Entanglement from Topology in Chern-Simons Theory

TL;DR

This paper explores how topology in 3D Chern-Simons theories encodes quantum entanglement, revealing that abelian states are stabilizer states and non-abelian states are approximately universal, with implications for quantum information and holography.

Contribution

It characterizes the set of quantum states prepared by Chern-Simons path integrals, linking topology to stabilizer states and demonstrating state universality in non-abelian theories.

Findings

Abelian Chern-Simons states are stabilizer states.

Explicit entanglement entropy formula for multi-torus subsystems.

Non-abelian states can approximate any quantum state.

Abstract

The way in which geometry encodes entanglement is a topic of much recent interest in quantum many-body physics and the AdS/CFT duality. This relation is particularly pronounced in the case of topological quantum field theories, where topology alone determines the quantum states of the theory. In this work, we study the set of quantum states that can be prepared by the Euclidean path integral in three-dimensional Chern-Simons theory. Specifically, we consider arbitrary 3-manifolds with a fixed number of torus boundaries in both abelian U(1) and non-abelian SO(3) Chern-Simons theory. For the abelian theory, we find that the states that can be prepared coincide precisely with the set of stabilizer states from quantum information theory. This constrains the multipartite entanglement present in this theory, but it also reveals that stabilizer states can be described by topology. In particular, we find an explicit expression for the entanglement entropy of a many-torus subsystem using only a single replica, as well as a concrete formula for the number of GHZ states that can be distilled from a tripartite state prepared through path integration. For the nonabelian theory, we find a notion of "state universality", namely that any state can be prepared to an arbitrarily good approximation. The manifolds we consider can also be viewed as toy models of multi-boundary wormholes in AdS/CFT.