Citation: Hastings, Matthew B. "Random mera states and the tightness of the brandao-horodecki entropy bound." arXiv preprint arXiv:1505.06468 (2015).
Web: https://arxiv.org/abs/1505.06468
Tags: Tensor-networks, Lieb-Robinson
This paper is a good starting point for a discussion of area laws for quantum entanglement in 1D. There are several new features of this paper, and its philosophical discussion is an extension of Part 1 of
> Hastings, M. B. "Notes on some questions in mathematical physics and quantum information." arXiv preprint arXiv:1404.4327 (2014).
It is known that gapped quantum systems in 1D satisfy an area-law, as proved Lieb-Robinson related techniques:
> Hastings, Matthew B. "An area law for one-dimensional quantum systems." Journal of statistical mechanics: theory and experiment 2007.08 (2007): P08024.
An immediate corollary of the Lieb-Robinson is a finite correlation length. Hence, we have the implication gap => finite correlation length, and gap => area law. The open problem was then whether or not finite correlation length => area law. This question was then settled by Brandao-Horodecki: yes, finite correlation length implies area law.
> Brandao, Fernando GSL, and Michal Horodecki. "Exponential decay of correlations implies area law." Communications in mathematical physics 333 (2015): 761-798.
Interestingly enough, though, this paper by Hastings constructs states which seem to have finite correlation length but don't have an area law. The way that these states escape the Brandao-Horodecki is extremely informative. The states Hastings constructs satisfy the following property: given any two spatially separated connected regions A,B, the commutator of any two operators supported on A,B respectively will have an exponential decay with respect to the distance between A and B. What Brandao-Horodecki assume is that the correlation between any two operators supported on two not necessarily connected seperated regions will have exponential decay.
This inclusion of non-connected regions is important, and can be understood via classical analogy. Consider an expander graph, with trivalent vertices and no nontrivial loops. Start with a point, and take a random walk of length N. Given the end-point, it is impossible to approximate the start-point. However, suppose we evolve forward in time by length N and backward in time by length N. Then, we can guess that the start-point is probably somewhere on the shortest path between the two end-points! This is very obvious when written out on a piece of paper. In this way, sandwiching an operator between operators both on its left AND on its right give tighter time evolution reconstructions than just having an operator on its right. In a way, this is more than an analogy. Lots of the counterexamples Brandao-Horodecki were trying to avoid were based on quantum analogues of expander graphs:
> Hastings, Matthew B. "Random unitaries give quantum expanders." Physical Review A—Atomic, Molecular, and Optical Physics 76.3 (2007): 032315.