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We consider stochastic and open quantum systems with a finite number ofstates, where a stochastic transition between two specific states is monitoredby a detector. The long-time counting statistics of the observed realizationsof the transition, parametrized by cumulants, is the only available informationabout the system. We present an analytical method for reconstructing generatorsof the time evolution of the system compatible with the observations. Thepracticality of the reconstruction method is demonstrated by the examples of alaser-driven atom and the kinetics of enzyme-catalyzed reactions. Moreover, wepropose cumulant-based criteria for testing the non-classicality andnon-Markovianity of the time evolution, and lower bounds for the systemdimension. Our analytical results rely on the close connection between thecumulants of the counting statistics and the characteristic polynomial of thegenerator, which takes the role of the cumulant generating function.

Inverse counting statistics for stochastic and open quantum systems: the characteristic polynomial approach