Proposition: Exact implementation criterion

The implementations agree on ${\mathcal{O}}_{\mathrm{out}}={⨁}_h{\mathcal{B}}_h$ for every hybrid input if and only if

The equality also holds on ${\mathcal{O}}_{\mathrm{out}}\otimes L(N)$ for every untouched reference $N$, using the same extended input in both implementations.

Proof. Choose an input supported in block $g$ and an output observable supported in block $h$. Equality of all state expectations is equality of the Hermitian adjoint images. Complex linearity gives (eq:implementation-adjoint). Conversely, summing these identities proves output agreement.

For a reference observable $B\otimes C$, the two adjoint images are equal by tensoring (eq:implementation-adjoint) with $C$. Such tensors span the reference-extended observation space.