Lemma: Complete-channel telescoping

For compatible complete channels ${\Phi }_j,{\tilde{\Phi }}_j$,

Tensor composition satisfies the analogous sum bound.

Proof. Subtract the products by replacing one factor at a time. CPTP pre- and postcomposition contract diamond distance, so each term is bounded by its local error. For tensor products, replace one tensor factor at a time and use that a channel has diamond norm one. Sum the resulting bounds.