Proposition: A normalized ReLU rule is not a channel on unknown states

Applying coordinatewise ReLU to amplitudes and normalizing does not define a well-defined quantum-state transformation on all input rays. More generally, an arbitrary nonlinear normalized state rule is not a quantum channel.

Proof. The vectors

represent the same physical ray. Normalized ReLU sends the first to $|0⟩$ and the second to $|1⟩$, so the rule is not ray invariant.

Channels are linear on density operators. For example, the nonlinear rule $T(\rho )={\rho }^2/\mathrm{Tr}({\rho }^2)$ sends $\mathrm{diag}(3/4,1/4)$ to $\mathrm{diag}(9/10,1/10)$, although it fixes the two endpoint pure states. It is therefore not affine on their mixture.