On a tracial version of Haemers bound
- URL: http://arxiv.org/abs/2107.02567v2
- Date: Thu, 19 May 2022 08:59:49 GMT
- Title: On a tracial version of Haemers bound
- Authors: Li Gao, Sander Gribling, Yinan Li
- Abstract summary: 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 call our bound the tracial Haemers bound, and we prove that it is multiplicative with respect to the strong product.
- Score: 20.98023024846862
- License: http://creativecommons.org/licenses/by/4.0/
- 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.
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