Lemma: Sequential reference-stable comparison

Let ${\Lambda }_t,{\tilde{\Lambda }}_t$ be channels on identified successive interfaces and suppose

Then

The same conclusion holds for complete history channels if histories, conditional residual states, auxiliary outputs, and failures are retained and each one-step comparison is proved on that augmented interface.

Proof. Telescope the channel difference by inserting and subtracting one implemented step at a time. CPTP maps have diamond norm one, so pre- and post-composition cannot increase the norm of a step difference. Sum the half-diamond allowances. The proof is unchanged on an augmented history register.