Black holes as mirrors: quantum information in random subsystems (0708.4025v2)

Abstract: We study information retrieval from evaporating black holes, assuming that the internal dynamics of a black hole is unitary and rapidly mixing, and assuming that the retriever has unlimited control over the emitted Hawking radiation. If the evaporation of the black hole has already proceeded past the "half-way" point, where half of the initial entropy has been radiated away, then additional quantum information deposited in the black hole is revealed in the Hawking radiation very rapidly. Information deposited prior to the half-way point remains concealed until the half-way point, and then emerges quickly. These conclusions hold because typical local quantum circuits are efficient encoders for quantum error-correcting codes that nearly achieve the capacity of the quantum erasure channel. Our estimate of a black hole's information retention time, based on speculative dynamical assumptions, is just barely compatible with the black hole complementarity hypothesis.

• The paper demonstrates that black holes rapidly mirror quantum information through unitary dynamics once over half the entropy is evaporated.
• It models black holes as random quantum circuits that efficiently thermalize infalling data, functioning like quantum error-correcting codes.
• The study implies that Hawking radiation quickly reveals hidden quantum information, offering a fresh perspective on black hole evaporation.

Quantum Information and Black Hole Evaporation

The paper "Black Holes as Mirrors: Quantum Information in Random Subsystems" addresses the intriguing question of whether information consumed by a black hole could be recoverable through the emitted Hawking radiation. This investigation is grounded on the assumption that the internal dynamics of a black hole are unitary and rapidly mixing, allowing for a thorough inspection of the quantum information content within the emitted radiation.

Summary and Analysis

In the framework provided by Hayden, the analysis of black holes begins with the idea that they can be modeled as quantum systems that process information rather than destroy it. Leveraging concepts from quantum information theory, the paper provides a compelling model for understanding how information could be encoded in the Hawking radiation emitted by black holes. The research hinges on several speculative yet plausible dynamical assumptions.

The conclusions are driven by two complementary analyses. Initially, it's postulated that a black hole’s internal dynamics operate via a random unitary transformation that rapidly thermalizes infalling quantum information. Under this transformation, it is shown that if a black hole has already evaporated more than half of its allowed entropy, any quantum information introduced into it is quickly apparent in the radiation within a few additional qubits being emitted. This concept is intricately linked to the known achievable rates for entanglement-assisted communication in quantum systems.

The investigation then turns to the thermalization time of black holes, assessed with speculative dynamical assumptions. It emerges that the internal state of the black hole becomes well-mixed within the Schwarzschild time of approximately $r_s\log(r_s/l_p)$, where $r_s$ is the Schwarzschild radius and $l_p$ is the Planck length. This efficient mixing relies on the nature of typical local quantum circuits which operate as efficient encoders for quantum error-correcting codes.

Key Results and Theoretical Implications

From the paper, it’s clear that if one assumes black holes encode information into their radiation, they appear almost like information mirrors. This challenges traditional concepts of black holes as insatiable information sinks, proposing instead that black holes, after reaching an entropy evaporation halfway point, could rapidly reveal previously concealed information.

This revelation has significant implications:

1. Black Hole Complementarity: The paper cautiously adopts the black hole complementarity hypothesis, speculating that no violations of quantum physics principles can be observed by any one particular observer. This reconciles the potential rapid escape of information with the perspective of infalling observers who (though theoretically possible) cannot utilize this information due to extreme conditions at the event horizon.
2. Quantum Error-Correction: The use of quantum error-correcting codes as a mechanism for information storage in black holes is revolutionary in linking black hole thermodynamics with information theory. It provides a potentially testable prediction that information absorbed by a black hole past a certain point could, in theory, be recoverable retroactively.
3. Dynamics and Thermalization: The speculative assignment of rapid thermalization times provides a tempo-spatial description that aligns with the notion of black holes rapidly processing and re-emitting quantum information akin to an efficient quantum circuit.

Future Directions

The speculative nature of the dynamical assumptions prompts a call for further inquiry into the quantum mechanics governing black holes. A more profound understanding of the interplay between local dynamics and global information retention could illuminate the role of black holes within quantum gravity. Additionally, considering the approximations made regarding quantum circuits, there exists potential for modeling these dynamics in a manner that might ultimately invite empirical verification. As theoretical tools and quantum technologies progress, these concepts could one day be reconciled with observable phenomena, providing a solid bridge between quantum information theory and gravitational physics.

In conclusion, this paper presents a well-articulated analysis that challenges conventional ideas, inviting further exploration into the quantum informational characteristics of black holes. With the prospect of unifying principles of quantum mechanics and general relativity, the implications of this work could extend far into the future of theoretical physics.