1920:
79:
dynamics of the total system, i.e. the system and the environment, to obtain information about the reduced system of interest by averaging the appropriate observables over the degrees of freedom of the environment. To model the dissipative effects due to the interaction with the environment, the
2510:
1773:
1281:
74:
interacting with the surrounding radiation field). A complete microscopic description of the degrees of freedom of the environment is typically too complicated. Hence, one looks for simpler descriptions of the dynamics of the open system. In principle, one should investigate the
466:
1443:
2087:
2668:
1915:{\displaystyle \operatorname {Dom} \left({\mathcal {L}}\right):=\left\{a\in {\mathcal {A}}~\left\vert ~\lim _{t\rightarrow 0}{\frac {{\mathcal {T}}_{t}(a)-a}{t}}=b{\text{ in }}\sigma {\text{-weak topology}}\right.\right\}}
2350:
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2005:
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2505:{\displaystyle {\mathcal {L}}\left(a\right)=i\left+\sum _{j}\left(V_{j}^{\dagger }aV_{j}-{\frac {1}{2}}\left\{V_{j}^{\dagger }V_{j},a\right\}\right)}
3137:
Gorini, Vittorio; Kossakowski, Andrzej; Sudarshan, Ennackal Chandy George (1976). "Completely positive dynamical semigroups of N-level systems".
223:
2988:
Carlen, Eric A.; Maas, Jan (September 2017). "Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance".
1276:{\displaystyle \lim _{\alpha }\langle u,(Tx_{\alpha })u\rangle =\sup _{\alpha }\langle u,(Tx_{\alpha })u\rangle =\langle u,(Tx)u\rangle }
2148:
975:
738:
3336:
Chebotarev, A.M; Fagnola, F (March 1998). "Sufficient
Conditions for Conservativity of Minimal Quantum Dynamical Semigroups".
2807:
Chebotarev, A.M; Fagnola, F (March 1998). "Sufficient
Conditions for Conservativity of Minimal Quantum Dynamical Semigroups".
2548:
92:
or a stochastic Schrödinger equation in which the infinite degrees of freedom of the environment are "synthesized" as a few
1734:
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1555:
1464:
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288:
3393:
3320:
1665:
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2202:
461:{\displaystyle {\mathcal {T}}_{t+s}\left(a\right)={\mathcal {T}}_{t}\left({\mathcal {T}}_{s}\left(a\right)\right)}
506:
336:
2949:
Fagnola, Franco; UmanitĂ , Veronica (2007-09-01). "Generators of detailed balance quantum markov semigroups".
51:
1928:
2518:
2242:
1668:
1438:{\displaystyle {\mathcal {T}}_{t}\left({\boldsymbol {1}}\right)={\boldsymbol {1}},\quad \forall t\geq 0,}
708:
1524:
1114:
1060:
916:
885:
834:
604:
541:
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471:
96:. Mathematically, time evolution in a Markovian open quantum system is no longer described by means of
66:
is not realistic because it should be completely isolated while, in practice, it is influenced by the
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2323:
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2124:
2096:
1981:
1710:
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951:
806:
655:
199:
175:
151:
124:
3222:
Chruściński, Dariusz; Pascazio, Saverio (September 2017). "A Brief
History of the GKLS Equation".
570:
16:
A kind of mathematical structure which describes the dynamics in a
Markovian open quantum system.
3102:
Kossakowski, A. (December 1972). "On quantum statistical mechanics of non-Hamiltonian systems".
2082:{\displaystyle \lim _{t\rightarrow 0^{+}}\left\|{\mathcal {T}}_{t}-{\mathcal {T}}_{0}\right\|=0}
1671:, but also in deducing regularity conditions for paths of classical Markov processes in view of
81:
47:
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2714:
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1660:) plays an important role not only in the proof of uniqueness and unitarity of solution of a
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to an environment, which typically has a large number of degrees of freedom (for example an
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8:
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3084: – Quantum mechanical system that interacts with a quantum-mechanical environment
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is called quantum Markov semigroup. Notice that, the identity-preserving property and
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28:
2663:{\displaystyle \sum _{j}V_{j}^{\dagger }V_{j}\in {\mathcal {B}}({\mathcal {H}})}
3409:
Dixmier, Jacques (1957). "Les algèbres d'opérateurs dans l'espace hilbertien".
63:
3253:
3011:
2972:
2932:
2915:
2864:
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1967:
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3066: – Markovian quantum master equation for density matrices (mixed states)
146:
93:
3078: – Description of physical properties at the atomic and subatomic scale
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3069:
3045:
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1973:
945:
275:{\displaystyle {\mathcal {T}}:=\left({\mathcal {T}}_{t}\right)_{t\geq 0}}
89:
3200:
2794:
2754:
3158:
2951:
Infinite
Dimensional Analysis, Quantum Probability and Related Topics
1369:
is said to be identity-preserving (or conservative, or
Markovian) if
1307:
3388:(Second ed.). New York: McGraw-Hill Science/Engineering/Math.
