On the quantum chromatic number of a graph

TL;DR

This paper explores the quantum chromatic number of graphs, establishing its relations with classical graph parameters, proving general properties, and demonstrating significant separations through random graphs and specific examples.

Contribution

It introduces new insights into quantum chromatic number, relates it to clique number and orthogonal representations, and provides examples of large separations from classical chromatic numbers.

Findings

Quantum chromatic number can differ significantly from classical chromatic number.

Large separations between clique number and quantum chromatic number are demonstrated.

Certain small quantum chromatic numbers (2 or 3) cannot differ from classical counterparts.

Abstract

We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince an interrogator with certainty that they have a colouring of the graph. After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is 2, nor if it is 3 in a restricted quantum model; on the other hand, we exhibit a graph on 18 vertices and 44 edges with chromatic number 5 and quantum chromatic number 4.