not just density operators. This illegal Wigner function corresponds to the operator $\hat{A}_\gamma$.) The specific formula for $P_{\alpha\gamma}$ given in Eq. (\[PXi\]) follows from Eq. (\[PfromE\]) when ${\mathcal E}$ can be expressed in the form (\[transformation\]).

Now, if we were allowed to choose $P_{\alpha\gamma}$ arbitrarily, even if we were to insist on the standard normalization condition $\sum_\alpha P_{\alpha\gamma} = 1$, we would easily be able to create an illegal quantum state from a legal one. For example, all points in phase space could be mapped with probability 1 to a specific point. Then the final state $W'$ would be a state that would produce a deterministic outcome for each of the $N+1$ mutually unbiased measurements associated with the striations, which is impossible. So we now ask this question: given a proposed set of transition probabilities $P_{\alpha\gamma}$, how does one know whether it corresponds to a valid quantum transformation?

We begin by inverting Eq. (\[Pdef\]) so as to express ${\mathcal B}_{\beta\delta}$ in terms of the $P$’s. For this, it will be helpful to refer to the following two properties of the $\hat{A}$’s, both of which follow from their orthogonality and their normalization: [^1]: for any $N \times N$ matrix $\hat{M}$, $$\label{properties}
