  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.

Quantum Physics;Strongly Correlated Electrons;High Energy Physics - Theory