  The theory of controlled quantum open systems describes quantum systems
interacting with quantum environments and influenced by external forces varying
according to given algorithms. It is aimed, for instance, to model quantum
devices which can find applications in the future technology based on quantum
information processing. One of the main problems making difficult the practical
implementations of quantum information theory is the fragility of quantum
states under external perturbations. The aim of this note is to present the
relevant results concerning ergodic properties of open quantum systems which
are useful for the optimization of quantum devices and noise (errors)
reduction. In particular we present mathematical characterization of the
so-called "decoherence-free subspaces" for discrete and continuous-time quantum
dynamical semigroups in terms of $C^*$-algebras and group representations. We
analyze the non-Markovian models also, presenting the formulas for errors in
the Born approximation. The obtained results are used to discuss the proposed
different strategies of error reduction.
