Generating random quantum channels

TL;DR

This paper explores three methods for generating random quantum channels, compares their mathematical properties and computational efficiency, and analyzes their spectral characteristics and invariant states.

Contribution

It introduces and compares three approaches to sampling quantum channels, establishing conditions for their equivalence and suitability for numerical studies.

Findings

The three sampling methods become equivalent under certain conditions.

The spectral gap and invariant states of random quantum channels are characterized.

Mean values of unitarity, output purity, and coherence are computed over the uniform measure.

Abstract

Several techniques of generating random quantum channels, which act on the set of $d$-dimensional quantum states, are investigated. We present three approaches to the problem of sampling of quantum channels and show under which conditions they become mathematically equivalent, and lead to the uniform, Lebesgue measure on the convex set of quantum operations. We compare their advantages and computational complexity and demonstrate which of them is particularly suitable for numerical investigations. Additional results focus on the spectral gap and other spectral properties of random quantum channels and their invariant states. We compute mean values of several quantities characterizing a given quantum channel, including its unitarity, the average output purity and the $2$-norm coherence of a channel, averaged over the entire set of the quantum channels with respect to the uniform measure. An ensemble of classical stochastic matrices obtained due to super-decoherence of random quantum stochastic maps is analyzed and their spectral properties are studied using the Bloch representation of a classical probability vector.