial differences which arise in the quantum regime in contrast to the classical setting. In our analysis, we primarily focus on the geometric approach to data inference problems, within which the aforementioned measures can be neatly interpreted as particular forms of divergences that quantify distances in the space of probability distributions or, when dealing with quantum systems, of density matrices. Moreover, with help of the standard language of Riemannian geometry we identify both the metrics such divergences must induce and the relations such metrics must then naturally inherit. Finally, we discuss exemplary applications of such a geometric approach to problems of quantum parameter estimation, "speed limits" and thermodynamics.