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:: Tue Apr 03 00:00:00 EDT 2018

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         = {Tue Apr 03 00:00:00 EDT 2018},

  month        = {Tue Apr 03 00:00:00 EDT 2018}

}

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Works referenced in this record:

Building up spacetime with quantum entanglement
journal, June 2010

Van Raamsdonk, Mark
General Relativity and Gravitation, Vol. 42, Issue 10
DOI: 10.1007/s10714-010-1034-0

Entanglement Equilibrium and the Einstein Equation
journal, May 2016

Jacobson, Ted
Physical Review Letters, Vol. 116, Issue 20
DOI: 10.1103/PhysRevLett.116.201101

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

Bousso, Raphael; Casini, Horacio; Fisher, Zachary
Physical Review D, Vol. 91, Issue 8
DOI: 10.1103/PhysRevD.91.084030

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

Van Raamsdonk, Mark
International Journal of Modern Physics D, Vol. 19, Issue 14
DOI: 10.1142/S0218271810018529

Nonlinear gravity from entanglement in conformal field theories
journal, August 2017

Faulkner, Thomas; Haehl, Felix M.; Hijano, Eliot
Journal of High Energy Physics, Vol. 2017, Issue 8
DOI: 10.1007/JHEP08(2017)057

Relative entropy and the Bekenstein bound
journal, September 2008

Casini, H.
Classical and Quantum Gravity, Vol. 25, Issue 20
DOI: 10.1088/0264-9381/25/20/205021

Glimmers of a Pre-geometric Perspective
journal, December 2009

Piazza, Federico
Foundations of Physics, Vol. 40, Issue 3
DOI: 10.1007/s10701-009-9387-5

On the architecture of spacetime geometry
journal, October 2014

Bianchi, Eugenio; Myers, Robert C.
Classical and Quantum Gravity, Vol. 31, Issue 21
DOI: 10.1088/0264-9381/31/21/214002

On spacetime entanglement
journal, June 2013

Myers, Robert C.; Pourhasan, Razieh; Smolkin, Michael
Journal of High Energy Physics, Vol. 2013, Issue 6
DOI: 10.1007/JHEP06(2013)013

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

Carroll, Sean M.; Remmen, Grant N.
Physical Review D, Vol. 93, Issue 12
DOI: 10.1103/PhysRevD.93.124052

Tensor Network States and Geometry
journal, June 2011

Evenbly, G.; Vidal, G.
Journal of Statistical Physics, Vol. 145, Issue 4
DOI: 10.1007/s10955-011-0237-4

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

Susskind, Leonard
Fortschritte der Physik, Vol. 64, Issue 1
DOI: 10.1002/prop.201500095

Thermodynamical aspects of gravity: new insights
journal, March 2010

Padmanabhan, T.
Reports on Progress in Physics, Vol. 73, Issue 4
DOI: 10.1088/0034-4885/73/4/046901

Holographic duality from random tensor networks
journal, November 2016

Hayden, Patrick; Nezami, Sepehr; Qi, Xiao-Liang
Journal of High Energy Physics, Vol. 2016, Issue 11
DOI: 10.1007/JHEP11(2016)009

Holographic entanglement entropy conjecture for general spacetimes
journal, October 2016

Sanches, Fabio; Weinberg, Sean J.
Physical Review D, Vol. 94, Issue 8
DOI: 10.1103/PhysRevD.94.084034

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

Orús, Román
Annals of Physics, Vol. 349
DOI: 10.1016/j.aop.2014.06.013

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

Croke, Christopher B.
International Journal of Mathematics, Vol. 03, Issue 04
DOI: 10.1142/S0129167X92000205

The large $N$ limit of superconformal field theories and supergravity
journal, January 1998

Maldacena, Juan
Advances in Theoretical and Mathematical Physics, Vol. 2, Issue 2
DOI: 10.4310/ATMP.1998.v2.n2.a1

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

Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel
Journal of High Energy Physics, Vol. 2017, Issue 2
DOI: 10.1007/JHEP02(2017)004

