The holographic shape of entanglement and Einstein’s equations
Abstract
We study shapedeformations 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 doubledeformation 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 HollandsIyerWald 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 RyuTakayanagi 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 RyuTakayanagi formula for arbitrary subregions, then it must necessarily satisfy the nonlinear Einstein equation.
 Authors:

 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:
 Research Org.:
 Duke Univ., Durham, NC (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1507703
 Grant/Contract Number:
 FG0205ER41367
 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 10298479
 Publisher:
 Springer Berlin
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AdSCFT 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.
Lewkowycz, Aitor, & Parrikar, Onkar. The holographic shape of entanglement and Einstein’s equations. United States. https://doi.org/10.1007/jhep05(2018)147
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.
@article{osti_1507703,
title = {The holographic shape of entanglement and Einstein’s equations},
author = {Lewkowycz, Aitor and Parrikar, Onkar},
abstractNote = {We study shapedeformations 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 doubledeformation 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 HollandsIyerWald 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 RyuTakayanagi 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 RyuTakayanagi formula for arbitrary subregions, then it must necessarily satisfy the nonlinear 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}
}
Web of Science
Figures / Tables:
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
Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime
journal, January 2015
 Engelhardt, Netta; Wall, Aron C.
 Journal of High Energy Physics, Vol. 2015, Issue 1
Shape Dependence of Holographic Rényi Entropy in Conformal Field Theories
journal, June 2016
 Dong, Xi
 Physical Review Letters, Vol. 116, Issue 25
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
Local subsystems in gauge theory and gravity
journal, September 2016
 Donnelly, William; Freidel, Laurent
 Journal of High Energy Physics, Vol. 2016, Issue 9
Averaged null energy condition from causality
journal, July 2017
 Hartman, Thomas; Kundu, Sandipan; Tajdini, Amirhossein
 Journal of High Energy Physics, Vol. 2017, Issue 7
On excited states in realtime AdS/CFT
journal, February 2016
 BottaCantcheff, Marcelo; Martínez, Pedro J.; Silva, Guillermo A.
 Journal of High Energy Physics, Vol. 2016, Issue 2
Towards a derivation of holographic entanglement entropy
journal, May 2011
 Casini, Horacio; Huerta, Marina; Myers, Robert C.
 Journal of High Energy Physics, Vol. 2011, Issue 5
Gravitational dynamics from entanglement “thermodynamics”
journal, April 2014
 Lashkari, Nima; McDermott, Michael B.; Van Raamsdonk, Mark
 Journal of High Energy Physics, Vol. 2014, Issue 4
Relative entropy and holography
journal, August 2013
 Blanco, David D.; Casini, Horacio; Hung, LingYan
 Journal of High Energy Physics, Vol. 2013, Issue 8
The Ryu–Takayanagi Formula from Quantum Error Correction
journal, May 2017
 Harlow, Daniel
 Communications in Mathematical Physics, Vol. 354, Issue 3
Deriving covariant holographic entanglement
journal, November 2016
 Dong, Xi; Lewkowycz, Aitor; Rangamani, Mukund
 Journal of High Energy Physics, Vol. 2016, Issue 11
Universality in the shape dependence of holographic Rényi entropy for general higher derivative gravity
journal, December 2016
 Chu, ChongSun; Miao, RongXin
 Journal of High Energy Physics, Vol. 2016, Issue 12
Bulk emergence and the RG flow of entanglement entropy
journal, May 2015
 Faulkner, Thomas
 Journal of High Energy Physics, Vol. 2015, Issue 5
Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence
journal, May 2006
 Ryu, Shinsei; Takayanagi, Tadashi
 Physical Review Letters, Vol. 96, Issue 18
Modular Hamiltonians of excited states, OPE blocks and emergent bulk fields
journal, January 2018
 Sárosi, Gábor; Ugajin, Tomonori
 Journal of High Energy Physics, Vol. 2018, Issue 1
Entanglement entropy for relevant and geometric perturbations
journal, February 2015
 Rosenhaus, Vladimir; Smolkin, Michael
 Journal of High Energy Physics, Vol. 2015, Issue 2
Thermodynamics of Spacetime: The Einstein Equation of State
journal, August 1995
 Jacobson, Ted
 Physical Review Letters, Vol. 75, Issue 7
Modular Hamiltonians for deformed halfspaces and the averaged null energy condition
journal, September 2016
 Faulkner, Thomas; Leigh, Robert G.; Parrikar, Onkar
 Journal of High Energy Physics, Vol. 2016, Issue 9
Shape dependence of holographic Rényi entropy in general dimensions
journal, November 2016
 Bianchi, Lorenzo; Chapman, Shira; Dong, Xi
 Journal of High Energy Physics, Vol. 2016, Issue 11
Bulk locality from modular flow
journal, July 2017
 Faulkner, Thomas; Lewkowycz, Aitor
 Journal of High Energy Physics, Vol. 2017, Issue 7
A covariant holographic entanglement entropy proposal
journal, July 2007
 Hubeny, Veronika E.; Rangamani, Mukund; Takayanagi, Tadashi
 Journal of High Energy Physics, Vol. 2007, Issue 07
A Conserved Current for Perturbations of EinsteinMaxwell SpaceTimes
journal, July 1990
 Burnett, G. A.; Wald, R. M.
 Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 430, Issue 1878
Generalized gravitational entropy
journal, August 2013
 Lewkowycz, Aitor; Maldacena, Juan
 Journal of High Energy Physics, Vol. 2013, Issue 8
Holographic entanglement entropy for general higher derivative gravity
journal, January 2014
 Dong, Xi
 Journal of High Energy Physics, Vol. 2014, Issue 1
Local phase space and edge modes for diffeomorphisminvariant theories
journal, February 2018
 Speranza, Antony J.
