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

Quantum focusing conjecture and the Page curve