Theorem: Exact schedule locality

For $A\in \mathcal{A}(Y)$ and $0\le s\le t\le T$:

1. $$\mathrm{supp}(({\Phi }_{s+1}^{\ast }\circ \cdots \circ {\Phi }_t^{\ast })(A))\subseteq N_{s,t}(Y);$$
2. $$N_{s,t}(Y)\subseteq \{x:d(x,Y)\le (t-s)R_{\mathrm{upd}}a\};$$

3. the output reduction on $Y$ depends on the state at tick $s$ only through its reduction to $N_{s,t}(Y)$.

Proof. Reverse-tick induction using lem:one-layer proves (i). For (ii), if $x$ enters through a block meeting the next region at $y$, then

For (iii), every output observable pulls back to the named input region. Equality of its expectations for all output observables is equality of the output density operators.