Lemma: Bounded propagator and composition estimates

For bounded self-adjoint generators,

Their channels, including ancillas and partial traces, have half-diamond error bounded by the same quantity, capped at one.

Compatible channel compositions telescope in half-diamond distance. Their error is at most the sum of the component errors, capped at one.

Proof. Variation of constants gives

Unitary factors have norm one. The purified-output vector comparison bounds trace distance, including references. Alternatively the complete induced trace norm of the channel difference is at most twice $\Vert \hat{U}-U\Vert$. The composition statement follows by inserting one changed factor at a time and using channel contraction.