# The holographic dual of Rényi relative entropy

## Abstract

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.

- Authors:

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

- Publication Date:

- Research Org.:
- Brookhaven National Lab. (BNL), Upton, NY (United States)

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

- OSTI Identifier:
- 1566870

- Report Number(s):
- BNL-212134-2019-JAAM

Journal ID: ISSN 1029-8479

- Grant/Contract Number:
- SC0012704

- 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: 2019; Journal Issue: 8; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; AdS-CFT Correspondence; Black Holes; Classical Theories of Gravity; Conformal Field Theory

### Citation Formats

```
Bao, Ning, Moosa, Mudassir, and Shehzad, Ibrahim. The holographic dual of Rényi relative entropy. United States: N. p., 2019.
Web. doi:10.1007/JHEP08(2019)099.
```

```
Bao, Ning, Moosa, Mudassir, & Shehzad, Ibrahim. The holographic dual of Rényi relative entropy. United States. doi:10.1007/JHEP08(2019)099.
```

```
Bao, Ning, Moosa, Mudassir, and Shehzad, Ibrahim. Tue .
"The holographic dual of Rényi relative entropy". United States. doi:10.1007/JHEP08(2019)099. https://www.osti.gov/servlets/purl/1566870.
```

```
@article{osti_1566870,
```

title = {The holographic dual of Rényi relative entropy},

author = {Bao, Ning and Moosa, Mudassir and Shehzad, Ibrahim},

abstractNote = {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.},

doi = {10.1007/JHEP08(2019)099},

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

number = 8,

volume = 2019,

place = {United States},

year = {2019},

month = {8}

}

Works referenced in this record:

##
Relative entropy and the Bekenstein bound

journal, September 2008

- Casini, H.
- Classical and Quantum Gravity, Vol. 25, Issue 20

##
Comment on the Bekenstein bound

journal, August 2018

- Longo, Roberto; Xu, Feng
- Journal of Geometry and Physics, Vol. 130

##
Modular Hamiltonians for deformed half-spaces 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

##
Relative entropy and the RG flow

journal, March 2017

- Casini, Horacio; Testé, Eduardo; Torroba, Gonzalo
- Journal of High Energy Physics, Vol. 2017, Issue 3

##
Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality

journal, July 2016

- Dong, Xi; Harlow, Daniel; Wall, Aron C.
- Physical Review Letters, Vol. 117, Issue 2

##
Bulk locality from modular flow

journal, July 2017

- Faulkner, Thomas; Lewkowycz, Aitor
- Journal of High Energy Physics, Vol. 2017, Issue 7

##
Nonlinear constraints on gravity from entanglement

journal, February 2015

- Banerjee, Shamik; Kaviraj, Apratim; Sinha, Aninda
- Classical and Quantum Gravity, Vol. 32, Issue 6

##
Locality of Gravitational Systems from Entanglement of Conformal Field Theories

journal, June 2015

- Lin, Jennifer; Marcolli, Matilde; Ooguri, Hirosi
- Physical Review Letters, Vol. 114, Issue 22

##
Relative Entropies in Conformal Field Theory

journal, July 2014

- Lashkari, Nima
- Physical Review Letters, Vol. 113, Issue 5

##
Relative entropy and holography

journal, August 2013

- Blanco, David D.; Casini, Horacio; Hung, Ling-Yan
- Journal of High Energy Physics, Vol. 2013, Issue 8

##
Entanglement entropy: a perturbative calculation

journal, December 2014

- Rosenhaus, Vladimir; Smolkin, Michael
- Journal of High Energy Physics, Vol. 2014, Issue 12

##
Universality in the geometric dependence of Rényi entropy

journal, January 2015

- Lewkowycz, Aitor; Perlmutter, Eric
- Journal of High Energy Physics, Vol. 2015, 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

##
Dynamics of the area law of entanglement entropy

journal, September 2016

- Leichenauer, Stefan; Moosa, Mudassir; Smolkin, Michael
- Journal of High Energy Physics, Vol. 2016, Issue 9

##
The gravity dual of Rényi entropy

journal, August 2016

- Dong, Xi
- Nature Communications, Vol. 7, Issue 1

##
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

##
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

##
Generalized gravitational entropy

journal, August 2013

- Lewkowycz, Aitor; Maldacena, Juan
- Journal of High Energy Physics, Vol. 2013, Issue 8

##
Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy

journal, July 2014

- Wilde, Mark M.; Winter, Andreas; Yang, Dong
- Communications in Mathematical Physics, Vol. 331, Issue 2

##
Monotonicity of a relative Rényi entropy

journal, December 2013

- Frank, Rupert L.; Lieb, Elliott H.
- Journal of Mathematical Physics, Vol. 54, Issue 12

##
Sandwiched Rényi divergence satisfies data processing inequality

journal, December 2013

- Beigi, Salman
- Journal of Mathematical Physics, Vol. 54, Issue 12

##
Quantum Hypothesis Testing and the Operational Interpretation of the Quantum Rényi Relative Entropies

journal, December 2014

- Mosonyi, Milán; Ogawa, Tomohiro
- Communications in Mathematical Physics, Vol. 334, Issue 3

##
Type of von Neumann Algebra Associated with Free Field

journal, December 1964

- Araki, Huzihiro
- Progress of Theoretical Physics, Vol. 32, Issue 6

##
On the modular structure of local algebras of observables

journal, March 1985

- Fredenhagen, Klaus
- Communications in Mathematical Physics, Vol. 97, Issue 1-2

##
Relative entropy of states of von Neumann algebras

journal, January 1975

- Araki, Huzihiro
- Publications of the Research Institute for Mathematical Sciences, Vol. 11, Issue 3

##
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

##
The holographic entropy zoo

journal, October 2018

- May, Alex; Hijano, Eliot
- Journal of High Energy Physics, Vol. 2018, Issue 10

##
Black hole entropy is the Noether charge

journal, October 1993

- Wald, Robert M.
- Physical Review D, Vol. 48, Issue 8

##
Local symmetries and constraints

journal, March 1990

- Lee, Joohan; Wald, Robert M.
- Journal of Mathematical Physics, Vol. 31, Issue 3

##
Stability of Black Holes and Black Branes

journal, December 2012

- Hollands, Stefan; Wald, Robert M.
- Communications in Mathematical Physics, Vol. 321, Issue 3

##
Expectations and entropy inequalities for finite quantum systems

journal, June 1974

- Lindblad, Göran
- Communications in Mathematical Physics, Vol. 39, Issue 2

##
Theory of Many-Particle Systems. I

journal, September 1959

- Martin, Paul C.; Schwinger, Julian
- Physical Review, Vol. 115, Issue 6

##
On the equilibrium states in quantum statistical mechanics

journal, June 1967

- Haag, R.; Hugenholtz, N. M.; Winnink, M.
- Communications in Mathematical Physics, Vol. 5, Issue 3

##
Anti de Sitter space and holography

journal, January 1998

- Witten, Edward
- Advances in Theoretical and Mathematical Physics, Vol. 2, Issue 2

##
The holographic entropy cone

journal, September 2015

- Bao, Ning; Nezami, Sepehr; Ooguri, Hirosi
- Journal of High Energy Physics, Vol. 2015, Issue 9