The holographic shape of entanglement and Einstein’s equations

Lewkowycz, Aitor; Parrikar, Onkar

doi:10.1007/jhep05(2018)147

Title: The holographic shape of entanglement and Einstein’s equations

Abstract

We study shape-deformations of the entanglement entropy and the modular Hamiltonian for an arbitrary subregion and state (with a smooth dual geometry) in a holographic conformal field theory. More precisely, we study a double-deformation comprising of a shape deformation together with a state deformation, where the latter corresponds to a small change in the bulk geometry. Using a purely gravitational identity from the Hollands-Iyer-Wald formalism together with the assumption of equality between bulk and boundary modular flows for the original, undeformed state and subregion, we rewrite a purely CFT expression for this double deformation of the entropy in terms of bulk gravitational variables and show that it precisely agrees with the Ryu-Takayanagi formula including quantum corrections. As a corollary, this gives a novel, CFT derivation of the JLMS formula for arbitrary subregions in the vacuum, without using the replica trick. Finally, we use our results to give an argument that if a general, asymptotically AdS spacetime satisfies the Ryu-Takayanagi formula for arbitrary subregions, then it must necessarily satisfy the non-linear Einstein equation.

Authors:

Lewkowycz, Aitor ^[1]; Parrikar, Onkar ^[2]

Stanford Univ., CA (United States). Stanford Inst. for Theoretical Physics, Dept. of Physics
Univ. of Pennsylvania, Philadelphia, PA (United States). David Rittenhouse Lab.

Publication Date:: 2018-05-23

Research Org.:: Duke Univ., Durham, NC (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1507703

Grant/Contract Number:: FG02-05ER41367

Resource Type:: Accepted Manuscript

Journal Name:: Journal of High Energy Physics (Online)

Additional Journal Information:: Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2018; Journal Issue: 5; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AdS-CFT Correspondence; Conformal Field Theory

Citation Formats


                    Lewkowycz, Aitor, and Parrikar, Onkar. The holographic shape of entanglement and Einstein’s equations.  United States: N. p., 2018. 
Web.  doi:10.1007/jhep05(2018)147.

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                    Lewkowycz, Aitor, & Parrikar, Onkar. The holographic shape of entanglement and Einstein’s equations.  United States.  https://doi.org/10.1007/jhep05(2018)147

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                    Lewkowycz, Aitor, and Parrikar, Onkar. Wed .  
"The holographic shape of entanglement and Einstein’s equations".  United States.  https://doi.org/10.1007/jhep05(2018)147.  https://www.osti.gov/servlets/purl/1507703.

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@article{osti_1507703,

  title        = {The holographic shape of entanglement and Einstein’s equations},

  author       = {Lewkowycz, Aitor and Parrikar, Onkar},

  abstractNote = {We study shape-deformations of the entanglement entropy and the modular Hamiltonian for an arbitrary subregion and state (with a smooth dual geometry) in a holographic conformal field theory. More precisely, we study a double-deformation comprising of a shape deformation together with a state deformation, where the latter corresponds to a small change in the bulk geometry. Using a purely gravitational identity from the Hollands-Iyer-Wald formalism together with the assumption of equality between bulk and boundary modular flows for the original, undeformed state and subregion, we rewrite a purely CFT expression for this double deformation of the entropy in terms of bulk gravitational variables and show that it precisely agrees with the Ryu-Takayanagi formula including quantum corrections. As a corollary, this gives a novel, CFT derivation of the JLMS formula for arbitrary subregions in the vacuum, without using the replica trick. Finally, we use our results to give an argument that if a general, asymptotically AdS spacetime satisfies the Ryu-Takayanagi formula for arbitrary subregions, then it must necessarily satisfy the non-linear Einstein equation.},

  doi          = {10.1007/jhep05(2018)147},

  journal      = {Journal of High Energy Physics (Online)},

  number       = 5,

  volume       = 2018,

  place        = {United States},

  year         = {2018},

  month        = {5}

}

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https://doi.org/10.1007/jhep05(2018)147

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Cited by: 11 works

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Figure 1: An illustration of the Euclidean path integral representation of the reduced density matrix. The solid blue line denotes the region R, the dashed blue line is R^c, and the solid black dot is $\partial$R. Also shown are the cylindrical tube surrounding the entangling surface $\partial$R_B, and the cutmore »

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The holographic dual of the entanglement wedge symplectic form
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Similar Records

Title: The holographic shape of entanglement and Einstein’s equations

Abstract

Citation Formats

Figures / Tables:

Building up spacetime with quantum entanglement journal, June 2010

Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime journal, January 2015

Shape Dependence of Holographic Rényi Entropy in Conformal Field Theories journal, June 2016

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

Local subsystems in gauge theory and gravity journal, September 2016

Averaged null energy condition from causality journal, July 2017

On excited states in real-time AdS/CFT journal, February 2016

Towards a derivation of holographic entanglement entropy journal, May 2011

Gravitational dynamics from entanglement “thermodynamics” journal, April 2014

Relative entropy and holography journal, August 2013

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

Deriving covariant holographic entanglement journal, November 2016

Universality in the shape dependence of holographic Rényi entropy for general higher derivative gravity journal, December 2016

