Theorem: Lüders compatibility

Under the sharp hypotheses and record compatibility, the following are equivalent:

1. $\mathcal{F}=\mathcal{G}$ as complete recorded residual channels;

2. $T{\mathcal{D}}_{\Pi }=T{\mathcal{D}}_P$;

3. $$T(P_{z}X{P}_{z^{\prime }})=0\text{whenever }z\ne z^{\prime },q(z)=q(z^{\prime }),X\in \mathcal{B}(H).$$

Thus full compatibility removes fine coherence within each coarse outcome. It need not remove coherence between different coarse outcomes.

Proof. By Lemma lem:sharp, the $w$-block of $\mathcal{G}(X)$ is $T({\Pi }_{w}X{\Pi }_w)$. Its difference from the corresponding $\mathcal{F}$-block consists exactly of the fine off-diagonal corners within ${\Pi }_w$.

Summing the $w$-blocks gives (ii). Conversely, their output supports lie in orthogonal $E_w$-corners, so a zero sum makes each corner zero. Testing a single $P_{z}X{P}_{z^{\prime }}$ proves necessity of (iii), and expansion into corners proves sufficiency.