Comparison of Quantum Channels by Superchannels

TL;DR

This paper extends the concept of conditional min-entropy to bipartite quantum channels, providing a framework to compare channels via superchannels, with applications in channel discrimination and resource theories.

Contribution

It introduces an extended conditional min-entropy for quantum channels, characterizes superchannel relations through semidefinite programming, and explores applications in quantum information processing.

Findings

Extended conditional min-entropy characterizes channel relations.

Superchannel relations form a pre-order extending quantum majorization.

Semidefinite programming characterizes channel transformations.

Abstract

We extend the definition of the conditional min-entropy from bipartite quantum states to bipartite quantum channels. We show that many of the properties of the conditional min-entropy carry over to the extended version, including an operational interpretation as a guessing probability when one of the subsystems is classical. We then show that the extended conditional min-entropy can be used to fully characterize when two bipartite quantum channels are related to each other via a superchannel (also known as supermap or a comb) that is acting on one of the subsystems. This relation is a pre-order that extends the definition of "quantum majorization" from bipartite states to bipartite channels, and can also be characterized with semidefinite programming. As a special case, our characterization provides necessary and sufficient conditions for when a set of quantum channels is related to another set of channels via a single superchannel. We discuss the applications of our results to channel discrimination, and to resource theories of quantum processes. Along the way we study channel divergences, entropy functions of quantum channels, and noise models of superchannels, including random unitary superchannels, and doubly-stochastic superchannels. For the latter we give a physical meaning as being completely-uniformity preserving.