Distance between states and statistical inference in quantum theory

Dennis Dieks, P.J. Veltkamp

Bernard Diu

Stanley Gudder

Masahito Hayashi

Hypothesis Testing Quantum Cramer-Rao Bound in Mixed States Model Quantum Cramer-Rao Bound in Pure States Model Group Symmetric Approach to Pure States Model Large Deviation Theory in Quantum Estimation Futher Topics on Quantum Statistical Inference.

Marcin Jarzyna, Jan Kolodynski

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.

Jean-Paul Marchand