Optimality Condition for the Petz Map

TL;DR

This paper establishes the necessary and sufficient conditions for the optimality of the Petz map in quantum error correction, enhancing understanding of its performance and providing practical criteria for its application.

Contribution

It introduces the first precise conditions for Petz map optimality based on entanglement fidelity, with efficient computability in certain cases.

Findings

Derived necessary and sufficient optimality conditions.

Characterized violations via a simple commutator.

Provided multiple illustrative examples.

Abstract

In quantum error correction, the Petz map serves as a perfect recovery map when the Knill-Laflamme conditions are satisfied. Notably, while perfect recovery is generally infeasible for most quantum channels of finite dimension, the Petz map remains a versatile tool with near-optimal performance in recovering quantum states. This work introduces and proves, for the first time, the necessary and sufficient conditions for the optimality of the Petz map in terms of entanglement fidelity. In some special cases, the violation of this condition can be easily characterized by a simple commutator that can be efficiently computed. We provide multiple examples that substantiate our new findings.