Multi-boundary entanglement in Chern-Simons theory and link invariants
Received: December
Multi-boundary entanglement in Chern-Simons theory and link invariants
Vijay Balasubramanian 0 1 2 4 5
Jackson R. Fliss 0 1 2 3 5
Robert G. Leigh 0 1 2 3 5
Onkar Parrikar 0 1 2 4 5
Open Access 0 1 2 5
c The Authors. 0 1 2 5
International Solvay Institutes,
0 1110 W. Green Street, Urbana, IL 61801 , U.S.A
1 Pleinlaan 2 , B-1050 Brussels , Belgium
2 209 S. 33rd Street, Philadelphia, PA 19104 , U.S.A
3 Department of Physics, University of Illinois
4 David Rittenhouse Laboratory, University of Pennsylvania
5 m) partition of a generic
We consider Chern-Simons theory for gauge group G at level k on 3-manifolds Mn with boundary consisting of n topologically linked tori. The Euclidean path integral on Mn de nes a quantum state on the boundary, in the n-fold tensor product of the torus Hilbert space. We focus on the case where Mn is the link-complement of some ncomponent link inside the three-sphere S3. The entanglement entropies of the resulting states de ne framing-independent link invariants which are sensitive to the topology of the chosen link. For the Abelian theory at level k (G = U(1)k) we give a general formula for the entanglement entropy associated to an arbitrary (mjn n-component link into sub-links. The formula involves the number of solutions to certain Diophantine equations with coe cients related to the Gauss linking numbers (mod k) between the two sublinks. This formula connects simple concepts in quantum information theory, knot theory, and number theory, and shows that entanglement entropy between sublinks vanishes if and only if they have zero Gauss linking (mod k). For G = SU(2)k, we study various two and three component links. We show that the 2-component Hopf link is maximally entangled, and hence analogous to a Bell pair, and that the Whitehead link, which has zero Gauss linking, nevertheless has entanglement entropy. Finally, we show that the Borromean rings have a \W-like" entanglement structure (i.e., tracing out one torus does not lead to a separable state), and give examples of other 3-component links which have \GHZ-like" entanglement (i.e., tracing out one torus does lead to a separable state).
Chern-Simons Theories; Topological Field Theories; Wilson; 't Hooft and
1 Introduction 2 3 4
The Abelian case: G = U(1)k
Two-component links
Three-component links
n-component links
Non-Abelian case: G = SU(2)k
Two-component states
Three-component states
Whitehead link
A Link invariants from monodromies of conformal blocks
B Relative entropies of links
B.3 Distinguishability of two component links
Multi-boundary states in Chern-Simons theory
Introduction
An important open question in quantum mechanics and quantum information theory is to
understand the possible patterns of entanglement that can arise naturally in
The local structure of wavefunctions is typically determined largely by the locality of
physical Hamiltonians because interactions create entanglement. However, entanglement is a
global property and very little is known about how it can be organized over long distances.
One way of thinking about this is to consider multiple disjoint regions that are su ciently
separated so that locality by itself will not prescribe the structure of entanglement. A
challenge is that there is no general prescription for even classifying the patterns of
entanglement between multiple disjoint entities. For three qubits, up to local operations,
or more precisely up to SLOCC (Stochastic Local Operations and Classical
Communication) transformations of the state, there are precisely two non-trivial classes of multipartite
property that tracing over one qubit disentangles the state, while in the W class,
reprequbits. A similar analysis of entanglement classes is not known in general for n qubits, or in
the more physical case of LOCC equivalence, let alone for disjoint regions of a eld theory.
Recently the AdS/CFT correspondence was proposed as a tool for studying
multipartitite etanglement.
The authors of [2, 3] examined the multi-boundary
threedimensional wormhole solutions of [4{10] and found non-trivial entanglement, computed
through the holographic Ryu-Takayangi formula [11], between subsets of boundary
components. One interesting result was that although there were regions of parameter space where
the entanglement between boundaries was entirely multi-partite, it was never of the GHZ
type. In special limits it was also possible to analyze the structure of the CFT wavefunction
in terms of the OPE coe cients. However, it was di cult to carry out a computation of
entanglement entropies in the eld theory at a generic point in the parameter space.
While the eld theory calculation of multi-boundary entanglement entropies is di cult
in general, one simple case where this can be done is in a topological quantum
theory [12{14] de ned on a manifold Mn, with boundary
n consisting of a union of n
disjoint components f 1; 2;
The Euclidean path integral for this theory as a
function (...truncated)