Space from Hilbert space: Recovering geometry from bulk entanglement
Abstract
We examine here 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 into a tensor product of factors, we consider a class of “redundancy-constrained states” in 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, yet the entropy (and the maximum change thereof) scales like the area of the boundary. A version of the 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.
- Authors:
-
- California Inst. of Technology (CalTech), Pasadena, CA (United States)
- Publication Date:
- Research Org.:
- California Institute of Technology (CalTech), Pasadena, CA (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP); National Science Foundation (NSF); Gordon and Betty Moore Foundation; John Simon Guggenheim Memorial Foundation
- Contributing Org.:
- Walter Burke Institute for Theoretical Physics
- OSTI Identifier:
- 1598344
- Alternate Identifier(s):
- OSTI ID: 1341309
- Grant/Contract Number:
- SC0011632; PHY-1125565; GBMF-2644
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review D
- Additional Journal Information:
- Journal Volume: 95; Journal Issue: 2; Journal ID: ISSN 2470-0010
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 79 ASTRONOMY AND ASTROPHYSICS; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Citation Formats
Cao, ChunJun, Carroll, Sean M., and Michalakis, Spyridon. Space from Hilbert space: Recovering geometry from bulk entanglement. United States: N. p., 2017.
Web. doi:10.1103/PhysRevD.95.024031.
Cao, ChunJun, Carroll, Sean M., & Michalakis, Spyridon. Space from Hilbert space: Recovering geometry from bulk entanglement. United States. https://doi.org/10.1103/PhysRevD.95.024031
Cao, ChunJun, Carroll, Sean M., and Michalakis, Spyridon. Fri .
"Space from Hilbert space: Recovering geometry from bulk entanglement". United States. https://doi.org/10.1103/PhysRevD.95.024031. https://www.osti.gov/servlets/purl/1598344.
@article{osti_1598344,
title = {Space from Hilbert space: Recovering geometry from bulk entanglement},
author = {Cao, ChunJun and Carroll, Sean M. and Michalakis, Spyridon},
abstractNote = {We examine here 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, yet 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.},
doi = {10.1103/PhysRevD.95.024031},
journal = {Physical Review D},
number = 2,
volume = 95,
place = {United States},
year = {Fri Jan 27 00:00:00 EST 2017},
month = {Fri Jan 27 00:00:00 EST 2017}
}
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