Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space) with $\dim H\ge 2$. Every projective symmetry $S$ is induced by either a unitary operator $U:H\to H$ or an antiunitary operator $A:H\to H$, in the sense that \begin{align*} S([\psi]) = [U\psi] \end{align*} or \begin{align*} S([\psi]) = [A\psi] \end{align*} for every unit vector $\psi\in H$. The implementing operator is unique up to multiplication by a complex phase of modulus $1$.