we can re-express Eq. (\[firsth\]) somewhat more elegantly as $$\label{secondh}
H_\beta - \frac{1}{N^2} \sum_\delta H_\delta = \frac{\hbar}{N^2} \sum_{\alpha\gamma} r_{\alpha\gamma} 
\hbox{Im}(\Gamma_{\alpha\beta\gamma}).$$

Now, given any candidate set of values $r_{\alpha\gamma}$, Eq. (\[secondh\]) will give us some function $H_\beta$ (up to an additive constant). But not every set of $r$ values actually arises from a Hamiltonian. To tell whether the given set does represent Hamiltonian evolution, we insert the $H_\beta$ of Eq. (\[secondh\]) back into Eq. (\[rdef\]) and see whether that equation yields the same $r$ values we started with. If so, then those values do arise from a Hamiltonian; otherwise they do not.

Carrying out this strategy, we arrive at the following condition characterizing those sets of values $r_{\alpha\gamma}$ that represent Hamiltonian evolution: $$\label{rcondition}
r_{\alpha\gamma} = \frac{2}{N^2} \sum_{\alpha'\gamma'} \bigg[ \sum_\beta \hbox{Im}(\Gamma_{\alpha\beta\gamma})
\hbox{Im}(\Gamma_{\alpha'\beta\gamma'}) \bigg] r_{\alpha'\gamma'}.$$ To write this condition more compactly, let us think of $r_{\alpha\gamma}$ as a column vector with $\alpha\gamma$ as its single index (taking $N^4$ values). Let us call this column vector $\vec{r}$. We also define a matrix $R$ in terms of its components as follows: $$\label{Rdef}
R_{\alpha\gamma,\alpha'\gamma'} = \frac{2}{N^2} \sum_\beta \hbox{Im}(\Gamma_{\alpha\beta\gamma})
\hbox{Im}(\Gamma_{\alpha'\beta\gamma'}).$$ Then the condition (\[rcondition\]) can be re-expressed simply as $$\label{rconditionshort}
\vec{r} = R\vec{r}.$$ In Appendix B we 