Relation to Page and Hawking

Page's random-state calculation concerns the entropy of a quantum subsystem of a pure bipartite state. For subsystem dimensions $m\le n$, its average entropy in bits is

Page proposed this formula and Sen supplied a proof PageAverage,Sen. Page's radiation argument adds a proposed black-hole interpretation to such information-theoretic considerations PageRadiation.

Neither this average nor a dimension crossing supplies a channel certificate for an entire variable quantum code. Hayden–Preskill retrieval of an additional diary, with early radiation available and specified mixing assumptions, is another task; it is not interchangeable with whole-code recovery from arbitrary classical counts HaydenPreskill.

Hawking's emission calculation supplies semiclassical field-theoretic information, not the complete endpoint channel used below Hawking. Cheung's En07-C keeps that separation explicit CheungEn07C. No Hawking occupation law, gravitational Hamiltonian, island formula, or quantum extremal surface is a premise of our finite proofs.