Direct proof of the algebraic relative-entropy Page-transition probe in type III factors

Prove that, for evaporating black holes described by type III von Neumann factors, the algebraic relative-entropy probe for the Page transition holds. Concretely, let V be the isometric encoding from the semiclassical code Hilbert space to the full quantum gravity Hilbert space, let \~Ψ and \~Φ be any pair of states on the semiclassical code space and Ψ = V\~Ψ, Φ = V\~Φ their images in the full theory. Show that the differences of Araki algebraic relative entropies D_Rad(t) = (\~Ψ|\~Φ; A_sc^{I∪R}) − (Ψ|Φ; A_qg^{Rad}) and D_BH(t) = (\~Ψ|\~Φ; A_sc^{X}) − (Ψ|Φ; A_qg^{BH}) furnish a valid Page-transition probe in type III factors, namely D_Rad(t) > δ and D_BH(t) > ε for t < t_P, D_Rad(t) = δ and D_BH(t) = ε at t = t_P, and D_Rad(t) ≤ δ and D_BH(t) ≤ ε for t > t_P, for suitable thresholds δ, ε (possibly zero), where (·|·; ·) denotes Araki’s algebraic relative entropy.

Background

The paper proposes an algebraic relative-entropy-based probe for the Page transition that is formulated and proved for von Neumann factors of type I/II, where algebraic von Neumann entropies are well-defined and the reconstruction theorems apply directly. The probe compares algebraic relative entropies computed on semiclassical code algebras and their images in the full quantum gravity description, yielding sign changes across the Page time that diagnose a phase transition in channel recoverability.

For type III factors, the standard island formulas are not directly applicable because the von Neumann entropy is ill-defined, but Araki’s algebraic relative entropy is well-defined for generic factors. The authors argue that the relative-entropy probe should extend to type III algebras and would provide a robust formulation of the Page transition in that setting, but they note that a direct proof in type III factors has not yet been obtained.