= Target no-signalling in a common-control unitary
{title2=$\rho_C'=\sum_b(\rho_B)_{bb}V_b\rho_CV_b^\dagger$}
Let $U=\sum_b U_b^A\otimes|b\rangle\langle b|_B\otimes V_b^C$, where the conditional target operators are unitary. For product input $\rho_A\otimes\rho_B\otimes\rho_C$,
$$
\rho_C'=\sum_b\langle b|\rho_B|b\rangle V_b\rho_CV_b^\dagger.
$$
Tracing over $B$ removes cross terms with unequal control labels. Tracing over $A$ leaves $\operatorname{Tr}(U_b\rho_AU_b^\dagger)=1$, 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.
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