\hspace{1cm} B_\beta = \frac{1}{N} \hbox{Tr}(\hat{U} \hat{A}_\beta),$$ from which it follows that $\sum_\beta |B_\beta|^2 = 1$.

In Ref. [@Wootters1987] it was shown that, among all real functions of two phase space points, those $P$’s that correspond to unitary transformations are completely characterized by the following two properties: $$\label{oldPcondition}
\begin{split}
&\hbox{(a)} \;\; \sum_\alpha P_{\alpha\rho} = 1. \\
&\hbox{(b)} \;\; \sum_{\rho\sigma\tau} P_{\alpha\rho} P_{\beta\sigma} P_{\gamma\tau} \Gamma_{\rho\sigma\tau} = \Gamma_{\alpha\beta\gamma},
\end{split}$$ where $\Gamma_{\alpha\beta\gamma}$ is the three-point structure function we defined in Eq. (\[Gamma\]). That is, in addition to the standard normalization condition, the $P$’s must leave $\Gamma$ unchanged.

In the spirit of Eq. (\[Pcondition\]), we can replace Eq. (\[oldPcondition\]b) with an alternative condition, so that the conditions for a unitary transformation become $$\label{newPcondition}
\begin{split}
&\hbox{(a)} \;\; \sum_\alpha P_{\alpha\gamma} = 1. \\
&\hbox{(b)} \;\; {\mathcal B}_{\beta\delta} = \frac{1}{N^2} \sum_{\alpha\gamma} \Xi_{\beta\alpha\delta\gamma} P_{\alpha\gamma} \;\; \hbox{has rank 1},
\end{split}$$ where again $\beta$ and $\delta$ are understood to be matrix indices. It is clear from Eq. (\[unitaryB\]) that statement (\[newPcondition\]b) is true for a unitary transformation. To see that (\[newPcondition\]a) and (\[newPcondition\]b) are also sufficient to certify unitarity, note first that the matrix ${\mathcal B}$ defined in Eq. (\[newPcondition\]b) is necessarily Hermitian, because of the symmetry (\[Xisymmetry\]) of $\Xi$. Moreover, the 