$\hat{\rho}$ to a final density matrix $\hat{\rho}'$ of the same system, can be expressed in the form $$\label{transformation}
\hat{\rho}' = \sum_j \hat{B}_j \hat{\rho} \hat{B}^\dag_j,$$ where the $N \times N$ Kraus matrices $\hat{B}_j$ satisfy the condition $$\label{Kraus}
\sum_j \hat{B}_j^\dag \hat{B}_j = \hat{I}.$$

It is a straightforward matter to re-express Eq. (\[transformation\]) as a transformation of the discrete Wigner function. For each $\hat{B}_j$, let us define the corresponding phase-space function $B^{(j)}_{\alpha}$ by $$B^{(j)}_{\alpha} = \frac{1}{N} \hbox{Tr}\, (\hat{A}_\alpha \hat{B}_j),$$ so that $\hat{B}_j = \sum_\alpha B^{(j)}_{\alpha} \hat{A}_\alpha$. The condition (\[Kraus\]) then becomes $$\label{Bcondition}
\sum_{\alpha\beta} {\mathcal B}_{\alpha\beta}  \hat{A}_\beta \hat{A}_\alpha = \hat{I},$$ where $$\label{Bdef}
{\mathcal B}_{\alpha\beta} = \sum_j {B}^{(j)}_{\alpha} {B}^{(j)*}_{\beta},$$ Expanding $\hat{\rho}$, $\hat{\rho}'$ and the $\hat{B}$’s in Eq. (\[transformation\]), we obtain $$W'_\alpha = \frac{1}{N} \sum_{\beta\gamma\delta} \hbox{Tr}(\hat{A}_\alpha \hat{A}_\beta \hat{A}_\gamma \hat{A}_\delta) {\mathcal B}_{\beta\delta} W_\gamma.$$ Thus by defining $P_{\alpha\gamma}$ to be $$\label{Pdef}
P_{\alpha\gamma} = \frac{1}{N} \sum_{\beta\delta} \hbox{Tr}(\hat{A}_\alpha \hat{A}_\beta \hat{A}_\gamma \hat{A}_\delta) {\mathcal B}_{\beta\delta},$$ we can express the evolution as $$\label{probprocess}
W'_\alpha = \sum_\gamma P_{\alpha\gamma} W_{\gamma}.$$ If we interpret $W_\gamma$ as the probability of finding the system at the phase space point $\gamma$, then $P_{\alpha\gamma}$ plays the role of the probability that a system at the point $\gamma$ will make a transition to $\alpha$ (but $P_{\alpha\gamma}$ can be negative). In what 