  We provide a general framework for the identification of open quantum
systems. By looking at the input-output behavior, we try to identify the system
inside a black box in which some Markovian time-evolution takes place. Due to
the generally irreversible nature of the dynamics, it is difficult to assure
full controllability over the system. Still, we show that the system is
identifiable up to similarity under a certain rank condition. The framework
also covers situations relevant to standard quantum process tomography, where
we do not have enough control over the system but have a tomographically
complete set of initial states and observables. Remarkably, the similarity
cannot in general be reduced to unitarity even for unitary systems, and the
spectra of Hamiltonians are not identifiable without additional knowledge.
