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Better Late than Never: Information Retrieval from Black Holes

Samuel L. Braunstein, Stefano Pirandola, Karol Życzkowski

TL;DR

The paper reframes black hole evaporation as a unitary, information-preserving process in which the black hole interior, its exterior neighborhood, and the emitted radiation form a tripartite system. Using a generalized decoupling framework and Haar-random encodings, it shows that the black hole’s thermodynamic entropy effectively equals its entanglement entropy across the horizon, and that information about in-fallen matter is encoded in tripartite correlations (a quantum one-time pad) and is retrieved only very late in the evaporation. The approach hinges on Rényi entropies and a generalized decoupling theorem to track information flow, highlighting a Planck-scale boundary where entanglement decays and correlations reappear in radiation. The results offer a resolution within unitary evolution and motivate a sum rule linking black hole entropy to entanglement across horizons, while distinguishing such behavior from non-entangled coal-like collapse.

Abstract

We show that, in order to preserve the equivalence principle until late times in unitarily evaporating black holes, the thermodynamic entropy of a black hole must be primarily entropy of entanglement across the event horizon. For such black holes, we show that the information entering a black hole becomes encoded in correlations within a tripartite quantum state, the quantum analogue of a one-time pad, and is only decoded into the outgoing radiation very late in the evaporation. This behavior generically describes the unitary evaporation of highly entangled black holes and requires no specially designed evolution. Our work suggests the existence of a matter-field sum rule for any fundamental theory.

Better Late than Never: Information Retrieval from Black Holes

TL;DR

The paper reframes black hole evaporation as a unitary, information-preserving process in which the black hole interior, its exterior neighborhood, and the emitted radiation form a tripartite system. Using a generalized decoupling framework and Haar-random encodings, it shows that the black hole’s thermodynamic entropy effectively equals its entanglement entropy across the horizon, and that information about in-fallen matter is encoded in tripartite correlations (a quantum one-time pad) and is retrieved only very late in the evaporation. The approach hinges on Rényi entropies and a generalized decoupling theorem to track information flow, highlighting a Planck-scale boundary where entanglement decays and correlations reappear in radiation. The results offer a resolution within unitary evolution and motivate a sum rule linking black hole entropy to entanglement across horizons, while distinguishing such behavior from non-entangled coal-like collapse.

Abstract

We show that, in order to preserve the equivalence principle until late times in unitarily evaporating black holes, the thermodynamic entropy of a black hole must be primarily entropy of entanglement across the event horizon. For such black holes, we show that the information entering a black hole becomes encoded in correlations within a tripartite quantum state, the quantum analogue of a one-time pad, and is only decoded into the outgoing radiation very late in the evaporation. This behavior generically describes the unitary evaporation of highly entangled black holes and requires no specially designed evolution. Our work suggests the existence of a matter-field sum rule for any fundamental theory.

Paper Structure

This paper contains 11 sections, 51 equations, 1 figure.

Table of Contents

  1. I. Structure and interpretation of decoupling theorems
  2. II. Decoupling in a classical setting
  3. III. Penalty for disentanglement across a hypothetical boundary
  4. IV. Uniform entanglement
  5. V. Formalism for General entanglement
  6. VI. Entanglement loss in the presence of in-fallen matter
  7. VII. Where's the Planck scale?
  8. VIII. Information retrieval from pure-state black holes
  9. IX. Information retrieval from an entangled black hole
  10. X. Heuristic flow via correlations
  11. XI. Black holes versus lumps of coal

Figures (1)

  • Figure 1: Correlations to the reference subsystem as a function of the number of qubits radiated ($\log_2 R$). Correlations between the reference (ref) subsystem and: (a) black hole interior, $B$; (b) radiation, $R$, and external ($\text{ext}$) neighborhood; (c) black hole interior and external neighborhood; and (d) radiation alone. Note that, as expected from Eq. (\ref{['monogamy']}), the sum of $C$'s in subplots (a) and (b) is a constant, as is that of subplots (c) and (d). In each subplot, the in-fallen matter consists of $S_{\text{matter}}= 10$ qubits and the black hole initially consists of $\log_2 BR = 100$ qubits with $\chi^{(q)}=0$. (Entropies are evaluated using base-two logarithms.)