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We study the concepts of complete positivity, positivity and non-Markovianityin a two-level open quantum system whose dynamics are governed by a time-localquantum master equation. We establish necessary and sufficient conditions onthe time-dependent relaxation rates to ensure complete positivity andpositivity of the dynamical map. We discuss their relations with thenon-Markovian behavior of the open system. We also analyze the long-timeasymptotic behavior of the dynamics as a function of the rates. We show underwhich conditions on the rates the system tends to the equilibrium state.Different examples illustrate this general study.

Complete positivity, positivity and long-time asymptotic behavior in a two-level open quantum system