Lemma: Remote invariance

Under $(\mathrm{NS}{)}^{\star }$, appending fresh zero-state factors outside $S$ does not change ${{\mathsf{P}}^{\mathsf{S}}}_{\psi }$. Applying a unitary outside $S$ does not change it either.

Proof. Compare a remote allocation-and-return program with a remote identity program, and schedule allocation before local readout. No-signaling equates the local laws. State sufficiency identifies the post-allocation law with the law at the enlarged global state.

For a remote gate, compare its gate-and-return program with the identity program and schedule the gate first.