Proposition: Composition modulo unobservable directions

For all admissible input pairs, agreement on $\mathcal{A}$ is preserved as agreement on $\mathcal{B}$ if and only if

Equivalently,

where ${\mathcal{C}}^{\perp }$ is the Hilbert–Schmidt orthogonal complement.

Proof. An admissible state sees $A$ only through ${\mathcal{E}}_{\mathcal{C}}(A)$. Therefore the displayed inclusion is sufficient.

For necessity, repeat the perturbation proof of thm:composition with Hermitian $H\in \mathcal{C}$ annihilating ${\mathcal{E}}_{\mathcal{C}}(\mathcal{A})$. The states $I/d\pm \epsilon H$ are admissible. The resulting orthogonality puts ${\mathcal{E}}_{\mathcal{C}}{\Theta }^{\ast }(B)$ in ${\mathcal{E}}_{\mathcal{C}}(\mathcal{A})$ for every Hermitian $B$, and then for every $B$. Finally, $\ker {\mathcal{E}}_{\mathcal{C}}={\mathcal{C}}^{\perp }$, proving the equivalent formulation.