Definition: Complete scalar-input output

For an encoding $\mathsf{Enc}:\mathbb{R}\to \mathcal{S}$ and a fixed output map $\Phi :\mathcal{S}\to \mathcal{Y}$, the scalar-input output is $\Phi \circ \mathsf{Enc}$. Its declared boundary includes all reported classical words and every returned quantum factor.

It is informative about $a$ if this composite is nonconstant. Orbit invariance means constancy on $\mathsf{Enc}(\lambda c_{\epsilon })$, $\lambda \in \mathbb{R}$.