The data processing inequality states that the quantum relative entropy between two states $ρ$ and $σ$ can never increase by applying the same quantum channel $\mathcal{N}$ to both states. This inequality can be strengthened with a remainder term in the form of a distance between $ρ$ and the closest recovered state $(\mathcal{R} \circ \mathcal{N})(ρ)$, where $\mathcal{R}$ is a recovery map with the property that $σ= (\mathcal{R} \circ \mathcal{N})(σ)$. We show the existence of an explicit recovery map that is universal in the sense that it depends only on $σ$ and the quantum channel $\mathcal{N}$ to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.