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 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.

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关于 Haemers Bound 的 Trac 版本

我们将图的量子独立数和量子香农容量的上限扩展到交换算子模型中的对应物。我们引入分数 Haemers 界的 von Neumann 代数推广（超过 $\mathbb {C}$ ) 并证明泛化上限是通勤量子独立数。我们称我们的界为 trac Haemers 界，我们证明它对于强乘积是乘法的。特别是，这使其成为香农容量的上限。trac Haemers 界无法与 Lovász theta 函数相提并论，后者是香农容量的另一个著名上界。我们表明，分离 trac 和分数 Haemers 界限将驳斥 Connes 的嵌入猜想。在此过程中，我们证明了 trac 秩和 trac Haemers 界是图的（交换量子）渐近谱的元素（Zuiddam，Combinatorica，2019）。我们还表明，惯性界（量子独立数的上限）是通勤量子独立数的上限。