Information Flow in Black Hole Evaporation

TL;DR

The paper studies how information about a black hole interior is encoded in Hawking radiation within the AEM$^4$Z doubly holographic framework (JT gravity coupled to a holographic BCFT bath, embedded in AdS$_3$). By analyzing quantum extremal surfaces and systematically excising segments of the radiation bath, the authors map a three-phase Page-curve evolution and reveal that interior data can be reconstructed from the bath, validating an explicit ER=EPR-like picture. They further show that the encoding is redundant and can be organized into a fractal, überholographic structure through iterative bath hole-punching, with a universal fractal dimension matching prior vacuum constructions. The work highlights that early radiation plays a crucial role for interior access, while late radiation adds redundancy, and it exposes deeper links between entanglement, replica/geodesic calculations, and information flow in evaporating black holes. These results offer a concrete, tractable realization of information recovery in black hole evaporation and suggest broad implications for holographic quantum error correction and the fractal structure of information localization in quantum gravity.

Abstract

Recently, new holographic models of black hole evaporation have given fresh insights into the information paradox [arXiv:1905.08255, arXiv:1905.08762, arXiv:1908.10996]. In these models, the black hole evaporates into an auxiliary bath space after a quantum quench, wherein the holographic theory and the bath are joined. One particularly exciting development is the appearance of "ER=EPR"-like wormholes in the (doubly) holographic model of [arXiv:1908.10996]. At late times, the entanglement wedge of the bath includes the interior of the black hole. In this paper, we employ both numerical and analytic methods to study how information about the black hole interior is encoded in the Hawking radiation. In particular, we systematically excise intervals from the bath from the system and study the corresponding Page transition. Repeating this process ad infinitum, we end up with a fractal structure on which the black hole interior is encoded, implementing the uberholography protocol of [arXiv:1612.00017].

Figures (22)

• Figure 1: In the AEM$^4$Z model, the holographic principle is invoked twice, resulting in three different pictures of the same physical system. In the top picture, there are two quantum mechanics systems (QM$_{\rm L}$ and QM$_{\rm R}$) as well as a field theory (CFT$_2$) vacuum state prepared on the half-line. The middle picture includes the 2D holographic geometry (JT gravity) dual to the entangled state of QM$_{\rm L}$ and QM$_{\rm R}$. The last picture contains the doubly-holographic description, with a bulk AdS$_3$ dual to the matter in the middle picture.
• Figure 2: A cartoon illustration of the three phases for the entanglement entropy of QM$_{\rm R}$ or QM$_{\rm L}$+bath, after the quench where QM$_{\rm R}$ is connected to the bath. The darker colors indicate the true generalized entropy, while the lighter colors indicate the general shape of each of the branches slightly beyond the regime where it provides the minimal value for the generalized entropy. Below the plot is a sketch of the shape of the extremal surfaces in AdS$_3$ which contribute to the generalized entropy in each phase.
• Figure 3: In the AEM$^4$Z model, the $\text{AdS}_2$ black hole is coupled to bath along the boundary $\sigma=0$ at time $\tau=0=t$. This results in the shock indicated by the yellow solid line. The evolution of quantum extremal surfaces is indicated by the solid blue curve. The first phase transition occurs when the QES jumps from the green point at $x^\pm = (\pi T_0)^{-1}$ to the other green point, and the second (Page) phase transition happens at the jump between the blue block. In this final phase, the QES tracks close to the new horizon.
• Figure 4: The entanglement entropy for an interval in a holographic BCFT on the upper half-plane has two branches. The dominant branch is determined by the cross ratio $\eta$ defined in eq. (\ref{['cross']}). The case illustrated here corresponds to a tensionless ETW brane in the bulk, or alternatively $\log g=0$ in the BCFT. For other choices of $\log g$, the ETW brane will be tensionful and intersect the UHP at some other angle.
• Figure 5: Motion of QES and other (non-minimal) extrema in the Quench and Scrambling Phases. The sub-figures show contour plots of generalized entropy as a function of $x_{\textrm{\tiny QES}}$ in the region bounded by the initial black hole horizon (solid black lines), a past null ray (dotted black line) emanating from the point $x_1$ on the AdS-bath boundary, and the shock (magenta lines); dark blue and bright yellow shading indicate low and high generalized entropies respectively. The blue curve marks points for which $\eta=1/2$. Three extrema of generalized entropy are shown: the bifurcation point (Q), a saddle point (S), and a maximum point (m). The QES (opaque point) in the Quench and Scrambling Phases is given respectively by Q and S. In order to make various qualitative features visible in this figure, we have chosen parameters differing from the baselines listed in table \ref{['tab:baseline']}; here, $\epsilon=\frac{1}{16}$, $c=16$, $k=\frac{1}{16}$, $T_0=\frac{2}{3\pi}$, $T_1=\frac{1}{\pi}$, $\phi_0=0$, and $\phi_r=\frac{1}{256}$.