  We identify the conditions that guarantee equivalence of the reduced dynamics
of an open quantum system (OQS) for two different types of environments - one a
continuous bosonic environment leading to a unitary system-environment
evolution and the other a discrete-mode bosonic environment resulting in a
system-mode (non-unitary) Lindbladian evolution. Assuming initial Gaussian
states for the environments, we prove that the two OQS dynamics are equivalent
if both the expectation values and two-time correlation functions of the
environmental interaction operators are the same at all times for the two
configurations. Since the numerical and analytical description of a
discrete-mode environment undergoing a Lindbladian evolution is significantly
more efficient than that of a continuous bosonic environment in a unitary
evolution, our result represents a powerful, non-perturbative tool to describe
complex and possibly highly non-Markovian dynamics. As a special application,
we recover and generalize the well-known pseudomodes approach to open system
dynamics.
