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

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:
 [1]
  1. Department of Computer Science, University of California, Irvine, California 2697-3435 (United States)
Publication Date:
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

Citation Formats

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