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arXiv:1801.01131 · Journal of High Energy Physics

Entanglement entropy and the colored Jones polynomial

Vijay Balasubramanian, Matthew P. DeCross, Jackson R. Fliss, Arjun Kar, Robert G. Leigh, Onkar Parrikar

Published May 1, 2018• 45 citations
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Abstract

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 $$ \mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{C}}\right) $$ 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.

DOI
10.1007/jhep05(2018)038

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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)\mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{C}}\right)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.
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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.
A bstract We study the multipartite entanglement structure of quantum states prepared by the Euclidean path integral over three-manifolds with multiple torus boundaries (the so-called link states) in both Abelian and non-Abelian Chern-Simons theories. For three-component link states in the Abelian theory, we derive an explicit formula for the Rényi multi-entropy in terms of linking numbers. We further show that the genuine multi-entropy faithfully quantifies the tripartite entanglement generated by GHZ-states, consistent with the fact that the prepared states are stabilizer states. General upper and lower bounds on the genuine tripartite multi-entropy are conjectured for finite-dimensional quantum states.
We investigate the topological entanglement entropy of quantum states arising in the context of three-dimensional Chern-Simons theory with compact gauge group GGG and Chern-Simons level kkk. We focus on the quantum states associated with the Tdm,dnT_{dm,dn}Tdm,dn​ torus link complements, which is a ddd-party pure quantum state, and analyze its large-party limit, i.e., d→∞d\to \inftyd→∞ 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∣\ln |Z_G|ln∣ZG​∣, where ∣ZG∣|Z_G|∣ZG​∣ is the order of the center of the group GGG. As an explicit example, we perform quantitative analysis for the simplest case of the SU(2) group and Td,dnT_{d,dn}Td,dn​ torus link to obtain the large-party limit of the entanglement entropy. We further investigate the semiclassical (k→∞k \to \inftyk→∞) limit of the entropies after taking the large-party limit for this particular example.
A path integral on a link complement of a three-sphere fixes a vector (the "link state") in Chern-Simons theory. The link state can be written in a certain basis with the colored link invariants as its coefficients. We use symmetric webs to systematically compute the colored link invariants, by which we can write down the multi-partite entangled state of any given link. It is still unknown if a product state necessarily implies that the corresponding components are unlinked, and we leave it as a conjecture.
Abstract The entanglement entropy of many quantum systems is difficult to compute in general. They are obtained as a limiting case of the Rényi entropy of index m, which captures the higher moments of the reduced density matrix. In this work, we study pure bipartite states associated with S3 complements of a two-component link which is a connected sum of a knot K \mathcal{K} K K and the Hopf link. For this class of links, the Chern-Simons theory provides the necessary setting to visualise the m-moment of the reduced density matrix as a three-manifold invariant Z( MKm {M}_{{\mathcal{K}}_m} MKm​​ M K m ), which is the partition function of MKm {M}_{{\mathcal{K}}_m} MKm​​ M K m . Here MKm {M}_{{\mathcal{K}}_m} MKm​​ M K m is a closed 3-manifold associated with the knot K \mathcal{K} K K m, where K \mathcal{K} K K m is a connected sum of m-copies of K \mathcal{K} K K (i.e., K \mathcal{K} K K # K \mathcal{K} K K . . . # K \mathcal{K} K K ) which mimics the well-known replica method. We analayse the partition functions Z( MKm {M}_{{\mathcal{K}}_m} MKm​​ M K m ) for SU(2) and SO(3) gauge groups, in the limit of the large Chern-Simons coupling k. For SU(2) group, we show that Z( MKm {M}_{{\mathcal{K}}_m} MKm​​ M K m ) can grow at most polynomially in k. On the contrary, we conjecture that Z( MKm {M}_{{\mathcal{K}}_m} MKm​​ M K m ) for SO(3) group shows an exponential growth in k, where the leading term of ln Z( MKm {M}_{{\mathcal{K}}_m} MKm​​ M K m ) is the hyperbolic volume of the knot complement S3\ K \mathcal{K} K K m. We further propose that the Rényi entropies associated with SO(3) group converge to a finite value in the large k limit. We present some examples to validate our conjecture and proposal.