 Postselection probability for Kerr-cat preparation

Normalize the initial polarization qubit and its analysing wave-plate transformation. For a conditional phase rotation of $\pi$, a vertical detection has the displayed probability and prepares $N(|\alpha ⟩+e^{i\varphi }|-\alpha ⟩)$. At $\alpha =0,\varphi =\pi$, that outcome has zero probability, explaining the singular normalization of the absent branch.