Hawking Radiation Correlations of Evaporating Black Holes in JT Gravity

TL;DR

This paper uses JT gravity with replica wormholes to study how Hawking radiation from an evaporating black hole encodes information. By computing the entropy of multiple bath intervals and exploring island saddles, the authors demonstrate long-range early-late correlations and establish both classical and quantum aspects of these correlations, including a lower bound on squashed entanglement. They show a delocalized purifier $R_B$ within the early radiation and discuss a breakdown of the Araki-Lieb triangle inequality when interior and exterior subsystems are treated separately, supporting the $A=R_B$ viewpoint. The work also connects to Hayden–Preskill-like information recovery, illustrating when and how diary information can be recovered from radiation and highlighting the role of islands behind horizons in observable bath physics.

Abstract

We consider the Hawking radiation emitted by an evaporating black hole in JT gravity and compute the entropy of arbitrary subsets of the radiation in the slow evaporation limit, and find a zoo of possible island saddles. The Hawking radiation is shown to have long range correlations. We compute the mutual information between early and late modes and bound from below their squashed entanglement. A small subset of late modes are shown to be correlated with modes in a suitably large subset of the radiation previously emitted as well as later modes. We show how there is a breakdown of the semi-classical approximation in the form of a violation of the Araki-Lieb triangle entropy inequality, if the interior of the black hole and the radiation are considered to be separate systems. Finally, we consider how much of the radiation must be collected, and how early, to recover information thrown into the black hole as it evaporates.

Figures (11)

• Figure 1: A Penrose diagram showing a shockwave inserted at the boundary of the AdS and bath regions with an in-going component that excites the extremal black hole, shifting the horizon out, leading to an evaporating black hole. Also shown is an interval $A$ in the bath and its island, the shaded area in the AdS region, the causal domain of the 2 QESs $p_{\hat{a}}$.
• Figure 2: The $w^+$ coordinates of the QESs determine when a null ray from the boundary is in the island of an interval in the bath. The shaded area is the island-$(21)$ of a single interval in the bath at time $t$. Note, also, the relation between the $w^-$ coordinates branch points in the bath and the $w^+$ coordinates of the QESs.
• Figure 3: A scenario to measure the mutual information of the early and late Hawking radiation emitted in time intervals $B=\langle 0,t_\text{Page}\rangle$ and $A=\langle t_\text{Page},t\rangle$ by collecting the radiation in appropriate spatial intervals on a Cauchy surface.
• Figure 4: Left: A plot of $S_{A|B}$ (dashed) and $S_{B|A}$ (solid) for the two temporal subsets $B = \langle 0, t \rangle$ and $A = \langle t , 4t_\text{Page}\rangle$. Notice that at least one of them is negative at each $t$ indicating the presence of quantum correlations. Right: for the same regions a plot of the upper and lower bounds on the squashed entanglement (assuming the UV cut off term is sufficiently small). Thus proves the existence of entanglement between $A$ and $B$.
• Figure 5: These plots show the mutual information $I_{A,B}$ (dotted) and conditional mutual information $I_{A,B|C}$ (continuous) for three intervals $A,B,C$ in the Hawking radiation collected at a fixed time $8 t_{\text{Page}}$. Left: with $A$ and $B$ fixed with $C$ moving and Right: with $A$ and $C$ fixed with $B$ moving. The shaded regions indicates where the intervals overlap. Notice that we get $I_{A,B|C}=0$ when $C=A$ or $B$, as expected from its definition. The non-negativity of $I_{A,B}$ and $I_{A,B|C}$ are checks of sub-additivity and strong sub-additivity, respectively.