  The data processing inequality states that the quantum relative entropy
between two states $\rho$ and $\sigma$ 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 $\rho$ and
the closest recovered state $(\mathcal{R} \circ \mathcal{N})(\rho)$, where
$\mathcal{R}$ is a recovery map with the property that $\sigma = (\mathcal{R}
\circ \mathcal{N})(\sigma)$. We show the existence of an explicit recovery map
that is universal in the sense that it depends only on $\sigma$ 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.
