Definition: Recovery tasks

The following tasks have different output specifications.

readout. For a specified logical POVM $\{F_y\}$ on $a$, a boundary measurement must implement the quantum-to-classical channel

Its output is classical. No quantum preservation of $a$ is required.

recovery. A channel ${\mathcal{D}}_A:\mathcal{B}({\mathcal{H}}_A)\to \mathcal{B}({\mathcal{H}}_a)$ must satisfy

recovery. For specified completely positive maps $\{{\mathcal{I}}_y:\mathcal{B}({\mathcal{H}}_a)\to \mathcal{B}({\mathcal{H}}_{a^{\prime }})\}$, whose sum is trace preserving, the required output channel is

A boundary programme must implement $\hat{\mathcal{I}}{\mathrm{Tr}}_b$, including its conditional quantum outputs, not only its outcome probabilities.