Approximate Petz recovery from the geometry of density operators

TL;DR

This paper establishes a new bound on the Petz map's effectiveness in approximate quantum error correction, linking it to the saturation of the data processing inequality for the second sandwiched Rényi relative entropy.

Contribution

It extends previous work on exact data processing inequality saturation to approximate cases using differential geometry and properties of the sandwiched Rényi entropy.

Findings

Petz map achieves order-ε error correction when the data processing inequality is nearly saturated.

The bound depends on the inverse Hilbert space dimension and the degree of saturation.

The second sandwiched Rényi entropy's properties are key to deriving the new bound.

Abstract

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 R\'{e}nyi relative entropy $\tilde{D}_{2}$. For large Hilbert spaces, our bound implies that the Petz map performs quantum error correction with order-$\epsilon$ accuracy whenever the data processing inequality for $\tilde{D}_{2}$ is saturated up to terms of order $\epsilon^2$ times the inverse Hilbert space dimension. Conceptually, our result is obtained by extending arXiv:2011.03473, 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 R\'{e}nyi relative entropy is quadratic in its first argument, and (ii) the observation that the second sandwiched R\'{e}nyi relative entropy satisfies the data processing inequality even when its first argument is a non-positive Hermitian operator.