Originating from the works of Bekenstein and Hawking on the entropy of black holes, area laws constitute a central tool for understanding entanglement and locality properties in quantum systems. Essentially, in a system that obeys an area law, the _entanglement_ entropy of a bounded region scales like its boundary
area, rather than its volume.
In 2007, in a seminal paper, Hastings proved that all 1D quantum spin systems with a constant spectral gap obey an area law in their ground state. The proof, however, uses rather involved analytical tools such as the Lieb-Robinson bounds and does not seem to be generalizable to higher dimensions. In this talk I will present a new, purely combinatorial, proof of the 1D area law for frustration-free systems. The proof gives an exponentially better bound on the entanglement entropy, and, in addition, might be generalizable to higher dimensions.
In the center of the proof lies a new tool called ``the detectability lemma'', which proves extremely useful for studying the ground states of frustration-free systems. I will describe this lemma, and also use it to present a very simple proof of another seminal result of Hastings: the exponential decay of correlations in the ground states of gapped spin systems (in any dimension).
The talk connects various works: Aharonov, Arad, Landau and Vazirani, 2008 & 2010 [3,4], as well as Hastings 2003 & 2007 [1,2].
Refs:
[1] http://arxiv.org/abs/cond-mat/0305505
[2] http://arxiv.org/abs/0705.2024
[3] http://arxiv.org/abs/0811.3412
[4] http://arxiv.org/abs/1011.3445
