Moments of quantum channel ensembles

TL;DR

This paper develops a framework for analyzing moments of quantum channel ensembles, introducing concepts like channel t-designs, and studies how different noise types affect these moments, with theoretical and numerical insights.

Contribution

It extends moment operator analysis from unitaries to quantum channels, introduces a hierarchy of ensembles, and defines channel t-designs with operational significance.

Findings

Noise can decrease moment operator norms (e.g., depolarizing noise).

Noise can increase moment operator norms (e.g., amplitude damping).

A block-orthogonal basis for permutations simplifies analysis.

Abstract

Moments of ensembles of unitaries play a central role in quantum information theory as they capture the statistical properties of dynamics of systems with some form of randomness. Indeed, concepts such as approximate $t$-designs arise when comparing how close an associated moment operator of a given unitary ensemble is to that of another, reference ensemble. Despite the importance of moment operators, their properties have not been as explored for quantum channels. In this work we develop a theoretical framework to compute moment operators for ensembles of quantum channels, for all moment orders $t$, with a special focus on determining ensembles that can be used as points of reference. By deriving hierarchies between ensembles, via inequalities of their moment operator norms, we give them operational meaning, and define useful concepts such as that of channel $t$-designs. Finally, we perform theoretical and numerical studies which show that different types of noise can decrease the norm of the moment operators (e.g., depolarizing noise), as well as increase it (e.g., amplitude damping), and generalize noise-induced concentration phenomena to channel-design-induced phenomena. Along the way, we find a block-orthogonal basis for permutations, which greatly simplifies our analyses, and may be of independent interest.