Large-party limit of topological entanglement entropy in Chern-Simons theory

TL;DR

This paper analyzes the behavior of topological entanglement entropy in Chern-Simons theory for large-party quantum states, revealing that only Abelian anyons contribute significantly in the large-party limit, with entropy bounded by the group's center.

Contribution

It provides a detailed analysis of the large-party limit of topological entanglement entropy in Chern-Simons theory, highlighting the dominance of Abelian sectors and establishing an upper bound related to the group's center.

Findings

Entanglement entropy is dominated by Abelian anyons in the large-party limit.

The entropy has an upper bound of  |Z_G|, the order of the group's center.

Explicit calculations for SU(2) and torus links illustrate the theoretical results.

Abstract

We investigate the topological entanglement entropy of quantum states arising in the context of three-dimensional Chern-Simons theory with compact gauge group $G$ and Chern-Simons level $k$. We focus on the quantum states associated with the $T_{dm,dn}$ torus link complements, which is a $d$-party pure quantum state, and analyze its large-party limit, i.e., $d\to \infty$ limit. We show that the entanglement measures in this limit will receive contributions only from the Abelian anyons, and non-Abelian sectors are suppressed in the large-party limit. Consequently, the large-party limiting value of the entanglement entropy has an upper bound of $\ln |Z_G|$, where $|Z_G|$ is the order of the center of the group $G$. As an explicit example, we perform quantitative analysis for the simplest case of the SU(2) group and $T_{d,dn}$ torus link to obtain the large-party limit of the entanglement entropy. We further investigate the semiclassical ($k \to \infty$) limit of the entropies after taking the large-party limit for this particular example.