# Bulk connectedness and boundary entanglement

## Abstract

Here we prove, for any state in a conformal field theory defined on a set of boundary manifolds with corresponding classical holographic bulk geometry, that for any bipartition of the boundary into two non-clopen sets, the density matrix cannot be a tensor product of the reduced density matrices on each region of the bipartition. In particular, there must be entanglement across the bipartition surface. We extend this no-go theorem to general, arbitrary partitions of the boundary manifolds into non-clopen parts, proving that the density matrix cannot be a tensor product. Finally, this result gives a necessary condition for states to potentially correspond to holographic duals.

- Authors:

- California Inst. of Technology (CalTech), Pasadena, CA (United States). Walter Burke Inst. for Theoretical Physics and Inst. for Quantum Information and Matter; Univ. of California, Berkeley, CA (United States)
- California Inst. of Technology (CalTech), Pasadena, CA (United States). Walter Burke Inst. for Theoretical Physics; Univ. of California, Berkeley, CA (United States)

- Publication Date:

- Research Org.:
- California Inst. of Technology (CalTech), Pasadena, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); National Science Foundation (NSF); Gordon and Betty Moore Foundation

- OSTI Identifier:
- 1597933

- Grant/Contract Number:
- [SC0011632; 82248-13067-44-PHPXH; DGE-1144469; 776]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Europhysics Letters

- Additional Journal Information:
- [ Journal Volume: 121; Journal Issue: 6]; Journal ID: ISSN 0295-5075

- Publisher:
- IOP Publishing

- Country of Publication:
- United States

- Language:
- English

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

### Citation Formats

```
Bao, Ning, and Remmen, Grant N. Bulk connectedness and boundary entanglement. United States: N. p., 2018.
Web. doi:10.1209/0295-5075/121/60007.
```

```
Bao, Ning, & Remmen, Grant N. Bulk connectedness and boundary entanglement. United States. doi:10.1209/0295-5075/121/60007.
```

```
Bao, Ning, and Remmen, Grant N. Fri .
"Bulk connectedness and boundary entanglement". United States. doi:10.1209/0295-5075/121/60007. https://www.osti.gov/servlets/purl/1597933.
```

```
@article{osti_1597933,
```

title = {Bulk connectedness and boundary entanglement},

author = {Bao, Ning and Remmen, Grant N.},

abstractNote = {Here we prove, for any state in a conformal field theory defined on a set of boundary manifolds with corresponding classical holographic bulk geometry, that for any bipartition of the boundary into two non-clopen sets, the density matrix cannot be a tensor product of the reduced density matrices on each region of the bipartition. In particular, there must be entanglement across the bipartition surface. We extend this no-go theorem to general, arbitrary partitions of the boundary manifolds into non-clopen parts, proving that the density matrix cannot be a tensor product. Finally, this result gives a necessary condition for states to potentially correspond to holographic duals.},

doi = {10.1209/0295-5075/121/60007},

journal = {Europhysics Letters},

number = [6],

volume = [121],

place = {United States},

year = {2018},

month = {5}

}

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Cited by: 2 works

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