Quantum Max-flow/Min-cut

TL;DR

This paper explores the quantum analog of the max-flow min-cut theorem in tensor networks, revealing that the classical equality does not always hold in quantum cases and establishing specific conditions where it does.

Contribution

It introduces the quantum max-flow and min-cut concepts, demonstrates their non-equivalence in general, and identifies conditions for their equality, connecting to quantum entanglement and satisfiability.

Findings

Quantum max-flow can differ from quantum min-cut in general cases.

Equality holds when capacities are powers of a fixed integer.

Connections to entanglement entropy and quantum satisfiability are established.

Abstract

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.