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

Bao, Ning; Carroll, Sean M.; Singh, Ashmeet
International Journal of Modern Physics D, Vol. 26, Issue 12
DOI: 10.1142/S0218271817430131

Boundary Rigidity and Holography
journal, January 2004

Porrati, M.; Rabadan, R.
Journal of High Energy Physics, Vol. 2004, Issue 01
DOI: 10.1088/1126-6708/2004/01/034

Proof of a quantum Bousso bound
journal, August 2014

Bousso, Raphael; Casini, Horacio; Fisher, Zachary
Physical Review D, Vol. 90, Issue 4
DOI: 10.1103/PhysRevD.90.044002

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

Cao, ChunJun; Carroll, Sean M.; Michalakis, Spyridon
Physical Review D, Vol. 95, Issue 2
DOI: 10.1103/PhysRevD.95.024031

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

Pretko, Michael
Physical Review D, Vol. 96, Issue 2
DOI: 10.1103/PhysRevD.96.024051

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

Monard, François
SIAM Journal on Imaging Sciences, Vol. 7, Issue 2
DOI: 10.1137/130938657

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

Eisert, J.; Cramer, M.; Plenio, M. B.
Reviews of Modern Physics, Vol. 82, Issue 1
DOI: 10.1103/RevModPhys.82.277

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

Gu, Zheng-Cheng; Wen, Xiao-Gang
Nuclear Physics B, Vol. 863, Issue 1
DOI: 10.1016/j.nuclphysb.2012.05.010

Relative entropy and holography
journal, August 2013

Blanco, David D.; Casini, Horacio; Hung, Ling-Yan
Journal of High Energy Physics, Vol. 2013, Issue 8
DOI: 10.1007/JHEP08(2013)060

Cool horizons for entangled black holes
journal, August 2013

Maldacena, J.; Susskind, L.
Fortschritte der Physik, Vol. 61, Issue 9
DOI: 10.1002/prop.201300020

The Large-N Limit of Superconformal Field Theories and Supergravity
journal, January 1999

Maldacena, Juan
International Journal of Theoretical Physics, Vol. 38, Issue 4, p. 1113-1133
DOI: 10.1023/A:1026654312961

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

Bao, Ning; Cao, ChunJun; Walter, Michael
Journal of High Energy Physics, Vol. 2015, Issue 9
DOI: 10.1007/JHEP09(2015)203

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

Harlow, Daniel
Communications in Mathematical Physics, Vol. 354, Issue 3
DOI: 10.1007/s00220-017-2904-z

Tensor networks from kinematic space
journal, July 2016

Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel
Journal of High Energy Physics, Vol. 2016, Issue 7
DOI: 10.1007/JHEP07(2016)100

Discrete radon transform
journal, February 1987

Beylkin, G.
IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 35, Issue 2
DOI: 10.1109/TASSP.1987.1165108

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

John, Fritz
Duke Mathematical Journal, Vol. 4, Issue 2
DOI: 10.1215/S0012-7094-38-00423-5

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

Bronstein, A. M.; Bronstein, M. M.; Kimmel, R.
Proceedings of the National Academy of Sciences, Vol. 103, Issue 5
DOI: 10.1073/pnas.0508601103

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

Verlinde, Erik
Journal of High Energy Physics, Vol. 2011, Issue 4
DOI: 10.1007/JHEP04(2011)029

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

Michel, Ren�
Inventiones Mathematicae, Vol. 65, Issue 1
DOI: 10.1007/BF01389295

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

Jacobson, Ted
Physical Review Letters, Vol. 75, Issue 7
DOI: 10.1103/PhysRevLett.75.1260

Emergent Gravity and the Dark Universe
journal, January 2017

Verlinde, Erik
SciPost Physics, Vol. 2, Issue 3
DOI: 10.21468/SciPostPhys.2.3.016

