Quantum Max-Flow Min-Cut theorem

TL;DR

This paper proves an asymptotic quantum max-flow min-cut theorem for tensor networks, showing that the quantum max-flow and min-cut become equal as the entanglement dimension grows infinitely large.

Contribution

It establishes a new asymptotic quantum max-flow min-cut theorem for tensor networks, resolving a previously disproven conjecture in the limit of large entanglement dimension.

Findings

Quantum max-flow equals quantum min-cut for infinitely many dimensions n.

The ratio of quantum max-flow to min-cut converges to 1 as n approaches infinity.

The asymptotic version of the open problem is validated.

Abstract

The max-flow min-cut theorem is a cornerstone result in combinatorial optimization. Calegari et al. (arXiv:0802.3208) initialized the study of quantum max-flow min-cut conjecture, which connects the rank of a tensor network and the min-cut. Cui et al. (arXiv:1508.04644) showed that this conjecture is false generally. In this paper, we establish a quantum max-flow min-cut theorem for a new definition of quantum maximum flow. In particular, we show that for any quantum tensor network, there are infinitely many $n$, such that quantum max-flow equals quantum min-cut, after attaching dimension $n$ maximally entangled state to each edge as ancilla. Our result implies that the ratio of the quantum max-flow to the quantum min-cut converges to $1$ as the dimension $n$ tends to infinity. As a direct application, we prove the validity of the asymptotical version of the open problem about the quantum max-flow and the min-cut, proposed in Cui et al. (arXiv:1508.04644 ).