Proposition: Explicit incompleteness

The direct amplitude report is not informationally complete for arbitrary qubit inputs. Even the direct and Hadamard report settings together are not informationally complete for arbitrary qubits.

The entire present coefficient package is also not informationally complete for the continuum scalar field.

Proof. The direct effects lie in $\mathrm{span}\{I,D\}$, so, for example, $|+⟩⟨+|$ and $|-⟩⟨-|$ have the same direct law. Adding the Hadamard setting gives effects in $\mathrm{span}\{I,D,X\}$. The distinct states $(I+Y)/2$ and $(I-Y)/2$ still have the same laws.

For the field claim choose a nonzero $v\in {C_c}^{\infty }(U\setminus (K_1\cup K_2))$. For every $\varphi$ and every real $t$,

All encodings depending only on these two values therefore agree on infinitely many distinct fields.