Proposition: One-live-qubit logical readout

In the ordered streaming interface, logical $X$-readout and logical $Z$-readout each admit peak one.

The rank-one logical $Z$ Lüders instrument, with its quantum post-measurement output, also admits peak one. None of these programmes is subsystem recovery.

Proof. For $X$, measure the physical observables $Z_1,X_2,Z_3$ sequentially and report the negative of their product. For $Z$, measure $Y_1,Z_2,Y_3$ and report the negative product. Equation (eq:reps) gives the required logical effects. An $X$-measurement uses $H$ before destructive readout; a $Y$-measurement uses $H{S}^{†}$. Each carrier is consumed before the next arrives.

For the logical $Z$ instrument, let the reported logical outcome be $\kappa \in \{0,1\}$. After consuming all boundary carriers, allocate $|0⟩$, apply $X^{\kappa }$, and return it. The resulting selective map has the correct effect and prepares the fixed state $|\kappa ⟩$. For a rank-one projector this is exactly

including its action on an external reference.

Forgetting the outcome instead gives logical $Z$-dephasing, not the identity channel. A logical Bell input detects the lost coherence.