An Area Law for One Dimensional Quantum Systems

Details An Area Law for One Dimensional Quantum Systems An Area Law for One Dimensional Quantum Systems

2007-05-14

arxiv.org/abs/0705.2024v3

JSTAT, P08024 (2007)

Physics

Quantum Physics

9 pages, 1 figure; typo fixed in Eq. 41; typo fixed in Eq. 4

Scientific paper

10.1088/1742-5468/2007/08/P08024

We prove an area law for the entanglement entropy in gapped one dimensional quantum systems. The bound on the entropy grows surprisingly rapidly with the correlation length; we discuss this in terms of properties of quantum expanders and present a conjecture on completely positive maps which may provide an alternate way of arriving at an area law. We also show that, for gapped, local systems, the bound on Von Neumann entropy implies a bound on R\'{e}nyi entropy for sufficiently large $\alpha<1$ and implies the ability to approximate the ground state by a matrix product state.

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