Lemma: Global composition and law errors

Suppose two complete protocols have corresponding CPTP stages ${\Phi }_j,{\hat{\Phi }}_j$ with $d_{\diamond }({\Phi }_j,{\hat{\Phi }}_j)\le {\eta }_j$. Then

If a protocol is sampled using input-independent laws $P,Q$ and the same conditional complete channels ${\Phi }_{\omega }$, then

If $M$ independent weight laws are each calibrated within total variation $\delta$, the product-law contribution is at most

Proof. Insert the stages one at a time and telescope the difference. Composition with CPTP maps contracts diamond norm. Adaptive classical controls can be included in the stages on the common classical–quantum interface.

For the law bound, take a maximal coupling of $P,Q$. On the agreement event the conditional channels coincide; on the disagreement event the half-diamond distance is at most one. Integrate. Independent maximal couplings on the edges give the product bound.