# Towards a bit threads derivation of holographic entanglement of purification

## Abstract

We apply the bit thread formulation of holographic entanglement entropy to reduced states describing only the geometry contained within an entanglement wedge. We argue that a certain optimized bit thread configuration, which we construct, gives a purification of the reduced state to a full holographic state obeying a precise set of conditional mutual information relations. When this purification exists, we establish, under certain assumptions, the conjectured *E _{P}* =

*E*relation equating the entanglement of purification with the area of the minimal cross section partitioning the bulk entanglement wedge. Along the way, we comment on minimal purifications of holographic states, geometric purifications, and black hole geometries.

_{W}- Authors:

- Univ. of California, Berkeley, CA (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Brookhaven National Lab. (BNL), Upton, NY (United States)
- KU Leuven, Leuven (Belgium)
- Univ. of British Columbia, Vancouver, BC (Canada)
- Univ. of California, Berkeley, CA (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Publication Date:

- Research Org.:
- Brookhaven National Lab. (BNL), Upton, NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (SC-21)

- OSTI Identifier:
- 1566868

- Report Number(s):
- BNL-212127-2019-JAAM

Journal ID: ISSN 1029-8479

- Grant/Contract Number:
- SC0012704

- Resource Type:
- Accepted Manuscript

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

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2019; Journal Issue: 7; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; AdS-CFT Correspondencel; Black Holes; Gauge-gravity correspondence

### Citation Formats

```
Bao, Ning, Chatwin-Davies, Aidan, Pollack, Jason, and Remmen, Grant N. Towards a bit threads derivation of holographic entanglement of purification. United States: N. p., 2019.
Web. doi:10.1007/JHEP07(2019)152.
```

```
Bao, Ning, Chatwin-Davies, Aidan, Pollack, Jason, & Remmen, Grant N. Towards a bit threads derivation of holographic entanglement of purification. United States. doi:10.1007/JHEP07(2019)152.
```

```
Bao, Ning, Chatwin-Davies, Aidan, Pollack, Jason, and Remmen, Grant N. Thu .
"Towards a bit threads derivation of holographic entanglement of purification". United States. doi:10.1007/JHEP07(2019)152. https://www.osti.gov/servlets/purl/1566868.
```

```
@article{osti_1566868,
```

title = {Towards a bit threads derivation of holographic entanglement of purification},

author = {Bao, Ning and Chatwin-Davies, Aidan and Pollack, Jason and Remmen, Grant N.},

abstractNote = {We apply the bit thread formulation of holographic entanglement entropy to reduced states describing only the geometry contained within an entanglement wedge. We argue that a certain optimized bit thread configuration, which we construct, gives a purification of the reduced state to a full holographic state obeying a precise set of conditional mutual information relations. When this purification exists, we establish, under certain assumptions, the conjectured EP = EW relation equating the entanglement of purification with the area of the minimal cross section partitioning the bulk entanglement wedge. Along the way, we comment on minimal purifications of holographic states, geometric purifications, and black hole geometries.},

doi = {10.1007/JHEP07(2019)152},

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

number = 7,

volume = 2019,

place = {United States},

year = {2019},

month = {7}

}

*Citation information provided by*

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