phase-space probability density in the sense of classical mechanics. The non-positivity of the WDF can be seen by integrating over all phase-space the product of two distributions built from different states $\psi$ and $\Phi$:

$$\int dx \int dv~ P_W[\psi](x,v) P_W[\Phi](x,v)  \propto \left| \left\langle \psi |\Phi\right\rangle\right|^2.$$

The right-hand side vanishes if the two states $\psi$, $\Phi$ are orthogonal which implies that the WDF cannot be positive everywhere. According to the Hudson theorem [@Hudson1974], the WDF of a pure state is point wise non-negative if and only if the state is Gaussian. If $\hat\rho$ is not a pure state, it can be represented as a convex combination of pure state operators, $\hat\rho =
\sum_n \lambda_n | \psi_n \rangle\langle \psi_n|$, in infinitely many ways. The WDF satisfies the so-called mixture property [@Ballentine1998], which is the requirement that the phase space distribution should depend only on the density operator $\hat\rho$ and not on the particular way it is represented as a mixture of some set of pure states $\left\lbrace \left|\psi_n\right\rangle\right\rbrace$. To summarize, the Wigner distribution function has many properties similar to the classical phase space distribution. Nevertheless it has been realized from the early days, that the concept of a joint probability at a 