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A Sharp Threshold Where Min-Cut Renyi Entropies Cease to Be a Spectrum

Zongming Liu

Published Dec 22, 2024• 0 citations
Publisher

Abstract

Because Trρk+1 are the moments of a probability measure on the unit interval, Hausdorff’s theorem makes complete monotonicity a complete criterion for a set of R ́enyi entropies to be the spectrum of a density matrix. Applied to the minimal-cut prescription of random tensor networks and monitored circuits, the criterion fails at a threshold with no free parameters: the sequence is a spectrum if and only if the competing cuts cross at R ́enyi index at most two, that is, when the multiplicity ratio does not exceed the bond dimension raised to the area difference. The obstruction is exact, and at small bond dimension it lies at a difference order below one hundred, so it is exhibitable by direct computation rather than only provable.

DOI
10.5281/zenodo.22258305

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Because Trρk+1 are the moments of a probability measure on the unit interval, Hausdorff’s theorem makes complete monotonicity a complete criterion for a set of R ́enyi entropies to be the spectrum of a density matrix. Applied to the minimal-cut prescription of random tensor networks and monitored circuits, the criterion fails at a threshold with no free parameters: the sequence is a spectrum if and only if the competing cuts cross at R ́enyi index at most two, that is, when the multiplicity ratio does not exceed the bond dimension raised to the area difference. The obstruction is exact, and at small bond dimension it lies at a difference order below one hundred, so it is exhibitable by direct computation rather than only provable.
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 study the evaluation of Rényi multi-entropies Sn(q)S^{(q)}_nSn(q)​ in Random Tensor Network (RTN) states in the large bond-dimension limit. For the case of Rényi index n=2n=2n=2 and arbitrary number of parties qqq, we prove that that multi-entropies are determined by minimal multiway cuts through the network. When the minimal multiway cut is degenerate, we characterize the full minimizer set via compatible families of minimal cuts and give a criterion for all minimizers to come from ordinary cut partitions. For n=2n=2n=2, this gives a natural generalization of the minimal cut description of bipartite entanglement to multipartite systems with arbitrarily many parties. For the case of integer n>2n>2n>2, we show that the minimal multiway cut conjecture is in general \emph{not true} by providing explicit counter examples for both the single random tensor and for the network built from isometric tilings. We discuss the implication for our results on the multipartite entanglement structures in RTN and holography.
A bstract In holographic theories, the reflected entropy has been shown to be dual to the area of the entanglement wedge cross section. We study the same problem in random tensor networks demonstrating an equivalent duality. For a single random tensor we analyze the important non-perturbative effects that smooth out the discontinuity in the reflected entropy across the Page phase transition. By summing over all such effects, we obtain the reflected entanglement spectrum analytically, which agrees well with numerical studies. This motivates a prescription for the analytic continuation required in computing the reflected entropy and its Rényi generalization which resolves an order of limits issue previously identified in the literature. We apply this prescription to hyperbolic tensor networks and find answers consistent with holographic expectations. In particular, the random tensor network has the same non-trivial tripartite entanglement structure expected from holographic states. We furthermore show that the reflected Rényi spectrum is not flat, in sharp contrast to the usual Rényi spectrum of these networks. We argue that the various distinct contributions to the reflected entanglement spectrum can be organized into approximate superselection sectors. We interpret this as resulting from an effective description of the canonically purified state as a superposition of distinct tensor network states. Each network is constructed by doubling and gluing various candidate entanglement wedges of the original network. The superselection sectors are labelled by the different cross-sectional areas of these candidate entanglement wedges.