Covariant quantum combinatorics with applications to zero-error communication

TL;DR

This paper develops a theory of covariant quantum relations and graphs in finite-dimensional systems with symmetry, applying it to zero-error quantum communication under symmetry constraints.

Contribution

It introduces covariant quantum relations and graphs, characterizes covariant channels, and classifies zero-error communication schemes with symmetry.

Findings

A necessary and sufficient condition for covariant quantum relations to be underlying channels.

Every quantum G-graph arises from a covariant channel.

Covariant channel reversibility is characterized by discrete confusability G-graphs.

Abstract

We develop the theory of quantum (a.k.a. noncommutative) relations and quantum (a.k.a. noncommutative) graphs in the finite-dimensional covariant setting, where all systems (finite-dimensional $C^*$-algebras) carry an action of a compact quantum group $G$, and all channels (completely positive maps preserving the canonical $G$-invariant state) are covariant with respect to the $G$-actions. We motivate our definitions by applications to zero-error quantum communication theory with a symmetry constraint. Some key results are the following: 1) We give a necessary and sufficient condition for a covariant quantum relation to be the underlying relation of a covariant channel. 2) We show that every quantum confusability graph with a $G$-action (which we call a quantum $G$-graph) arises as the confusability graph of a covariant channel. 3) We show that a covariant channel is reversible precisely when its confusability $G$-graph is discrete. 4) When $G$ is quasitriangular (this includes all compact groups), we show that covariant zero-error source-channel coding schemes are classified by covariant homomorphisms between confusability $G$-graphs.