Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy
Marius Junge, Renato Renner, David Sutter, Mark M. Wilde, Andreas Winter · Annales Henri Poincaré · 2018
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.