The holographic dual of Rényi relative entropy

Bao, Ning; Moosa, Mudassir; Shehzad, Ibrahim

doi:10.1007/JHEP08(2019)099

Title: The holographic dual of Rényi relative entropy

Journal Article · Tue Aug 20 00:00:00 EDT 2019 · Journal of High Energy Physics (Online)

DOI:https://doi.org/10.1007/JHEP08(2019)099· OSTI ID:1566870

Bao, Ning ^[1];

^[2]; Shehzad, Ibrahim ^[2]

Berkeley Center for Theoretical Physics, Berkeley, CA (United States); Brookhaven National Lab. (BNL), Upton, NY (United States)
Cornell Univ., Ithaca, NY (United States)

The relative entropy is a measure of the distinguishability of two quantum states. A great deal of progress has been made in the study of the relative entropy between an excited state and the vacuum state of a conformal field theory (CFT) reduced to a spherical region. For example, when the excited state is a small perturbation of the vacuum state, the relative entropy is known to have a universal expression for all CFT’s. Specifically, the perturbative relative entropy can be written as the symplectic flux of a certain scalar field in an auxiliary AdS-Rindler spacetime. Moreover, if the CFT has a semi-classical holographic dual, the relative entropy is known to be related to conserved charges in the bulk dual spacetime. In this paper, we introduce a one-parameter generalization of the relative entropy which we call refined Rényi relative entropy. We study this quantity in CFT’s and find a one-parameter generalization of the aforementioned known results about the relative entropy. We also discuss a new family of positive energy theorems in asymptotically locally AdS spacetimes that arises from the holographic dual of the refined Rényi relative entropy.

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Research Organization:: Brookhaven National Laboratory (BNL), Upton, NY (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (SC-21)

Grant/Contract Number:: SC0012704

OSTI ID:: 1566870

Report Number(s):: BNL-212134-2019-JAAM; TRN: US2001007

Journal Information:: Journal of High Energy Physics (Online), Vol. 2019, Issue 8; ISSN 1029-8479

Publisher:: Springer BerlinCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 6 works

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References (80)

