  The focusing theorem fails for evaporating black holes because the null
energy condition is violated by quantum effects. The quantum focusing
conjecture is proposed so that it is satisfied even if the null energy
condition is violated. The conjecture states that the derivative of the sum of
the area of a cross-section of the null geodesic congruence and the
entanglement entropy of matters outside it is non-increasing. Naively, it is
expected that the quantum focusing conjecture is violated after the Page time
as both the area of the horizon and the entanglement entropy of the Hawking
radiation are decreasing. We calculate the entanglement entropy after the Page
time by using the island rule, and find the following results: (i) the page
time is given by an approximately null surface, (ii) the entanglement entropy
is increasing along the outgoing null geodesic even after the Page time, and
(iii) the quantum focusing conjecture is not violated.

High Energy Physics - Theory;General Relativity and Quantum Cosmology