Low entropy output states for products of random unitary channels

TL;DR

This paper investigates the output states of products of random unitary channels acting on entangled states, revealing low entropy states and eigenvalue behaviors using free probability and extended Weingarten calculus.

Contribution

It introduces a systematic approach to analyze eigenvalue distributions of outputs from products of random unitary channels, extending graphical Weingarten calculus.

Findings

Eigenvalues of fixed number of unitaries are computed explicitly.

Eigenvalue distribution converges to a specific shape as the number of unitaries grows linearly with dimension.

The study highlights differences between random unitary channels and generic random channels.

Abstract

In this paper, we study the behaviour of the output of pure entangled states after being transformed by a product of conjugate random unitary channels. This study is motivated by the counterexamples by Hastings and Hayden-Winter to the additivity problems. In particular, we study in depth the difference of behaviour between random unitary channels and generic random channels. In the case where the number of unitary operators is fixed, we compute the limiting eigenvalues of the output states. In the case where the number of unitary operators grows linearly with the dimension of the input space, we show that the eigenvalue distribution converges to a limiting shape that we characterize with free probability tools. In order to perform the required computations, we need a systematic way of dealing with moment problems for random matrices whose blocks are i.i.d. Haar distributed unitary operators. This is achieved by extending the graphical Weingarten calculus introduced in Collins and Nechita (2010).