Limiting spectral distribution of random self-adjoint quantum channels

TL;DR

This paper investigates the spectral distribution of random quantum channels with Hermitian Kraus operators, revealing a semi-circle law when the Kraus rank grows with system size and an explicit law otherwise.

Contribution

It establishes the limiting spectral distribution for quantum channels with random Hermitian Kraus operators, including the semi-circle law and an explicit alternative law for fixed Kraus rank.

Findings

Semi-circle distribution when Kraus rank grows with n

Explicit spectral law for fixed Kraus rank

Connection to free probability theory

Abstract

We study the limiting spectral distribution of quantum channels whose Kraus operators are sampled as $n\times n$ random Hermitian matrices satisfying certain assumptions. We show that when the Kraus rank goes to infinity with n, the limiting spectral distribution (suitably rescaled) of the corresponding quantum channel coincides with the semi-circle distribution. When the Kraus rank is fixed, the limiting spectral distribution is no longer the semi-circle distribution. It corresponds to an explicit law, which can also be described using tools from free probability.