The optimized quantum f-divergences form a family of distinguishability measures that includes the quantum relative entropy and the sandwiched Rényi 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 Rényi 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]. Joint work with Li Gao.
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