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Beyond $2$-to-$2$: Geometrization of Entanglement Wedge Connectivity in Holographic Scattering

Bowen Zhao

TL;DR

The paper addresses how multipartite boundary scattering in AdS/CFT is reflected in the connectivity of entanglement wedges. It extends the 2-to-2 Connected Wedge Theorem to general $n$-to-$n$ scattering, proving a weaker necessary condition for wedge connectivity and introducing a new independent sufficient condition, along with criteria for a nonempty entanglement wedge intersection ${\mathcal S}_E$. It also analyzes generalized bulk scattering regions ${\mathcal S}_E$ and develops a multipartite information-theoretic perspective via mutual information concepts, clarifying the holographic dictionary for multipartite entanglement. The work highlights that multipartite connectivity is governed by more intricate geometric and information-theoretic constraints than in the $2$-to-$2$ case, and points toward a full characterization based on genuinely multipartite information measures in future research.

Abstract

We extend recent discussions on generalization of the Connected Wedge Theorem about $2$-to-$2$ holographic scattering problem to $n$-to-$n$ scatterings ($n>2$). In this broader setting, our theorem provides a weaker necessary condition for the connectedness of boundary entanglement wedges than previously identified. Besides, we prove a novel sufficient condition for this connectedness. We also present a analysis of the criteria ensuring a non-empty entanglement wedge intersection region $\mathcal{S}_E$. These results refine the holographic dictionary between geometric connectivity and quantum entanglement for general multi-particle scattering.

Beyond $2$-to-$2$: Geometrization of Entanglement Wedge Connectivity in Holographic Scattering

TL;DR

The paper addresses how multipartite boundary scattering in AdS/CFT is reflected in the connectivity of entanglement wedges. It extends the 2-to-2 Connected Wedge Theorem to general -to- scattering, proving a weaker necessary condition for wedge connectivity and introducing a new independent sufficient condition, along with criteria for a nonempty entanglement wedge intersection . It also analyzes generalized bulk scattering regions and develops a multipartite information-theoretic perspective via mutual information concepts, clarifying the holographic dictionary for multipartite entanglement. The work highlights that multipartite connectivity is governed by more intricate geometric and information-theoretic constraints than in the -to- case, and points toward a full characterization based on genuinely multipartite information measures in future research.

Abstract

We extend recent discussions on generalization of the Connected Wedge Theorem about -to- holographic scattering problem to -to- scatterings (). In this broader setting, our theorem provides a weaker necessary condition for the connectedness of boundary entanglement wedges than previously identified. Besides, we prove a novel sufficient condition for this connectedness. We also present a analysis of the criteria ensuring a non-empty entanglement wedge intersection region . These results refine the holographic dictionary between geometric connectivity and quantum entanglement for general multi-particle scattering.

Paper Structure

This paper contains 14 sections, 14 theorems, 53 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notations and Assumptions
  3. Review and Preliminary
  4. Boundary setup of n-to-n scattering
  5. Causal Anchoring Principle
  6. Intersections among wedge horizons
  7. Characterization of connected entanglement wedges
  8. Generalizing the n-to-n Connected Wedge Theorem
  9. An improvement of the n-to-n connected wedge theorem
  10. Consequences of connected entanglement wedges
  11. Generalized bulk scattering regions
  12. Conclusion and Discussion
  13. An observation about different generalizations of the 2-to-2 connected wedge theorem
  14. Future Direction

Key Result

Theorem 1.1

Under standard assumptions, for a $2$-to-$2$ bulk-only scattering configuration, the entanglement wedge of $V_1 \cup V_2$ is connected.

