Entanglement Entropy and the Colored Jones Polynomial

TL;DR

This paper explores the entanglement structure of states in Chern-Simons theory linked to knot theory, revealing how entanglement relates to topological features and hyperbolic geometry of links, with implications for quantum gravity.

Contribution

It provides a detailed analysis of entanglement in Chern-Simons states, connecting topological properties of links to quantum entanglement patterns and extending to hyperbolic structures and gravity.

Findings

Torus links exhibit GHZ-like entanglement structure.

Hyperbolic links show W-like entanglement, with non-separable partial traces.

Entanglement entropy bounds relate to surface genus in link complements.

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 $S^3$ 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.