Quantum Max-Flow Min-Cut theorem
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916840947580928 |
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| author | Yu, Nengkun |
| author_facet | Yu, Nengkun |
| contents | 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 ). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_00905 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quantum Max-Flow Min-Cut theorem Yu, Nengkun Quantum Physics Information Theory 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 ). |
| title | Quantum Max-Flow Min-Cut theorem |
| topic | Quantum Physics Information Theory |
| url | https://arxiv.org/abs/2110.00905 |