  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 ).
