Entanglement from Topology in Chern-Simons Theory

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.

7 pages, 5 figures, many donuts. Version 2: updated affiliations and funding sources, other minor changes

References in corpus (4)

• Non-Abelian Anyons and Topological Quantum Computation
• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Genons, twist defects, and projective non-Abelian braiding statistics
• Copenhagen vs Everett, Teleportation, and ER=EPR

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• Comments on a state-operator correspondence for the torus
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• Topological Link Models of Multipartite Entanglement
• A Gap Between the Hypergraph and Stabilizer Entropy Cones
• TQFT gravity and ensemble holography
• Renormalization group and approximate error correction
• Entanglement Classification from a Topological Perspective
• Entanglement on multiple $S^2$ boundaries in Chern-Simons theory
• Galois Conjugation and Multiboundary Entanglement Entropy
• Connectomes and Properties of Quantum Entanglement
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• Entanglement in Typical States of Chern-Simons Theory
• Multipartite Entanglement Structure of Fibered Link States
• Clifford operators in SU(N)1; N not odd prime
• Homological Tools for the Quantum Mechanic
• Hypergraph States in SU(N)1, N odd prime, Chern-Simons Theory
• SU(N)1 Chern-Simons theory, the Clifford group, and Entropy Cone
• Large-party limit of topological entanglement entropy in Chern-Simons theory
• Topological Entanglement Entropy and Braids in Chern-Simons Theory