Optical Wigner function ($W(x,y)$)

For $X_1=x$, $X_2=y$ with $[X_1,X_2]=i/2$, normalize the phase-space function by $\int W(x,y)dxdy=1$. A coherent state of amplitude $\alpha$ has $W=2e^{-2|x+iy-\alpha {|}^2}/\pi$. Quadrature marginals are physical probability distributions, but the full function can be negative. Cross terms of an optical cat state give interference fringes, even when low-order moments approach those of a mixture.