Quantum focusing conjecture and the Page curve

Abstract: 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.