The thesis is dedicated to the investigation of the property of synchronizability of road-colored directed graphs, which can also be understood as deterministic finite automata. In this context, Markov chains, a special class of stochastic processes, are considered. They have a canonical representation as directed graphs and, conversely, a directed graph with a probability distribution on the outgoing edges of its vertices induces a Markov chain. Such an identification is also possible between Markov chains and road-colored directed graphs. Especially with regard to the property of synchronizability, this connection has already led to interesting results in both areas, the deepening and expansion of which is one of the main goals of this work.

Moreover, it has already been shown in the field of quantum probability theory that the notion of asymptotic completeness of certain quantum Markov processes, motivated by scattering theory, is closely related to the classical idea of synchronizability. Indeed, when considering commutative systems, the synchronizability of a road-colored graph provides an equivalent characterization of the asymptotic completeness of the associated quantum Markov process. In this framework, the present thesis deals with possible generalizations of the classical concept of synchronizability in a quantum mechanical context.