Fault-tolerant quantum computation by anyons
journal, January 2003

Kitaev, A. Yu.
Annals of Physics, Vol. 303, Issue 1
DOI: 10.1016/S0003-4916(02)00018-0

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

Elliott, R. J.; Wood, C.
Journal of Physics C: Solid State Physics, Vol. 4, Issue 15
DOI: 10.1088/0022-3719/4/15/023

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

Pastawski, Fernando; Yoshida, Beni; Harlow, Daniel
Journal of High Energy Physics, Vol. 2015, Issue 6
DOI: 10.1007/JHEP06(2015)149

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

Pfeuty, P.; Elliott, R. J.
Journal of Physics C: Solid State Physics, Vol. 4, Issue 15
DOI: 10.1088/0022-3719/4/15/024

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

Sharafutdinov, Vladimir
Journal of Geometric Analysis, Vol. 17, Issue 1
DOI: 10.1007/BF02922087

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

Burago, Dmitri; Ivanov, Sergei
Annals of Mathematics, Vol. 171, Issue 2
DOI: 10.4007/annals.2010.171.1183

Toward a holographic theory for general spacetimes
journal, April 2017

Nomura, Yasunori; Salzetta, Nico; Sanches, Fabio
Physical Review D, Vol. 95, Issue 8
DOI: 10.1103/PhysRevD.95.086002

Integral geometry and holography
journal, October 2015

Czech, Bartłomiej; Lamprou, Lampros; McCandlish, Samuel
Journal of High Energy Physics, Vol. 2015, Issue 10
DOI: 10.1007/JHEP10(2015)175

Quantum corrections to holographic entanglement entropy
journal, November 2013

Faulkner, Thomas; Lewkowycz, Aitor; Maldacena, Juan
Journal of High Energy Physics, Vol. 2013, Issue 11
DOI: 10.1007/JHEP11(2013)074

Gravitation from entanglement in holographic CFTs
journal, March 2014

Faulkner, Thomas; Guica, Monica; Hartman, Thomas
Journal of High Energy Physics, Vol. 2014, Issue 3
DOI: 10.1007/JHEP03(2014)051

Spacetime equals entanglement
journal, December 2016

Nomura, Yasunori; Salzetta, Nico; Sanches, Fabio
Physics Letters B, Vol. 763
DOI: 10.1016/j.physletb.2016.10.045

Rigidity and the distance between boundary points
journal, January 1991

Croke, Christopher B.
Journal of Differential Geometry, Vol. 33, Issue 2
DOI: 10.4310/jdg/1214446326

Entanglement renormalization and holography
journal, September 2012

Swingle, Brian
Physical Review D, Vol. 86, Issue 6
DOI: 10.1103/PhysRevD.86.065007

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

Abstract

Citation Formats

Building up spacetime with quantum entanglement journal, June 2010

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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On spacetime entanglement journal, June 2013

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

Tensor Network States and Geometry journal, June 2011

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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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

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The Large-N Limit of Superconformal Field Theories and Supergravity journal, January 1999

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

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

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

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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

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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

Building up spacetime with quantum entanglement
journal, June 2010

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

Nonlinear gravity from entanglement in conformal field theories
journal, August 2017

Relative entropy and the Bekenstein bound
journal, September 2008

Glimmers of a Pre-geometric Perspective
journal, December 2009

On the architecture of spacetime geometry
journal, October 2014

On spacetime entanglement
journal, June 2013

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

Tensor Network States and Geometry
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Entanglement is not enough: Entanglement is not enough
journal, January 2016

Thermodynamical aspects of gravity: new insights
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Holographic duality from random tensor networks
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Holographic entanglement entropy conjecture for general spacetimes
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A practical introduction to tensor networks: Matrix product states and projected entangled pair states
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journal, January 1998

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
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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
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
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The Large-N Limit of Superconformal Field Theories and Supergravity
journal, January 1999

Holographic entropy inequalities and gapped phases of matter
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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

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journal, April 2011

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

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journal, August 1995

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
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
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Rigidity and the distance between boundary points
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