The Asymptotics of Quantum Max-Flow Min-Cut

Abstract

The quantum max-flow min-cut conjecture relates the rank of a tensor network to the minimum cut in the case that all tensors in the network are identical in Calegari et al. (J Am Math Soc 23(1):107–188, 2010). This conjecture was shown to be false in Cui et al. (J Math Phys 57:062206, 2016) by an explicit counter-example. Here, we show that the conjecture is almost true, in that the ratio of the quantum max-flow to the quantum min-cut converges to 1 as the dimension N of the degrees of freedom on the edges of the network tends to infinity. The proof is based on estimating moments of the singular values of the network. We introduce a generalization of “rainbow diagrams” to tensor networks to estimate the dominant diagrams. A direct comparison of second and fourth moments lower bounds the ratio of the quantum max-flow to the quantum min-cut by a constant. To show the tighter bound that the ratio tends to 1, we consider higher moments. In addition, we show that the limiting moments as N → ∞ agree with that in a different ensemble where tensors in the network are chosen independently; this is used to show that the distributions of singular values in the two different ensembles weakly converge to the same limiting distribution. We present also a numerical study of one particular tensor network, which shows a surprising dependence of the rank deficit on N mod 4 and suggests further conjecture on the limiting behavior of the rank.