Random Bures mixed states and the distribution of their purity

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.