Kochen-Specker Sets and the Rank-1 Quantum Chromatic Number

TL;DR

This paper investigates the rank-1 quantum chromatic number of graphs, establishing new conditions for its relation to classical graph parameters using Kochen-Specker sets, and constructs a family of such sets with increasing dimension.

Contribution

It provides necessary and sufficient conditions relating the rank-1 quantum chromatic number, chromatic number, and orthogonal representations, and constructs new Kochen-Specker sets of increasing dimension.

Findings

Characterization of when q(G) equals (G)

Identification of graphs where (G) < ^{(1)}(G)

Conditions for ^{(1)}(G) < (G)

Abstract

The quantum chromatic number of a graph $G$ is sandwiched between its chromatic number and its clique number, which are well known NP-hard quantities. We restrict our attention to the rank-1 quantum chromatic number $\chi_q^{(1)}(G)$, which upper bounds the quantum chromatic number, but is defined under stronger constraints. We study its relation with the chromatic number $\chi(G)$ and the minimum dimension of orthogonal representations $\xi(G)$. It is known that $\xi(G) \leq \chi_q^{(1)}(G) \leq \chi(G)$. We answer three open questions about these relations: we give a necessary and sufficient condition to have $\xi(G) = \chi_q^{(1)}(G)$, we exhibit a class of graphs such that $\xi(G) < \chi_q^{(1)}(G)$, and we give a necessary and sufficient condition to have $\chi_q^{(1)}(G) < \chi(G)$. Our main tools are Kochen-Specker sets, collections of vectors with a traditionally important role in the study of noncontextuality of physical theories, and more recently in the quantification of quantum zero-error capacities. Finally, as a corollary of our results and a result by Avis, Hasegawa, Kikuchi, and Sasaki on the quantum chromatic number, we give a family of Kochen-Specker sets of growing dimension.