Abstract
The are several non-equivalent notions of Markovian quantum evolution. In this paper we show that the one based on the so-called CP-divisibility of the corresponding dynamical map enjoys the following stability property: the dynamical map Λ_t is CP-divisible iff the second tensor power Λ_t⊗Λ_t is CP-divisible as well. Moreover, the P-divisibility of the map Λ_t⊗Λ_t is equivalent to the CP-divisibility of the map Λ_t. Interestingly, the latter property is no longer true if we replace the P-divisibility of Λ_t⊗Λ_t by simple positivity and the CP-divisibility of Λ_t by complete positivity. That is, unlike when Λ_t has a time-independent generator, positivity of Λ_t⊗Λ_t does not imply complete positivity of Λ_t.