Proposition: Logical readout is not full quantum return

On the authorized triple $123$, logical $Z$-readout can be implemented by sequentially measuring $Y_1,Z_2,Y_3$ and reporting their negative parity. Measuring and then preparing the reported logical $Z$-eigenstate realizes the logical rank-one Lüders instrument. Its unconditional channel is

Thus exact logical $Z$-statistics, and even the corresponding rank-one selective instrument, do not certify full-code recovery.

Proof. The measured product is $R_Z$, so Lemma lem:triple gives the logical projective effects. For a rank-one projector $P_t=|t⟩⟨t|$, the measure-and-prepare map is

including its action on a reference. Summing gives dephasing. Its Pauli mixture gives the upper error $1/2$, while the Bell overlap $1/2$ gives the matching lower bound.