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The Lindblad equation describes the time evolution of a density matrix of aquantum mechanical system. Stationary solutions are obtained by time-averagingthe solution, which will in general depend on the initial state. We provide ananalytical expression for the steady states of the Lindblad equation using thequantum jump unraveling, a version of an ergodic theorem, and the stationaryprobabilities of the corresponding discret-time Markov chains. Our result isvalid when the number of states appearing the in quantum trajectory is finite.The classical case of a Markov jump-process is recovered as a special case, anddifferences between the two are discussed.

Explicit expressions for stationary states of the Lindblad equation for a finite state space