# Linearity of holographic entanglement entropy

## Abstract

Here, we consider the question of whether the leading contribution to the entanglement entropy in holographic CFTs is truly given by the expectation value of a linear operator as is suggested by the Ryu-Takayanagi formula. We investigate this property by computing the entanglement entropy, via the replica trick, in states dual to superpositions of macroscopically distinct geometries and find it consistent with evaluating the expectation value of the area operator within such states. However, we find that this fails once the number of semi-classical states in the superposition grows exponentially in the central charge of the CFT. Moreover, in certain such scenarios we find that the choice of surface on which to evaluate the area operator depends on the density matrix of the entire CFT. This nonlinearity is enforced in the bulk via the homology prescription of Ryu-Takayanagi. We thus conclude that the homology constraint is not a linear property in the CFT. We also discuss the existence of entropy operators in general systems with a large number of degrees of freedom.

- Authors:

- Stanford Univ., Stanford, CA (United States)
- Institute for Advanced Study, Princeton, NJ (United States)

- Publication Date:

- Research Org.:
- Stanford Univ., CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22); USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); National Science Foundation (NSF)

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

- OSTI Identifier:
- 1358372

- Alternate Identifier(s):
- OSTI ID: 1356081

- Grant/Contract Number:
- SC0009988; PHY-1316699

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Volume: 2017; Journal Issue: 2; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; level-spacing distributions; black-holes; airy kernel; quantum; system; states; AdS-CFT correspondence; black holes in string theory; gauge-gravity correspondence; conformal field theory; AdS-CFT Correspondence

### Citation Formats

```
Almheiri, Ahmed, Dong, Xi, and Swingle, Brian.
```*Linearity of holographic entanglement entropy*. United States: N. p., 2017.
Web. doi:10.1007/JHEP02(2017)074.

```
Almheiri, Ahmed, Dong, Xi, & Swingle, Brian.
```*Linearity of holographic entanglement entropy*. United States. doi:10.1007/JHEP02(2017)074.

```
Almheiri, Ahmed, Dong, Xi, and Swingle, Brian. Tue .
"Linearity of holographic entanglement entropy". United States. doi:10.1007/JHEP02(2017)074. https://www.osti.gov/servlets/purl/1358372.
```

```
@article{osti_1358372,
```

title = {Linearity of holographic entanglement entropy},

author = {Almheiri, Ahmed and Dong, Xi and Swingle, Brian},

abstractNote = {Here, we consider the question of whether the leading contribution to the entanglement entropy in holographic CFTs is truly given by the expectation value of a linear operator as is suggested by the Ryu-Takayanagi formula. We investigate this property by computing the entanglement entropy, via the replica trick, in states dual to superpositions of macroscopically distinct geometries and find it consistent with evaluating the expectation value of the area operator within such states. However, we find that this fails once the number of semi-classical states in the superposition grows exponentially in the central charge of the CFT. Moreover, in certain such scenarios we find that the choice of surface on which to evaluate the area operator depends on the density matrix of the entire CFT. This nonlinearity is enforced in the bulk via the homology prescription of Ryu-Takayanagi. We thus conclude that the homology constraint is not a linear property in the CFT. We also discuss the existence of entropy operators in general systems with a large number of degrees of freedom.},

doi = {10.1007/JHEP02(2017)074},

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

issn = {1029-8479},

number = 2,

volume = 2017,

place = {United States},

year = {2017},

month = {2}

}

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