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A quantum Markov semigroup can be represented via classical diffusionprocesses solving a stochastic Schr\"odinger equation. In this paper we firstprove that a quantum Markov semigroup is irreducible if and only if classicaldiffusion processes are total in the Hilbert space of the system. Then we studythe relationship between irreducibility of a quantum Markov semigroup andproperties of these diffusions such as accessibility, the Lie algebra rankcondition, and irreducibility. We prove that all these properties are, ingeneral, weaker than irreducibility of the quantum Markov semigroup,nevertheless, they are equivalent for some important classes of semigroups.

On the relationship between a quantum Markov semigroup and its representation via linear stochastic Schroedinger equations