  The optimized quantum $f$-divergences form a family of distinguishability
measures that includes the quantum relative entropy and the sandwiched R\'enyi
relative quasi-entropy as special cases. In this paper, we establish physically
meaningful refinements of the data-processing inequality for the optimized
$f$-divergence. In particular, the refinements state that the absolute
difference between the optimized $f$-divergence and its channel-processed
version is an upper bound on how well one can recover a quantum state acted
upon by a quantum channel, whenever the recovery channel is taken to be a
rotated Petz recovery channel. Not only do these results lead to physically
meaningful refinements of the data-processing inequality for the sandwiched
R\'enyi relative entropy, but they also have implications for perfect
reversibility (i.e., quantum sufficiency) of the optimized $f$-divergences.
Along the way, we improve upon previous physically meaningful refinements of
the data-processing inequality for the standard $f$-divergence, as established
in recent work of Carlen and Vershynina [arXiv:1710.02409, arXiv:1710.08080].
Finally, we extend the definition of the optimized $f$-divergence, its
data-processing inequality, and all of our recoverability results to the
general von Neumann algebraic setting, so that all of our results can be
employed in physical settings beyond those confined to the most common
finite-dimensional setting of interest in quantum information theory.
