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A simple proof of the detectability lemma and spectral gap amplification

Anshu, Anurag and Arad, Itai and Vidick, Thomas (2016) A simple proof of the detectability lemma and spectral gap amplification. Physical Review B, 93 (20). Art. No. 205142. ISSN 1098-0121. doi:10.1103/PhysRevB.93.205142.

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The detectability lemma is a useful tool for probing the structure of gapped ground states of frustration-free Hamiltonians of lattice spin models. The lemma provides an estimate on the error incurred by approximating the ground space projector with a product of local projectors. We provide a simpler proof for the detectability lemma which applies to an arbitrary ordering of the local projectors, and show that it is tight up to a constant factor. As an application, we show how the lemma can be combined with a strong converse by Gao to obtain local spectral gap amplification: We show that by coarse graining a local frustration-free Hamiltonian with a spectral gap γ>0 to a length scale O(γ^(−1/2)), one gets a Hamiltonian with an Ω(1) spectral gap.

Item Type:Article
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Vidick, Thomas0000-0002-6405-365X
Additional Information:Published by the American Physical Society. This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. (Received 2 March 2016; revised manuscript received 17 April 2016; published 23 May 2016) We thank Zeph Landau for many insightful discussions, and Mark Wilde for bringing Ref. [23] to our attention. We also thank an anonymous referee for pointing out minor imprecisions in an earlier draft of this paper. T.V. was partially supported by the IQIM, an NSF Physics Frontiers Center (NFS Grant No. PHY-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-12500028). A.A. was supported by Core grants of Centre for Quantum Technologies, Singapore. Research at the Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation, also through the Tier 3 Grant random numbers from quantum processes.
Group:Institute for Quantum Information and Matter
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Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
Gordon and Betty Moore FoundationGBMF-12500028
Centre for Quantum TechnologiesUNSPECIFIED
Ministry of Education (Singapore)UNSPECIFIED
National Research Foundation (Singapore)UNSPECIFIED
Issue or Number:20
Record Number:CaltechAUTHORS:20160318-153303794
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Official Citation:Simple proof of the detectability lemma and spectral gap amplification Anurag Anshu, Itai Arad, and Thomas Vidick Phys. Rev. B 93, 205142 (2016)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:65491
Deposited By: Tony Diaz
Deposited On:18 Mar 2016 22:52
Last Modified:10 Nov 2021 23:46

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