The Choi-Jamiolkowski isomorphism and covariant quantum channels

TL;DR

This paper extends the Choi-Jamiolkowski isomorphism to a broader class of quantum channels, especially covariant channels under symmetry groups, facilitating their analysis in quantum information theory.

Contribution

It introduces a generalized isomorphism for completely positive maps, emphasizing covariant channels under various symmetry groups, including compact and Euclidean groups.

Findings

Simplifies analysis of covariant quantum channels

Demonstrates application to phase-shift-covariant channels

Extends framework to channels with Euclidean symmetry

Abstract

A generalization of the Choi-Jamiolkowski isomorphism for completely positive maps between operator algebras is introduced. Particular emphasis is placed on the case of normal unital completely positive maps defined between von Neumann algebras. This generalization is applied especially to the study of maps which are covariant under actions of a symmetry group. We highlight with the example of, e.g., phase-shift-covariant quantum channels the ease of this method in particular in the case of a compact symmetry group. We also discuss the case of channels which are covariant under actions of the Euclidean group of rigid motions in 3 dimensions.