Figures (5)

  • Figure 1: Boundary set-up of $3$-to-$3$ scattering process. Points $c_1, \cdots, c_n$ denote inputs while points $r_1,\cdots, r_n$ denote outputs. The point $\alpha_i$ is the conjugate point of $c_i$ while the point $\beta_j$ is the conjugate point of $r_j$. Inpute regions $V_i$, with spacelike boundary points $a_i$ and $b_I$, and output regions $W_j$, with spacelike boundary points $e_j$ and $f_j$, are also shown. The causal domain $\tilde{Y}_1$ is marked for later reference.
  • Figure 2: Illustration of the surface $\mathcal{Z}_{in}$ formed by future null sheets emanating from $RT(V_i)$ for $n=3$. Green curves in $\Sigma_1$ and $\Sigma_2$ lie on $\partial J^+[\mathcal{E}(V_i)]$ while red curves in $\Sigma_1$ and $\Sigma_2$ lie on $\partial J^-[\mathcal{E}(W_j)]$. The ridges $\mathcal{R}_{V_i,V_{i+1}}=\partial J^+[{\mathcal{E}(V_i)}]\cap \partial J^+[{\mathcal{E}(V_{i+1})}]$ are labeld explicitly. Note that $n+1=1$ because of the $S^1$ topology.
  • Figure 3: Illustration of focusing calculation when $n=3$. Panel (a) shows the geometric structure $\mathcal{Z}_{in}$ cut by $\partial J^-[\mathcal{E}(W_i)]$ (compare with Figure \ref{['fig:ZZin']}), where future null sheets emanating from $RT(V_i)$ are shown in green and past null sheets emanating from $RT(W_i)$ are shown in red. The curves $\mathcal{C}_i=\partial J^-[\mathcal{E}(W_i)]\cap \mathcal{Z}_{in}$ is shown as $b_i-d_i-a_{i+1}$ (counting modulo $n$). Panel $(b)$ show the $\mathcal{Z}_{in}$ in a flaten fashion where $S_{W_i}$ is the intersectio of null sheet $\partial J^-[\mathcal{E}(W_i)]$ with $\Sigma_1$, i.e. $S_{W_i}=\partial J^-[\mathcal{E}(W_i)]\cap \Sigma_1$. Arrows indicate the direction along which null expansion $\theta$ decreases. Panels (c) and (d) are similar to panels $(a)$ and $(b)$ but for partially connected scenarios.
  • Figure 4: Illustration of the geometric structure used in deriving consequences of connected entanglement wedge $\mathcal{E}(V_1\cup \cdots \cup V_n)$. Panel (a) shows the geometric structure $\mathcal{Z}_{in}$ (future horizon of $\mathcal{E}(V_1\cup \cdots \cup V_n)$) cut by $\partial J^-[\mathcal{E}(Y_i)]$, where future null sheets emanating from $RT(X_i)$ are shown in red and past null sheets emanating from $RT(Y_j)$ are shown in green. The curves $\mathcal{C}_i=\partial J^-[\mathcal{E}(Y_i)]\cap \mathcal{Z}_{in}$ is shown as $a_i-d_i-b_i$. Panel $(b)$ show the $\mathcal{Z}_{in}$ in a flaten fashion where $S_{V_i}$ is the intersectio of null sheet $\partial J^-[\mathcal{E}(Y_i)]$ with $\Sigma_1$, i.e. $S_{V_i}=\partial J^-[\mathcal{E}(Y_i)]\cap \Sigma_1$. Arrows indicate the direction along which null expansion $\theta$ decreases. Panels $(c)$ and $(d)$ are similar to panels $(a)$ and $(b)$ but for enlarged input regions.
  • Figure 5: Illustration of modified input and output regions for $n=3$. Panel (a) show that $RT(Y_2)\cup RT(Y_3)$ can be regarded as HRRT surfaces of $\mathcal{E}(W_2\cup \tilde{W}_3)$, where $\tilde{W}_3 \supseteq W_3\cup Y_1\cup W_1$. We get an effective $c_2,c_3\to r_2,\tilde{r}_3$ scattering. Panel (b) shows that $RT(X_1) \cup RT(X_2)$ can be regarded as HRRT surfaces of $\mathcal{E}(\tilde{V}_1 \cup V_2)$, where $\tilde{V}_1 \supseteq V_3 \cup X_3 \cup V_1$. We get an effective $\tilde{c}_1,c_2\to r_1, r_2$ scattering. Note that in this case output regions are enlarged although output points $r_1,r_2$ remain unchanged.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Remark 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • ...and 15 more