Random MERA States and the Tightness of the Brandao-Horodecki Entropy Bound

TL;DR

This paper constructs a random MERA state with variable bond dimension to explore the tightness of the Brandao-Horodecki entropy bound, showing that certain entanglement and mutual information properties are achievable and providing evidence for the bound's tightness.

Contribution

It introduces a novel random MERA construction with variable bond dimension to analyze the entropy bound and demonstrates the bound's tightness through theoretical and numerical evidence.

Findings

Existence of states with linear mutual information at all scales

The entropy bound is shown to be tight for certain states

Numerical evidence supports the conjecture of exponential correlation decay

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\cite{bh} 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.