On a tracial version of Haemers bound

TL;DR

This paper introduces the tracial Haemers bound, a new algebraic upper bound on the Shannon capacity of graphs in the commuting operator model, and explores its properties and implications for quantum graph parameters.

Contribution

It generalizes the fractional Haemers bound to the von Neumann algebra setting and establishes the tracial Haemers bound as a multiplicative upper bound on Shannon capacity.

Findings

The tracial Haemers bound is multiplicative under the strong product.

It is incomparable with the Lovász theta function.

Separating the tracial and fractional Haemers bounds would refute Connes' embedding conjecture.

Abstract

We extend upper bounds on the quantum independence number and the quantum Shannon capacity of graphs to their counterparts in the commuting operator model. We introduce a von Neumann algebraic generalization of the fractional Haemers bound (over $\mathbb{C}$) and prove that the generalization upper bounds the commuting quantum independence number. We call our bound the tracial Haemers bound, and we prove that it is multiplicative with respect to the strong product. In particular, this makes it an upper bound on the Shannon capacity. The tracial Haemers bound is incomparable with the Lov\'asz theta function, another well-known upper bound on the Shannon capacity. We show that separating the tracial and fractional Haemers bounds would refute Connes' embedding conjecture. Along the way, we prove that the tracial rank and tracial Haemers bound are elements of the (commuting quantum) asymptotic spectrum of graphs (Zuiddam, Combinatorica, 2019). We also show that the inertia bound (an upper bound on the quantum independence number) upper bounds the commuting quantum independence number.