Proposition: Complete noisy channel and unchanged populations

A Kraus family for ${\mathcal{N}}_{\eta ,\lambda }$ is

Its complete complement is

The marginal ${\mathcal{N}}_{\eta ,\lambda }({\rho }_M)$ is (eq:rhoR) with $\eta z$ replaced by $\lambda \eta z$, and its adjoint entry replaced accordingly. All occupation probabilities and $⟨N_R⟩$ are unchanged for every input.

Proof. Since $a_{\lambda }+b_{\lambda }=1$ and ${S_R}^2=I$, the Kraus effects sum to $I_M$. Taking both partial traces of ${\tilde{V}}_{\eta ,\lambda }$ gives the channels and complement. Conjugation by $S_R$ leaves diagonal entries unchanged and negates the $00,11$ coherence. The weighted sum multiplies that coherence by $a_{\lambda }-b_{\lambda }=\lambda$.