Information theoretic parameters of non-commutative graphs and convex corners

TL;DR

This paper develops a comprehensive framework for non-commutative convex corners and graphs, introducing new parameters, entropy measures, and tensor product behaviors, extending classical graph theory into quantum settings.

Contribution

It generalizes graph theoretic quantities to non-commutative convex corners, introduces quantum entropy measures, and explores tensor product properties in this new framework.

Findings

Established a second anti-blocker theorem for non-commutative convex corners.

Defined quantum versions of fractional chromatic number and clique covering number.

Proved continuity of quantum graph entropy and computed parameters for specific cases.

Abstract

We establish a second anti-blocker theorem for non-commutative convex corners, show that the anti-blocking operation is continuous on bounded sets of convex corners, and define optimisation parameters for a given convex corner that generalise well-known graph theoretic quantities. We define the entropy of a state with respect to a convex corner, characterise its maximum value in terms of a generalised 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 behaviour of the introduced parameters with respect to tensoring. Specialising to non-commutative graphs, we obtain quantum versions of the fractional chromatic number and the clique covering number, as well as a notion of non-commutative 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 non-commutative graph and compute the values of our parameters in some specific cases.