{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T07:37:01Z","timestamp":1774597021495,"version":"3.50.1"},"reference-count":36,"publisher":"World Scientific Pub Co Pte Ltd","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Open Syst. Inf. Dyn."],"published-print":{"date-parts":[[2024,6]]},"abstract":"<jats:p> We generalize the result of Gorini, Kossakowski, and Sudarshan [J. Math. Phys. 17:821, 1976] that every generator of a quantum-dynamical semigroup decomposes uniquely into a closed and a dissipative part, assuming the trace of both vanishes. More precisely, we show that given any generator [Formula: see text] of a completely positive dynamical semigroup and any matrix [Formula: see text] there exists a unique matrix [Formula: see text] and a unique completely positive map [Formula: see text] such that (i) [Formula: see text], (ii) the superoperator [Formula: see text] has trace zero, and (iii) [Formula: see text] is a real number. The key to proving this is the relation between the trace of a completely positive map, the trace of its Kraus operators, and expectation values of its Choi matrix. Moreover, we show that the above decomposition is orthogonal with respect to some [Formula: see text]-weighted inner product. <\/jats:p>","DOI":"10.1142\/s1230161224500070","type":"journal-article","created":{"date-parts":[[2024,7,2]],"date-time":"2024-07-02T05:40:40Z","timestamp":1719898840000},"source":"Crossref","is-referenced-by-count":5,"title":["Understanding and Generalizing Unique Decompositions of Generators of Dynamical Semigroups"],"prefix":"10.1142","volume":"31","author":[{"given":"Frederik","family":"vom Ende","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, Arnimallee 14, 14195 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2024,6,28]]},"reference":[{"key":"S1230161224500070BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-05187-6"},{"key":"S1230161224500070BIB002","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0653-8"},{"key":"S1230161224500070BIB003","volume-title":"The Theory of Open Quantum Systems","author":"Breuer H.-P.","year":"2002"},{"key":"S1230161224500070BIB004","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(75)90075-0"},{"key":"S1230161224500070BIB005","doi-asserted-by":"publisher","DOI":"10.1142\/S1230161214400046"},{"key":"S1230161224500070BIB006","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.127.050401"},{"key":"S1230161224500070BIB007","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.3.023006"},{"key":"S1230161224500070BIB008","doi-asserted-by":"publisher","DOI":"10.22331\/q-2021-11-25-590"},{"key":"S1230161224500070BIB009","doi-asserted-by":"publisher","DOI":"10.1016\/0034-4877(80)90065-8"},{"key":"S1230161224500070BIB010","doi-asserted-by":"publisher","DOI":"10.1016\/S0034-4877(09)90022-2"},{"key":"S1230161224500070BIB011","unstructured":"Freiberger, T. and  Matthieu, M. , in:  Elementary Operators & Applications,  179\u2013187,  World Scientific,  Singapore,  1992."},{"key":"S1230161224500070BIB012","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.107.054115"},{"key":"S1230161224500070BIB013","doi-asserted-by":"publisher","DOI":"10.1063\/1.522979"},{"key":"S1230161224500070BIB014","volume-title":"The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement","author":"Heinosaari T.","year":"2012"},{"key":"S1230161224500070BIB015","volume-title":"Lie Groups, Convex Cones, and Semigroups","author":"Hilgert J.","year":"1989"},{"key":"S1230161224500070BIB016","doi-asserted-by":"publisher","DOI":"10.1515\/9783110273403"},{"key":"S1230161224500070BIB017","volume-title":"Matrix Analysis","author":"Horn R.","year":"1987"},{"key":"S1230161224500070BIB018","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511840371"},{"key":"S1230161224500070BIB019","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.104.052219"},{"key":"S1230161224500070BIB020","doi-asserted-by":"publisher","DOI":"10.1140\/epjqt\/s40507-022-00138-x"},{"key":"S1230161224500070BIB021","first-page":"1021","volume":"20","author":"Kossakowski A.","year":"1972","journal-title":"Bull. 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