Information theoretic parameters of noncommutative graphs and convex corners

Abstract

We establish a second anti-blocker theorem for noncommutative convex corners, show that the anti-blocking operation is continuous on bounded sets of convex corners, and define optimization parameters for a given convex corner that generalize well-known graph theoretic quantities. We define the entropy of a state with respect to a convex corner, characterize its maximum value in terms of a generalized fractional chromatic number and establish entropy splitting results that demonstrate the entropic complementarity between a convex corner and its anti-blocker. We identify two extremal tensor products of convex corners and examine the behavior of the introduced parameters with respect to tensoring. Specializing to noncommutative graphs, we obtain quantum versions of the fractional chromatic number and the clique covering number, as well as a notion of noncommutative graph entropy of a state, which we show to be continuous with respect to the state and the graph. We define the Witsenhausen rate of a noncommutative graph and compute the values of our parameters in some specific cases.