Theorem: Scheduled locality

For layers ${\Phi }_j={⨂}_{B\in {\mathcal{B}}_j}{\Phi }_{j,B}$, where ${\mathcal{B}}_j$ is a partition and every block map is CPTP, define

Then

Equal input reductions on that ancestor region give equal output reductions on $Y$. For a joint output with reference $E$, the required equal input reductions are on the joint ancestor–reference region.

If every block has diameter at most $\Delta$, then

Proof. The adjoint of a channel is unital. Blocks disjoint from an observable's support act on identity and do not change it. A block meeting that support can enlarge it only within the block. Reverse induction proves the support inclusion. The metric bound follows by adding at most one block diameter per layer. Testing all output observables proves the reduced state statements; carry $\mathcal{B}({\mathcal{H}}_E)$ unchanged for the reference version.