Fractional quantum channels, scrambling, and black hole information recovery

The paper develops a rigorous CPTP framework for simulating closed timelike curves (CTCs) in finite-dimensional quantum systems, governed by a Caputo fractional master equation in Lindblad form. The time evolution is defined through the Mittag-Leffler superoperator Eα(t) = Eα,1(t^α L), whose complete positivity and trace preservation are proved via the Choi-Jamiołkowski isomorphism for all t ≥ 0 and ε > 0. The protocol replaces the ill-defined operator inversion of D-CTCs and the exponential overhead of P-CTCs with a Hayden-Preskill-inspired decoding map acting on a composite system of one message qubit S and an n-qubit auxiliary register A. The main recovery theorem establishes |ρSʳᵉᶜ - ρS(0)|₁ ≤ C₁λ² + C₂ε + C₃ · 2-n/2, with explicit, independently controllable constants corresponding to Bob's perturbation, Lindblad dissipation, and finite-size scrambling error respectively. The framework is then applied to the black hole information paradox through a precise dictionary identifying the fractional order α with the memory exponent of the horizon's stress-energy correlator, the auxiliary register with Hawking radiation modes, and the decoding map with an external observer reconstructing the infalling state. Key results include a fractional Page curve governed by Eα,1(-(t/tPage)^α) predicting a smooth entropy turnover rather than a sharp kink, an explicit information recovery bound for Hawking radiation (Corollary IX.6), a circumvention of the firewall argument through fractional memory correlations encoded in the off-diagonal elements of the Choi matrix without high-energy horizon excitations, and a correspondence between the k ≥ 2 terms of the Mittag-Leffler series and replica wormhole contributions to the island formula. Price's law for Schwarzschild perturbations yields the falsifiable prediction α ≈ 0.33. Fractional OTOC decay, always below the Maldacena-Shenker-Stanford bound, is experimentally accessible in sonic BEC analog black holes and near-term quantum processors.