# The Holographic Entropy Cone

## Abstract

We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with smooth holographic dual geometries. For 2, 3, and 4 regions, we prove that the strong subadditivity and the monogamy of mutual information give the complete set of inequalities. This is in contrast to the situation for generic quantum systems, where a complete set of entropy inequalities is not known for 4 or more regions. We also find an infinite new family of inequalities applicable to 5 or more regions. The set of all holographic entropy inequalities bounds the phase space of Ryu-Takayanagi entropies, defining the holographic entropy cone. We characterize this entropy cone by reducing geometries to minimal graph models that encode the possible cutting and gluing relations of minimal surfaces. We find that, for a fixed number of regions, there are only finitely many independent entropy inequalities. To establish new holographic entropy inequalities, we introduce a combinatorial proof technique that may also be of independent interest in Riemannian geometry and graph theory.

- Authors:

- Publication Date:

- Research Org.:
- SLAC National Accelerator Lab., Menlo Park, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC); High Energy Physics (HEP)

- OSTI Identifier:
- 1183696

- Report Number(s):
- SLAC-PUB-16294

arXiv:1505.07839

- Grant/Contract Number:
- AC02-76SF00515

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics

- Additional Journal Information:
- Journal Volume: 2015

- Country of Publication:
- United States

- Language:
- English

- Subject:
- Math and Math Physics, General Physics, Theory-HEP,HEPTH

### Citation Formats

```
Bao, Ning, Nezami, Sepehr, Ooguri, Hirosi, Stoica, Bogdan, Sully, James, and Walter, Michael.
```*The Holographic Entropy Cone*. United States: N. p., 2015.
Web. doi:10.1007/JHEP09(2015)130.

```
Bao, Ning, Nezami, Sepehr, Ooguri, Hirosi, Stoica, Bogdan, Sully, James, & Walter, Michael.
```*The Holographic Entropy Cone*. United States. doi:10.1007/JHEP09(2015)130.

```
Bao, Ning, Nezami, Sepehr, Ooguri, Hirosi, Stoica, Bogdan, Sully, James, and Walter, Michael. Mon .
"The Holographic Entropy Cone". United States. doi:10.1007/JHEP09(2015)130. https://www.osti.gov/servlets/purl/1183696.
```

```
@article{osti_1183696,
```

title = {The Holographic Entropy Cone},

author = {Bao, Ning and Nezami, Sepehr and Ooguri, Hirosi and Stoica, Bogdan and Sully, James and Walter, Michael},

abstractNote = {We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with smooth holographic dual geometries. For 2, 3, and 4 regions, we prove that the strong subadditivity and the monogamy of mutual information give the complete set of inequalities. This is in contrast to the situation for generic quantum systems, where a complete set of entropy inequalities is not known for 4 or more regions. We also find an infinite new family of inequalities applicable to 5 or more regions. The set of all holographic entropy inequalities bounds the phase space of Ryu-Takayanagi entropies, defining the holographic entropy cone. We characterize this entropy cone by reducing geometries to minimal graph models that encode the possible cutting and gluing relations of minimal surfaces. We find that, for a fixed number of regions, there are only finitely many independent entropy inequalities. To establish new holographic entropy inequalities, we introduce a combinatorial proof technique that may also be of independent interest in Riemannian geometry and graph theory.},

doi = {10.1007/JHEP09(2015)130},

journal = {Journal of High Energy Physics},

number = ,

volume = 2015,

place = {United States},

year = {2015},

month = {9}

}

*Citation information provided by*

Web of Science

Web of Science