Hayden--Preskill Model via Local Quenches

TL;DR

This work develops a continuum model of the Hayden–Preskill information‑recovery protocol in two‑dimensional CFTs using local joining quenches. It contrasts a free Dirac fermion CFT, where mutual information $I(N:B')$ displays quasi‑particle revivals and piecewise linear decay, with holographic CFTs, where a bounded‑slit geometry yields a sharp transition: once the late radiation interval $|R|$ becomes comparable to or larger than the reference size $|N|$, $I(N:B')$ vanishes at late times, signaling HP recovery. The analysis combines twist‑field replica methods on the upper half‑plane (single slit) and annulus mappings (bounded slit) with holographic RT/BCFT geodesics to reveal how integrable versus chaotic scrambling affects information transfer. The results identify a geometrical HP recovery threshold and underscore fundamental differences between quasi‑particle–driven and fast‑scrambling dynamics in continuum 2d CFT realizations of black‑hole information processing.

Abstract

We model the Hayden--Preskill (HP) information recovery protocol in 2d CFTs via local joining quenches. Euclidean path integrals with slits prepare the HP subsystems: the message $M$, its reference $N$, the Page-time black hole $B$, the early radiation $E$, and the late radiation $R$; the remaining black hole after emitting $R$ is denoted as $B'$. The single-slit geometry provides an analytically tractable toy model, while the bounded-slit geometry more closely captures the HP setup. In the free Dirac fermion 2d CFT, the mutual information $I(N\!:\!B')$ shows quasi-particle dynamics with partial or full revivals, whereas that in holographic 2d CFTs, which are expected to be maximally chaotic, exhibit sharp transitions: in the bounded-slit case, when the size of the late radiation becomes comparable to that of the reference $N$, $I(N\!:\!B')$ does vanish at late time, otherwise it remains finite. This contrast between free CFTs and holographic CFTs gives a clear characterization of the HP recovery threshold.

Figures (15)

• Figure 1.0.1: Setup of the Hayden-Preskill protocol (left) and its modeling in terms of a two dimensional CFT (right). In the left picture, we initially decompose the full Hilbert space into $E$ (early radiation), $B$ (black hole), $M$ (message) and $N$ (the reference of the message), where $E$ and $M$ are maximally entangled with $B$ and $N$, respectively. After the time evolution of $B$ and $M$ described by the unitary, the new radiation $R$ is collected. The remaining subsystem is called $B'$. In the right picture, the two CFT states are realized on the upper and lower horizontal line, which are entangled with each other as a thermofield double state.
• Figure 2.1.1: Single-slit setup: two Euclidean thermal cylinders are glued at $\tau = \beta/2$, with joining point $\alpha$ and identified boundaries $\tau = 0 \sim \beta$.
• Figure 2.1.2: The single-slit quench model in the $w$ and $\xi$ coordinates, with the slit explicitly shown in black. The blue region denotes the $N$ subsystem, the orange double interval corresponds to $B'$, the gray interval corresponds to $E$ and the red single interval represents $R$. The left panel is plotted in the $w$ coordinate, while the right panel is in the $\xi$ coordinate.
• Figure 2.1.3: Bounded-slit setup: two Euclidean thermal cylinders are glued at $\tau = \beta/2$, with joining point $\alpha$ and identified boundaries $\tau = 0 \sim \beta$.
• Figure 2.1.4: The bounded-slit quench model in the $w$ ,$z$ , and $\zeta$ coordinates, with the slit explicitly shown in black. The blue region denotes the $N$ subsystem, the orange double interval corresponds to $B'$, the gray interval corresponds to $E$ and the red single interval represents $R$. The left panel is plotted in the $w$ coordinate, the middle panel is in the $z$ coordinate, while the right panel is in the $\zeta$ coordinate.