Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy

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 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 (R ∘ N)(ρ ) , where R is a recovery map with the property that σ = (R ∘ 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 N to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.