Revisiting the equality conditions of the data-processing inequality for the sandwiched Rényi divergence

We provide a transparent, simple, and unified treatment of recent results on the equality conditions for the data-processing inequality of the sandwiched quantum Rényi divergence, including the statement that the equality in the data-processing implies recoverability via the Petz recovery map for the full range of the Rényi parameter α recently proven by Jenčová [J. Phys. A: Math. Theor. 50, 085303 (2017)]. We also obtain a new set of equality conditions, generalizing a previous result by Leditzky et al. [Lett. Math. Phys. 107, 61 (2017)].