Proposition: A nonlinear expectation law is not an affine channel

A proposed output probability

on all qubit states is not an outcome probability of a fixed single-input quantum instrument.

Proof. Instrument probabilities are affine: $p(\rho )=\mathrm{Tr}(E\rho )$ for an effect $E$. The proposed law equals $1$ on both $|0⟩⟨0|$ and $|1⟩⟨1|$, but equals $0$ on their equal mixture. An affine law would equal $1$ on that mixture.