Spectral lower bounds for the quantum chromatic number of a graph

TL;DR

This paper establishes that many spectral lower bounds for the classical chromatic number also apply to the quantum chromatic number, using linear algebra techniques to connect the two concepts.

Contribution

It introduces a combinatorial approach to show spectral bounds for the quantum chromatic number, extending classical bounds to the quantum setting.

Findings

Spectral bounds for classical chromatic number apply to quantum chromatic number

Uses linear algebra techniques like pinching and twirling

Provides examples illustrating the bounds

Abstract

The quantum chromatic number, $\chi_q(G)$, of a graph $G$ was originally defined as the minimal number of colors necessary in a quantum protocol in which two provers that cannot communicate with each other but share an entangled state can convince an interrogator with certainty that they have a coloring of the graph. We use an equivalent purely combinatorial definition of $\chi_q(G)$ to prove that many spectral lower bounds for the chromatic number, $\chi(G)$, are also lower bounds for $\chi_q(G)$. This is achieved using techniques from linear algebra called pinching and twirling. We illustrate our results with some examples.