Page curve

The Page curve is the time profile of the entanglement (von Neumann) entropy of the radiation emitted by an evaporating black hole, as it must look if evaporation is unitary. Introduced by Don Page in 1993, it rises while the radiation is the smaller subsystem, reaches a maximum when roughly half the black hole's entropy has been radiated — the Page time — and then falls back to zero as the hole disappears, leaving the final radiation in a pure state.

Its shape follows from a simple bound: the entropy of a subsystem cannot exceed the entropy of the smaller of the two complementary pieces. Early in evaporation the radiation holds less information than the hole, so its entropy grows; late in evaporation the shrinking hole, whose capacity is its Bekenstein–Hawking entropy S ∝ M², holds less, so the radiation's entropy must track the hole's down to zero. The curve is approximately min(f, 1−f), where f is the fraction of the hole's entropy already radiated.

The Page curve is the benchmark for any proposed resolution of the information paradox. Hawking's original calculation gives a monotonically rising line, ending at maximum entropy (information lost); a unitary theory must instead reproduce the turnover. For decades no semiclassical computation could. In 2019, calculations using gravitational 'islands' and replica wormholes reproduced the falling branch directly from the path integral, deriving the Page curve rather than assuming it — widely regarded as the most significant progress on the paradox in a generation.

Don Page introduced the curve in two 1993 papers, building on his result for the average entropy of a subsystem of a random pure state. It reframed the information paradox from a verbal dispute into a quantitative prediction. The curve gained renewed prominence in 2019–2020 when Penington and, independently, Almheiri, Engelhardt, Marolf, and Maxfield derived it from the gravitational path integral using quantum extremal surfaces and replica wormholes.