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Every open-system dynamics can be associated to infinitely many stochasticpictures, called unravelings, which have proved to be extremely useful inseveral contexts, both from the conceptual and the practical point of view.Here, focusing on quantum-jump unravelings, we demonstrate that there existsinherent freedom in how to assign the terms of the underlying master equationto the deterministic and jump parts of the stochastic description, which leadsto a number of qualitatively different unravelings. As relevant examples, weshow that a fixed basis of post-jump states can be selected under some definiteconditions, or that the deterministic evolution can be set by a chosentime-independent non-Hermitian Hamiltonian, even in the presence of externaldriving. Our approach relies on the definition of rate operators, whosepositivity equips each unraveling with a continuous-measurement scheme and isrelated to a long known but so far not widely used property to classify quantumdynamics, known as dissipativity. Starting from formal mathematical concepts,our results allow us to get fundamental insights into open quantum systemdynamics and to enrich their numerical simulations.

How to design quantum-jump trajectories via distinct master equation representations