$\bra{l_i}\mathcal{L}=0$ for more than one pair of states, causing an initial state dependence of the stationary state. Generically, we can still choose a basis such that $\bra{l_1}=\bra{-}$ and $\left\langle l_i|r_j\right\rangle=\delta_{ij}$ within this null eigenspace, allowing us to use nearly the same proof scheme as above. The reset rates break the zero eigenvalues degeneracy and provides a unique stationary state, with the remainder of the null eigenspace shifted to an eigenvalue of $-\Gamma$. In Eq. the unique steady state $\ket{P_{\text{ss}}}$ is replaced by the state-dependent steady state that would be reached from the reset state under the original dynamics.

Stationary current {#DetailedBalance}
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The current associated with the stationary state $\ket{P^\Gamma_\text{ss}}$ of the reset process is defined, for any given link or transition $c\rightarrow c'$, by $$J^\Gamma_{c\rightarrow c'}=\left\langle c|P^\Gamma_{\text{ss}}\right\rangle{\mathcal{L}}^\Gamma_{c'c}-\left\langle c'|P^\Gamma_{\text{ss}}\right\rangle{\mathcal{L}}^\Gamma_{cc'},$$ where ${\mathcal{L}}^\Gamma_{ij}$ is the $(i,j)$ component of ${\mathcal{L}}^\Gamma$. Substituting the expression of the stationary state, found in Eq. , together with the expression of the generator ${\mathcal{L}}^\Gamma$, we can decompose the current into three parts as $$\begin{aligned}
J^\Gamma_{c\rightarrow c'}&=J_{c\rightarrow c'}
    +\sum_{i=2}^{D}\frac{\Gamma\left\langle l_i|C_0\right\rangle}{\Gamma-\lambda_i}J^{i}_{c\rightarrow c'}\nonumber\\
    &\quad+\Gamma\left(\left\langle c|P^\Gamma_{\text{ss}}\right\rangle{\delta}_{c'c_0}-\left\langle c'|P^\Gamma_{\text{ss}}\right\rangle{\delta}_{c_0c'}\right),
\label{eqcurr1}\end{aligned}$$ where $$J^{i}_{c\rightarrow c'}=\left\langle c|r_i\right\rangle{\mathcal{L}}_{c'c}-\left\langle c'|r_i\right\rangle{\mathcal{L}}_{cc'}.$$ The first term on the right-hand side of Eq.  is the current of the original reset-free process, while the 