  The classical max-flow min-cut theorem describes transport through certain
idealized classical networks. We consider the quantum analog for tensor
networks. By associating an integral capacity to each edge and a tensor to each
vertex in a flow network, we can also interpret it as a tensor network, and
more specifically, as a linear map from the input space to the output space.
The quantum max flow is defined to be the maximal rank of this linear map over
all choices of tensors. The quantum min cut is defined to be the minimum
product of the capacities of edges over all cuts of the tensor network. We show
that unlike the classical case, the quantum max-flow=min-cut conjecture is not
true in general. Under certain conditions, e.g., when the capacity on each edge
is some power of a fixed integer, the quantum max-flow is proved to equal the
quantum min-cut. However, concrete examples are also provided where the
equality does not hold.
  We also found connections of quantum max-flow/min-cut with entropy of
entanglement and the quantum satisfiability problem. We speculate that the
phenomena revealed may be of interest both in spin systems in condensed matter
and in quantum gravity.
