Let $A$ be a unital $C^*$-algebra, and let $S\subset S(A)$ be a separating family of states, meaning that for every nonzero element $x\in A$ there exists $\phi\in S$ such that $\phi(x)\neq 0$. For each $\phi\in S$, let $(H_\phi,\pi_\phi,\xi_\phi)$ be the GNS triple associated to $\phi$. Let