An area law for entanglement from exponential decay of correlations

Abstract

Area laws for entanglement in quantum many-body systems give useful information about their low-temperature behaviour and are tightly connected to the possibility of good numerical simulations. An intuition from quantum many-body physics suggests that an area law should hold whenever there is exponential decay of correlations in the system, a property found, for instance, in non-critical phases of matter. However, the existence of quantum data-hiding states—that is, states having very small correlations, yet a volume scaling of entanglement—was believed to be a serious obstruction to such an implication. Here we prove that notwithstanding the phenomenon of data hiding, one-dimensional quantum many-body states satisfying exponential decay of correlations always fulfil an area law. To obtain this result we combine several recent advances in quantum information theory, thus showing the usefulness of the field for addressing problems in other areas of physics.

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Figure 1: EDC intuitively suggests an area law.
Figure 2: Data hiding as an obstruction.
Figure 3: Revealing correlations: main steps of the proof.

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Acknowledgements

We would like to thank D. Aharonov, I. Arad and A. Harrow for interesting discussions on area laws and related subjects and M. Hastings for useful correspondence. F.G.S.L.B. acknowledges support from EPSRC, and the Swiss National Science Foundation, through the National Centre of Competence in Research QSIT M.H. acknowledges the support of EC IP QESSENCE, ERC QOLAPS, and National Science Centre, grant no. DEC-2011/02/A/ST2/00305. Part of this work was done at the National Quantum Information Centre of Gdansk. F.G.S.L.B. and M.H. acknowledge the hospitality of Institute Mittag Leer within the programme Quantum Information Science (2010), where part of this work was done.

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Correspondence to Fernando G. S. L. Brandão.

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Brandão, F., Horodecki, M. An area law for entanglement from exponential decay of correlations. Nature Phys 9, 721–726 (2013). https://doi.org/10.1038/nphys2747

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