Quantum Random Evolutions

In this work, we develop a mathematical framework to model a quantum system whose time evolution may depend on the state of a randomly changing environment that evolves according to a Markovian process. When the environment changes its state, three possible things may occur: the quantum system starts evolving according to a new Hamiltonian, it may suffer an instantaneous perturbation that changes its state or both things may happen simultaneously. We consider the case of quantum systems with finite dimensional Hilbert state space, in which case the observables are described by Hermitian matrices. We show how to average over the environment to predict the expected value of the density matrix with which one can compute the expected values of the observables of interest.