 Journal of High Energy Physics, Vol. 2018, Issue 2
Generalized entropy and higher derivative gravity
journal, March 2014
 Camps, Joan
 Journal of High Energy Physics, Vol. 2014, Issue 3
Comparison of the Noether charge and Euclidean methods for computing the entropy of stationary black holes
journal, October 1995
 Iyer, Vivek; Wald, Robert M.
 Physical Review D, Vol. 52, Issue 8
Metric perturbations of extremal surfaces
journal, January 2018
 Mosk, Benjamin
 Classical and Quantum Gravity, Vol. 35, Issue 4
Quantum corrections to holographic entanglement entropy
journal, November 2013
 Faulkner, Thomas; Lewkowycz, Aitor; Maldacena, Juan
 Journal of High Energy Physics, Vol. 2013, Issue 11
Stability of Black Holes and Black Branes
journal, December 2012
 Hollands, Stefan; Wald, Robert M.
 Communications in Mathematical Physics, Vol. 321, Issue 3
Relative entropy equals bulk relative entropy
journal, June 2016
 Jafferis, Daniel L.; Lewkowycz, Aitor; Maldacena, Juan
 Journal of High Energy Physics, Vol. 2016, Issue 6
Gravitation from entanglement in holographic CFTs
journal, March 2014
 Faulkner, Thomas; Guica, Monica; Hartman, Thomas
 Journal of High Energy Physics, Vol. 2014, Issue 3
Entropy, extremality, euclidean variations, and the equations of motion
journal, January 2018
 Dong, Xi; Lewkowycz, Aitor
 Journal of High Energy Physics, Vol. 2018, Issue 1
Rényi entropy and conformal defects
journal, July 2016
 Bianchi, Lorenzo; Meineri, Marco; Myers, Robert C.
 Journal of High Energy Physics, Vol. 2016, Issue 7
Entanglement entropy: a perturbative calculation
journal, December 2014
 Rosenhaus, Vladimir; Smolkin, Michael
 Journal of High Energy Physics, Vol. 2014, Issue 12
Modular Hamiltonians on the null plane and the Markov property of the vacuum state
journal, August 2017
 Casini, Horacio; Testé, Eduardo; Torroba, Gonzalo
 Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 36
Nonlinear Gravity from Entanglement in Conformal Field Theories
text, January 2017
 Faulkner, Thomas; Haehl, Felix M.; Hijano, Eliot
 arXiv
Works referencing / citing this record:
The holographic dual of the entanglement wedge symplectic form
journal, January 2020
 Kirklin, Josh
 Journal of High Energy Physics, Vol. 2020, Issue 1
Nearly AdS2 holography in quantum CGHS model
journal, January 2020
 Hirano, Shinji; Lei, Yang
 Journal of High Energy Physics, Vol. 2020, Issue 1
Holographic excited states in AdS black holes
journal, April 2019
 BottaCantcheff, Marcelo; Martínez, Pedro J.; Silva, Guillermo A.
 Journal of High Energy Physics, Vol. 2019, Issue 4
Towards bulk metric reconstruction from extremal area variations
journal, August 2019
 Bao, Ning; Cao, ChunJun; Fischetti, Sebastian
 Classical and Quantum Gravity, Vol. 36, Issue 18
Surface theory: the classical, the quantum, and the holographic
journal, September 2019
 Engelhardt, Netta; Fischetti, Sebastian
 Classical and Quantum Gravity, Vol. 36, Issue 20
The holographic dual of the entanglement wedge symplectic form
text, January 2020
 Kirklin, Josh
 Apollo  University of Cambridge Repository
The holographic dual of the entanglement wedge symplectic form
text, January 2020
 Kirklin, Josh
 Apollo  University of Cambridge Repository
Towards Bulk Metric Reconstruction from Extremal Area Variations
text, January 2019
 Bao, Ning; Cao, ChunJun; Fischetti, Sebastian
 arXiv
The Holographic Dual of the Entanglement Wedge Symplectic Form
text, January 2019
 Kirklin, Josh
 arXiv
Nearly AdS2 holography in quantum CGHS model
journal, January 2020
 Hirano, Shinji; Lei, Yang
 Journal of High Energy Physics, Vol. 2020, Issue 1
Surface theory: the classical, the quantum, and the holographic
journal, September 2019
 Engelhardt, Netta; Fischetti, Sebastian
 Classical and Quantum Gravity, Vol. 36, Issue 20
More of the bulk from extremal area variations
journal, December 2020
 Bao, Ning; Cao, ChunJun; Fischetti, Sebastian
 Classical and Quantum Gravity, Vol. 38, Issue 4
The holographic dual of the entanglement wedge symplectic form
text, January 2020
 Kirklin, Josh
 Apollo  University of Cambridge Repository
Secondorder Lovelock Gravity from Entanglement in Conformal Field Theories
text, January 2019
 Fan, Cunwei; La Nave, Gabriele; Phillips, Philip W.
 arXiv
Holographic scattering requires a connected entanglement wedge
text, January 2019
 May, Alex; Penington, Geoff; Sorce, Jonathan
 arXiv
Modular Hamiltonians for Euclidean Path Integral States
preprint, January 2020
 Balakrishnan, Srivatsan; Parrikar, Onkar
 arXiv
Endpoint contributions to excitedstate modular Hamiltonians
text, January 2020
 Kabat, Daniel; Lifschytz, Gilad; Nguyen, Phuc
 arXiv
Bit threads, Einstein's equations and bulk locality
text, January 2020
 Agón, Cesar A.; Cáceres, Elena; Pedraza, Juan F.
 arXiv
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