Measurement of linear photon polarization ($M(\varphi )=\cos 2\varphi Z+\sin 2\varphi X$)

With $|V⟩=|0⟩$ and $|H⟩=|1⟩$, an analyzer oriented at angle $\varphi$ from vertical has positive-outcome state $\cos \varphi |0⟩+\sin \varphi |1⟩$. Subtracting the two orthogonal rank-one projection operators gives $M(\varphi )=\cos 2\varphi Z+\sin 2\varphi X$, where $X,Z$ are Pauli operators. Physical polarization angles are therefore half their corresponding Bloch sphere angles.