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

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... truction. 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. 1 This is a slight over-simplification. In reality, part of the Hawking radiation will be reflected back into the black hole, adding 'greybody factors' to the radiation that escapes. 2 The Page time is commonly called the 'halfway point' in the black hole evaporation, although, because of the thermodynamic irreversibility of the evaporation and the time dependence of the black hole temperature, it does not occur halfway through the evaporation either by time or by horizon area/entropy [7] . 3 In this formula, SBH is the Bekenstein-Hawking entropy of the black hole, and β is the black hole inverse temperature. 6 This is slightly ahistorical. The Page curve was conjectured well before AdS/CFT was known. However the conjecture was still based on the assumption that the black hole evaporation could be modelled by a Haar random unitary. 7 We do, of course, need to assume the Ryu-Takayanagi formula and entanglement wedge reconstruction, which are both fundamentally holographic ideas. The Page curve cannot be found using the semiclassical bulk description alone, because it results from the build-up of non-perturbatively small effects. See section 3.3 for more details. 12 By this we mean a boundary region B, whose boundary ∂B is empty. 13 One might worry that the classical Ryu-Takayanagi surface might not be spacelike separated from the boundary at some boundary time. However, this cannot happen, assuming the null energy condition [21] .