Theorem: Compositional observation boundary

For a channel $\Phi$ and observation spaces ${\mathcal{O}}_{\mathrm{in}},{\mathcal{O}}_{\mathrm{out}}$, the following are equivalent:

and

If this inclusion holds successively, compatible continuations preserve the declared agreement.

The reference-stable comparison of two extended inputs requires agreement on ${\mathcal{O}}_{\mathrm{in}}\otimes \mathcal{B}(E)$, not merely agreement of their unreferenced marginals.

Proof. Sufficiency follows by pulling each output observable back. For necessity, take a Hermitian matrix $H$ annihilating ${\mathcal{O}}_{\mathrm{in}}$ under the trace pairing. Since $I\in {\mathcal{O}}_{\mathrm{in}}$, $\mathrm{Tr}H=0$. For a faithful state $\omega$, both $\omega +\epsilon H$ and $\omega -\epsilon H$ are states for sufficiently small $\epsilon >0$. They agree on ${\mathcal{O}}_{\mathrm{in}}$, hence

Finite-dimensional double orthogonal complementation gives the inclusion. Complex linearity completes the proof. Tensoring the inclusion proves the reference statement.