Purity distribution for generalized random Bures mixed states

TL;DR

This paper analyzes the distribution of purity in random mixed quantum states under the Bures measure, revealing phase transitions and connecting to the O(n) model, with implications for quantum information theory.

Contribution

It introduces a generalized Bures measure for random states and computes the full purity distribution using random matrix theory and Coulomb gas methods, uncovering phase transitions.

Findings

Identified three regimes with two phase transitions in the Coulomb gas model.

Discovered a first-order phase transition characterized by a charge detachment.

Linked Bures states to the O(n) model, generalizing previous results.

Abstract

We compute the distribution of the purity for random density matrices (i.e.random mixed states) in a large quantum system, distributed according to the Bures measure. The full distribution of the purity is computed using a mapping to random matrix theory and then a Coulomb gas method. We find three regimes that correspond to two phase transitions in the associated Coulomb gas. The first transition is characterized by an explosion of the third derivative on the left of the transition point. The second transition is of first order, it is characterized by the detachement of a single charge of the Coulomb gas. A key remark in this paper is that the random Bures states are closely related to the O(n) model for n=1. This actually led us to study "generalized Bures states" by keeping $n$ general instead of specializing to n=1.