Tensor power of dynamical maps and P- vs. CP-divisibility

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 $\Lambda_t$ is CP-divisible iff the second tensor power $\Lambda_t\otimes\Lambda_t$ is CP-divisible as well. Moreover, the P-divisibility of the map $\Lambda_t\otimes\Lambda_t$ 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 $\Lambda_t\otimes\Lambda_t$ 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 $\Lambda_t\otimes\Lambda_t$ does not imply complete positivity of $\Lambda_t$.