# Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space

## Abstract

We consider the emergence from quantum entanglement of spacetime geometry in a bulk region. For certain classes of quantum states in an appropriately factorized Hilbert space, a spatial geometry can be defined by associating areas along codimension-one surfaces with the entanglement entropy between either side. We show how radon transforms can be used to convert these data into a spatial metric. Under a particular set of assumptions, the time evolution of such a state traces out a four-dimensional spacetime geometry, and we argue using a modified version of Jacobson’s “entanglement equilibrium” that the geometry should obey Einstein’s equation in the weak-field limit. Here, we also discuss how entanglement equilibrium is related to a generalization of the Ryu-Takayanagi formula in more general settings, and how quantum error correction can help specify the emergence map between the full quantum-gravity Hilbert space and the semiclassical limit of quantum fields propagating on a classical spacetime.

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1431154

- Alternate Identifier(s):
- OSTI ID: 1501451

- Grant/Contract Number:
- SC0011632

- 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 (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTRONOMY AND ASTROPHYSICS

### Citation Formats

```
Cao, ChunJun, and Carroll, Sean M. Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space. United States: N. p., 2018.
Web. doi:10.1103/PhysRevD.97.086003.
```

```
Cao, ChunJun, & Carroll, Sean M. Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space. United States. doi:10.1103/PhysRevD.97.086003.
```

```
Cao, ChunJun, and Carroll, Sean M. Tue .
"Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space". United States. doi:10.1103/PhysRevD.97.086003.
```

```
@article{osti_1431154,
```

title = {Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space},

author = {Cao, ChunJun and Carroll, Sean M.},

abstractNote = {We consider the emergence from quantum entanglement of spacetime geometry in a bulk region. For certain classes of quantum states in an appropriately factorized Hilbert space, a spatial geometry can be defined by associating areas along codimension-one surfaces with the entanglement entropy between either side. We show how radon transforms can be used to convert these data into a spatial metric. Under a particular set of assumptions, the time evolution of such a state traces out a four-dimensional spacetime geometry, and we argue using a modified version of Jacobson’s “entanglement equilibrium” that the geometry should obey Einstein’s equation in the weak-field limit. Here, we also discuss how entanglement equilibrium is related to a generalization of the Ryu-Takayanagi formula in more general settings, and how quantum error correction can help specify the emergence map between the full quantum-gravity Hilbert space and the semiclassical limit of quantum fields propagating on a classical spacetime.},

doi = {10.1103/PhysRevD.97.086003},

journal = {Physical Review D},

number = 8,

volume = 97,

place = {United States},

year = {2018},

month = {4}

}

DOI: 10.1103/PhysRevD.97.086003

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