We study different characterizations of positive (P) divisibility in two-level open quantum systems whose dynamics are governed by a time-local master equation with time-dependent relaxation rates. In the case of phase-covariant dynamics, necessary and sufficient conditions for the P-divisibility of the dynamical map are given in terms of inequalities on such relaxation rates. We propose a geometric P-divisibility criterion based on the Bloch coordinates. The connection to the Breuer-Laine-Piilo (BLP) measure of non-Markovianity is also established. We show how these results can be extended to a Redfield master equation. As an application of such characterizations, we study the open dynamics of a qubit interacting with a single bosonic mode. More precisely, we characterize the properties of the local map of the qubit generated by its interaction with the bosonic mode, playing the role of an extremely reduced bath. Interesting observations are made, opening perspectives for a deeper physical understanding of complete positive divisibility, P-divisibility, and BLP measure.

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• Open quantum systems
• Open quantum systems & decoherence
• Quantum formalism
• Non-Markovian processes