Covariant Quantum Combinatorics with Applications to Zero-Error Communication

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∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras) carry an action of a compact quantum group G, and all channels (completely positive maps preserving a certain G-invariant functional) 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.