Proposition: Affine obstruction to nonconstant definite labels

Let $\mathcal{S}$ be a convex state set. If an affine channel assigns each $\rho \in \mathcal{S}$ a classical distribution concentrated at one label $\ell (\rho )$, and distinct labels are distinguishable, then $\ell$ is constant.

Proof. If $\ell (\rho )\ne \ell (\sigma )$, affinity forces the classical output on $a\rho +(1-a)\sigma$, $0<a<1$, to be

This distribution is not concentrated at one label, a contradiction.