$ \begin{aligned} p(r|z,\phi) &= \frac{1}{\sqrt{2\pi\sigma^2}} e^{-(r-z\cos(\phi))^2/2\sigma^2}\\ \sqrt{p(r|z,\phi)} &= \frac{1}{\sqrt[4]{2\pi\sigma^2}} e^{-(r-z\cos(\phi))^2/4\sigma^2}\\ \sigma^2 &= \tau/dt \\ \ket{\psi} &= \begin{pmatrix}\sqrt{p(z=0)} e^{-i\theta/2} \\ \sqrt{p(z=1)}e^{i\theta/2}\end{pmatrix}\\ \ket{\psi'} &= \begin{pmatrix}\sqrt{p(z=0|r)} e^{-i\theta/2} \\ \sqrt{p(z=1|r)}e^{i\theta/2}\end{pmatrix} \\ &= \frac{1}{\sqrt{p(r)}}\begin{bmatrix}\sqrt{p(r|z=0)} & 0 \\ 0 & \sqrt{p(r|z=1)}\end{bmatrix} \ket \psi\\ \hat R(r|\phi) &= \frac{1}{\sqrt[4]{2\pi\sigma^2}} e^{-(1 + r^2)/4\tau + re^{-i\phi}\sigma_z/2\tau} \end{aligned} $