Entanglement Wedge Reconstruction and the Information Paradox (1905.08255v3)

Abstract: When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT, we show that there is a phase transition in the location of the quantum Ryu-Takayanagi surface, at precisely the Page time. The new RT surface lies slightly inside the event horizon, at an infalling time approximately the scrambling time $\beta/2\pi \log S_{BH}$ into the past. We can immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-Preskill decoding criterion, using entanglement wedge reconstruction. Because part of the interior is now encoded in the early Hawking radiation, the decreasing entanglement entropy of the black hole is exactly consistent with the semiclassical bulk entanglement of the late-time Hawking modes, despite the absence of a firewall. By studying the entanglement wedge of highly mixed states, we can understand the state dependence of the interior reconstructions. A crucial role is played by the existence of tiny, non-perturbative errors in entanglement wedge reconstruction. Directly after the Page time, interior operators can only be reconstructed from the Hawking radiation if the initial state of the black hole is known. As the black hole continues to evaporate, reconstructions become possible that simultaneously work for a large class of initial states. Using similar techniques, we generalise Hayden-Preskill to show how the amount of Hawking radiation required to reconstruct a large diary, thrown into the black hole, depends on both the energy and the entropy of the diary. Finally we argue that, before the evaporation begins, a single, state-independent interior reconstruction exists for any code space of microstates with entropy strictly less than the Bekenstein-Hawking entropy, and show that this is sufficient state dependence to avoid the AMPSS typical-state firewall paradox.

• The paper establishes that the quantum RT surface undergoes a critical transition at the Page time, providing a framework to resolve black hole evaporation information loss.
• It precisely locates the quantum RT surface slightly inside the event horizon at the scrambling time, clarifying how interior information is encoded in Hawking radiation.
• The study rigorously derives the Page curve, confirming that early Hawking radiation encodes interior degrees of freedom and upholds unitarity in gravitational systems.

Entanglement Wedge Reconstruction and the Information Paradox: A Technical Overview

This paper, authored by Geoffrey Penington, thoroughly investigates the interplay between entanglement wedge reconstruction and the information paradox within the framework of AdS/CFT. The paper elucidates how the entanglement structure in holographic systems can resolve longstanding questions about the fate of information in black hole evaporation. Central to this exploration is the concept of a quantum Ryu-Takayanagi (RT) surface and its time-dependent behavior, leading to significant insights into the AdS/CFT duality and its implications for quantum gravity.

Key Observations and Results

1. Non-Empty Quantum RT Surface and the Page Time: The paper establishes that in holographic systems, the quantum RT surface undergoes a critical transition precisely at the Page time. At this point, the quantum RT surface shifts from being empty to non-empty, reflecting a transition in the entanglement structure that underpins black hole evaporation.
2. Location of the Quantum RT Surface: The surface is determined to lie slightly inside the event horizon, at an infalling time approximately the scrambling time ($\beta/2\pi \log S_{BH}$) into the past. This derives from assumptions about the slow evolution of an evaporating black hole, approximated by a Vaidya metric. This precise location marks a shift in how the interior information is encoded within the boundary theory and the Hawking radiation.
3. Derivation of the Page Curve: The paper presents a rigorous derivation of the Page curve using entanglement wedge reconstruction principles. The decreasing entropy of the black hole, despite ongoing Hawking radiation, aligns with semi-classical expectations and respects unitarity.
4. Interior Information Encoding and Reconstruction: At the Page time, some interior degrees of freedom become encoded in early Hawking radiation. Initially state-dependent, this encoding becomes increasingly generic as the black hole evaporates, eventually requiring only partial state knowledge for decoding—consistent with the criteria posited by Hayden and Preskill.
5. State Dependence in Interior Reconstruction: An integral part of Penington's argument involves addressing state dependence in entanglement reconstruction. The paper posits that while reconstructions become feasible for mixed states with enough information, minimally state-dependent reconstructions might be required in case of larger code spaces.
6. Finite Temperature Efacts and Dynamic Evaporation Considerations: Through intricate considerations, the paper extends the treatment to cases where the infalling matter has non-zero temperature or constant energy density. Crucially, the presence of such matter influences the RT surface location and entails a nuanced dynamics consistent with expected unitarity conditions.

Theoretical and Practical Implications

The implications of this paper are manifold. The findings prominently reinforce the notion that all relevant information about initial black hole configurations becomes accessible post-Page time, supporting the non-destructive nature of quantum information in gravitational systems. Additionally, the paper provides a robust theoretical framework to integrate state dependence and entanglement wedge reconstruction into descriptions of black hole physics.

On a practical front, these insights could spur further exploration into quantum error correction in holographic contexts, as the non-permanent state dependency offers a fresh perspective on how entanglement can encode gravitational phenomena. Moreover, the paper enhances our grasp of holographic duality as not only a perceptual bridge but a tangible tool for resolving deep quantum gravitational puzzles.

Concluding Remarks

Penington's work underscores the profound depth and utility of entanglement wedge reconstruction in understanding black hole information retention and retrieval. While the paper remains grounded in AdS/CFT, it hints at broader applicability, potentially providing a template for addressing information paradoxes in various quantum gravity contexts. With meticulous calculation and lucid exposition, it advances our understanding of how quantum and gravitational frameworks may eventually coalesce into a coherent narrative of black hole entropy dynamics.