sections, reset occurred with equal probability from every state at times distributed exponentially with rate $\Gamma$. We can construct a similar kind of reset dynamics for quantum open systems by adding jump operators $$J^\Gamma_i=\sqrt{\Gamma}\ket{\psi}\bra{\phi_i}$$ to the Lindbladian . Here $\ket{\phi_i}$ form a complete orthonormal basis, i.e., $\left\langle\phi_i|\phi_j\right\rangle=\delta_{ij}$, and $\ket{\psi}$ is the reset state. This modifies the Lindblad generator to $$\label{ModifiedLindblad}
{\mathcal{L}}_{\Gamma}(\rho) = {\mathcal{L}}(\rho)+\mathcal{V}_{\Gamma}(\rho)-\Gamma\rho,$$ where $$\mathcal{V}_{\Gamma}(\rho) = \Gamma\,\text{Tr}(\rho)\ket{\psi}\bra{\psi}.$$ To check that this construction has state-independent resets, as desired, we can consider the quantum jump Monte Carlo (QJMC) approach to simulating individual trajectories of the system’s evolution (see, e.g., [@plenio1998]). When the system undergoes a stochastic dissipative change (a “jump”), the probability of the change being a reset is $$P_\Gamma(\ket{\phi})\propto\sum_k^{\text{dim}(\mathcal{H})}\bra{\phi}{J^\Gamma_k}^\dagger J^\Gamma_k\ket{\phi}=\Gamma\left\langle\phi|\phi\right\rangle,$$ and is thus state independent as required.

To analyse the spectrum of the model with resets, we make the same assumptions as in Sec. \[ResetMarkov\]: we assume that the stationary state of the model without resets is unique, and that it is diagonalizable (i.e., there are no non-trivial Jordan blocks). The uniqueness of the stationary state implies $\text{Tr}(R_i)=0$ and hence $${\mathcal{L}}_\Gamma(R_i)=(\lambda_i-\Gamma)R_i$$ for $i\in\{2,...,\text{dim}({{\mathcal{H}}}^2)\}$. This shows, similarly to the classical case, that the $R_i$’s remain eigenmodes of the model with resets, with 