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 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 Wlike 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.

Interfaces and the extended Hilbert space of Chern-Simons theory

“…While this is the generic story for local spatial subsystems, there are other possible non-local partitions of the Hilbert space that appear to be perfectly well-defined even in gauge theories (for example the multiboundary setups in[51,52]. )…”

Topological Entanglement and Knots

“…Technically, to calculate (16), one computes the quantum trace over the Wilson line with the insertion of R-matrices in all the crossings (in a 2d-knot projection). The R-matrix always acts on the product of two representations, and it can be defined either as the universal R-matrix from the quantum group or as a solution for the Yang-Baxter equation:…”

“…A totally different possibility one can consider is to act on the Wilson loop and/or manifold as a whole [15,16]. A distinct basis in such a Hilbert space can be introduced by the unknot (circle) along the non-contractible cycle in the bulk of a solid torus, colored with an integrable representation:…”

Topological string entanglement

“…Note in particular, that our considerations are different from those of[39] who consider physical gauge/gravity examples and extract topological entanglement contribution from the RT prescription 6. One can extract more interesting information by considering states involving Wilson lines along various knots and links as discussed in[42][43][44]. For instance,[45] argues for a relation between topological entanglement and the BTZ black hole entropy.…”