 Equatorial qubit measurement

The orthonormal quantum states $|{\alpha }_s⟩=(|0⟩+(-1{)}^s{e}^{-i\alpha }|1⟩)/\sqrt{2}$, $s=0,1$, define an equatorial projective measurement. The measured observable is $\cos \alpha X-\sin \alpha Y$, with eigenvalue $(-1{)}^s$. Thus $\alpha =0$ measures $X$, and $\alpha =\pi /2$ measures $-Y$ with these outcome labels. The sign convention matters when comparing different definitions of equatorial bases.