Monotonicity of optimized quantum f-divergence

Optimized quantum f-divergence was first introduced by Wilde and further explored by Li and Wilde later. Wilde raised the question of whether the monotonicity of optimized quantum f-divergence can be generalized to maps that are not quantum channels. In this paper, we answer this question by generalizing the monotonicity of optimized quantum f-divergences to positive trace preserving maps satisfying a Schwarz inequality. Any 2-positive maps satisfy such a Schwarz inequality. The main tool in this paper is the Petz recovery map.