Relating the Bures measure to the Cauchy two-matrix model

TL;DR

This paper explores the relationship between the Bures metric in quantum states and the Cauchy two-matrix model, revealing how their associated point processes are interconnected and enabling easier computation of quantum state statistics.

Contribution

It provides an explicit connection between the Pfaffian point process of the Bures ensemble and the determinantal process of the Cauchy two-matrix model, facilitating statistical analysis.

Findings

Derived the kernels relating Bures and Cauchy two-matrix models.

Established how a Pfaffian process can originate from a determinantal process.

Enabled calculation of Bures ensemble level statistics using Cauchy model results.

Abstract

The Bures metric is a natural choice in measuring the distance of density operators representing states in quantum mechanics. In the past few years a random matrix ensemble and the corresponding joint probability density function of its eigenvalues was identified. Moreover a relation with the Cauchy two-matrix model was discovered but never thoroughly investigated, leaving open in particular the following question: How are the kernels of the Pfaffian point process of the Bures random matrix ensemble related to the ones of the determinantal point process of the Cauchy two-matrix model and moreover, how can it be possible that a Pfaffian point process derives from a determinantal point process? We give a very explicit answer to this question. The aim of our work has a quite practical origin since the calculation of the level statistics of the Bures ensemble is highly mathematically involved while we know the statistics of the Cauchy two-matrix ensemble. Therefore we solve the whole level statistics of a density operator drawn from the Bures prior.