Entanglement Entropy and the Colored Jones Polynomial

TL;DR

This work develops a quantum-information perspective on the colored Jones polynomial by interpreting link-complement path integrals in Chern-Simons theory as multi-boundary quantum states. It shows that $U(1)$ theories yield stabilizer/weighted graph states with entanglement governed by linking data, that SU(2) torus links exhibit GHZ-like entanglement while many hyperbolic links exhibit W-like entanglement diagnosed via negativity, and that extending to $SL(2,C)$ ties entanglement to hyperbolic geometry through the Neumann-Zagier potential in a controlled asymptotic expansion. The results connect topological properties such as minimal-genus separating surfaces and torus/hyperbolic link types to multipartite entanglement structures, and motivate future exploration of geometric, holographic, and tensor-network interpretations. The work thus bridges knot theory, quantum information, and quantum gravity-inspired models, suggesting new avenues for classifying links and understanding holographic dualities in a topological setting.

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.

Figures (15)

• Figure 1: (a) The typical setup for studying entanglement entropy in quantum field theory involves choosing a connected spatial slice $\Sigma$ and partitioning it into two subregions $A$ (the shaded disc) and its complement $\bar{A}$. (b) In the present paper, we are interested in considering disconnected Cauchy surfaces $\Sigma = \Sigma_1 \cup \Sigma_2 \cup \cdots$ and studying the entanglement between these various disconnected components.
• Figure 2: The spatial manifold $\Sigma_n$ for $n=3$ is the disjoint union of three tori. $M_n$ is a 3-manifold such that $\partial M_n = \Sigma_n$.
• Figure 3: The link complement (the shaded region) of a 3-component link (bold lines) inside the three-sphere. The white region indicates a tubular neighbourhood of the link which has been drilled out of the 3-sphere.
• Figure 4: We can compute the entanglement between the two sublinks $\mathcal{L}^m_A$ (blue) and $\mathcal{L}^{n-m}_{\bar{A}}$ (orange) of $\mathcal{L}^n$ by tracing out the factor corresponding to $A$ in the full state $|\mathcal{L}^n\rangle$ and computing the von Neumann entropy of the resulting reduced density matrix.
• Figure 5: A four-component link and its associated weighted graph. Each knot corresponds to one vertex in the graph. The weight of an edge (depicted here by the number of edges connecting two vertices) is the linking number between the circles corresponding to the vertices.