allow ourselves to assume that $r_{\alpha\gamma}$ is antisymmetric under interchange of $\alpha$ and $\gamma$, we can combine the two terms in Eq. (\[rcondition4\]) to get $$\label{rcondition5}
r_{\alpha\gamma} = \frac{2}{N^2} \sum_\zeta r_{\alpha + \zeta, \gamma + \zeta} \cos \left[ \frac{4 \pi}{N} \langle\alpha - \gamma , \zeta\rangle \right].$$ Thus we can take as our condition on the $r$’s either Eq. (\[rcondition4\]) by itself, which [*implies*]{} that $r_{\gamma\alpha} = - r_{\alpha\gamma}$, or Eq. (\[rcondition5\]) [*together with*]{} the condition $r_{\gamma\alpha} = - r_{\alpha\gamma}$. Either of these statements serves to characterize precisely those sets of transition rates that correspond to Hamiltonian dynamics.

The case of a single qubit, with $N=2$, is simpler. In that case, one finds that Eq. (\[rconditionshort\]) is equivalent to a combination of two conditions on the $r$’s that we have already encountered: $$\label{rconditionqubit}
\begin{split}
&\hbox{(a)} \;\; \sum_\gamma r_{\alpha\gamma} = 0. \\
&\hbox{(b)} \;\; r_{\gamma\alpha} = - r_{\alpha\gamma}. 
\end{split}$$ In fact one can prove that these conditions are sufficient just by counting the number of free parameters. One finds that of the 16 possible values of an unconstrained $r_{\alpha\gamma}$, only three remain after we impose the conditions in Eq. (\[rconditionqubit\]). This number is the same as the rank of the projection operator 