# Entanglement entropy and the colored Jones polynomial

## Abstract

We study the multi-party entanglement structure of states in Chern-Simons theory created by performing the path integral on 3-manifolds with linked torus boundaries, called link complements. For gauge group SU(2), the wavefunctions of these states (in a particular basis) are the colored Jones polynomials of the corresponding links. We first review the case of U(1) Chern-Simons theory where these are stabilizer states, a fact we use to re-derive an explicit formula for the entanglement entropy across a general link bipartition. We then present the following results for SU(2) Chern-Simons theory: (i) The entanglement entropy for a bipartition of a link gives a lower bound on the genus of surfaces in the ambient S3 separating the two sublinks. (ii) All torus links (namely, links which can be drawn on the surface of a torus) have a GHZ-like entanglement structure — i.e., partial traces leave a separable state. By contrast, through explicit computation, we test in many examples that hyperbolic links (namely, links whose complements admit hyperbolic structures) have W-like entanglement — i.e., partial traces leave a non-separable state. (iii) Finally, we consider hyperbolic links in the complexified SL(2,C) Chern-Simons theory, which is closely related to 3d Einstein gravity with a negativemore »

- Authors:

- Univ. of Pennsylvania, Philadelphia, PA (United States). David Rittenhouse Lab.; Vrije Univ. Brussel (VUB) and International Solvay Institutes, Brussels (Belgium). Theoretische Natuurkunde
- Univ. of Pennsylvania, Philadelphia, PA (United States). David Rittenhouse Lab.
- Univ. of Illinois, Urbana, IL (United States). Dept. of Physics

- Publication Date:

- Research Org.:
- Duke Univ., Durham, NC (United States); Univ. of Illinois at Urbana-Champaign, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1507583

- Grant/Contract Number:
- FG02-05ER41367; SC0015655

- Resource Type:
- Journal Article: Accepted Manuscript

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

- Additional Journal Information:
- Journal Volume: 2018; Journal Issue: 5; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Chern-Simons Theories; Topological Field Theories; Wilson; 't Hooft and Polyakov loops; Conformal Field Theory

### Citation Formats

```
Balasubramanian, Vijay, DeCross, Matthew, Fliss, Jackson, Kar, Arjun, Leigh, Robert G., and Parrikar, Onkar.
```*Entanglement entropy and the colored Jones polynomial*. United States: N. p., 2018.
Web. doi:10.1007/jhep05(2018)038.

```
Balasubramanian, Vijay, DeCross, Matthew, Fliss, Jackson, Kar, Arjun, Leigh, Robert G., & Parrikar, Onkar.
```*Entanglement entropy and the colored Jones polynomial*. United States. doi:10.1007/jhep05(2018)038.

```
Balasubramanian, Vijay, DeCross, Matthew, Fliss, Jackson, Kar, Arjun, Leigh, Robert G., and Parrikar, Onkar. Mon .
"Entanglement entropy and the colored Jones polynomial". United States. doi:10.1007/jhep05(2018)038. https://www.osti.gov/servlets/purl/1507583.
```

```
@article{osti_1507583,
```

title = {Entanglement entropy and the colored Jones polynomial},

author = {Balasubramanian, Vijay and DeCross, Matthew and Fliss, Jackson and Kar, Arjun and Leigh, Robert G. and Parrikar, Onkar},

abstractNote = {We study the multi-party entanglement structure of states in Chern-Simons theory created by performing the path integral on 3-manifolds with linked torus boundaries, called link complements. For gauge group SU(2), the wavefunctions of these states (in a particular basis) are the colored Jones polynomials of the corresponding links. We first review the case of U(1) Chern-Simons theory where these are stabilizer states, a fact we use to re-derive an explicit formula for the entanglement entropy across a general link bipartition. We then present the following results for SU(2) Chern-Simons theory: (i) The entanglement entropy for a bipartition of a link gives a lower bound on the genus of surfaces in the ambient S3 separating the two sublinks. (ii) All torus links (namely, links which can be drawn on the surface of a torus) have a GHZ-like entanglement structure — i.e., partial traces leave a separable state. By contrast, through explicit computation, we test in many examples that hyperbolic links (namely, links whose complements admit hyperbolic structures) have W-like entanglement — i.e., partial traces leave a non-separable state. (iii) Finally, we consider hyperbolic links in the complexified SL(2,C) Chern-Simons theory, which is closely related to 3d Einstein gravity with a negative cosmological constant. In the limit of small Newton constant, we discuss how the entanglement structure is controlled by the Neumann-Zagier potential on the moduli space of hyperbolic structures on the link complement.},

doi = {10.1007/jhep05(2018)038},

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

issn = {1029-8479},

number = 5,

volume = 2018,

place = {United States},

year = {2018},

month = {5}

}