Theorem: Hybrid observation criterion

For observation spaces in finite direct sums, preservation of agreement for all states is equivalent to

For ${\mathcal{O}}_{\mathrm{in}}={⨁}_g{\mathcal{A}}_g$ and ${\mathcal{O}}_{\mathrm{out}}={⨁}_h{\mathcal{B}}_h$, this is equivalent to

In particular, successor probabilities depend only on $\rho {|}_{{\mathcal{A}}_g}$ exactly when $E_{h|g}\in {\mathcal{A}}_g$ for every $h$.

Proof. Use the faithful-state perturbation argument of Theorem thm:composition within the direct-sum algebra. An output observable can have just one nonzero block, proving necessity of every branch inclusion. Summation proves sufficiency. Apply the same argument to the single effect $E_{h|g}$ for the final statement.