Relative entropy and the Bekenstein bound Casini, H. Classical and Quantum Gravity, Vol. 25, Issue 20 https://doi.org/10.1088/0264-9381/25/20/205021	journal	September 2008
The gravity dual of Rényi entropy Dong, Xi Nature Communications, Vol. 7, Issue 1 https://doi.org/10.1038/ncomms12472	journal	August 2016
On the modular structure of local algebras of observables Fredenhagen, Klaus Communications in Mathematical Physics, Vol. 97, Issue 1-2 https://doi.org/10.1007/BF01206179	journal	March 1985
Local symmetries and constraints Lee, Joohan; Wald, Robert M. Journal of Mathematical Physics, Vol. 31, Issue 3 https://doi.org/10.1063/1.528801	journal	March 1990
Relative entropy and the RG flow Casini, Horacio; Testé, Eduardo; Torroba, Gonzalo Journal of High Energy Physics, Vol. 2017, Issue 3 https://doi.org/10.1007/JHEP03(2017)089	journal	March 2017
Towards a derivation of holographic entanglement entropy Casini, Horacio; Huerta, Marina; Myers, Robert C. Journal of High Energy Physics, Vol. 2011, Issue 5 https://doi.org/10.1007/JHEP05(2011)036	journal	May 2011
Relative entropy and holography Blanco, David D.; Casini, Horacio; Hung, Ling-Yan Journal of High Energy Physics, Vol. 2013, Issue 8 https://doi.org/10.1007/JHEP08(2013)060	journal	August 2013
Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy Wilde, Mark M.; Winter, Andreas; Yang, Dong Communications in Mathematical Physics, Vol. 331, Issue 2 https://doi.org/10.1007/s00220-014-2122-x	journal	July 2014
Theory of Many-Particle Systems. I Martin, Paul C.; Schwinger, Julian Physical Review, Vol. 115, Issue 6 https://doi.org/10.1103/PhysRev.115.1342	journal	September 1959
Sandwiched Rényi divergence satisfies data processing inequality Beigi, Salman Journal of Mathematical Physics, Vol. 54, Issue 12 https://doi.org/10.1063/1.4838855	journal	December 2013
Relative entropy of states of von Neumann algebras Araki, Huzihiro Publications of the Research Institute for Mathematical Sciences, Vol. 11, Issue 3 https://doi.org/10.2977/prims/1195191148	journal	January 1975
Monotonicity of a relative Rényi entropy Frank, Rupert L.; Lieb, Elliott H. Journal of Mathematical Physics, Vol. 54, Issue 12 https://doi.org/10.1063/1.4838835	journal	December 2013
Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence Ryu, Shinsei; Takayanagi, Tadashi Physical Review Letters, Vol. 96, Issue 18 https://doi.org/10.1103/PhysRevLett.96.181602	journal	May 2006
Anti de Sitter space and holography Witten, Edward Advances in Theoretical and Mathematical Physics, Vol. 2, Issue 2 https://doi.org/10.4310/ATMP.1998.v2.n2.a2	journal	January 1998
Entanglement entropy for relevant and geometric perturbations Rosenhaus, Vladimir; Smolkin, Michael Journal of High Energy Physics, Vol. 2015, Issue 2 https://doi.org/10.1007/JHEP02(2015)015	journal	February 2015
Modular Hamiltonians for deformed half-spaces and the averaged null energy condition Faulkner, Thomas; Leigh, Robert G.; Parrikar, Onkar Journal of High Energy Physics, Vol. 2016, Issue 9 https://doi.org/10.1007/JHEP09(2016)038	journal	September 2016
The holographic entropy zoo May, Alex; Hijano, Eliot Journal of High Energy Physics, Vol. 2018, Issue 10 https://doi.org/10.1007/JHEP10(2018)036	journal	October 2018
Bulk locality from modular flow Faulkner, Thomas; Lewkowycz, Aitor Journal of High Energy Physics, Vol. 2017, Issue 7 https://doi.org/10.1007/JHEP07(2017)151	journal	July 2017
A covariant holographic entanglement entropy proposal Hubeny, Veronika E.; Rangamani, Mukund; Takayanagi, Tadashi Journal of High Energy Physics, Vol. 2007, Issue 07 https://doi.org/10.1088/1126-6708/2007/07/062	journal	July 2007
Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality Dong, Xi; Harlow, Daniel; Wall, Aron C. Physical Review Letters, Vol. 117, Issue 2 https://doi.org/10.1103/PhysRevLett.117.021601	journal	July 2016
