Efficient decoding for the Hayden-Preskill protocol

TL;DR

The paper addresses the Hayden–Preskill information-recovery problem by modeling a black hole and its entangled partner with $n$ EPR pairs and a Haar-random scrambling unitary $U$. It introduces two decoding strategies: a probabilistic postselected teleportation decoder and a deterministic Grover-based decoder, both leveraging scrambling and late-time OTOTOCs to recover the diary state with high fidelity. The authors derive a fidelity bound $F \ge 1/(1+\delta)$ with $\delta = d_A d_R \Delta - 1$ and show that decoding succeeds when $d_D \gg \sqrt{d_A d_R}$, with the probabilistic method succeeding with probability $\approx 1/(d_A d_R)$ and the deterministic method requiring $\mathcal{O}(\sqrt{d_A d_R}\,\mathcal{C})$ queries to $U$. They connect the OTOC-based quantities to Rényi-2 mutual information and discuss higher-order OTOCs in the Grover-based protocol, offering a bridge between quantum information and holographic perspectives on black-hole information transfer. The results elucidate concrete, scalable decoding procedures in highly scrambling systems and highlight the role of scrambling dynamics in enabling practical state recovery from Hawking radiation.

Abstract

We present two particular decoding procedures for reconstructing a quantum state from the Hawking radiation in the Hayden-Preskill thought experiment. We work in an idealized setting and represent the black hole and its entangled partner by $n$ EPR pairs. The first procedure teleports the state thrown into the black hole to an outside observer by post-selecting on the condition that a sufficient number of EPR pairs remain undisturbed. The probability of this favorable event scales as $1/d_{A}^2$, where $d_A$ is the Hilbert space dimension for the input state. The second procedure is deterministic and combines the previous idea with Grover's search. The decoding complexity is $\mathcal{O}(d_{A}\mathcal{C})$ where $\mathcal{C}$ is the size of the quantum circuit implementing the unitary evolution operator $U$ of the black hole. As with the original (non-constructive) decoding scheme, our algorithms utilize scrambling, where the decay of out-of-time-order correlators (OTOCs) guarantees faithful state recovery.

Figures (3)

• Figure 1: The Hayden-Preskill decoding problem (a) and its variant with a reference system (b).
• Figure 2: Probabilistic decoding by postselecting on an EPR state. The actual decoder, denoted by $V$, corresponds to the shaded area.
• Figure 3: The deterministic decoder (a) and the Grover rotation (b).