know that for a given set of $P$ values, there exist a set of complex numbers ${\mathcal B}_{\beta\delta}$ such that the $P$’s can be expressed in that form? (In asking this question we are not yet insisting that the ${\mathcal B}$’s arise from a legitimate set of $\hat{B}_j$ operators.) In fact this is not a problem. For any numbers $P_{\alpha\gamma}$, if we insert the ${\mathcal B}$’s of Eq. (\[inverse\]) back into Eq. (\[Pdef\]), we find that we arrive again at the values of $P_{\alpha\gamma}$ that we started with. This is because $$\frac{1}{N^2} \sum_{\beta\delta} \Xi_{\alpha\beta\gamma\delta} \Xi_{\beta\sigma\delta\tau} = \delta_{\alpha\sigma}\delta_{\gamma\tau}\; ,$$ as can be shown directly using the properties of the $A$’s given in Eq. (\[properties\]). Thus any set of $P$’s is consistent with Eq. (\[Pdef\]) if we allow the ${\mathcal B}$’s to be entirely unconstrained.

Our first constraint on the $P$’s comes from Eq. (\[Bcondition\]), which places a condition on ${\mathcal B}$. Let us use Eq. (\[inverse\]) to express this condition in terms of the transition probabilities. We have $$\begin{split}
\hat{I} &= \sum_{\mu\nu}{\mathcal B}_{\mu\nu} \hat{A}_\nu \hat{A}_\mu = \frac{1}{N^3}\sum_{\mu\nu\alpha\gamma}
P_{\alpha\gamma} \hbox{Tr}(\hat{A}_\mu \hat{A}_\alpha \hat{A}_\nu \hat{A}_\gamma) \hat{A}_\nu \hat{A}_\mu \\
&=\frac{1}{N^2}\sum_{\nu\alpha\gamma} P_{\alpha\gamma} \hat{A}_\nu \hat{A}_\alpha \hat{A}_\nu \hat{A}_\gamma \hspace{1cm} \hbox{[by Eq.~(\ref{properties}a)]} \\
&=\frac{1}{N}\sum_{\alpha\gamma} P_{\alpha\gamma} (\hbox{Tr}\, \hat{A}_\alpha ) \hat{A}_\gamma 