Estimating quantum chromatic numbers

TL;DR

This paper advances the understanding of quantum chromatic numbers by introducing new parameters, proving computational properties, and establishing connections with classical graph invariants through semidefinite programming techniques.

Contribution

It introduces the tracial rank, a new lower bound for the commuting quantum chromatic number, and develops SDP algorithms for computing quantum chromatic parameters.

Findings

The commuting quantum chromatic number is computable via SDP.

The tracial rank is multiplicative and provides lower bounds.

The tracial rank of an odd cycle is precisely determined.

Abstract

We develop further the new versions of quantum chromatic numbers of graphs introduced by the first and fourth authors. We prove that the problem of computation of the commuting quantum chromatic number of a graph is solvable by an SDP algorithm and describe an hierarchy of variants of the commuting quantum chromatic number which converge to it. We introduce the tracial rank of a graph, a parameter that gives a lower bound for the commuting quantum chromatic number and parallels the projective rank, and prove that it is multiplicative. We describe the tracial rank, the projective rank and the fractional chromatic numbers in a unified manner that clarifies their connection with the commuting quantum chromatic number, the quantum chromatic number and the classical chromatic number, respectively. Finally, we present a new SDP algorithm that yields a parameter larger than the Lov\'asz number and is yet a lower bound for the tracial rank of the graph. We determine the precise value of the tracial rank of an odd cycle.