Lemma: Complete restoration and stability

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

Proof. For a fixed code state $\omega$, the channel

is CP, trace preserving, and satisfies ${\mathcal{L}}_V{\mathrm{Ad}}_V={\mathrm{id}}_C$. Isometric embeddings preserve nonzero singular values, also after tensoring with $I_Q$. Trace-distance contraction on referenced states gives diamond-distance contraction. Insert any decoder into the triangle inequality, take infima, and interchange the compared channels to obtain the stability bound.