Covariant quantum combinatorics with applications to zero-error communication

• URL: http://arxiv.org/abs/2302.07776v3
• Date: Thu, 2 Nov 2023 22:02:01 GMT
• Title: Covariant quantum combinatorics with applications to zero-error communication
• Authors: Dominic Verdon
• Abstract summary: We develop the theory of quantum relations and quantum (a.k.a. noncommutative) graphs in the finite-dimensional covariant setting. We motivate our definitions by applications to zero-error quantum communication theory with a symmetry constraint.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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