Theorem: Exact Lüders residual criterion

Assume that the fine and coarse measurements are PVMs, the coarse instrument is Lüders, and $T^{\ast }(E_w)={\Pi }_w$ for every $w$. Then the following statements are equivalent:

1. $\mathcal{F}=\mathcal{G}$, equivalently they agree on ${\mathcal{O}}_{L({\mathcal{H}}_{\mathrm{c}})}$ for every input;

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

3. for every $X$ and every distinct $z,z^{\prime }$ with $q(z)=q(z^{\prime })$,

   $$T(P_{z}X{P}_{z^{\prime }})=0.$$

Proof. By lem:intertwining, the difference of outcome maps is

Each such difference is supported in the $E_w$-corner. Its sum over $w$ is

Since the $E_w$ are orthogonal, this sum vanishes if and only if every corner vanishes: multiply the sum on both sides by $E_w$. This proves the equivalence of (i) and (ii).

Condition (iii) makes every summand in eq:block-difference zero, proving (i). Conversely, assume (ii) and take $X=P_{z}Y{P}_{z^{\prime }}$, with $z\ne z^{\prime }$ in the same coarse outcome. Then ${\mathcal{D}}_{\Pi }(X)=X$ and ${\mathcal{D}}_P(X)=0$, so $T(X)=0$. This proves (iii).