Quantum Max-Flow Min-Cut theorem

• URL: http://arxiv.org/abs/2110.00905v1
• Date: Sun, 3 Oct 2021 02:11:39 GMT
• Title: Quantum Max-Flow Min-Cut theorem
• Authors: Nengkun Yu
• Abstract summary: We establish a quantum max-flow min-cut theorem for a new definition of quantum maximum flow. 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.
• Score: 11.98034899127065
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 ).

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