Universal recovery maps and approximate sufficiency of quantum relative entropy

TL;DR

This paper proves the existence of a universal recovery map for quantum states that depends only on the target state and the channel, providing a new perspective on approximate quantum error correction.

Contribution

It introduces an explicit, universal recovery map that depends solely on the target state and the channel, advancing the understanding of quantum error correction.

Findings

Existence of a universal recovery map independent of the initial state.

Provides an information-theoretic characterization of approximate quantum error correction.

Strengthens the data processing inequality with a universal recovery approach.

Abstract

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.