Random quantum channels I: graphical calculus and the Bell state phenomenon

TL;DR

This paper introduces a graphical calculus for analyzing random quantum channels, demonstrating eigenvalue convergence and improving bounds on the largest eigenvalue, with implications for entropy inequalities.

Contribution

It develops a novel graphical tool for computing moments of random quantum channels and applies it to prove eigenvalue convergence and improve bounds.

Findings

Eigenvalues of certain random matrix models converge almost surely

Sharp bounds on the largest eigenvalue are obtained

Applications to minimal output entropy inequalities

Abstract

This paper is the first of a series where we study quantum channels from the random matrix point of view. We develop a graphical tool that allows us to compute the expected moments of the output of a random quantum channel. As an application, we study variations of random matrix models introduced by Hayden \cite{hayden}, and show that their eigenvalues converge almost surely. In particular we obtain for some models sharp improvements on the value of the largest eigenvalue, and this is shown in a further work to have new applications to minimal output entropy inequalities.