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

Approximate recoverability and the quantum data processing inequality