Theorem: Complete schedule locality

For the complete scheduled channels,

The terminal reduction to $Y$ factors through the initial reduction to $N_0(Y)$. The factorization is reference preserving. Changing a complete gate outside the ancestor subnetwork does not change the terminal reduction, when wiring and included gates are held fixed.

Proof. At one backward step, an output observable is the identity on every block disjoint from its current support. The adjoint of a trace-preserving channel is unital, so such a block remains an identity. Other blocks can enlarge support only to their union. Induction gives (eq:adjoint-locality).

To obtain a reduced channel, append any fixed state on the initial complement of $N_0(Y)$, apply the complete evolution, and take the output reduction to $Y$. The adjoint inclusion shows that this result is independent of the appended state and agrees with the actual output for every input, including correlated inputs. The same adjoint argument tensored with ${\mathrm{id}}_E$ proves the reference version.

On an excluded gate all pulled-back observables are identities. Any replacement complete channel preserves those identities.