Proposition: Affine obstruction to nonconstant definite packets

Let $\mathcal{S}$ be a convex set of quantum states. Suppose an affine map assigns to every $\rho \in \mathcal{S}$ a probability distribution concentrated at one classical label:

Then $\gamma$ is constant on $\mathcal{S}$.

The same conclusion holds for packets in any classical outcome space in which distinct packets can be separated by a measurable event.

Proof. If $\gamma (\rho )\ne \gamma (\sigma )$, then for $0<t<1$, affinity gives

This distribution is not concentrated at one label. The mixture belongs to $\mathcal{S}$, giving a contradiction. In a general outcome space, an event separating the two packets has probability strictly between zero and one on that mixture, with the same contradiction.