Lemma: Complete restoration and contraction

Every isometric channel has a CPTP left inverse and preserves the trace norm of every referenced operator. CPTP pre- and postprocessing contract $d_{\diamond }$. Consequently

Proof. For a fixed state ${\omega }_C$, the map

is CP, trace preserving, and left-inverts ${\mathrm{Ad}}_V$. An isometric embedding preserves all nonzero singular values, also after reference extension. Trace-distance contraction on referenced states gives diamond contraction. Insert an arbitrary decoder in the triangle inequality, take infima, and interchange the compared channels to prove stability.