The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole

TL;DR

The paper investigates how bulk quantum fields influence the generalized entropy and the entanglement wedge in evaporating black holes using a two-boundary JT gravity model coupled to a bath. By tracking quantum extremal surfaces through evaporation, it demonstrates a Page-time phase transition and a Hayden-Preskill-like information recovery through a quantum surface that moves away from the classical horizon, consistent with unitarity. The work reveals a rich QES structure, including a novel surface that arises from quantum effects rather than a small perturbation of a classical surface, and analyzes left-right QES gaps in the context of information recovery from the bath. These results provide a holographic realization of information scrambling and recovery in a controlled low-dimensional setting, with implications for higher-dimensional black holes and the interpretation of entanglement wedges in evaporating spacetimes.

Abstract

Bulk quantum fields are often said to contribute to the generalized entropy $\frac{A}{4G_N} +S_{\mathrm{bulk}}$ only at $O(1)$. Nonetheless, in the context of evaporating black holes, $O(1/G_N)$ gradients in $S_{\mathrm{bulk}}$ can arise due to large boosts, introducing a quantum extremal surface far from any classical extremal surface. We examine the effect of such bulk quantum effects on quantum extremal surfaces (QESs) and the resulting entanglement wedge in a simple two-boundary $2d$ bulk system defined by Jackiw-Teitelboim gravity coupled to a 1+1 CFT. Turning on a coupling between one boundary and a further external auxiliary system which functions as a heat sink allows a two-sided otherwise-eternal black hole to evaporate on one side. We find the generalized entropy of the QES to behave as expected from general considerations of unitarity, and in particular that ingoing information disappears from the entanglement wedge after a scambling time $\fracβ{2π} \ln ΔS + O(1)$ in accord with expectations for holographic implementations of the Hayden-Preskill protocol. We also find an interesting QES phase transition at what one might call the Page time for our process.

Figures (10)

• Figure 1: Our two-sided AdS$_2$ system initially has reflecting boundary conditions (solid vertical lines) on its right boundary. An independent copy $B$ of our CFT on the right half of Minkowski space which will play the role of the bath also begins with reflecting boundary conditions. At some finite time (orange dot), the right-AdS$_2$ boundary conditions become transparent, coupling the AdS$_2$ CFT to the Bath CFT.
• Figure 2: After the evaporation of the right CFT $R$ into the left $L$, the quantum extremal surface moves in a spacelike direction towards the right boundary from $X_\mathrm{old}$ to $X_\mathrm{new}$. A message sent into $R$ in the past will escape the new entanglement wedge of $R$ and enter that of $L$.
• Figure 3: Left: we prepare the state at $t=0$ (dashed line) in both AdS$_2$ and bath in the half-line vacuum, given by the Euclidean path integral in the lower half, but for subsequent time evolution must identify boundary times with the function $u=f(t)$. Right: after making a diffeomorphism in the AdS half, the state is prepared by a path integral with deformed boundary, but time evolution of the coupled system is given simply by the Hamiltonian of the CFT on the line. The lower and upper parts of the right diagram represent the Euclidean and Lorentzian pieces of the path integral in the $y$ coordinates.
• Figure 4: Penrose diagrams of the identification for the time-symmetric coupled system. Only the future-half of the diagram is relevant to the physical system in which the coupling is switched on only at $t=0$. At $t=0$ the bath is prepared in the Minkowski half-line vacuum. Dash-dotted lines denote future and past null infinity in the relevant Minkowski space. The remaining bath boundary (dotted) is identified with the right boundary of AdS$_2$ using the diffeomorphism $f$. We describe the state of the matter fields in the patch of AdS$_2$ shown, bounded by the dash-dotted lines denoting future and past Cauchy horizons. The dashed lines denote the event horizons of the right boundary. The resulting state is the half-line vacuum in the Minkowski half-space with auxiliary metric $dwd\bar{w}$, for which the Penrose diagram is shown on the right. In the physically relevant limit $w_0\to 0$, the worldline of the joined boundaries is pushed to the left in the $w$ coordinates, becoming nearly null.
• Figure 5: The von Neumann entropy of the black hole, as computed from the generalized entropy on the quantum extremal surface, at early times after coupling to the bath. In this plot, we have $T_0=\frac{1}{2\pi}$ and $\frac{T_1-T_0}{T_0}=10^{-10}$.