Communication Complexity of Graph Isomorphism, Coloring, and Distance Games

TL;DR

This paper explores the communication complexity in nonlocal graph games, introducing a new vertex distance game and analyzing how non-signalling strategies can distinguish graph properties more finely than classical or quantum strategies.

Contribution

It introduces the vertex distance game and D-fractional isomorphisms, providing new insights into nonlocal games and the role of non-signalling strategies in graph theory.

Findings

Perfect non-signalling strategies collapse communication complexity under certain conditions.

D-fractional isomorphisms characterize strategies for the vertex distance game.

Non-signalling strategies distinguish the new game more finely than classical or quantum strategies.

Abstract

In quantum information, nonlocal games are particularly useful for differentiating classical, quantum, and non-signalling correlations. An example of differentiation is given by the principle of no-collapse of communication complexity, which is often interpreted as necessary for a feasible physical theory. It is satisfied by quantum correlations but violated by some non-signalling ones. In this work, we investigate this principle in the context of three nonlocal games related to graph theory, starting from the well-known graph isomorphism and graph coloring games, and introducing a new game, the vertex distance game, with a parameter $D\in\mathbb N$, that generalizes the former two to some extent. For these three games, we prove that perfect non-signalling strategies collapse communication complexity under favorable conditions. We also define a refinement of fractional isomorphism of graphs, namely D-fractional isomorphisms, and we show that this characterizes perfect non-signalling strategies for the vertex distance game. Surprisingly, we observe that non-signalling strategies provide a finer distinction for the new game compared to classical and quantum strategies since the parameter D is visible only in the non-signalling setting.