Citation: Aharonov, Dorit, et al. "Quantum Hamiltonian complexity and the detectability lemma." arXiv preprint arXiv:1011.3445 (2010).
Web: https://arxiv.org/abs/0811.3412
Tags: Spin-chains, Information-theory
This paper uses the detectability lemma for frustration-free local Hamiltonians to the same effect that the Lieb-Robinson bound can be used for general local Hamiltonians. Namely, the authors use the detectability lemma to prove an area law for 1D systems and to prove an exponential decay of correlations. The idea is to apply a circuit of local projections, and argue that it well-approximates the ground state projection up to exponential corrections. This circuit of projections has an obvious lightcone associated to it, which enforces a notion of locality to the system.
The detectability was introduces a year earlier, by the same authors:
> Aharonov, Dorit, et al. "The detectability lemma and quantum gap amplification." Proceedings of the forty-first annual ACM symposium on Theory of computing. 2009.
This older paper used the detectability lemma for Hamiltonian complexity. The present paper gave a shorter proof of the detectability lemma, removed assumptions on its statement, and applied it to a more correct usecase. This is certainly the correct paper to read to get introduced to the lemma.
The authors argue that this alternative to the Lieb-Robinson approach is necessary because it is much more combinatorial, and hence is accessible to people with a computer-science background. In their own words: "These analytic tools constitute a major barrier for a fuller participation by computer scientists in this important aspect of Hamiltonian complexity".
An alternative version of this paper, which is arguably nicer to read and includes some earlier work, can be found here:
> Aharonov, Dorit, et al. "The 1d area law and the complexity of quantum states: A combinatorial approach." 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science. IEEE, 2011.