Geometric approach to quantum statistical inference

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.

20 pages (+ bibliography), accepted for publication in IEEE Journal on Selected Areas in Information Theory

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• Lightcone Bounds for Quantum Circuit Mapping via Uncomplexity
• Quantum dichotomies and coherent thermodynamics beyond first-order asymptotics
• Bayesian quantum thermometry based on thermodynamic length
• Evolved Quantum Boltzmann Machines
• Quantum natural gradient with thermal-state initialization
• Natural gradient and parameter estimation for quantum Boltzmann machines
• Information geometry of bosonic Gaussian thermal states