Central Limit Theorem for Tensor Products of Free Variables via Bi-Free Independence

TL;DR

This paper establishes a Central Limit Theorem for tensor products of free variables using bi-free independence, linking bi-free probability with quantum channels and random matrices asymptotics.

Contribution

It introduces a novel bi-free probability approach to analyze the spectral distribution and asymptotics of quantum channels and random matrices.

Findings

Spectral distributions tend to an averaged sum of bi-free products.

A new CLT for operators in bi-free probability is proven.

Connections between bi-free probability and quantum information are demonstrated.

Abstract

In this paper, a connection between bi-free probability and the asymptotics of random quantum channels and tensor products of random matrices is established. Using bi-free matrix models, it is demonstrated that the spectral distribution of certain self-adjoint quantum channels and tensor products of random matrices tend to a distribution that can be obtained by an averaged sum of products of bi-freely independent pairs. Subsequently, using bi-free techniques, a Central Limit Theorem for such operator is established.