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To extend the classical concept of Markovianity to an open quantum system,different notions of the divisibility of its dynamics have been introduced.Here we analyze this issue by five complementary approaches: equations ofmotion, real-time diagrammatics, Kraus-operator sums, as well as time-local(TCL) and nonlocal (Nakajima-Zwanzig) quantum master equations. As a case studyfeaturing several types of divisible dynamics, we examine in detail an exactlysolvable noninteracting fermionic resonant level coupled arbitrarily stronglyto a fermionic bath at arbitrary temperature in the wideband limit. Inparticular, the impact of divisibility on the time-dependence of the observablelevel occupation is investigated and compared with typical Markovianapproximations. We find that the loss of semigroup-divisibility is accompaniedby a prominent reentrant behavior: Counter to intuition, the level occupationmay temporarily \emph{increase} significantly in order to reach a stationarystate with \emph{smaller} occupation, implying a reversal of the measurabletransport current. In contrast, the loss of the so-called completely-positivedivisibility is more subtly signaled by the \emph{prohibition} of such currentreversals in specific time-intervals. Experimentally, it can be detected in thefamily of transient currents obtained by varying the initial occupation. Toquantify the nonzero footprint left by the system in its effective environment,we determine the exact time-dependent state of the latter as well as relatedinformation measures such as entropy, exchange entropy and coherentinformation.

Five approaches to exact open-system dynamics: Complete positivity, divisibility and time-dependent observables