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arXiv:1505.06468 · Quantum Information and Computation

Random MERA states and the tightness of the Brandao-Horodecki entropy bound

Matthew B. Hastings

Published Oct 1, 2016• 7 citations
arXivHTMLPDF

Abstract

We construct a random MERA state with a bond dimension that varies with the level of the MERA. This causes the state to exhibit a very different entanglement structure from that usually seen in MERA, with neighboring intervals of length l exhibiting a mutual information proportional to epsilon l for some constant epsilon, up to a length scale exponentially large in epsilon. We express the entropy of a random MERA in terms of sums over cuts through the MERA network, with the entropy in this case controlled by the cut minimizing bond dimensions cut through. One motivation for this construction is to investigate the tightness of the Brandao-Horodecki[8] entropy bound relating entanglement to correlation decay. Using the random MERA, we show that at least part of the proof is tight: there do exist states with the required property of having linear mutual information between neighboring intervals at all length scales. We conjecture that this state has exponential correlation decay and that it demonstrates that the Brandao-Horodecki bound is tight (at least up to constant factors), and we provide some numerical evidence for this as well as a sketch of how a proof of correlation decay might proceed.

DOI
10.26421/qic16.13-14-8

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We construct a random MERA state with a bond dimension that varies with the level of the MERA. This causes the state to exhibit a very different entanglement structure from that usually seen in MERA, with neighboring intervals of length l exhibiting a mutual information proportional to epsilon l for some constant epsilon, up to a length scale exponentially large in epsilon. We express the entropy of a random MERA in terms of sums over cuts through the MERA network, with the entropy in this case controlled by the cut minimizing bond dimensions cut through. One motivation for this construction is to investigate the tightness of the Brandao-Horodecki[8] entropy bound relating entanglement to correlation decay. Using the random MERA, we show that at least part of the proof is tight: there do exist states with the required property of having linear mutual information between neighboring intervals at all length scales. We conjecture that this state has exponential correlation decay and that it demonstrates that the Brandao-Horodecki bound is tight (at least up to constant factors), and we provide some numerical evidence for this as well as a sketch of how a proof of correlation decay might proceed.
A remarkable feature of typical ground states of strongly-correlated many-body systems is that the entanglement entropy is not an extensive quantity. In one dimension, there exists a proof that a finite correlation length sets a constant upper-bound on the entanglement entropy, called the area law. However, the known bound exists only in a hypothetical limit, rendering its physical relevance highly questionable. In this paper, we give a simple proof of the area law for entanglement entropy in one dimension under the condition of exponentially decaying correlations. Our proof dramatically reduces the previously known bound on the entanglement entropy, bringing it, for the first time, into a realistic regime. The proof is composed of several simple and straightforward steps based on elementary quantum information tools. We discuss the underlying physical picture, based on a renormalization-like construction underpinning the proof, which transforms the entanglement entropy of a continuous region into a sum of mutual informations in different length scales and the entanglement entropy at the boundary.
Entanglement is a key quantum phenomena and understanding transitions between phases of matter with different entanglement properties are an interesting probe of quantum mechanics. We numerically study a model of a 2D tensor network proposed to have an entanglement entropy transition first considered by Vasseur et al.[Phys. Rev. B 100, 134203 (2019)]. We find that by varying the bond dimension of the tensors in the network we can observe a transition between an area and volume phase with a logarithmic critical point around D≈2D\approx 2D≈2. We further characterize the critical behavior measuring a critical exponent using entanglement entropy and the tripartite quantum mutual information, observe a crossover from a `nearly pure' to entangled area law phase using the the distributions of the entanglement entropy and find a cubic decay of the pairwise mutual information at the transition. We further consider the dependence of these observables for different Rényi entropy. This work helps further validate and characterize random tensor networks as a paradigmatic examples of an entanglement transition.
Abstract For pure states of multi-dimensional quantum lattice systems, which in a convenient computational basis have amplitude and phase structure of sufficiently rapid decorrelation, we construct high fidelity approximations of relatively low complexity. These are used for a conditional proof of area-law bounds for the states’ entanglement entropy. The condition is also shown to imply exponential decay of the state’s mutual information between disjoint regions, and hence exponential clustering of local observables. The applicability of the general results is demonstrated on the quantum Ising model in transverse field. Combined with available model-specific information on spin-spin correlations, we establish an area-law type bound on the entanglement in the model’s subcritical ground states, valid in all dimensions and up to the model’s quantum phase transition.
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.
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The complexity of quantum many-body systems is manifested in the vast diversity of their correlations, making it challenging to distinguish the generic from the atypical features.This can be addressed by analyzing correlations through ensembles of random states, chosen to faithfully embody the relevant physical properties.Here, we focus on spins with local interactions, whose correlations are extremely well captured by tensor network states.Adopting an operational perspective, we define ensembles of random tensor network states in one and two spatial dimensions that admit a sequential generation.As such, they directly correspond to outputs of quantum circuits with a sequential architecture and random gates.In one spatial dimension, the ensemble explores the entire family of matrix product states, while in two spatial dimensions, it corresponds to random isometric tensor network states.We extract the scaling behavior of the average correlations between two subsystems as a function of their distance.Using elementary concentration results, we then deduce the typical case for measures of correlation such as the von Neumann mutual information and a measure arising from the Hilbert-Schmidt norm.We find for all considered cases that the typical behavior is an exponential decay (for both one and two spatial dimensions).We observe the consistent emergence of a correlation length that depends only on the underlying spatial dimension and not the considered measure.Remarkably, increasing the bond dimension leads to a higher correlation length in one spatial dimension but has the opposite effect in two spatial dimensions.