Strengthening the quantum data processing inequality for maximal f-divergences via unraveling

Maximal f-divergences, among them the Belavkin--Staszewski relative entropy, are non-increasing under completely positive trace-preserving maps, which is the quantum data processing inequality (DPI). We quantify this decrease for Lindblad evolutions with Hermitian jump operators by lifting the problem to a stochastic unraveling, in which the two states become measures on pure states evolving under a classical Fokker--Planck equation. The lift rests on the identity that the infimum of the classical f-divergence over ensembles realizing two states is equal to their maximal f-divergence. This infimum is attained on a common basis of pure states, which is admissible initial data for the unraveling. The result is a strengthened DPI, with the loss bounded below by the time integral of a relative f-Fisher information. Given a two-sided bound on the unraveling's transition kernel, we derive two exponential decay estimates. For bracket-generating jump operators, the rate reaches twice the spectral gap of the symmetrized generator after an explicit delay. Without bracket generation, the kernel bound alone gives any rate below twice the spectral gap of the full generator.