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arXiv:1508.04644 · Journal of Mathematical Physics

Quantum Max-flow/Min-cut

Shawn Xingshan Cui, Michael Hartley Freedman, Or Sattath, Richard Stong, Greg Minton

Published Jun 1, 2016• 26 citations

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.

DOI: 10.1063/1.4954231

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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 ).

Efficient and reliable entanglement distribution forms the core of quantum network architecture. The Quantum Max-Flow concept defines the upper boundary for efficiently transmitting entanglement within quantum tensor networks (Calegari, Freedman, and Walker, Journal of the American Mathematical Society, 2010). Despite the general invalidity of the Quantum analog of the Max-Flow Min-Cut theorem, this paper introduces a precise Quantum Max-Flow Min-Cut Theorem tailored to quantum tensor networks enhanced by ancilla entanglement. Specifically, our research establishes infinite ’n’ values for any quantum tensor network, where the Quantum Max-Flow matches the Quantum Min-Cut by attaching a maximally entangled state with dimension ’n’ to each edge. Our protocol exclusively relies on quantum teleportation and guarantees unambiguous success. This work highlights the potential of quantum repeaters, enabled by ancilla entanglement, to efficiently distribute entanglement throughout quantum networks.

The \emph{max-flow min-cut theorem} has been recently used in the theory of random tensor networks in quantum information theory, where it is helpful for computing the behavior of important physical quantities, such as the entanglement entropy. In this paper, we extend the max-flow min-cut theorem to a relation among different \emph{partial orders} on the set of vertices of a network and introduce a new partial order for the vertices based on the \emph{min-cut structure} of the network. We apply the extended max-flow min-cut theorem to random tensor networks and find that the \emph{finite correction} to the entanglement Rényi entropy arising from the degeneracy of the min-cuts is given by the number of \emph{order morphisms} from the min-cut partial order to the partial order induced by non-crossing partitions on the symmetric group. Moreover, we show that the number of order morphisms corresponds to moments of a graph-dependent measure which generalizes the free Bessel law in some special cases in free probability theory.

We discuss quantum capacities for two types of entanglement networks: Q for the quantum repeater network with free classical communication, and R for the tensor network as the rank of the linear operation represented by the tensor network. We find that Q always equals R in the regularized case for the same network graph. However, the relationships between the corresponding one-shot capacities Q1and R1are more complicated, and the min-cut upper bound is in general not achievable. We show that the tensor network can be viewed as a stochastic protocol with the quantum repeater network, such that R1is a natural upper bound of Q1. We analyze the possible gap between R1and Q1for certain networks, and compare them with the one-shot classical capacity of the corresponding classical network.

In this note, we discuss the geometry of matrix product states with periodic boundary conditions and provide three infinite sequences of examples where the quantum max-flow is strictly less than the quantum min-cut. In the first, we fix the underlying graph to be a 4-cycle and verify a prediction of Hastings that inequality occurs for infinitely many bond dimensions. In the second, we generalize this result to a 2d-cycle. In the third, we show that the 2d-cycle with periodic boundary conditions gives inequality for all d when all bond dimensions equal two, namely, a gap of at least 2d−2 between the quantum max-flow and the quantum min-cut.

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