Proposition: Environmental composition and implementation errors

Finite compositions of the declared environmental, reduction, and nonselective measurement channels are CPTP. If corresponding implemented channels have half-diamond errors $e_1,\dots ,e_k$, the composite error is at most

In finite dimensions, sums of bounded environmental and reduction generators can also be obtained as the norm limit of their alternating short-time channel products.

Proof. Composition preserves complete positivity and trace. For the error, expand the difference of two composites into a telescoping sum and use channel contraction.

For bounded finite generators ${\mathcal{L}}_1,{\mathcal{L}}_2$,

At $h=s/n$, telescoping against $e^{h({\mathcal{L}}_1+{\mathcal{L}}_2)}$ gives an error $O(n{h}^2)=O(n^{-1})$.