Bulk metric reconstruction from entanglement data via minimal surface area variations

TL;DR

The work addresses reconstructing asymptotically AdS bulk geometries from boundary entanglement data for ball-shaped regions. It develops explicit inversion formulas that relate variations in entanglement entropy to perturbations of the bulk metric, starting with δf(z) around AdS and extending to boundary-coordinate dependent deformations via higher-order linearization. An iterative reconstruction scheme is proposed to recover the full bulk metric from area data, with rigorous treatment in the linearized regime and concrete examples such as AdS-Schwarzschild, plus a plan for nonlinear convergence near a conformal fixed point. By connecting RT/HRT-inspired bulk reconstruction to modern inverse-problem techniques, the paper advances a mathematically grounded pathway for emergent spacetime from entanglement data. The results have potential implications for probing bulk geometry from boundary information in strongly coupled systems and for developing robust, data-driven bulk-reconstruction algorithms.

Abstract

We investigate the reconstruction of asymptotically anti-de Sitter (AdS) bulk geometries from boundary entanglement entropy data for ball-shaped entangling regions. By deriving an explicit inversion formula, we relate variations in entanglement entropy to deviations of the bulk metric about a fixed background. Applying this formula, we recover the Schwarzschild-AdS spacetime in the low-temperature regime to first order. We further extend our analysis to include deformations of the bulk geometry with nontrivial dependence on boundary directions, and propose an iterative reconstruction scheme aimed at recovering the full spacetime starting close to a conformal fixed point. We do this by building on recent advances in the mathematics of inverse problems by introducing the higher-order linearization method as a new tool in the context of holographic bulk reconstruction.

Figures (4)

• Figure 1: Minimal surfaces anchored on the same boundary disk $\partial \mathcal{A}$ in pure AdS ($\Sigma_0$) and perturbed AdS ($\Sigma_s$). This deformation could arise from dialing some external parameter in the dual field theory or in an attempt to capture say contribution from additional degrees of freedom to the entanglement entropy such as flavors Karch:2014ufaChalabi:2020tlwJokela:2024cxb. Here $z$ is the holographic coordinate so that the boundary is at $z=0$.
• Figure 2: Illustration of the boundary deformation $\delta(\partial{\cal A})$ of the entangling region $\cal A$ enclosed by the black curve $\partial{\cal A} = \partial\Sigma$. We will limit on deforming $\partial{\cal A}=S^1$, however.
• Figure 3: Our convention for the spherical coordinate system. Here $\theta=\pi/2$ corresponds to the conformal boundary.
• Figure 4: Linearized solutions $u_k$ to the minimal surface equation around the exact hemisphere solution in pure AdS.