Dynamics of the area law of entanglement entropy Leichenauer, Stefan; Moosa, Mudassir; Smolkin, Michael Journal of High Energy Physics, Vol. 2016, Issue 9 https://doi.org/10.1007/JHEP09(2016)035	journal	September 2016
Generalized gravitational entropy Lewkowycz, Aitor; Maldacena, Juan Journal of High Energy Physics, Vol. 2013, Issue 8 https://doi.org/10.1007/JHEP08(2013)090	journal	August 2013
Type of von Neumann Algebra Associated with Free Field Araki, Huzihiro Progress of Theoretical Physics, Vol. 32, Issue 6 https://doi.org/10.1143/PTP.32.956	journal	December 1964
Locality of Gravitational Systems from Entanglement of Conformal Field Theories Lin, Jennifer; Marcolli, Matilde; Ooguri, Hirosi Physical Review Letters, Vol. 114, Issue 22 https://doi.org/10.1103/PhysRevLett.114.221601	journal	June 2015
Black hole entropy is the Noether charge Wald, Robert M. Physical Review D, Vol. 48, Issue 8 https://doi.org/10.1103/PhysRevD.48.R3427	journal	October 1993
Comment on the Bekenstein bound Longo, Roberto; Xu, Feng Journal of Geometry and Physics, Vol. 130 https://doi.org/10.1016/j.geomphys.2018.03.004	journal	August 2018
Nonlinear constraints on gravity from entanglement Banerjee, Shamik; Kaviraj, Apratim; Sinha, Aninda Classical and Quantum Gravity, Vol. 32, Issue 6 https://doi.org/10.1088/0264-9381/32/6/065006	journal	February 2015
Relative Entropies in Conformal Field Theory Lashkari, Nima Physical Review Letters, Vol. 113, Issue 5 https://doi.org/10.1103/PhysRevLett.113.051602	journal	July 2014
Stability of Black Holes and Black Branes Hollands, Stefan; Wald, Robert M. Communications in Mathematical Physics, Vol. 321, Issue 3 https://doi.org/10.1007/s00220-012-1638-1	journal	December 2012
On the equilibrium states in quantum statistical mechanics Haag, R.; Hugenholtz, N. M.; Winnink, M. Communications in Mathematical Physics, Vol. 5, Issue 3 https://doi.org/10.1007/BF01646342	journal	June 1967
Quantum Hypothesis Testing and the Operational Interpretation of the Quantum Rényi Relative Entropies Mosonyi, Milán; Ogawa, Tomohiro Communications in Mathematical Physics, Vol. 334, Issue 3 https://doi.org/10.1007/s00220-014-2248-x	journal	December 2014
The holographic entropy cone Bao, Ning; Nezami, Sepehr; Ooguri, Hirosi Journal of High Energy Physics, Vol. 2015, Issue 9 https://doi.org/10.1007/JHEP09(2015)130	journal	September 2015
Entanglement entropy: a perturbative calculation Rosenhaus, Vladimir; Smolkin, Michael Journal of High Energy Physics, Vol. 2014, Issue 12 https://doi.org/10.1007/JHEP12(2014)179	journal	December 2014
On the Modular Structure of Local Algebras of Observables Fredenhagen, Klaus Quantum Field Theory https://doi.org/10.1007/978-3-642-70307-2_5	book	January 1985
Nonlinear gravity from entanglement in conformal field theories Faulkner, Thomas; Haehl, Felix M.; Hijano, Eliot Journal of High Energy Physics, Vol. 2017, Issue 8 https://doi.org/10.1007/JHEP08(2017)057	journal	August 2017
Gravitational positive energy theorems from information inequalities Lashkari, Nima; Lin, Jennifer; Ooguri, Hirosi Progress of Theoretical and Experimental Physics, Vol. 2016, Issue 12 https://doi.org/10.1093/ptep/ptw139	journal	December 2016
Proof of the generalized second law for rapidly evolving Rindler horizons Wall, Aron C. Physical Review D, Vol. 82, Issue 12 https://doi.org/10.1103/PhysRevD.82.124019	journal	December 2010
Proof of the generalized second law for rapidly changing fields and arbitrary horizon slices Wall, Aron C. Physical Review D, Vol. 85, Issue 10 https://doi.org/10.1103/PhysRevD.85.104049	journal	May 2012
Entropy on a null surface for interacting quantum field theories and the Bousso bound Bousso, Raphael; Casini, Horacio; Fisher, Zachary Physical Review D, Vol. 91, Issue 8 https://doi.org/10.1103/PhysRevD.91.084030	journal	April 2015
