Lemma: Complete branches define the tested channel

Equation (eq:branchsum) is CPTP. For device Kraus path maps $K_{zh}$, completeness is

A successful branch alone need not be trace preserving.

Proof. Compositions and sums preserve complete positivity. Trace preservation follows from trace preservation of every $D_z$ and of ${\sum }_z{\mathcal{M}}_z$. The Kraus identity is the trace-preservation condition for the complete instrument. Selecting only some branches generally removes terms from this identity.