Entanglement wedge reconstruction and the information paradox

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.

Massive islands

“…Despite most of the recent activity on black holes being focused on low dimensions, it is believed that similar arguments also work in higher dimensions [3]. One of the few explicit calculations of black hole entanglement in this context has been performed by ref.…”

“…One of the key ingredients in both the Page curve calculations of [3,4] and the construction of entanglement islands in higher dimensions [17] is the use of a fully quantum version [27] of the Ryu-Takayanagi (RT) entropy formula [28]. For a gravitational theory in AdS, one can calculate entanglement entropies in the dual field theory by constructing a RT surface in the bulk whose boundary is the entangling surface in the dual CFT.…”

“…The true quantum RT surfaces minimizes the sum of area and matter entanglement terms. One key insight in the calculation of the Page curve in [3,4] is that the true quantum entangling surface jumps in the process of black hole evaporation.…”

“…Since AdS effectively acts as a box, large black holes in AdS do not evaporate. In order to get a tractable model of an evaporating black hole in AdS, the authors of [3,4] impose transparent boundary conditions on AdS, allowing radiation reaching the boundary of AdS to couple to an external heat bath.…”

“…Certainly the numeric agreement is striking. 3 We will keep on referring to the boundary of the BCFT as the defect, since the bulk spacetime of the BCFT itself is the boundary of AdS d+1 . We do not study defect CFTs in the sense that the field theory continues past the defect.…”