Local modular Hamiltonians from the quantum null energy condition Koeller, Jason; Leichenauer, Stefan; Levine, Adam Physical Review D, Vol. 97, Issue 6 https://doi.org/10.1103/PhysRevD.97.065011	journal	March 2018
Energy density from second shape variations of the von Neumann entropy Leichenauer, Stefan; Levine, Adam; Shahbazi-Moghaddam, Arvin Physical Review D, Vol. 98, Issue 8 https://doi.org/10.1103/PhysRevD.98.086013	journal	October 2018
Distinguishability of black hole microstates Bao, Ning; Ooguri, Hirosi Physical Review D, Vol. 96, Issue 6 https://doi.org/10.1103/PhysRevD.96.066017	journal	September 2017
Relative entropy equals bulk relative entropy Jafferis, Daniel L.; Lewkowycz, Aitor; Maldacena, Juan Journal of High Energy Physics, Vol. 2016, Issue 6 https://doi.org/10.1007/JHEP06(2016)004	journal	June 2016
Entanglement Wedge Reconstruction via Universal Recovery Channels Cotler, Jordan; Hayden, Patrick; Penington, Geoffrey Physical Review X, Vol. 9, Issue 3 https://doi.org/10.1103/PhysRevX.9.031011	journal	July 2019
Gravitational dynamics from entanglement “thermodynamics” Lashkari, Nima; McDermott, Michael B.; Van Raamsdonk, Mark Journal of High Energy Physics, Vol. 2014, Issue 4 https://doi.org/10.1007/JHEP04(2014)195	journal	April 2014
Gravitation from entanglement in holographic CFTs Faulkner, Thomas; Guica, Monica; Hartman, Thomas Journal of High Energy Physics, Vol. 2014, Issue 3 https://doi.org/10.1007/JHEP03(2014)051	journal	March 2014
Constraining gravity using entanglement in AdS/CFT Banerjee, Shamik; Bhattacharyya, Arpan; Kaviraj, Apratim Journal of High Energy Physics, Vol. 2014, Issue 5 https://doi.org/10.1007/JHEP05(2014)029	journal	May 2014
Canonical energy is quantum Fisher information Lashkari, Nima; Van Raamsdonk, Mark Journal of High Energy Physics, Vol. 2016, Issue 4 https://doi.org/10.1007/JHEP04(2016)153	journal	April 2016
Bulk emergence and the RG flow of entanglement entropy Faulkner, Thomas Journal of High Energy Physics, Vol. 2015, Issue 5 https://doi.org/10.1007/JHEP05(2015)033	journal	May 2015
Entanglement entropy, planar surfaces, and spectral functions Rosenhaus, Vladimir; Smolkin, Michael Journal of High Energy Physics, Vol. 2014, Issue 9 https://doi.org/10.1007/JHEP09(2014)119	journal	September 2014
Entanglement entropy across a deformed sphere Mezei, Márk Physical Review D, Vol. 91, Issue 4 https://doi.org/10.1103/PhysRevD.91.045038	journal	February 2015
On the shape dependence of Entanglement Entropy Carmi, Dean Journal of High Energy Physics, Vol. 2015, Issue 12 https://doi.org/10.1007/JHEP12(2015)043	journal	December 2015
Deriving covariant holographic entanglement Dong, Xi; Lewkowycz, Aitor; Rangamani, Mukund Journal of High Energy Physics, Vol. 2016, Issue 11 https://doi.org/10.1007/JHEP11(2016)028	journal	November 2016
On quantum Rényi entropies: A new generalization and some properties Müller-Lennert, Martin; Dupuis, Frédéric; Szehr, Oleg Journal of Mathematical Physics, Vol. 54, Issue 12 https://doi.org/10.1063/1.4838856	journal	December 2013
APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory Witten, Edward Reviews of Modern Physics, Vol. 90, Issue 4 https://doi.org/10.1103/RevModPhys.90.045003	journal	October 2018
Constraining quantum fields using modular theory Lashkari, Nima Journal of High Energy Physics, Vol. 2019, Issue 1 https://doi.org/10.1007/JHEP01(2019)059	journal	January 2019
Some properties of the Noether charge and a proposal for dynamical black hole entropy Iyer, Vivek; Wald, Robert M. Physical Review D, Vol. 50, Issue 2 https://doi.org/10.1103/PhysRevD.50.846	journal	July 1994
General definition of “conserved quantities” in general relativity and other theories of gravity Wald, Robert M.; Zoupas, Andreas Physical Review D, Vol. 61, Issue 8 https://doi.org/10.1103/PhysRevD.61.084027	journal	March 2000
Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems Kubo, Ryogo Journal of the Physical Society of Japan, Vol. 12, Issue 6 https://doi.org/10.1143/JPSJ.12.570	journal	June 1957
From Euclidean sources to Lorentzian spacetimes in holographic conformal field theories Marolf, Donald; Parrikar, Onkar; Rabideau, Charles Journal of High Energy Physics, Vol. 2018, Issue 6 https://doi.org/10.1007/JHEP06(2018)077	journal	June 2018
Gauge theory correlators from non-critical string theory Gubser, S. S.; Klebanov, I. R.; Polyakov, A. M. Physics Letters B, Vol. 428, Issue 1-2 https://doi.org/10.1016/S0370-2693(98)00377-3	journal	May 1998
Holographic Renyi entropy from quantum error correction Akers, Chris; Rath, Pratik Journal of High Energy Physics, Vol. 2019, Issue 5 https://doi.org/10.1007/JHEP05(2019)052	journal	May 2019
Stability of gravity with a cosmological constant Abbott, L. F.; Deser, S. Nuclear Physics B, Vol. 195, Issue 1 https://doi.org/10.1016/0550-3213(82)90049-9	journal	February 1982
The positivity of energy for asymptotically anti-de Sitter spacetimes Woolgar, E. Classical and Quantum Gravity, Vol. 11, Issue 7 https://doi.org/10.1088/0264-9381/11/7/022	journal	July 1994
Quantum focusing conjecture Bousso, Raphael; Fisher, Zachary; Leichenauer, Stefan Physical Review D, Vol. 93, Issue 6 https://doi.org/10.1103/PhysRevD.93.064044	journal	March 2016
Modular Hamiltonians on the null plane and the Markov property of the vacuum state Casini, Horacio; Testé, Eduardo; Torroba, Gonzalo Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 36 https://doi.org/10.1088/1751-8121/aa7eaa	journal	August 2017
Renyi relative entropies and renormalization group flows Casini, Horacio; Medina, Raimel; Landea, Ignacio Salazar Journal of High Energy Physics, Vol. 2018, Issue 9 https://doi.org/10.1007/JHEP09(2018)166	journal	September 2018
A Covariant Holographic Entanglement Entropy Proposal Hubeny, Veronika E.; Rangamani, Mukund; Takayanagi, Tadashi arXiv https://doi.org/10.48550/arxiv.0705.0016	text	January 2007
Relative entropy and the Bekenstein bound Casini, H. arXiv https://doi.org/10.48550/arxiv.0804.2182	text	January 2008
Towards a derivation of holographic entanglement entropy Casini, Horacio; Huerta, Marina; Myers, Robert C. arXiv https://doi.org/10.48550/arxiv.1102.0440	text	January 2011
Generalized gravitational entropy Lewkowycz, Aitor; Maldacena, Juan arXiv https://doi.org/10.48550/arxiv.1304.4926	text	January 2013
Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy Wilde, Mark M.; Winter, Andreas; Yang, Dong arXiv https://doi.org/10.48550/arxiv.1306.1586	text	January 2013
Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies Mosonyi, Milan; Ogawa, Tomohiro arXiv https://doi.org/10.48550/arxiv.1309.3228	text	January 2013
Entanglement Entropy: A Perturbative Calculation Rosenhaus, Vladimir; Smolkin, Michael arXiv https://doi.org/10.48550/arxiv.1403.3733	text	January 2014
Entanglement Entropy for Relevant and Geometric Perturbations Rosenhaus, Vladimir; Smolkin, Michael arXiv https://doi.org/10.48550/arxiv.1410.6530	text	January 2014
Dynamics of the Area Law of Entanglement Entropy Leichenauer, Stefan; Moosa, Mudassir; Smolkin, Michael arXiv https://doi.org/10.48550/arxiv.1604.00388	text	January 2016
Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition Faulkner, Thomas; Leigh, Robert G.; Parrikar, Onkar arXiv https://doi.org/10.48550/arxiv.1605.08072	text	January 2016
The holographic entropy zoo May, Alex; Hijano, Eliot arXiv https://doi.org/10.48550/arxiv.1806.06077	text	January 2018
Black Hole Entropy is Noether Charge Wald, Robert M. arXiv https://doi.org/10.48550/arxiv.gr-qc/9307038	text	January 1993
Anti De Sitter Space And Holography Witten, Edward arXiv https://doi.org/10.48550/arxiv.hep-th/9802150	text	January 1998

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