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 three-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 non-Abelian 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 multiboundary wormholes in AdS/CFT.
We consider Chern-Simons theory for gauge group G at level k on 3-manifolds M n with boundary consisting of n topologically linked tori. The Euclidean path integral on M n defines a quantum state on the boundary, in the n-fold tensor product of the torus Hilbert space. We focus on the case where M n is the link-complement of some n-component link inside the three-sphere S 3. The entanglement entropies of the resulting states define 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 (m|n − m) partition of a generic n-component link into sub-links. The formula involves the number of solutions to certain Diophantine equations with coefficients 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).
Abstract We study the multi-party entanglement structure of states in Chern-Simons theory created by performing the path integral on 3-manifolds with linked torus boundaries, called link complements. For gauge group SU(2), the wavefunctions of these states (in a particular basis) are the colored Jones polynomials of the corresponding links. We first review the case of U(1) Chern-Simons theory where these are stabilizer states, a fact we use to re-derive an explicit formula for the entanglement entropy across a general link bipartition. We then present the following results for SU(2) Chern-Simons theory: (i) The entanglement entropy for a bipartition of a link gives a lower bound on the genus of surfaces in the ambient S 3 separating the two sublinks. (ii) All torus links (namely, links which can be drawn on the surface of a torus) have a GHZ-like entanglement structure — i.e., partial traces leave a separable state. By contrast, through explicit computation, we test in many examples that hyperbolic links (namely, links whose complements admit hyperbolic structures) have W-like entanglement — i.e., partial traces leave a non-separable state. (iii) Finally, we consider hyperbolic links in the complexified
SL(2,C) S L 2 ℂ Chern-Simons theory, which is closely related to 3d Einstein gravity with a negative cosmological constant. In the limit of small Newton constant, we discuss how the entanglement structure is controlled by the Neumann-Zagier potential on the moduli space of hyperbolic structures on the link complement.
We study genuine multi-entropy in Abelian
U(1)k Chern-Simons theory. For key-ring link states, where only the linking numbers between one distinguished component
K and the remaining
q−1 components are nonzero, we derive an exact closed-form expression for the
q-partite Rényi multi-entropy for general
q, level
k, and Rényi index
n. For
q=4, this shows that the genuine multi-entropy
GMn(4) collapses exactly onto the tripartite information
I3,n for all
n, while for
q=5 it is likewise completely determined, for all
n, by a linear combination of tripartite and bipartite Rényi multi-entropies. We then go beyond the key-ring class and study general four-component link states with arbitrary pairwise linking numbers. A numerical scan over Chern--Simons levels
2≤k≤24 shows that the all-
n collapse found analytically for key-ring states does not survive for generic link states. Remarkably, the collapse remains exact at
n=2 for every level examined. At
n=3, violations occur, within the scanned range, only when
3∣k, with a further dependence on the
3-adic valuation of
k. At
n=4 and
n=5, violations occur for every level examined, with rates that vary strongly with
k. These results show that the breakdown is not controlled simply by the zero-divisor structure of composite
Zk, but instead exhibits a nontrivial joint dependence on the Rényi index and the arithmetic structure of the Chern--Simons level.
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
Tdm,dn torus link complements, which is a
d-party pure quantum state, and analyze its large-party limit, i.e.,
d→∞ 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∣ZG∣, where
∣ZG∣ 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
Td,dn torus link to obtain the large-party limit of the entanglement entropy. We further investigate the semiclassical (
k→∞) limit of the entropies after taking the large-party limit for this particular example.
Abstract We study the multi-boundary entanglement structure of the states prepared in (1+1) and (2+1) dimensional Chern-Simons theory with finite discrete gauge group G. The states in (1+1)-d are associated with Riemann surfaces of genus g with multiple S 1 boundaries and we use replica trick to compute the entanglement entropy for such states. In (2+1)-d, we focus on the states associated with torus link complements which live in the tensor product of Hilbert spaces associated with multiple T 2. We present a quantitative analysis of the entanglement structure for both abelian and non-abelian groups. For all the states considered in this work, we find that the entanglement entropy for direct product of groups is the sum of entropy for individual groups, i.e. EE(G 1 × G 2) = EE(G 1) + EE(G 2). Moreover, the reduced density matrix obtained by tracing out a subset of the total Hilbert space has a positive semidefinite partial transpose on any bi-partition of the remaining Hilbert space.