Theory {#secresqtheory}
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We consider a quantum system in a Hilbert space ${\mathcal{H}}$ of dimension $\text{dim}({\mathcal{H}})$ with density matrix $\rho$, whose evolution is given by $$\frac{d\rho}{dt}={\mathcal{L}}(\rho),
\label{eqlindeq}$$ where $$\label{LindbladOp}
{\mathcal{L}}(\rho) = -i[H,\rho]+\sum_{j}\left[{J}_{j}\rho{J}_{j}^{\dagger}-\frac{1}{2}\{{J}_{j}^{\dagger}{J}_{j},\rho\}\right],$$ is the Lindblad master operator. Here $H$ is the Hamiltonian of the system and the jump operators ${J}_{j}$ mediate the system-bath interaction, providing coupling of the system to the surrounding environment.

Since ${\mathcal{L}}$ acts linearly on the density matrix, the evolution can be understood in terms of its eigenvalues and eigenmatrices. Let us denote the eigenvalues of ${\mathcal{L}}$ by ${\lambda}_{k}$ and order them such that $\text{Re}({\lambda}_{k})\geq\text{Re}({\lambda}_{k+1})$. As in the classical case, we have $\text{Re}(\lambda_k)\leq 0$, with $\lambda_1=0$ corresponding to the stationary state, due to the fact that ${\mathcal{L}}$ is completely positive and trace-preserving. Moreover, as ${\mathcal{L}}$ is in general not Hermitian, it has right and left eigenmatrices denoted by $${\mathcal{L}}(R_k)=\lambda_k R_k$$ and $${\mathcal{L}}^\dagger(L_k)=\lambda_k^* L_k,$$ respectively. These are normalised such that $$\text{Tr}({L}^\dagger_{k}{R}_{k'})={\delta}_{kk'}.$$ Generally, the stationary state ${\rho}_{\rm ss}$ is unique. Normalising it, as usual, by $\text{Tr}({\rho}_{\rm ss})=1$, we then have $L_1=\mathds{1}$. Defining $${c}_{k}=\text{Tr}[{L}^\dagger_{k}\rho(0)]$$ for an initial state $\rho(0)$, the system’s state at time $t$ is given by $$\label{Expansion}
\rho(t)={e}^{t{\mathcal{L}}}[\rho(0)]={\rho}_{\rm ss}+\sum_{k}{c}_{k}{e}^{t{\lambda}_{k}}{R}_{k}.$$

For the classical stochastic processes discussed in the previous 