Approximate recoverability and the quantum data processing inequality

In this paper, we discuss the quantum data processing inequality and its refinements that are physically meaningful in the context of approximate recoverability. An important conjecture regarding this due to Seshadreesan et. al. in J. Phys. A: Math. Theor. 48 (2015) is disproved. We prove some inequalities capturing universal approximate recoverability with the Petz recovery map for the sandwiched quasi and R\'enyi relative entropies for the parameter $t=2$. We also obtain convexity theorems on some parametrized versions of the relative entropy and fidelity, which can be of independent interest.