Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space

Cao, ChunJun; Carroll, Sean M.

doi:10.1103/PhysRevD.97.086003

Title: Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space

Abstract

We consider the emergence from quantum entanglement of spacetime geometry in a bulk region. For certain classes of quantum states in an appropriately factorized Hilbert space, a spatial geometry can be defined by associating areas along codimension-one surfaces with the entanglement entropy between either side. We show how radon transforms can be used to convert these data into a spatial metric. Under a particular set of assumptions, the time evolution of such a state traces out a four-dimensional spacetime geometry, and we argue using a modified version of Jacobson’s “entanglement equilibrium” that the geometry should obey Einstein’s equation in the weak-field limit. Here, we also discuss how entanglement equilibrium is related to a generalization of the Ryu-Takayanagi formula in more general settings, and how quantum error correction can help specify the emergence map between the full quantum-gravity Hilbert space and the semiclassical limit of quantum fields propagating on a classical spacetime.

Authors:: Cao, ChunJun; Carroll, Sean M.

Publication Date:: 2018-04-03

Research Org.:: California Institute of Technology (CalTech), Pasadena, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), High Energy Physics (HEP)

OSTI Identifier:: 1431154

Alternate Identifier(s):: OSTI ID: 1501451

Grant/Contract Number:: SC0011632

Resource Type:: Published Article

Journal Name:: Physical Review D

Additional Journal Information:: Journal Name: Physical Review D Journal Volume: 97 Journal Issue: 8; Journal ID: ISSN 2470-0010

Publisher:: American Physical Society (APS)

Country of Publication:: United States

Language:: English

Subject:: 79 ASTRONOMY AND ASTROPHYSICS

Citation Formats


                    Cao, ChunJun, and Carroll, Sean M. Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space.  United States: N. p., 2018. 
Web.  doi:10.1103/PhysRevD.97.086003.

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                    Cao, ChunJun, & Carroll, Sean M. Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space.  United States.  https://doi.org/10.1103/PhysRevD.97.086003

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                    Cao, ChunJun, and Carroll, Sean M. Tue .  
"Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space".  United States.  https://doi.org/10.1103/PhysRevD.97.086003.

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@article{osti_1431154,

  title        = {Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space},

  author       = {Cao, ChunJun and Carroll, Sean M.},

  abstractNote = {We consider the emergence from quantum entanglement of spacetime geometry in a bulk region. For certain classes of quantum states in an appropriately factorized Hilbert space, a spatial geometry can be defined by associating areas along codimension-one surfaces with the entanglement entropy between either side. We show how radon transforms can be used to convert these data into a spatial metric. Under a particular set of assumptions, the time evolution of such a state traces out a four-dimensional spacetime geometry, and we argue using a modified version of Jacobson’s “entanglement equilibrium” that the geometry should obey Einstein’s equation in the weak-field limit. Here, we also discuss how entanglement equilibrium is related to a generalization of the Ryu-Takayanagi formula in more general settings, and how quantum error correction can help specify the emergence map between the full quantum-gravity Hilbert space and the semiclassical limit of quantum fields propagating on a classical spacetime.},

  doi          = {10.1103/PhysRevD.97.086003},

  journal      = {Physical Review D},

  number       = 8,

  volume       = 97,

  place        = {United States},

  year         = {2018},

  month        = {4}

}

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Title: Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space

Abstract

Citation Formats

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Entanglement Equilibrium and the Einstein Equation journal, May 2016

Entropy on a null surface for interacting quantum field theories and the Bousso bound journal, April 2015

Building up Space–Time with Quantum Entanglement journal, December 2010

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What is the entropy in entropic gravity? journal, June 2016

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Entanglement is not enough: Entanglement is not enough journal, January 2016

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Holographic duality from random tensor networks journal, November 2016

Holographic entanglement entropy conjecture for general spacetimes journal, October 2016

A practical introduction to tensor networks: Matrix product states and projected entangled pair states journal, October 2014

Volumes of Balls in Manifolds Without Conjugate Points journal, August 1992

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Equivalent equations of motion for gravity and entropy journal, February 2017

The Hilbert space of quantum gravity is locally finite-dimensional journal, October 2017

Boundary Rigidity and Holography journal, January 2004

Proof of a quantum Bousso bound journal, August 2014

Space from Hilbert space: Recovering geometry from bulk entanglement journal, January 2017

