Ground state entanglement in one-dimensional translationally invariant quantum systems

Journal Article · Mon Feb 15 04:00:00 EST 2010 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3254321· OSTI ID:21335899

Irani, Sandy ^[1]

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

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}.

• OSTI ID: 21335899
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 2 Vol. 51; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

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