Entanglement measures and approximate quantum error correction

TL;DR

This paper establishes that small entanglement loss along a quantum channel enables approximate quantum error correction, with explicit bounds for various entanglement measures and insights into their relationships.

Contribution

It generalizes the conditions for approximate quantum error correction based on entanglement measures and provides bounds relating entanglement of formation and distillable entanglement.

Findings

Small entanglement loss implies approximate error correction.

Explicit bounds for entanglement of formation and distillable entanglement.

A bound on the gap between entanglement measures for finite systems.

Abstract

It is shown that, if the loss of entanglement along a quantum channel is sufficiently small, then approximate quantum error correction is possible, thereby generalizing what happens for coherent information. Explicit bounds are obtained for the entanglement of formation and the distillable entanglement, and their validity naturally extends to other bipartite entanglement measures in between. Robustness of derived criteria is analyzed and their tightness compared. Finally, as a byproduct, we prove a bound quantifying how large the gap between entanglement of formation and distillable entanglement can be for any given finite dimensional bipartite system, thus providing a sufficient condition for distillability in terms of entanglement of formation.