Recovery and the Data Processing Inequality for Quasi-Entropies

We prove a number of quantitative stability bounds for the cases of equality in Petz's monotonicity theorem for quasi-relative entropies S_f (p||σ) defined in terms of an operator monotone decreasing functions f and, in particular, the Rényi relative entropies. Included in our results are bounds in terms of the Petz recovery map, but we obtain more general results. The present treatment is entirely elementary and developed in the context of finite dimensional von Neumann algebras, where the results are already non-trivial and of interest in quantum information theory.