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Space from Hilbert Space: Recovering Geometry from Bulk Entanglement

Cao, ChunJun and Carroll, Sean M. and Michalakis, Spyridon (2017) Space from Hilbert Space: Recovering Geometry from Bulk Entanglement. Physical Review D, 95 (2). Art. No. 024031. ISSN 2470-0010. https://resolver.caltech.edu/CaltechAUTHORS:20160704-200753575

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Abstract

We examine how to construct a spatial manifold and its geometry from the entanglement structure of an abstract quantum state in Hilbert space. Given a decomposition of Hilbert space H into a tensor product of factors, we consider a class of “redundancy-constrained states” in H that generalize the area-law behavior for entanglement entropy usually found in condensed-matter systems with gapped local Hamiltonians. Using mutual information to define a distance measure on the graph, we employ classical multidimensional scaling to extract the best-fit spatial dimensionality of the emergent geometry. We then show that entanglement perturbations on such emergent geometries naturally give rise to local modifications of spatial curvature which obey a (spatial) analog of Einstein’s equation. The Hilbert space corresponding to a region of flat space is finite-dimensional and scales as the volume, though the entropy (and the maximum change thereof) scales like the area of the boundary. A version of the ER=EPR conjecture is recovered, in that perturbations that entangle distant parts of the emergent geometry generate a configuration that may be considered as a highly quantum wormhole.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1103/PhysRevD.95.024031DOIArticle
http://journals.aps.org/prd/abstract/10.1103/PhysRevD.95.024031PublisherArticle
http://arxiv.org/abs/1606.08444arXivDiscussion Paper
ORCID:
AuthorORCID
Carroll, Sean M.0000-0002-4226-5758
Additional Information:© 2017 American Physical Society. Received 7 July 2016; published 27 January 2017. We would like to thank Ning Bao, Aidan Chatwin-Davies, Bartek Czech, Nick Hunter-Jones, Shaun Maguire, Hirosi Ooguri, John Preskill, Jason Pollack, and Brian Swingle for helpful discussions. C. C. would like to thank Ning Bao copiously for his suggestions and support throughout the course of this project. C. C. also thanks the organizers of the YITP long term workshop on “Quantum Information in String Theory and Many-body Systems.” This research is funded in part by the Walter Burke Institute for Theoretical Physics at Caltech, by DOE Grant No. DE-SC0011632, by the Foundational Questions Institute, by the Gordon and Betty Moore Foundation through Grant No. 776 to the Caltech Moore Center for Theoretical Cosmology and Physics, and by the John Simon Guggenheim Memorial Foundation. S. M. acknowledges funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant No. PHY-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-2644).
Group:Walter Burke Institute for Theoretical Physics, Moore Center for Theoretical Cosmology and Physics, IQIM, Institute for Quantum Information and Matter
Funders:
Funding AgencyGrant Number
Walter Burke Institute for Theoretical Physics, CaltechUNSPECIFIED
Department of Energy (DOE)DE-SC0011632
Foundational Questions Institute (FQXI)UNSPECIFIED
Gordon and Betty Moore Foundation776
John Simon Guggenheim Memorial FoundationUNSPECIFIED
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
NSFPHY-1125565
Gordon and Betty Moore FoundationGBMF-2644
Other Numbering System:
Other Numbering System NameOther Numbering System ID
CALT-TH2016-15
Issue or Number:2
Record Number:CaltechAUTHORS:20160704-200753575
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160704-200753575
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:68821
Collection:CaltechAUTHORS
Deposited By: Joy Painter
Deposited On:05 Jul 2016 17:21
Last Modified:03 Oct 2019 10:16

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