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

Approximate Petz recovery from the geometry of density operators