# The gravity dual of Rényi entropy

## Abstract

A remarkable yet mysterious property of black holes is that their entropy is proportional to the horizon area. This area law inspired the holographic principle, which was later realized concretely in gauge-gravity duality. In this context, entanglement entropy is given by the area of a minimal surface in a dual spacetime. However, discussions of area laws have been constrained to entanglement entropy, whereas a full understanding of a quantum state requires Re´nyi entropies. Here we show that all Rényi entropies satisfy a similar area law in holographic theories and are given by the areas of dual cosmic branes. This geometric prescription is a one-parameter generalization of the minimal surface prescription for entanglement entropy. Applying this we provide the first holographic calculation of mutual Re´nyi information between two disks of arbitrary dimension. Our results provide a framework for efficiently studying Re´nyi entropies and understanding entanglement structures in strongly coupled systems and quantum gravity.

- Authors:

- Inst. for Advanced Study, Princeton, NJ (United States). School of Natural Sciences

- Publication Date:

- Research Org.:
- Princeton Univ., NJ (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC); National Science Foundation (NSF)

- OSTI Identifier:
- 1358548

- Grant/Contract Number:
- SC0009988

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Nature Communications

- Additional Journal Information:
- Journal Volume: 7; Journal ID: ISSN 2041-1723

- Publisher:
- Nature Publishing Group

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Dong, Xi.
```*The gravity dual of Rényi entropy*. United States: N. p., 2016.
Web. doi:10.1038/ncomms12472.

```
Dong, Xi.
```*The gravity dual of Rényi entropy*. United States. doi:10.1038/ncomms12472.

```
Dong, Xi. Fri .
"The gravity dual of Rényi entropy". United States.
doi:10.1038/ncomms12472. https://www.osti.gov/servlets/purl/1358548.
```

```
@article{osti_1358548,
```

title = {The gravity dual of Rényi entropy},

author = {Dong, Xi},

abstractNote = {A remarkable yet mysterious property of black holes is that their entropy is proportional to the horizon area. This area law inspired the holographic principle, which was later realized concretely in gauge-gravity duality. In this context, entanglement entropy is given by the area of a minimal surface in a dual spacetime. However, discussions of area laws have been constrained to entanglement entropy, whereas a full understanding of a quantum state requires Re´nyi entropies. Here we show that all Rényi entropies satisfy a similar area law in holographic theories and are given by the areas of dual cosmic branes. This geometric prescription is a one-parameter generalization of the minimal surface prescription for entanglement entropy. Applying this we provide the first holographic calculation of mutual Re´nyi information between two disks of arbitrary dimension. Our results provide a framework for efficiently studying Re´nyi entropies and understanding entanglement structures in strongly coupled systems and quantum gravity.},

doi = {10.1038/ncomms12472},

journal = {Nature Communications},

number = ,

volume = 7,

place = {United States},

year = {Fri Aug 12 00:00:00 EDT 2016},

month = {Fri Aug 12 00:00:00 EDT 2016}

}

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