Bulk quantum fields are often said to contribute to the generalized entropy A 4G N`S bulk only at Op1q. Nonetheless, in the context of evaporating black holes, Op1{G N q gradients in S bulk can arise due to large boosts, introducing a quantum extremal surface far from any classical extremal surface. We examine the effect of such bulk quantum effects on quantum extremal surfaces (QESs) and the resulting entanglement wedge in a simple two-boundary 2d bulk system defined by Jackiw-Teitelboim gravity coupled to a 1+1 CFT. Turning on a coupling between one boundary and a further external auxiliary system which functions as a heat sink allows a two-sided otherwise-eternal black hole to evaporate on one side. We find the generalized entropy of the QES to behave as expected from general considerations of unitarity, and in particular that ingoing information disappears from the entanglement wedge after a scambling time β 2π log ∆S`Op1q in accord with expectations for holographic implementations of the Hayden-Preskill protocol. We also find an interesting QES phase transition at what one might call the Page time for our process.

We propose that holographic entanglement entropy can be calculated at arbitrary orders in the bulk Planck constant using the concept of a "quantum extremal surface": a surface which extremizes the generalized entropy, i.e. the sum of area and bulk entanglement entropy. At leading order in bulk quantum corrections, our proposal agrees with the formula of Faulkner, Lewkowycz, and Maldacena, which was derived only at this order; beyond leading order corrections, the two conjectures diverge. Quantum extremal surfaces lie outside the causal domain of influence of the boundary region as well as its complement, and in some spacetimes there are barriers preventing them from entering certain regions. We comment on the implications for bulk reconstruction.

A future holographic screen is a hypersurface of indefinite signature, foliated by marginally trapped surfaces with area A(r). We prove that A(r) grows strictly monotonically. Future holographic screens arise in gravitational collapse. Past holographic screens exist in our own universe; they obey an analogous area law. Both exist more broadly than event horizons or dynamical horizons. Working within classical General Relativity, we assume the null curvature condition and certain generiticity conditions. We establish several nontrivial intermediate results. If a surface σ divides a Cauchy surface into two disjoint regions, then a null hypersurface N that contains σ splits the entire spacetime into two disjoint portions: the future-and-interior, K + ; and the past-and-exterior, K − . If a family of surfaces σ(r) foliate a hypersurface, while flowing everywhere to the past or exterior, then the future-and-interior K + (r) grows monotonically under inclusion. If the surfaces σ(r) are marginally trapped, we prove that the evolution must be everywhere to the past or exterior, and the area theorem follows. A thermodynamic interpretation as a Second Law is suggested by the Bousso bound, which relates A(r) to the entropy on the null slices N (r) foliating the spacetime. In a companion letter, we summarize the proof and discuss further implications.

We report a new area law in General Relativity. A future holographic screen is a hypersurface foliated by marginally trapped surfaces. We show that their area increases monotonically along the foliation. Future holographic screens can easily be found in collapsing stars and near a big crunch. Past holographic screens exist in any expanding universe and obey a similar theorem, yielding the first rigorous area law in big bang cosmology. Unlike event horizons, these objects can be identified at finite time and without reference to an asymptotic boundary. The Bousso bound is not used, but it naturally suggests a thermodynamic interpretation of our result.

When a black hole forms from collapse in a holographic theory, the information in the black hole interior remains encoded in the boundary. We prove that the area of the black hole's apparent horizon is precisely the entropy associated to coarse graining over the information in its interior, subject to knowing the exterior geometry. This is the maximum holographic entanglement entropy that is compatible with all classical measurements conducted outside of the apparent horizon. We identify the boundary dual to this entropy and explain why it obeys a Second Law of Thermodynamics.