Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy

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