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The theoretical description of quantum dynamics in an intriguing way does notnecessarily imply the underlying dynamics is indeed intriguing. Here we showhow a known very interesting master equation with an always negative decay rate[eternal non-Markovianity (ENM)] arises from simple stochastic Schr\"odingerdynamics (random unitary dynamics). Equivalently, it may be seen as arisingfrom a mixture of Markov (semi-group) open system dynamics. Both theseapproaches lead to a more general family of CPT maps, characterized by a pointwithin a parameter triangle. Our results show how ENM quantum dynamics can berealised easily in the laboratory. Moreover, we find a quantumtime-continuously measured (quantum trajectory) realisation of the dynamics ofthe ENM master equation based on unitary transformations and projectivemeasurements in an extended Hilbert space, guided by a classical Markovprocess. Furthermore, a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL)representation of the dynamics in an extended Hilbert space can be found, witha remarkable property: there is no dynamics in the ancilla state. Finally,analogous constructions for two qubits extend these results fromnon-CP-divisible to non-P-divisible dynamics.

Eternal non-Markovianity: from random unitary to Markov chain realisations