Let $H$ be a [Hilbert space](/page/Hilbert%20Space), let $M \subseteq \mathcal{L}(H)$ be a unital self-adjoint subalgebra, and let $M'$ denote the commutant of $M$ in $\mathcal{L}(H)$. For $\xi \in H$, the vector $\xi$ is cyclic for $M$, meaning $\overline{\{S\xi:S\in M\}}=H$, if and only if $\xi$ is separating for $M'$, meaning that for every $T\in M'$, $T\xi=0$ implies $T=0$.