Random Quantum Operations

TL;DR

This paper introduces a natural ensemble of random quantum operations, analyzes their spectral properties, and demonstrates their convergence behavior, connecting quantum maps with random matrix theory.

Contribution

It defines a new ensemble of quantum maps, investigates their spectral properties, and establishes a quantum analogue of the Frobenius-Perron theorem.

Findings

Spectral properties follow a distribution related to the Ginibre ensemble.

Generic quantum maps exhibit exponential convergence to invariant states.

A general formula for eigenvalue density of quantum maps is derived.

Abstract

We define a natural ensemble of trace preserving, completely positive quantum maps and present algorithms to generate them at random. Spectral properties of the superoperator Phi associated with a given quantum map are investigated and a quantum analogue of the Frobenius-Perron theorem is proved. We derive a general formula for the density of eigenvalues of Phi and show the connection with the Ginibre ensemble of real non-symmetric random matrices. Numerical investigations of the spectral gap imply that a generic state of the system iterated several times by a fixed generic map converges exponentially to an invariant state.