The are several nonequivalent 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 $\Lambda_t$ is CP divisible if and only if the second tensor power $\LAmbda_t ⊗ \Lambda_t$ is CP divisible as well. Moreover, the P divisibility of the map $\Lambda_t ⊗ \Lambda_t$ is equivalent to the CP divisibility of the map $\Lambda_$t . Interestingly, the latter property is no longer true if we replace the P divisibility of $\Lambda_t ⊗ \Lambda_t$ by simple positivity and the CP divisibility of $\Lambda_t$ by complete positivity. That is, unlike when $\Lambda_t$ has a time-independent generator, positivity of $\Lambda_t ⊗ \Lambda_t$ does not imply complete positivity of $\Lambda_t$ .
