Ground state entanglement in one-dimensional translationally invariant quantum systems

doi:https://doi.org/10.1063/1.3254321

Title: Ground state entanglement in one-dimensional translationally invariant quantum systems

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Abstract

We examine whether it is possible for one-dimensional translationally invariant Hamiltonians to have ground states with a high degree of entanglement. We present a family of translationally invariant Hamiltonians (H{sub n}) for the infinite chain. The spectral gap of H{sub n} is {omega}(1/poly(n)). Moreover, for any state in the ground space of H{sub n} and any m, there are regions of size m with entanglement entropy {omega}(min(m,n)). A similar construction yields translationally invariant Hamiltonians for finite chains that have unique ground states exhibiting high entanglement. The area law proven by Hastings ['An area law for one dimensional quantum systems', J. Stat. Mech.: Theory Exp. 2007 (08024)] gives a constant upper bound on the entanglement entropy for one-dimensional ground states that is independent of the size of the region but exponentially dependent on 1/{delta}, where {delta} is the spectral gap. This paper provides a lower bound, showing a family of Hamiltonians for which the entanglement entropy scales polynomially with 1/{delta}. Previously, the best known such bound was logarithmic in 1/{delta}.

Authors:

Irani, Sandy ^[1]

Department of Computer Science, University of California, Irvine, California 2697-3435 (United States)

Publication Date:: 2010-02-15

OSTI Identifier:: 21335899

Resource Type:: Journal Article

Journal Name:: Journal of Mathematical Physics

Additional Journal Information:: Journal Volume: 51; Journal Issue: 2; Other Information: DOI: 10.1063/1.3254321; (c) 2010 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUND STATE; ENTROPY; GROUND STATES; HAMILTONIANS; ONE-DIMENSIONAL CALCULATIONS; QUANTUM ENTANGLEMENT

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                    Irani, Sandy. Ground state entanglement in one-dimensional translationally invariant quantum systems.  United States: N. p., 2010. 
        Web.  doi:10.1063/1.3254321.

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                    Irani, Sandy. Ground state entanglement in one-dimensional translationally invariant quantum systems.  United States.  https://doi.org/10.1063/1.3254321

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                    Irani, Sandy. Mon .  
        "Ground state entanglement in one-dimensional translationally invariant quantum systems".  United States.  https://doi.org/10.1063/1.3254321.

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@article{osti_21335899,

  title        = {Ground state entanglement in one-dimensional translationally invariant quantum systems},

  author       = {Irani, Sandy},

  abstractNote = {We examine whether it is possible for one-dimensional translationally invariant Hamiltonians to have ground states with a high degree of entanglement. We present a family of translationally invariant Hamiltonians (H{sub n}) for the infinite chain. The spectral gap of H{sub n} is {omega}(1/poly(n)). Moreover, for any state in the ground space of H{sub n} and any m, there are regions of size m with entanglement entropy {omega}(min(m,n)). A similar construction yields translationally invariant Hamiltonians for finite chains that have unique ground states exhibiting high entanglement. The area law proven by Hastings ['An area law for one dimensional quantum systems', J. Stat. Mech.: Theory Exp. 2007 (08024)] gives a constant upper bound on the entanglement entropy for one-dimensional ground states that is independent of the size of the region but exponentially dependent on 1/{delta}, where {delta} is the spectral gap. This paper provides a lower bound, showing a family of Hamiltonians for which the entanglement entropy scales polynomially with 1/{delta}. Previously, the best known such bound was logarithmic in 1/{delta}.},

  doi          = {10.1063/1.3254321},

  url          = {https://www.osti.gov/biblio/21335899},
  journal      = {Journal of Mathematical Physics},

  issn         = {0022-2488},

  number       = 2,

  volume       = 51,

  place        = {United States},

  year         = {2010},

  month        = {2}

}

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