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We present a new proof for the 1D area law for frustration-free systems with a constant gap, which exponentially improves the entropy bound in Hastings' 1D area law, and which is tight to within a polynomial factor.
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2011
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D. Aharonov, I. Arad, Z. Landau, and U. Vazirani, in 2011 IEEE 52st Annual Symposium on Foundations of Computer Science (IEEE, 2011) pp. 324–333
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2009
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M. B. Hastings, Phys. Rev. B 76
Cited in the paper.
The examples in these papers are of a frustrated Hamiltonians, whereas our results apply for frustration-free systems. However, the construction in Ref. \rev@citealpnum
Cited in the paper.
Without loss of generality, we may absorb any phases of μ \mu into | ϕ ⟩ {|{\phi}\delimiter 86414091} and assume μ \mu is real and positive
Cited in the paper.
At first sight it seems that the shrinking should be of Δ 0 ( k + 1 ) / 2 \Delta_{0}^{(k+1)/2} , but actually we can do much better by “duplicating” the projections in the middle. For example, for k = 3 k=3 , we get Π e v e n Π o d d Π e v e n Π o d d = ( Π e v e n Π o d d ) ( Π o d d Π e v e n ) ( Π e v e n Π o d d ) \Pi_{even}\Pi_{odd}\Pi_{even}\Pi_{odd}=(\Pi_{even}\Pi_{odd})(\Pi_{odd}\Pi_{even})(\Pi_{even}\Pi_{odd}) . Every bracket contributes a Δ 0 \Delta_{0} factor, so overall we get Δ 0 3 \Delta_{0}^{3}
Cited in the paper.
S. Irani, Private communication (2011)
2011
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