Bulk emergence and the RG flow of entanglement entropy journal, May 2015

Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence journal, May 2006

Modular Hamiltonians of excited states, OPE blocks and emergent bulk fields journal, January 2018

Entanglement entropy for relevant and geometric perturbations journal, February 2015

Universality in the geometric dependence of Rényi entropy journal, January 2015

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

Modular Hamiltonians for deformed half-spaces and the averaged null energy condition journal, September 2016

Shape dependence of holographic Rényi entropy in general dimensions journal, November 2016

Bulk locality from modular flow journal, July 2017

A covariant holographic entanglement entropy proposal journal, July 2007

A Conserved Current for Perturbations of Einstein-Maxwell Space-Times journal, July 1990

Generalized gravitational entropy journal, August 2013

Holographic entanglement entropy for general higher derivative gravity journal, January 2014

Local phase space and edge modes for diffeomorphism-invariant theories journal, February 2018

Entanglement temperature and entanglement entropy of excited states journal, December 2013

Generalized entropy and higher derivative gravity journal, March 2014

Comparison of the Noether charge and Euclidean methods for computing the entropy of stationary black holes journal, October 1995

Metric perturbations of extremal surfaces journal, January 2018

Quantum corrections to holographic entanglement entropy journal, November 2013

Stability of Black Holes and Black Branes journal, December 2012

Relative entropy equals bulk relative entropy journal, June 2016

Gravitation from entanglement in holographic CFTs journal, March 2014

Entropy, extremality, euclidean variations, and the equations of motion journal, January 2018

Rényi entropy and conformal defects journal, July 2016

Entanglement entropy: a perturbative calculation journal, December 2014

The holographic dual of the entanglement wedge symplectic form journal, January 2020

Nearly AdS2 holography in quantum CGHS model journal, January 2020

Holographic excited states in AdS black holes journal, April 2019

Einstein gravity from ANEC correlators journal, August 2019

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

Surface theory: the classical, the quantum, and the holographic journal, September 2019

The holographic dual of the entanglement wedge symplectic form text, January 2020

The holographic dual of the entanglement wedge symplectic form text, January 2020

Towards Bulk Metric Reconstruction from Extremal Area Variations text, January 2019

The Holographic Dual of the Entanglement Wedge Symplectic Form text, January 2019

Nearly AdS$_2$ holography in quantum CGHS model text, January 2019

Building up spacetime with quantum entanglement
journal, June 2010

Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime
journal, January 2015

Shape Dependence of Holographic Rényi Entropy in Conformal Field Theories
journal, June 2016

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

Local subsystems in gauge theory and gravity
journal, September 2016

Averaged null energy condition from causality
journal, July 2017

On excited states in real-time AdS/CFT
journal, February 2016

Towards a derivation of holographic entanglement entropy
journal, May 2011

Gravitational dynamics from entanglement “thermodynamics”
journal, April 2014

Relative entropy and holography
journal, August 2013

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

Deriving covariant holographic entanglement
journal, November 2016

Universality in the shape dependence of holographic Rényi entropy for general higher derivative gravity
journal, December 2016

Bulk emergence and the RG flow of entanglement entropy
journal, May 2015

Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence
journal, May 2006

Modular Hamiltonians of excited states, OPE blocks and emergent bulk fields
journal, January 2018

Entanglement entropy for relevant and geometric perturbations
journal, February 2015

Universality in the geometric dependence of Rényi entropy
journal, January 2015

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

Modular Hamiltonians for deformed half-spaces and the averaged null energy condition
journal, September 2016

Shape dependence of holographic Rényi entropy in general dimensions
journal, November 2016

Bulk locality from modular flow
journal, July 2017

A covariant holographic entanglement entropy proposal
journal, July 2007

A Conserved Current for Perturbations of Einstein-Maxwell Space-Times
journal, July 1990

Generalized gravitational entropy
journal, August 2013

Holographic entanglement entropy for general higher derivative gravity
journal, January 2014

Local phase space and edge modes for diffeomorphism-invariant theories
journal, February 2018

Entanglement temperature and entanglement entropy of excited states
journal, December 2013

Generalized entropy and higher derivative gravity
journal, March 2014

Comparison of the Noether charge and Euclidean methods for computing the entropy of stationary black holes
journal, October 1995

Metric perturbations of extremal surfaces
journal, January 2018

Quantum corrections to holographic entanglement entropy
journal, November 2013

Stability of Black Holes and Black Branes
journal, December 2012

Relative entropy equals bulk relative entropy
journal, June 2016

Gravitation from entanglement in holographic CFTs
journal, March 2014

Entropy, extremality, euclidean variations, and the equations of motion
journal, January 2018

Rényi entropy and conformal defects
journal, July 2016

Entanglement entropy: a perturbative calculation
journal, December 2014

The holographic dual of the entanglement wedge symplectic form
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Nearly AdS2 holography in quantum CGHS model
journal, January 2020

Holographic excited states in AdS black holes
journal, April 2019

Einstein gravity from ANEC correlators
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Towards bulk metric reconstruction from extremal area variations
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Surface theory: the classical, the quantum, and the holographic
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