On sufficient conditions for holographic scattering

TL;DR

The paper probes how boundary entanglement structure governs bulk scattering in holography, focusing on the connected wedge theorem (CWT) and Leutheusser–Liu's generalized proposal with a converse (GCW). It proves the forward direction: if the boundary input and output regions lie in phases with connected entanglement wedges, then the bulk intersection region $\\mathcal{S}_E = \\mathcal{E}[\\hat{\\mathcal{V}}_1 \cup \\hat{\\mathcal{V}}_2] \cap \\mathcal{E}[\\hat{\\mathcal{W}}_1 \cup \\hat{\\mathcal{W}}_2]$ is non-empty, and derives a bound on mutual information via a bulk ridge $\\mathfrak{r}_E$. It then shows a weaker GCW-with-converse: a non-empty bulk region $\\tilde{\\mathcal{S}}_E$ corresponds to boundary mutual informations $I(\\hat{\\mathcal{V}}_1: \\hat{\\mathcal{V}}_2)$ and $I(\\hat{\\mathcal{W}}_1: \\hat{\\mathcal{W}}_2)$ being $O(1/G)$; however, extensive counterexamples in conical defect AdS$_3$ demonstrate that most stronger logics relating wedge connectivity to $\\mathcal{S}_E$ do not hold. The work reframes the connection between boundary correlation and bulk locality in terms of algebraic and information‑theoretic properties, suggesting new directions in boundary QI interpretations and potential extensions to more general spacetimes and scattering problems.

Abstract

Holography implies scattering in the bulk can be mediated by entanglement on the boundary. The connected wedge theorem (CWT) of May, Penington, and Sorce is a concrete example where bulk scattering implies correlation between certain boundary regions. However the converse does not hold. We investigate a recent proposal of Leutheusser and Liu for a generalization of the CWT with converse. We prove the forward direction: having pairs of CFT ``input'' (and likewise ``output'') regions in a phase with connected entanglement wedge implies that a particular bulk subregion (the intersection of ``input'' and ``output'' entanglement wedges) is non-empty. We then establish a modified version of the proposal which has a converse, and identify counter-examples to the stronger conjecture.

Figures (10)

• Figure 1: Boundary geometrical set-up for the connected wedge theorem. The rectangle represents a fixed time interval of the CFT; since this is on a cylinder, the left and right edges are identified. Input points $c_{1}, c_{2}$ are red, while output points $r_{1}, r_{2}$ are blue. The decision regions $\hat{\mathcal{V}}_{1}, \hat{\mathcal{V}}_{2}$ are shaded light red, while the decision regions $\hat{\mathcal{W}}_{1}, \hat{\mathcal{W}}_{2}$ are shaded light blue. The boundary scattering region $\hat{\mathcal{S}} \equiv \hat{\mathcal{J}}^{+}[c_{1}] \cap \hat{\mathcal{J}}^{+}[c_{2}] \cap \hat{\mathcal{J}}^{-}[r_{1}] \cap \hat{\mathcal{J}}^{-}[r_{2}]$ is empty.
• Figure 2: Lift $\lambda$ (red) and slope $\sigma$ (blue), defined in \ref{['eq:lift']} and \ref{['eq:slope']} and constituting the null membrane. We also indicate the boundary regions $\hat{\mathcal{X}}_{1}, \hat{\mathcal{X}}_{2}$, the slice $\Sigma$, and the ridge $r$.
• Figure 3: Cross-section of spacetime highlighting its decomposition into the two wedges in \ref{['eq:endpoint_wedges']} (blue) and the two wedges in \ref{['eq:nonendpoint_wedges']} (red). The darker shaded blue regions correspond to $\mathcal{E}[\hat{\mathcal{Y}}_{1}]$ and $\mathcal{E}[\hat{\mathcal{Y}}_{2}]$, while the darker shaded red region corresponds to $\mathcal{E}[\hat{\mathcal{W}}_{1} \cup \hat{\mathcal{W}}_{2}]$. We argue that the trajectory of the extended ridge $\rho_{\hat{\mathcal{X}}}^{+}$ must correspond to the solid green line rather than the dashed green line, since $\mathcal{S}_{E} = \emptyset$.
• Figure 4: Depiction of the null membrane $\tilde{\mathcal{N}}$ and scattering region $\mathcal{S}_{E}$ for the case that $\rho_{\hat{\mathcal{Y}}}^{-} \cap \mathcal{J}^{-}[\Sigma] = \emptyset$. The lift $\tilde{\lambda}$ and slope $\tilde{\sigma}$ are depicted in red and blue respectively, while the scattering region $\mathcal{S}_{E}$ is illustrated as a yellow tetrahedron, and the ridge $\mathfrak{r}_{E}$ is highlighted in green. Focusing upward along the slope and downward along the lift results in the upper bound \ref{['eq:lbmi']}.
• Figure 5: Boundary set-up for scattering problem counter-examples. The dashed blue lines represent the past of $r_1$ (for $\hat{\mathcal{W}}_1$) and $r_2$ (for $\hat{\mathcal{W}}_2$), while the dashed red lines represent the future of $c_1$ (for $\hat{\mathcal{V}}_1$) and $c_2$ (for $\hat{\mathcal{V}}_2$). We label the left and right-most points of $\hat{\mathcal{W}}_1$, $\hat{\mathcal{W}}_2$ and $\hat{\mathcal{V}}_2$, as they will be used in our analysis. The region $\hat{\mathcal{V}}_1$ will be kept fixed. Notice that we are interested in the regime in which $\hat{\mathcal{W}}_1$ has angular length strictly larger than $\pi$.

Theorems & Definitions (3)

• Theorem 1
• Proposition 3
• proof