Multiboundary Wormholes and Holographic Entanglement

TL;DR

The paper investigates multipartite entanglement in AdS3/CFT2 by studying states |Σ⟩ defined via Euclidean path integrals on Riemann surfaces with n boundaries, and analyzes how bulk connectivity and boundary moduli shape entanglement. It develops a detailed framework connecting CFT data (via n-point functions and OPE coefficients) to bulk geometries, including puncture-limit simplifications where V_a become diagonal in energy. Using holographic entanglement entropy (HRT) and factorization limits, the authors map out phase structures where entanglement is predominantly bipartite or multipartite, and prove intrinsic n-partite entanglement for even n (and n−1 for odd n) in various regimes. A random-state model is shown to capture many leading-order entanglement features, offering a unifying lens on how geometry encodes quantum information in multiboundary wormholes and highlighting the nuanced distinction between correlation and entanglement in holographic contexts.

Abstract

The AdS/CFT correspondence relates quantum entanglement between boundary Conformal Field Theories and geometric connections in the dual asymptotically Anti-de Sitter space-time. We consider entangled states in the n-fold tensor product of a 1+1 dimensional CFT Hilbert space defined by the Euclidean path integral over a Riemann surface with n holes. In one region of moduli space, the dual bulk state is a black hole with n asymptotically AdS_3 regions connected by a common wormhole, while in other regions the bulk fragments into disconnected components. We study the entanglement structure and compute the wave function explicitly in the puncture limit of the Riemann surface in terms of CFT n-point functions. We also use AdS minimal surfaces to measure entanglement more generally. In some regions of the moduli space the entanglement is entirely multipartite, though not of the GHZ type. However, even when the bulk is completely connected, in some regions of the moduli space the entanglement is almost entirely bipartite: significant entanglement occurs only between pairs of CFTs. We develop new tools to analyze intrinsically n-partite entanglement, and use these to show that for some wormholes with n similar sized horizons there is intrinsic entanglement between at least n-1 parties, and that the distillable entanglement between the asymptotic regions is at least (n+1)/2 partite.

Figures (9)

• Figure 1: The $t=0$ surface $\Sigma$ in non-rotating BTZ as a quotient of the Poincaré disc. The two marked geodesics (blue in the colour version, and located symmetrically above and below the center of the figure) are identified by the action of $\gamma$. The region between them provides a fundamental domain for the quotient. $B_1$, $B_2$ become the two circular boundaries of $\mathbb{H}^2/\Gamma$. There is a minimal geodesic $H$, which coincides with the bifurcation surface of the BTZ event horizon; the length $L$ of this geodesic fully characterizes the geometry of $\Sigma$.
• Figure 2: The $t=0$ surface $\Sigma$ in the pair of pants wormhole as a quotient of the Poincaré disc. The pairs of labeled geodesics (blue and red in colour version) are identified by the action of $\Gamma$. The region of the Poincaré disc bounded by these geodesics provides a fundamental domain for the quotient. $B_1$, $B_2$ and $B_3 \cup B_3'$ become the desired three circular boundaries. There are corresponding minimal closed geodesics $H_1$, $H_2$ and $H_3 \cup H_3'$, each lying at a bifurcation surface associated with past and future event horizons for the corresponding asymptotic boundary. The lengths $L_a$ of these geodesic fully characterize the geometry of $\Sigma$.
• Figure 3: The $t=0$ surface $\Sigma$ in the wormhole with four boundaries as a quotient of the Poincaré disc. The pairs of marked geodesics (blue, green and red in colour version) are identified by the action of $\Gamma$. The region they bound has four asymptotic boundaries and is a fundamental domain for $\mathbb{H}^2/\Gamma$. It may be formed by sewing together two pairs of pants along $H_{14}$. The addition of a twist in the sewing would imply that we can no longer choose a reflection-symmetric representation of the geometry. While the first and second pair of identified geodesics are as before, the new identification introduces three new parameters; the centers $\alpha_3$, $\alpha_3'$ of the two identified geodesics are independent, although they can be taken to have the same opening angle, $\psi_3 = \psi_3'$. The identified geodesics are thus labeled by six parameters, corresponding to the moduli space of genus zero surfaces with four boundaries. In terms of geodesic lengths, we can take as independent parameters the lengths $L_a$ of the four horizons, and two additional moduli characterizing the geometry of the interior region. These are naturally chosen to be the length $L_{14}$ of the minimal geodesic $H_{14}$ in the center and the twist $\theta_{14}$ applied along this geodesic (which is related to $\alpha_3 + \alpha_3'$). There are also similar geodesics $H_{13}$, $H_{12}$ corresponding to the different ways of splitting the surface with four boundaries into two pairs of pants. But the lengths of $H_{13}$, $H_{12}$ are not independent; they are determined by the moduli above.
• Figure 4: Different decompositions of the genus two Riemann surface which preserve a $\mathbb Z_2$ reflection symmetry. In each case the boundary of the handlebody is decomposed into two pairs of pants. If we take the pair of pants $\Sigma$ to have the same geometry, the geometry of the resulting genus two surface is different in each case, but the different decompositions describe the same family of bulk handlebody geometries up to diffeomorphism. In the bulk handlebody, the cycles indicated in the left figure are contractible, while those indicated in the second are not. Slicing the bulk handlebody along the cycles in the left figure thus provides a bulk initial data surface consisting of three disconnected discs, corresponding to three copies of global AdS. The second slicing produces a bulk initial data surface for a three-boundary wormhole. The final slicing, along the cycles indicated in the right figure, yields a bulk initial data surface consisting of an annulus and a disconnected disc, corresponding to BTZ and a disconnected copy of global AdS.
• Figure 5: The limit of small $L_a$ for the surface $\Sigma$, showing the region $I$ between the minimal geodesics. Here the identifications are drawn in a different presentation to make it obvious that all the minimal geodesics go off to infinite proper distance in the limit of small $L_a$.