Black hole as a multipartite entangler: multi-entropy in AdS${}_3$/CFT${}_2$

TL;DR

This work analyzes multipartite entanglement in holographic states dual to pure BTZ black holes using multi-entropy and its genuine version. It demonstrates that genuine tripartite entanglement exhibits a volume-law in high-temperature BTZ backgrounds, in sharp contrast to universal, size-independent behavior in vacuum AdS$_3$, and identifies a half-system phase where leading genuine tripartite entanglement vanishes, aligning with Haar-random state expectations. By incorporating finite radial cutoffs, the authors reveal nontrivial size dependence of genuine multi-entropy and uncover an area-law contribution in AdS$_3$, suggesting that UV conformal structure and IR backreaction shape multipartite entanglement across scales. Overall, the results indicate that black holes act as potent multipartite entanglers, imprinting distinctive signatures on multi-entropy and informing holographic tensor-network models of black-hole interiors.

Abstract

We study multipartite entanglement in typical pure states holographically dual to pure BTZ black holes, using multi-entropy and its ``genuine'' version. In the bulk, these quantities are computed by minimal geodesic networks (so-called Steiner trees). We find that at sufficiently high temperature, the genuine tripartite multi-entropy exhibits a volume-law scaling in sharp contrast to vacuum AdS$_3$, where the genuine contribution is universal and size-independent. Moreover, we find another phase: once one subsystem exceeds half of the total system, the leading genuine tripartite entanglement vanishes and reduces to that for global AdS${}_3$. This transition is indeed consistent with recent arguments for distillable EPR pairs in tripartite Haar-random states. Motivated by finite-cutoff holography, we further study the radial cutoff dependence of multi-entropy and show that genuine multi-entropy acquires nontrivial size dependence even for the tripartite case in AdS${}_3$. As a byproduct, we also observe an intriguing ``area-law'' contribution to multi-entropy that is relevant to vacuum AdS${}_3$.

Figures (11)

• Figure 1: Bulk dual of $S^{(3)}(A:B:C)$ in the case where region $A$ is disconnected and composed of $A_1$ and $A_2$. Left: When $A_1$ and $A_2$ are sufficiently large compared with the total boundary region, $\mathcal{W}$ is given by the RT surfaces associated with regions $B$ and $C$. Right: When $A_1$ and $A_2$ are sufficiently small compared with the total boundary region, $\mathcal{W}$ consists of a set of geodesic segments that allow for two branch points.
• Figure 2: Schematic picture of the computation of the holographic tripartite multi-entropy. For later convenience, we take the large-$r_+$ limit here. The orange solid lines represent the Steiner tree $\mathcal{W}$ corresponding to the contribution to $S^{(3)}$. The dashed lines denote the RT surfaces used to compute the holographic entanglement entropy of each subsystem. Depending on the subsystem sizes, there are up to three distinct phases, characterized by which region ($\mathcal{D}_{A,B,C}$), separated by the tree $\mathcal{W}$, contains the black hole. The large contribution for the genuine tripartite multi-entropy ${\rm GM}^{(3)}$, proportional to the black hole entropy $S_{BH}$, arises from the difference between the paths of the Steiner tree and the RT surfaces. For the definition of $\mathcal{W}$ and $\mathcal{D}_{A,B,C}$, refer to Section \ref{['subsec:review_hol']}.
• Figure 3: Three phases in the computation of ${\rm GM}^{(3)}$ around Eq.\ref{['eq:tri_eq']}. The left panel corresponds to $0<|A|<\frac{2\pi}{3}$. For presentation purposes, we slightly break the symmetry $|B|=|C|$ by taking $|B|<|C|$, so that the black hole lies in the domain $\mathcal{D}_{C}$. The central panel corresponds to $\frac{2\pi}{3}<|A|<\pi$, where the black hole enters $\mathcal{D}_{A}$ as $A$ becomes the largest subsystem among the three. The right panel corresponds to $\pi<|A|<2\pi$. In this regime, the RT surface computing $S(A)$ undergoes a phase transition. As a result, the computation of ${\rm GM}^{(3)}$ effectively reduces to that in global AdS${}_3$, and hence its leading contribution vanishes in the large-$r_+$ limit.
• Figure 4: A tree-level Witten-like diagram relevant for computing $S^{(3)}$ with a disconnected interval or $S^{(4)}$ with adjacent intervals. While we have possibly four phases for each channel depending on which of the four partitioned regions includes the black hole, we will focus on the symmetric setup where we do not need to care about this.
• Figure 5: Length of geometry $\mathcal{W}(r,\alpha)$ as a function of $r$ and $\alpha$. Here we set $\frac{|A|}{2} = \pi/6 \approx 0.52$, $r_b = 10^4$ and $r_+ = 10^2$. Then $r_+/\ell$ is large enough, and it can be seen that the slope changes from one constant value to another with opposite sign around $\alpha \approx 0.52$.