Generalized Bures products from free probability

TL;DR

This paper introduces new generalized Bures products derived from free probability theory, analyzing their eigenvalue and singular value distributions in large matrices, supported by simulations and conjectures.

Contribution

It develops three new random matrix products as generalizations of the Bures model using free probability, and derives their spectral density properties.

Findings

Eigenvalue and singular value densities computed in the thermodynamic limit

Identification of divergences at zero in the spectral densities

Conjectures on the relationship between densities and spectral domain shapes

Abstract

Inspired by the theory of quantum information, I use two non-Hermitian random matrix models - a weighted sum of circular unitary ensembles and a product of rectangular Ginibre unitary ensembles - as building blocks of three new products of random matrices which are generalizations of the Bures model. I apply the tools of both Hermitian and non-Hermitian free probability to calculate the mean densities of their eigenvalues and singular values in the thermodynamic limit, along with their divergences at zero; the results are supported by Monte Carlo simulations. I pose and test conjectures concerning the relationship between the two densities (exploiting the notion of the N-transform), the shape of the mean domain of the eigenvalues (an extension of the single ring theorem), and the universal behavior of the mean spectral density close to the domain's borderline (using the complementary error function).