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Soft mode and interior operator in Hayden-Preskill thought experiment

Beni Yoshida

TL;DR

This work addresses whether a smooth black-hole horizon can coexist with Hayden-Preskill-type recovery under energy-conserving scrambling. It introduces two toy-models based on a $U(1)$-symmetric Haar random unitary to distinguish soft (low-energy) and heavy (high-energy) modes, showing that recovery is possible via soft modes while heavy modes produce classical correlations and hinder recovery. By connecting interior operator reconstruction to the Hayden-Preskill recovery protocol, the paper demonstrates that interior partners can be written explicitly and that only a small subset of early-radiation qubits is needed to render the interior robust to perturbations. The results support a code-like, scrambling-driven picture of black hole interiors and offer a concrete mechanism to reconcile horizon smoothness with information recovery, with implications for AMPS-type puzzles and the role of soft modes as codewords.

Abstract

We study the smoothness of the black hole horizon in the Hayden-Preskill thought experiment by using two particular toy models based on variants of Haar random unitary. The first toy model corresponds to the case where the coarse-grained entropy of a black hole is larger than its entanglement entropy. We find that, while the outgoing mode and the remaining black hole are entangled, the Hayden-Preskill recovery cannot be performed. The second toy model corresponds to the case where the system consists of low energy soft modes and high energy heavy modes. We find that the Hayden-Preskill recovery protocol can be carried out via soft modes whereas heavy modes give rise to classical correlations between the outgoing mode and the remaining black hole. We also point out that the procedure of constructing the interior partners of the outgoing soft mode operators can be interpreted as the Hayden-Preskill recovery, and as such, the known recovery protocol enables us to explicitly write down the interior operators. Hence, while the infalling mode needs to be described jointly by the remaining black hole and the early radiation in our toy model, adding a few extra qubits from the early radiation is sufficient to reconstruct the interior operators.

Soft mode and interior operator in Hayden-Preskill thought experiment

TL;DR

This work addresses whether a smooth black-hole horizon can coexist with Hayden-Preskill-type recovery under energy-conserving scrambling. It introduces two toy-models based on a -symmetric Haar random unitary to distinguish soft (low-energy) and heavy (high-energy) modes, showing that recovery is possible via soft modes while heavy modes produce classical correlations and hinder recovery. By connecting interior operator reconstruction to the Hayden-Preskill recovery protocol, the paper demonstrates that interior partners can be written explicitly and that only a small subset of early-radiation qubits is needed to render the interior robust to perturbations. The results support a code-like, scrambling-driven picture of black hole interiors and offer a concrete mechanism to reconcile horizon smoothness with information recovery, with implications for AMPS-type puzzles and the role of soft modes as codewords.

Abstract

We study the smoothness of the black hole horizon in the Hayden-Preskill thought experiment by using two particular toy models based on variants of Haar random unitary. The first toy model corresponds to the case where the coarse-grained entropy of a black hole is larger than its entanglement entropy. We find that, while the outgoing mode and the remaining black hole are entangled, the Hayden-Preskill recovery cannot be performed. The second toy model corresponds to the case where the system consists of low energy soft modes and high energy heavy modes. We find that the Hayden-Preskill recovery protocol can be carried out via soft modes whereas heavy modes give rise to classical correlations between the outgoing mode and the remaining black hole. We also point out that the procedure of constructing the interior partners of the outgoing soft mode operators can be interpreted as the Hayden-Preskill recovery, and as such, the known recovery protocol enables us to explicitly write down the interior operators. Hence, while the infalling mode needs to be described jointly by the remaining black hole and the early radiation in our toy model, adding a few extra qubits from the early radiation is sufficient to reconstruct the interior operators.

Paper Structure

This paper contains 22 sections, 61 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Summary of diagrammatic techniques
  3. Hayden-Preskill with code subspaces
  4. Haar integral
  5. Recoverability
  6. Smoothness
  7. Physical interpretation
  8. Hayden-Preskill with $U(1)$ symmetry
  9. $U(1)$-symmetric system
  10. Haar integral
  11. Recoverability
  12. Smoothness
  13. Physical interpretation
  14. Recovery via soft mode
  15. Construction of interior operator
  16. ...and 7 more sections

Figures (3)

  • Figure 1: The Hilbert space structure.
  • Figure 2: $U(1)$-symmetric Haar random unitary as a toy model of energy conserving dynamics.
  • Figure 3: The AMPS puzzle and scrambling. Alice, an infalling observer, can distill an EPR pair by accessing $\bar{B}_{0}C$ while Bob, an outside observer, can distill an EPR pair by accessing $\bar{B}=\bar{B}_{0}\bar{B}_{1}$. Note that $\bar{B}_{0}$ can be any subsystem of the early radiation $\bar{B}$ as long as $|\bar{B}_{0}|\gtrapprox |D|$ due to scrambling property of $U$.