Tensor power of dynamical maps and positive versus completely positive divisibility

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 ${\mathrm{\Lambda }}_t$ is CP divisible if and only if the second tensor power ${\mathrm{\Lambda }}_t\otimes {\mathrm{\Lambda }}_t$ is CP divisible as well. Moreover, the P divisibility of the map ${\mathrm{\Lambda }}_t\otimes {\mathrm{\Lambda }}_t$ is equivalent to the CP divisibility of the map ${\mathrm{\Lambda }}_t$. Interestingly, the latter property is no longer true if we replace the P divisibility of ${\mathrm{\Lambda }}_t\otimes {\mathrm{\Lambda }}_t$ by simple positivity and the CP divisibility of ${\mathrm{\Lambda }}_t$ by complete positivity. That is, unlike when ${\mathrm{\Lambda }}_t$ has a time-independent generator, positivity of ${\mathrm{\Lambda }}_t\otimes {\mathrm{\Lambda }}_t$ does not imply complete positivity of ${\mathrm{\Lambda }}_t$.