# New constraints for holographic entropy from maximin: A no-go theorem

## Abstract

The Ryu-Takayanagi (RT) formula for static spacetimes arising in the AdS/CFT correspondence satisfies inequalities that are not yet proven in the case of the Rangamani-Hubeny-Takayanagi (HRT) formula, which applies to general dynamical spacetimes. Wall’s maximin construction is the only known technique for extending inequalities of holographic entanglement entropy from the static to dynamical case. We show that this method currently has no further utility when dealing with inequalities for five or fewer regions. Despite this negative result, we propose the validity of one new inequality for covariant holographic entanglement entropy for five regions. This inequality, while not maximin provable, is much weaker than many of the inequalities satisfied by the RT formula and should therefore be easier to prove. If it is valid, then there is strong evidence that holographic entanglement entropy plays a role in general spacetimes including those that arise in cosmology. Our new inequality is obtained by the assumption that the HRT formula satisfies every known balanced inequality obeyed by the Shannon entropies of classical probability distributions. This is a property that the RT formula has been shown to possess and which has been previously conjectured to hold for quantum mechanics in general.

- Authors:

- Publication Date:

- Research Org.:
- Univ. of California, Santa Barbara, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); Foundational Questions Inst. (United States)

- OSTI Identifier:
- 1434555

- Alternate Identifier(s):
- OSTI ID: 1498657

- Grant/Contract Number:
- SC0011702; FQXi-RFP-1507

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 97 Journal Issue: 8; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; gauge-gravity dualities; quantum entanglement; quantum gravity

### Citation Formats

```
Rota, Massimiliano, and Weinberg, Sean J. New constraints for holographic entropy from maximin: A no-go theorem. United States: N. p., 2018.
Web. doi:10.1103/PhysRevD.97.086013.
```

```
Rota, Massimiliano, & Weinberg, Sean J. New constraints for holographic entropy from maximin: A no-go theorem. United States. doi:10.1103/PhysRevD.97.086013.
```

```
Rota, Massimiliano, and Weinberg, Sean J. Wed .
"New constraints for holographic entropy from maximin: A no-go theorem". United States. doi:10.1103/PhysRevD.97.086013.
```

```
@article{osti_1434555,
```

title = {New constraints for holographic entropy from maximin: A no-go theorem},

author = {Rota, Massimiliano and Weinberg, Sean J.},

abstractNote = {The Ryu-Takayanagi (RT) formula for static spacetimes arising in the AdS/CFT correspondence satisfies inequalities that are not yet proven in the case of the Rangamani-Hubeny-Takayanagi (HRT) formula, which applies to general dynamical spacetimes. Wall’s maximin construction is the only known technique for extending inequalities of holographic entanglement entropy from the static to dynamical case. We show that this method currently has no further utility when dealing with inequalities for five or fewer regions. Despite this negative result, we propose the validity of one new inequality for covariant holographic entanglement entropy for five regions. This inequality, while not maximin provable, is much weaker than many of the inequalities satisfied by the RT formula and should therefore be easier to prove. If it is valid, then there is strong evidence that holographic entanglement entropy plays a role in general spacetimes including those that arise in cosmology. Our new inequality is obtained by the assumption that the HRT formula satisfies every known balanced inequality obeyed by the Shannon entropies of classical probability distributions. This is a property that the RT formula has been shown to possess and which has been previously conjectured to hold for quantum mechanics in general.},

doi = {10.1103/PhysRevD.97.086013},

journal = {Physical Review D},

number = 8,

volume = 97,

place = {United States},

year = {2018},

month = {4}

}

DOI: 10.1103/PhysRevD.97.086013

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