Lemma: Complete-channel telescoping

If ${\Lambda }_m\cdots {\Lambda }_1$ and ${\hat{\Lambda }}_m\cdots {\hat{\Lambda }}_1$ are compatible compositions of channels and $d_{\diamond }({\hat{\Lambda }}_j,{\Lambda }_j)\le e_j$, then their half-diamond distance is at most $\min \{1,{\sum }_j{e}_j\}$. Every step interface must include retained quantum systems and classical histories.

Proof. Insert and subtract compositions differing at one factor. Pre- and postcomposition by channels contract the complete trace norm. Apply the triangle inequality.