Approximate Petz Recovery from the Geometry of Density Operators

We derive a new bound on the effectiveness of the Petz map as a universal recovery channel in approximate quantum error correction using the second sandwiched Rényi relative entropy D~2. For large Hilbert spaces, our bound implies that the Petz map performs quantum error correction with order-ϵ accuracy whenever the data processing inequality for D~2 is saturated up to terms of order ϵ2 times the inverse Hilbert space dimension. Conceptually, our result is obtained by extending (Cree and Sorce in J Phys A Math Theor, 2022. http://iopscience.iop.org/article/10.1088/1751-8121/ac5648), in which we studied exact saturation of the data processing inequality using differential geometry, to the case of approximate saturation. Important roles are played by (i) the fact that the exponential of the second sandwiched Rényi relative entropy is quadratic in its first argument, and (ii) the observation that the second sandwiched Rényi relative entropy satisfies the data processing inequality even when its first argument is a non-positive Hermitian operator.