Theorem: Amplitude visibility or erasure

For the family (eq:encoding), the following are equivalent:

1. the complete output is invariant on $a(\lambda )=\lambda c_{\epsilon }$;

2. $B_y=C_y=0$ for every reported output $y$;

3. the complete output is independent of $a$ on $\mathbb{R}$.

For the classical law alone the corresponding criterion is ${\beta }_y={\gamma }_y=0$ for every $y$.

Proof. Differentiation of (eq:selective) gives

Different classical blocks cannot cancel one another. At $\lambda =0$, a zero derivative forces $B_y=0$. At any nonzero $\lambda$, $st\ne 0$, forcing $C_y=0$. The converse follows from (eq:selective). The same argument after taking traces proves the law criterion.