On the asymptotic distribution of block-modified random matrices

TL;DR

This paper analyzes the asymptotic eigenvalue distribution of block-modified random matrices using operator-valued free probability, providing a unifying framework, new distributions, and a numerical algorithm applicable in quantum information theory.

Contribution

It introduces a method to compute the limiting eigenvalue distribution of block-modified matrices using operator-valued free probability and subordination, unifying and extending previous results.

Findings

Derived explicit formulas for asymptotic eigenvalue distributions

Developed a numerical algorithm for general cases

Unified existing results and obtained new distributions

Abstract

We study random matrices acting on tensor product spaces which have been transformed by a linear block operation. Using operator-valued free probability theory, under some mild assumptions on the linear map acting on the blocks, we compute the asymptotic eigenvalue distribution of the modified matrices in terms of the initial asymptotic distribution. Moreover, using recent results on operator-valued subordination, we present an algorithm that computes, numerically but in full generality, the limiting eigenvalue distribution of the modified matrices. Our analytical results cover many cases of interest in quantum information theory: we unify some known results and we obtain new distributions and various generalizations.