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Here, focusing on quantum-jump unravelings, we demonstrate that there exists inherent freedom in how to assign the terms of the underlying master equation to the deterministic and jump parts of the stochastic description, which leads to a number of qualitatively different unravelings. As relevant examples, we show that a fixed basis of post-jump states can be selected under some definite conditions, or that the deterministic evolution can be set by a chosen time-independent non-Hermitian Hamiltonian, even in the presence of external driving. Our approach relies on the definition of rate operators, whose positivity equips each unraveling with a continuous-measurement scheme and is related to a long known but so far not widely used property to classify quantum dynamics, known as dissipativity. Starting from formal mathematical concepts, our results allow us to get fundamental insights into open quantum system dynamics and to enrich their numerical simulations.<\/jats:p>","DOI":"10.22331\/q-2022-10-13-835","type":"journal-article","created":{"date-parts":[[2022,10,13]],"date-time":"2022-10-13T14:21:46Z","timestamp":1665670906000},"page":"835","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":["How to design quantum-jump trajectories via distinct master equation representations"],"prefix":"10.22331","volume":"6","author":[{"given":"Dariusz","family":"Chru\u015bci\u0144ski","sequence":"first","affiliation":[{"name":"Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5\/7, 87-100 Toru\u0144, Poland"}]},{"given":"Kimmo","family":"Luoma","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Theoretische Physik, Technische Universit\u00e4t Dresden, D-01062, Dresden, Germany"},{"name":"Turku Center for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014, Turun Yliopisto, Finland"}]},{"given":"Jyrki","family":"Piilo","sequence":"additional","affiliation":[{"name":"Turku Center for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014, Turun Yliopisto, Finland"}]},{"given":"Andrea","family":"Smirne","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica ``Aldo Pontremoli&apos;&apos;, Universit\u00e0 degli Studi di Milano, Via Celoria 16, I-20133 Milan, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16, I-20133 Milan, Italy"}]}],"member":"9598","published-online":{"date-parts":[[2022,10,13]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"H.-P. 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