\hspace{1.3cm}
\hbox{[by Eq.~(\ref{properties}b)]} \\
&= \frac{1}{N} \sum_\gamma \left( \sum_\alpha P_{\alpha\gamma} \right) \hat{A}_\gamma.
\end{split}$$ This condition will be satisfied as long as the $P$’s are normalized in the sense that $\sum_\alpha P_{\alpha\gamma} = 1$ for every $\gamma$. Moreover, the equation [*implies*]{} this normalization condition, as can be seen by multiplying both sides by $\hat{A}_\tau$ and taking the trace. Thus the condition (\[Bcondition\]) is equivalent to the natural normalization condition on the $P$’s.

We get a more restrictive condition on the $P$’s from the form of the definition of ${\mathcal B}_{\beta\delta}$. Regarded as a matrix with $\beta$ and $\delta$ as the matrix indices, we can see from Eq. (\[Bdef\]) that ${\mathcal B}_{\beta\delta}$ must be positive semidefinite: any matrix that can be written in this form is positive semidefinite, and any positive semidefinite matrix can be written in this form. Thus we arrive at our criteria for determining whether a given set of transition probabilities $P_{\alpha\gamma}$ represents a legitimate quantum process: $$\label{Pcondition}
\begin{split}
&\hbox{(a)} \;\; \sum_\alpha P_{\alpha\gamma} = 1 \hspace{2mm} \hbox{for every $\gamma$} \\
&\hbox{(b)} \;\; \sum_{\alpha\gamma} \Xi_{\beta\alpha\delta\gamma} P_{\alpha\gamma}\hspace{2mm} \hbox{is positive semidefinite},
\end{split}$$ where the mathematical expression in Eq. (\[Pcondition\]b) is understood to be a matrix with indices $\beta$ and $\delta$.

As an 