Optimality Condition for the Transpose Channel

In quantum error correction, the Petz transpose channel serves as a perfect recovery map when the Knill-Laflamme condition is satisfied. Notably, while perfect recovery is generally infeasible for most quantum channels of finite dimension, the transpose channel 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 condition under which the transpose channel is strictly optimal in terms of channel fidelity. The violation of this condition can be easily characterized by a simple commutator that can be efficiently computed. We provide multiple examples that validate our new findings.