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Published January 15, 2017 | Submitted + Published
Journal Article Open

Space from Hilbert Space: Recovering Geometry from Bulk Entanglement


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.

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).

Attached Files

Published - PhysRevD.95.024031.pdf

Submitted - 1606.08444v3.pdf


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