On the Quantum Information Content of a Hawking Pair

TL;DR

This work identifies a macroscopic geometric transition at $\tau_c=β/8$ that governs when Hawking-pair information is extracted from a black hole in a holographic setting. By constructing a Hawking-pair state $|\Phi\rangle$ on top of a microcanonical CFT state $|\Psi\rangle$ and evaluating its von Neumann entropy difference $ΔS$, the authors show that gravitational backreaction on the horizon drives the information transfer, with a dominant negative gravitational contribution $S_{grav}$ and a subleading quantum relative-entropy $S_{rel}$ that is confined to a narrow transition region. The analysis uses three complementary methods (thermodynamics, replica trick, and JT gravity) and connects to modern holographic insights based on type II$_\infty$ von Neumann algebras and crossed products, clarifying how information is encoded in bulk geometry rather than in local QFT alone. The results yield a Page-like behavioral curve and highlight the central role of holographic algebraic structure in resolving the black hole information content transfer problem.

Abstract

We introduce a new probe designed to keep track of the quantum information content of a Hawking pair as a function of the distance from the black hole horizon. We compute the entropy content of this Hawking pair probe via a semi-classical replica method that relies on free field Wick contractions and their leading order gravitational back reaction on the black hole horizon area. We find that the information transfer from the black hole state to the Hawking pair is triggered by a geometric transition that, somewhat surprisingly, takes place at a macroscopic distance from the horizon. We relate our computation to recent insights about the role of von Neumann algebras in holography.

Figures (8)

• Figure 1: The hybrid space-time with a euclidean black hole transitioning to a lorentzian black hole. A Hawking pair is created at points on the past boundary, placed on opposite sides of the horizon. The red dot and line on the left denote an end-of-the-world brane.
• Figure 2: The red and blue graph depict the energy $E$ and $E'$ as a function of $\tau$ running from 0 to $\beta/4$, with $\beta$ kept constant. The entropy difference $\Delta S$ vanishes in the region where $E'>E$ and equals $\beta \Delta E$ in the region where $E'<E$.
• Figure 3: The entropy difference $\Delta S_{\rm grav}$ at fixed temperature rises in region $I$ where $E'>E$ and decreases to zero in the region where $E'<E$.
• Figure 4: The replica geometry for the relative entropy, here shown for $n=2$, is a branched cover of the euclidean black hole spacetime with total excess angle $2\pi n$ located at the horizon. The boundary-to-boundary propagators equal the thermal two-point function at inverse temperature $\beta$. Depending on $\tau$, either the left or right Wick contraction dominates.
• Figure 5: Graph of the gravitational contribution $\Delta S_{grav}$ (solid black) and the relative entropy $\Delta S_{\rm rel}$ (dashed brown) as a function of the euclidean time $\tau$. The red and blue lines indicate the probability $p_0$ and $p_1$ that the particle number of the exterior state is $0$ or $1$.