Citation: Arad, Itai, Zeph Landau, and Umesh Vazirani. "Improved one-dimensional area law for frustration-free systems." Physical Review B—Condensed Matter and Materials Physics 85.19 (2012): 195145.
Web: https://arxiv.org/abs/1111.2970
Tags: Lieb-Robinson, Spin-chains, Information-theory
In this paper, the authors arrive at an improved version of the area law for one dimensional spin chains. Namely, they bound the entanglement across every cut by a factor which grows polynomially in both the dimensionality of the underlying qudits and the size of the gap. This is an exponential improvement on Hasting's result, and is within polynomial factors of counterexamples:
> Gottesman, Daniel, and Matthew B. Hastings. "Entanglement versus gap for one-dimensional spin systems." New journal of physics 12.2 (2010): 025002.
> Irani, Sandy. "Ground state entanglement in one-dimensional translationally invariant quantum systems." Journal of Mathematical Physics 51.2 (2010).
This paper is building off of an earlier detectability-lemma argument for area laws:
> Aharonov, Dorit, et al. "Quantum Hamiltonian complexity and the detectability lemma." arXiv preprint arXiv:1011.3445 (2010).
The insight of this paper is to use the Chebyshev polynomials in the same way that they are used in classical approximation theory. This technique adds crucial quantitative improvements which make new techniques possible. It is an instructive technique to be aware of. The tagline is that the Chebyshev polynomials can be used to make approximations of the ground state projector in a way which respects locality in an almost optimally.