Graph Homomorphisms for Quantum Players

Abstract

A homomorphism from a graph $X$ to a graph $Y$ is an adjacency preserving mapping $f:V(X) \rightarrow V(Y)$. We consider a nonlocal game in which Alice and Bob are trying to convince a verifier with certainty that a graph $X$ admits a homomorphism to $Y$. This is a generalization of the well-studied graph coloring game. Via systematic study of quantum homomorphisms we prove new results for graph coloring. Most importantly, we show that the Lov\'{a}sz theta number of the complement lower bounds the quantum chromatic number, which itself is not known to be computable. We also show that some of our newly introduced graph parameters, namely quantum independence and clique numbers, can differ from their classical counterparts while others, namely quantum odd girth, cannot. Finally, we show that quantum homomorphisms closely relate to zero-error channel capacity. In particular, we use quantum homomorphisms to construct graphs for which entanglement-assistance increases their one-shot zero-error capacity.