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Such evolutions preserve the equivalence of ensembles and therefore are free from problems with signaling. We show that if family of linear non-trace-preserving maps satisfies the semigroup property then the generated family of convex quasi-linear operations also possesses the semigroup property. Next we generalize the Gorini-Kossakowski-Sudarshan-Lindblad type equation for the considered evolution. As examples we discuss the general qubit evolution in our model as well as an extension of the Jaynes-Cummings model. We apply our formalism to spin density matrix of a charged particle moving in the electromagnetic field as well as to flavor evolution of solar neutrinos.<\/jats:p>","DOI":"10.22331\/q-2021-03-23-420","type":"journal-article","created":{"date-parts":[[2021,3,23]],"date-time":"2021-03-23T11:45:24Z","timestamp":1616499924000},"page":"420","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":18,"title":["Nonlinear extension of the quantum dynamical semigroup"],"prefix":"10.22331","volume":"5","author":[{"given":"Jakub","family":"Rembieli\u0144ski","sequence":"first","affiliation":[{"name":"Department of Theoretical Physics, Faculty of Physics and Applied Informatics, University of Lodz, Pomorska 149\/153, 90-236 \u0141\u00f3d\u017a, Poland"}]},{"given":"Pawe\u0142","family":"Caban","sequence":"additional","affiliation":[{"name":"Department of Theoretical Physics, Faculty of Physics and Applied Informatics, University of Lodz, Pomorska 149\/153, 90-236 \u0141\u00f3d\u017a, Poland"}]}],"member":"9598","published-online":{"date-parts":[[2021,3,23]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"V. 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