  We determine the conditions for the equivalence between the multi-time
expectation values of a general finite-dimensional open quantum system when
interacting with, respectively, an environment undergoing a free unitary
evolution or a discrete environment under a free evolution fixed by a proper
Gorini-Kossakowski-Lindblad-Sudarshan generator. We prove that the equivalence
holds if both environments are bosonic and Gaussian and if the one- and
two-time correlation functions of the corresponding interaction operators are
the same at all times. This result leads to a non-perturbative evaluation of
the multi-time expectation values of operators and maps of open quantum systems
interacting with a continuous set of bosonic modes by means of a limited number
of damped modes, thus setting the ground for the investigation of open-system
multi-time quantities in fully general regimes.
