# On a tracial version of Haemers bound

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 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 Lovász 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.

## Authors

• 4 publications
• 5 publications
• 16 publications
• ### On a fractional version of Haemers' bound

In this note, we present a fractional version of Haemers' bound on the S...
02/01/2018 ∙ by Boris Bukh, et al. ∙ 0

• ### A Bound on the Shannon Capacity via a Linear Programming Variation

We prove an upper bound on the Shannon capacity of a graph via a linear ...
04/16/2018 ∙ by Sihuang Hu, et al. ∙ 0

• ### Shannon capacity, Chess, DNA and Umbrellas

A vexing open problem in information theory is to find the Shannon capac...
08/11/2021 ∙ by Oliver Knill, et al. ∙ 0

• ### Topological Bounds on the Dimension of Orthogonal Representations of Graphs

An orthogonal representation of a graph is an assignment of nonzero real...
11/28/2018 ∙ by Ishay Haviv, et al. ∙ 0

• ### Probabilistic refinement of the asymptotic spectrum of graphs

The asymptotic spectrum of graphs, introduced by Zuiddam (arXiv:1807.001...
03/05/2019 ∙ by Péter Vrana, et al. ∙ 0

• ### Quantum asymptotic spectra of graphs and non-commutative graphs, and quantum Shannon capacities

We study several quantum versions of the Shannon capacity of graphs and ...
10/01/2018 ∙ by Yinan Li, et al. ∙ 0