Definition: Three output specifications

The following tasks have different types.

1. Observable readout: for a specified POVM $\{F_y\}$, the target is the classical channel

   $$\rho ⟼\sum _y\mathrm{Tr}(F_y\rho )|y⟩⟨y|.$$

   No quantum preservation of the message is required.

2. Deterministic quantum recovery: the output is a qubit, and performance is measured by $d_{\diamond }(\mathcal{DN},{\mathrm{id}}_M)$.

3. Instrument output: for CP maps $\{{\mathcal{I}}_y\}$ whose sum is trace preserving, the output includes both a classical outcome and a conditional quantum state:

   $$\hat{\mathcal{I}}(X)=\sum _y|y⟩⟨y|\otimes {\mathcal{I}}_y(X).$$

   Matching only the outcome probabilities does not match this quantum–classical channel.

An instrument distance requires a target with the same output type. A flagged qubit output cannot be compared directly with an unflagged identity channel.