  We derive a new bound on the effectiveness of the Petz map as a universal
recovery channel in approximate quantum error correction using the second
sandwiched R\'{e}nyi relative entropy $\tilde{D}_{2}$. For large Hilbert
spaces, our bound implies that the Petz map performs quantum error correction
with order-$\epsilon$ accuracy whenever the data processing inequality for
$\tilde{D}_{2}$ is saturated up to terms of order $\epsilon^2$ times the
inverse Hilbert space dimension. Conceptually, our result is obtained by
extending arXiv:2011.03473, in which we studied exact saturation of the data
processing inequality using differential geometry, to the case of approximate
saturation. Important roles are played by (i) the fact that the exponential of
the second sandwiched R\'{e}nyi relative entropy is quadratic in its first
argument, and (ii) the observation that the second sandwiched R\'{e}nyi
relative entropy satisfies the data processing inequality even when its first
argument is a non-positive Hermitian operator.
