Approximate quantum error correction

TL;DR

This work analyzes when quantum error correction can achieve high fidelity given a noisy quantum channel. It establishes that perfect correction is possible exactly when the coherent information $I$ equals the initial entanglement entropy $S^Q$, and it provides a constructive recovery scheme based on a partial measurement and a conditional unitary. For small deficits $S^Q - I < \varepsilon$, it proves near-perfect correction: there exists a Q-side recovery with entanglement fidelity $F_e > (1 - \sqrt{\varepsilon})^2$, i.e., fidelity at least $1 - 2\sqrt{\varepsilon}$, by relating entropic closeness to fidelity through $S(\rho||\sigma)$ and $D(\rho,\sigma)$. The results give a robust, quantitative bridge from entropy-based criteria to operational performance, with implications for quantum channel capacities and approximate QEC schemes.

Abstract

The errors that arise in a quantum channel can be corrected perfectly if and only if the channel does not decrease the coherent information of the input state. We show that, if the loss of coherent information is small, then approximate error correction is possible.

Figures (1)

• Figure 1: Composite system $RQ$ is initially in the entangled state $\left | \Psi^{RQ} \right \rangle$. The evolution of system $Q$ includes interaction with the environment $E$.