Random Bures mixed states and the distribution of their purity

TL;DR

This paper introduces an efficient method to generate random quantum states according to the Bures measure and analyzes the distribution of their purity, revealing connections to integrability and Painlevé equations.

Contribution

It presents a new algorithm for sampling Bures-distributed states and derives the moments of their purity distribution using integrability techniques.

Findings

Efficient algorithm for Bures measure sampling

Explicit moments of purity distribution derived

Connection to Painlevé equations established

Abstract

Ensembles of random density matrices determined by various probability measures are analysed. A simple and efficient algorithm to generate at random density matrices distributed according to the Bures measure is proposed. This procedure may serve as an initial step in performing Bayesian approach to quantum state estimation based on the Bures prior. We study the distribution of purity of random mixed states. The moments of the distribution of purity are determined for quantum states generated with respect to the Bures measure. This calculation serves as an exemplary application of the "deform-and-study" approach based on ideas of integrability theory. It is shown that Painlev\'e equation appeared as a part of the presented theory.