Pulling Out the Island with Modular Flow

TL;DR

This work addresses how information hidden in the island inside a black hole interior can be retrieved using only the radiation degrees of freedom. It leverages the equivalence between boundary and bulk modular flow and the island formula, demonstrating a concrete, operator-level decoding via the microscopic modular Hamiltonian $\boldsymbol{K}$ in JT gravity setups. By solving the $(1+1)$-D free massless fermion case and extending to nonzero temperature, it presents explicit modular-flow prescriptions and a clear gravity–duality picture for pulling island content into the bath, with implications for entanglement wedge reconstruction and Hayden–Preskill-style decoding. The findings suggest that gravity's nonperturbative structure naturally encodes island information into boundary data, offering a general, causality-consistent route to unitarity in black hole evaporation scenarios and guiding potential generalizations to higher dimensions.

Abstract

Recent works have suggested that the entanglement wedge of Hawking radiation coming from an AdS black hole, will include an island inside the black hole interior after the Page time. In this paper, we propose a concrete way to extract the information from the island by acting only on the radiation degrees of freedom, building on the equivalence between the boundary and bulk modular flow. We consider examples with black holes in JT gravity coupled to baths. In the case that the bulk conformal fields contain free massless fermion field, we provide explicit bulk picture of the information extraction process, where we find that one can almost pull out an operator from the island to the bath with modular flow.

Figures (10)

• Figure 1: Left: the Penrose diagram of the system. Dynamical gravity only lives in the green region. Right: the quantum mechanical description of the system, which involves a (0+1) dimensional system coupled to a CFT on a half infinite line.
• Figure 2: The entanglement wedge of region $\boldsymbol{[a_2 ,b_2]}$ in the quantum mechanical description contains the region $[a_2 ,b_2]$ itself plus an island $[a_1 ,b_1]$ (the blue regions in the left figure).
• Figure 3: In the discussion of free chiral fermions, it is convenient to think about the intervals in terms of light-cone coordinates.
• Figure 4: An example with the region $[-1,-0.1] \cup [0.1,1]$ and $z(0) \approx 0.336$. The causal diamonds of the two intervals are shaded in blue. Under the modular flow, the two fermion operators travel along the red curve while mixing with each other. The positive $\tau$ direction is marked by the arrow. Note that the red curves are not generated by the conformal Killing vectors inside each diamond.
• Figure 5: The trajectories of the left-moving and right-moving operators during the flow in fig. \ref{['fig:flowexample']}. Note that from the definition of $u_{\pm}$ in (\ref{['defregion']}), $u_-$ is larger at the left end-point, while $u_+$ is larger at the right end-point, thus under the modular flow in fig.\ref{['fig:flowexample']}, the left-moving operator is pushed to the right, while the right-moving operator is pushed to the left. Also note that the regions $1$ and $2$ are also labeled oppositely for operators with different chiralities.