Quantum Max-flow/Min-cut

Journal Article · Wed Jun 15 04:00:00 EDT 2016 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4954231· OSTI ID:22596848

Sattath, Or ^[1]; Stong, Richard; Minton, Greg ^[2]

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1. Computer Science Division, University of California, Berkeley, California 94720 (United States)
2. Center for Communications Research, La Jolla, California 92121 (United States)

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.

• OSTI ID: 22596848
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 6 Vol. 57; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ENTROPY
INTEGRALS
QUANTUM ENTANGLEMENT
QUANTUM GRAVITY
TENSORS