Target no-signalling in a common-control unitary (${\rho }_C^{\prime }={\sum }_b({\rho }_B{)}_{bb}{V}_b{\rho }_C{V}_b^{†}$)

Let $U={\sum }_b{U}_b^A\otimes |b⟩⟨b{|}_B\otimes V_b^C$, where the conditional target operators are unitary. For product input ${\rho }_A\otimes {\rho }_B\otimes {\rho }_C$,Tracing over $B$ removes cross terms with unequal control labels. Tracing over $A$ leaves $\mathrm{Tr}(U_b{\rho }_A{U}_b^{†})=1$, proving the formula and its independence of ${\rho }_A$. Exchanging $A$ and $C$ proves the corresponding reverse independence. Thus the two targets cannot signal to each other through this fixed common-control interaction, despite phase kickback allowing a target to affect its control.

Two controlled-NOT gates with a common control and distinct targets are an example. They commute; their chronological order can be exchanged, giving an equivalent causal proof. Intervening control rotations or additional interactions can change the conclusion.