Corbeel & De Haro: replica-wormhole results reproduce the Page bound, but not yet for an ordinary Schwarzschild black hole

physicsphilosophy-of-physicsblack-holesquantum-informationpage-timeblack-hole-information-paradoxarxiv

Don Page's 1993 "Page time paradox" follows from the central dogma: if a black hole is a unitarily-evolving quantum statistical system, the von Neumann entropy of its emitted Hawking radiation must satisfy a bound imposed by the Bekenstein-Hawking entropy, the "Page bound."

"Page demonstrated that, if the central dogma is true and the black hole is a quantum statistical mechanical object, respecting unitarity results in a bound, imposed by the Bekenstein-Hawking entropy, on the number of Hawking modes entangled with the black hole degrees of freedom."

This conflicts with Stephen Hawking's semiclassical calculation, which gives an ever-increasing radiation entropy past the "Page time" (roughly half the evaporation time) — the core tension the quantum-information programme in black hole physics, including the Hayden–Preskill model, is built to address. Corbeel & De Haro report that more recent work (Almheiri et al. 2019, 2020a/b; Penington 2020; Penington et al. 2022), using "replica wormhole" saddles in the gravitational path integral, reproduces an entropy curve respecting the Page bound — but flag this as an incomplete resolution:

"These arguments rely on idealisations, such as working in low spacetime dimensions or in AdS, so there is not yet a fully definitive solution for an ordinary Schwarzschild black hole."

This is the paper's clearest statement of what remains unverified in the broader "quantum information resolves the paradox" narrative: the Page-bound-respecting entropy curve has been derived, but only in idealized settings, not for the ordinary black holes the paradox was originally posed about.