unique stationary state $\ket{P_\text{ss}}$, corresponding from to the eigenvalue $\lambda_1=0$, so that $\ket{P_\text{ss}}=\ket{r_1}$. We also assume that ${\mathcal{L}}$ contains no Jordan blocks, corresponding to non-exponentially decaying modes, so we do not need to consider generalized eigenvectors [^1]. The normalization of the stationary state can be expressed as $$\sum_C \left\langle C|P_\text{ss}\right\rangle=\left\langle-|P_\text{ss}\right\rangle=1,$$ where we have introduced the “flat” state $$\bra{-}=\sum_C \bra{C}.$$ Conservation of probability also requires $\bra{-}{\mathcal{L}}=0$, which implies $\bra{l_1}=\bra{-}$ and hence $\left\langle -|r_i \right\rangle=0$ for all $ i \neq 1$. From the ergodicity assumption, all other eigenvalues are possibly complex but have real parts less than zero, that is, $\text{Re}(\lambda_i)<0$ for all $i\neq1$. Both this and the Jordan block assumption can be relaxed to arrive at similar but slightly more general results.

The generator ${\mathcal{L}}$ defines our original process. The reset version of that process is constructed simply by adding new transitions at a rate $\Gamma$ from every configuration to a target or reset state, denoted by $C_0$. The generator of the reset process is thus given by $$\begin{aligned}
{\mathcal{L}}^\Gamma={\mathcal{L}}+\Gamma\sum_{C\neq C_0} \ket{C_0}\bra{C}-\Gamma\sum_{C\neq C_0}\ket{C}\bra{C},\end{aligned}$$ where the additional terms can be absorbed into the old jump operators and escape rate operator to give shifted transition and escape rates. Note that we can add 