Emergent gravity of fractons: Mach’s principle revisited journal, July 2017

Numerical Implementation of Geodesic X-Ray Transforms and Their Inversion journal, January 2014

Colloquium : Area laws for the entanglement entropy journal, February 2010

Emergence of helicity ±2 modes (gravitons) from qubit models journal, October 2012

Relative entropy and holography journal, August 2013

Cool horizons for entangled black holes journal, August 2013

Holographic entropy inequalities and gapped phases of matter journal, September 2015

The Ryu–Takayanagi Formula from Quantum Error Correction journal, May 2017

Tensor networks from kinematic space journal, July 2016

Discrete radon transform journal, February 1987

The ultrahyperbolic differential equation with four independent variables journal, June 1938

Generalized multidimensional scaling: A framework for isometry-invariant partial surface matching journal, January 2006

On the origin of gravity and the laws of Newton journal, April 2011

Sur la rigidit� impos�e par la longueur des g�od�siques journal, February 1981

Thermodynamics of Spacetime: The Einstein Equation of State journal, August 1995

Emergent Gravity and the Dark Universe journal, January 2017

Fault-tolerant quantum computation by anyons journal, January 2003

The Ising model with a transverse field. I. High temperature expansion journal, October 1971

Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence journal, June 2015

The Ising model with a transverse field. II. Ground state properties journal, October 1971

Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds journal, March 2007

Boundary rigidity and filling volume minimality of metrics close to a flat one journal, January 2010

Toward a holographic theory for general spacetimes journal, April 2017

Integral geometry and holography journal, October 2015

Quantum corrections to holographic entanglement entropy journal, November 2013

Gravitation from entanglement in holographic CFTs journal, March 2014

Spacetime equals entanglement journal, December 2016

Rigidity and the distance between boundary points journal, January 1991

Entanglement renormalization and holography journal, September 2012

Quantum-First Gravity journal, February 2019

The multiple realizability of general relativity in quantum gravity journal, September 2019

Towards bulk metric reconstruction from extremal area variations journal, August 2019

Black holes in the quantum universe journal, November 2019

Von Neumann Entropy from Unitarity journal, May 2019

Building up spacetime with quantum entanglement
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Entanglement Equilibrium and the Einstein Equation
journal, May 2016

Entropy on a null surface for interacting quantum field theories and the Bousso bound
journal, April 2015

Building up Space–Time with Quantum Entanglement
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Nonlinear gravity from entanglement in conformal field theories
journal, August 2017

Relative entropy and the Bekenstein bound
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Glimmers of a Pre-geometric Perspective
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On the architecture of spacetime geometry
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On spacetime entanglement
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What is the entropy in entropic gravity?
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Holographic duality from random tensor networks
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A practical introduction to tensor networks: Matrix product states and projected entangled pair states
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Volumes of Balls in Manifolds Without Conjugate Points
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The large $N$ limit of superconformal field theories and supergravity
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Equivalent equations of motion for gravity and entropy
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The Hilbert space of quantum gravity is locally finite-dimensional
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Boundary Rigidity and Holography
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Proof of a quantum Bousso bound
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Space from Hilbert space: Recovering geometry from bulk entanglement
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Emergent gravity of fractons: Mach’s principle revisited
journal, July 2017

Numerical Implementation of Geodesic X-Ray Transforms and Their Inversion
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Colloquium : Area laws for the entanglement entropy
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Emergence of helicity ±2 modes (gravitons) from qubit models
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Relative entropy and holography
journal, August 2013

Cool horizons for entangled black holes
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Holographic entropy inequalities and gapped phases of matter
journal, September 2015

The Ryu–Takayanagi Formula from Quantum Error Correction
journal, May 2017

Tensor networks from kinematic space
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Discrete radon transform
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The ultrahyperbolic differential equation with four independent variables
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Generalized multidimensional scaling: A framework for isometry-invariant partial surface matching
journal, January 2006

On the origin of gravity and the laws of Newton
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Sur la rigidit� impos�e par la longueur des g�od�siques
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Emergent Gravity and the Dark Universe
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Fault-tolerant quantum computation by anyons
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The Ising model with a transverse field. I. High temperature expansion
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Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence
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The Ising model with a transverse field. II. Ground state properties
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Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds
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Boundary rigidity and filling volume minimality of metrics close to a flat one
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Toward a holographic theory for general spacetimes
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