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

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 ${\Lambda }_t$ 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 ${\Lambda }_t$.

arXiv comment5 pages

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• Absolutely separating quantum maps and channels
• Witnessing non-Markovian dynamics through correlations
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