Proposition: Channel-model error and recovery stability

Suppose two accessible channels with the same types obey

For any fixed CPTP decoder $\mathcal{D}$,

Their optimal recovery errors also satisfy

For every common referenced input, their output states have trace distance at most $\xi$. For the corresponding accessible marginals,

and

For a fixed decoder, the Bell fidelities differ by at most $\xi$.

Proof. Postprocessing contraction and the reverse triangle inequality prove the fixed-decoder statement. Apply it for every decoder and take infima in both directions to obtain optimal-error stability. The referenced state bound is the definition of channel distance.

For purity,

Since the state difference has trace zero, center the occupation operator:

because ${\Vert N_R-I\Vert }_{\infty }=1$. For distinct basis vectors $u,v$ and Hermitian $X$,

This follows by testing against a phase-aligned Hermitian off-diagonal operator of norm one. It gives the coherence-entry estimate. The Bell-projector effect bound gives the fidelity estimate.