Entanglement Wedge Reconstruction and the Information Paradox

TL;DR

The paper shows that, in evaporating black holes within AdS/CFT and absorbing boundary conditions, a non-empty quantum extremal surface emerges inside the horizon at the Page time, driving a Page curve consistent with unitarity. Using entanglement wedge reconstruction, it derives Hayden-Preskill decoding criteria and explains how interior information becomes encoded in early Hawking radiation, thereby resolving the firewall paradox without a firewall. It emphasizes that state dependence of interior reconstructions is essential, introduces the notion of minimal state dependence to avoid AMPSS, and extends the analysis to greybody factors and large diaries, with insights extending to toy models and potential generalizations beyond AdS/CFT. The work argues that nonperturbatively small reconstruction errors are crucial for consistency and that the bulk-to-boundary map remains linear even when interior reconstructions are state-dependent. Altogether, it provides a coherent, bulk-based account of unitary black hole evaporation, its Page curve, and interior encoding across a broad range of scenarios.

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 $β/2π\log S_{BH}$ 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.

Figures (17)

• Figure 1: With reflecting boundary conditions, outgoing modes in a surface ending on the boundary at time $t_1$ may become ingoing modes on a surface ending at time $t_2$, but the same degrees of freedom will always be contained in each surface. The bulk entropy cannot depend on the boundary time. In contrast, with absorbing boundary conditions, outgoing modes at time $t_1$ may escape the bulk in the reservoir $\mathcal{H}_\text{rad}$ by time $t_2$, and so no longer be contained in a surface ending at time $t_2$. The bulk entropy, and hence the notion of quantum extremality, depends on the boundary time.
• Figure 2: Schematic drawings of Cauchy slices through the black hole interior, both before the Page time (left) and after the Page time (right). The blue lines indicate entanglement between the interior of the black hole and the reservoir $\mathcal{H}_\text{rad}$. Before the Page time, there exist Cauchy slices where the empty surface is the surface homologous to the boundary with minimal generalised entropy. It is therefore the Ryu-Takayanagi surface $\chi$. For illustrative purposes, we draw this surface cutting the entanglement between the interior and the reservoir. After the Page time, however, no such Cauchy slice exists. Within any Cauchy slice, there will always exist a surface, near the horizon and homologous to the boundary, with smaller generalised entropy. The Ryu-Takayanagi surface must become non-empty at the Page time.
• Figure 3: The bulk entropy of the region, shown in blue, between the Ryu-Takayanagi surface and the boundary can be decomposed into the entropy of the ingoing and outgoing modes. The ingoing modes are in the infalling vacuum, and the bulk region includes ingoing modes spread over approximately the scrambling time, which diverges in the semiclassical limit. This means that the gradient, in units of infalling time, of the entropy of the infalling modes tends to zero. In contrast, as the RT surface approaches the past lightcone of the boundary, there will be a negative logarithmic divergence in the (renormalised) entropy of the outgoing modes. This divergence should stabilise the location of the quantum extremal surface a small distance away from the outgoing lightcone.
• Figure 4: The quantum Ryu-Takayanagi surface $\chi_q$, the classical maximin surface $\chi_c$, and the entanglement wedges $\mathcal{E}_\text{rad}$ and $\mathcal{E}_\text{CFT}$ of the reservoir and CFT, in Eddington-Finkelstein coordinates (left) and in a Penrose diagram (right). In the interests of simplicity, the Penrose diagram does not include the post-evaporation region, which would be in the top right. The classical maximin surfaces lies at the intersection of the past lightcone (dashed) with the apparent horizon $r_s$ (dotted), which is outside the event horizon. The quantum RT surface, in contrast, lies slightly inside the event horizon. Much of the interior is in the entanglement wedge $\mathcal{E}_\text{rad}$ of the reservoir (green), although part of the interior still lies in the entanglement wedge $\mathcal{E}_\text{CFT}$ of the CFT (blue). As the black hole continues to evaporate, the RT surface moves forward in infalling time along a spacelike trajectory, following the red arrow. On timescales that are small compared to the evaporation time, it remains a fixed radial distance inside the event horizon.
• Figure 5: Because the absorbing boundary conditions are only deterministic when evolving forwards in time, the domain of dependence of the boundary is the future boundary. The entanglement wedge $\mathcal{E}_\text{CFT}$ of the CFT (light blue) contains the causal wedge $\mathcal{C}_\text{CFT}$ (dark blue). When time is reversed using standard reflective boundary conditions, the entanglement wedge still contains the causal wedge because the backreaction on the geometry creates a white hole.