© 2018, The Author(s). 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.
Junge, M., Renner, R., Sutter, D., Wilde, M. M., & Winter, A. (2018). Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy. Annales Henri Poincare, 19, 2955-2978. https://doi.org/10.1007/s00023-018-0716-0