3236:
3002:
2963:
2916:"Classification and decomposition of Quantum Markov Semigroups"
3402:
3036: – Topologies on the set of operators on a Hilbert space
2189:{\displaystyle \mathrm {Dom} ({\mathcal {L}})={\mathcal {A}}}
1683:
The infinitesimal generator of a quantum dynamical semigroup
121:, so the dimensionality of the system could be infinite. Let
1968:
Characterization of generators of uniformly continuous QMSs
1904:
117:
In general, quantum dynamical semigroups can be defined on
71:
3136:
1005:{\displaystyle T:{\mathcal {A}}\rightarrow {\mathcal {A}}}
831:
Under the condition of complete positivity, the operators
3042: – *-algebra of bounded operators on a Hilbert space
776:{\displaystyle t\mapsto {\mathcal {T}}_{t}\left(a\right)}
2848:"Transience and recurrence of quantum Markov semigroups"
3059:
Pages displaying short descriptions of redirect targets
3050:
Pages displaying short descriptions of redirect targets
2588:{\displaystyle V_{j}\in {\mathcal {B}}({\mathcal {H}})}
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1760:{\displaystyle \operatorname {Dom} ({\mathcal {L}})}
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3313:
Operator algebras and quantum statistical mechanics
3221:
3177:"On the generators of quantum dynamical semigroups"
2706:{\displaystyle H\in {\mathcal {B}}({\mathcal {H}})}
2320:Under such assumption, the infinitesimal generator
1590:{\displaystyle \left\|{\mathcal {T}}_{t}\right\|=1}
1486:{\displaystyle {\boldsymbol {1}}\in {\mathcal {A}}}
3072: – Random process independent of past history
3057: – Generalization of the exponential function
3048: – Generalization of the exponential function
2846:Fagnola, Franco; Rebolledo, Rolando (2003-06-01).
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1050:{\displaystyle \left(x_{\alpha }\right)_{\alpha }}
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326:{\displaystyle {\mathcal {T}}_{0}\left(a\right)=a}
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2002:is uniformly continuous in addition, which means
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1340:
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2010:
1831:
1200:
1153:
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3311:Bratteli, Ola; Robinson, Derek William (1987).
2800:
34:. The axiomatic definition of the prototype of
2948:
3280:"Quantum Markov semigroups and quantum flows"
3170:
3168:
1012:is said to be normal if for every increasing
2786:{\displaystyle \left\{\cdot ,\cdot \right\}}
2232:{\displaystyle t\mapsto {\mathcal {T}}_{t}a}
1270:
1246:
1240:
1209:
1193:
1162:
100:of unitary maps, but one needs to introduce
3315:(2nd ed.). New York: Springer-Verlag.
3101:
2239:will automatically be continuous for every
529:{\displaystyle \forall a\in {\mathcal {A}}}
359:{\displaystyle \forall a\in {\mathcal {A}}}
3379:
3377:
3165:
3095:
2987:
2883:"Decoherence of quantum Markov semigroups"
42:in 1972, and then developed by V. Gorini,
3349:
3295:
3235:
3001:
2962:
2931:
2880:
2863:
2820:
1493:is the identity element. For simplicity,
3273:
3271:
3174:
1669:quantum stochastic differential equation
196:is a collection of bounded operators on
3408:
3374:
3277:
3224:Open Systems & Information Dynamics
2913:
3423:
3215:
3181:Communications in Mathematical Physics
3130:
2887:Annales de l'Institut Henri Poincaré B
3383:
3268:
2920:Probability Theory and Related Fields
2852:Probability Theory and Related Fields
1957:{\displaystyle {\mathcal {L}}(a):=b}
1371:
2538:{\displaystyle a\in {\mathcal {A}}}
2262:{\displaystyle a\in {\mathcal {A}}}
728:{\displaystyle a\in {\mathcal {A}}}
172:, a quantum dynamical semigroup on
145:be a von Neumann algebra acting on
13:
3297:10.22199/S07160917.1999.0003.00002
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1632:
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1545:{\displaystyle {\mathcal {T}}_{t}}
1531:
1502:
1478:
1420:
1384:
1354:
1322:
1135:{\displaystyle {\mathcal {A}}_{+}}
1121:
1081:{\displaystyle {\mathcal {A}}_{+}}
1067:
997:
987:
957:
937:{\displaystyle {\mathcal {A}}_{+}}
923:
906:{\displaystyle {\mathcal {T}}_{t}}
892:
882:-weakly continuous if and only if
855:{\displaystyle {\mathcal {T}}_{t}}
841:
812:
783:is continuous with respect to the
751:
720:
661:
625:{\displaystyle {\mathcal {T}}_{t}}
611:
562:{\displaystyle {\mathcal {T}}_{t}}
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340:
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282:, with the following properties:
245:
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3411:Mathematical Reviews (MathSciNet)
913:are normal. Recall that, letting
496:{\displaystyle \forall s,t\geq 0}
113:Quantum dynamical semigroup (QDS)
2914:UmanitĂ , Veronica (April 2006).
1978:If the quantum Markov semigroup
1469:
1412:
1400:
3139:Journal of Mathematical Physics
3104:Reports on Mathematical Physics
1419:
652:-weakly continuous operator in
3338:Journal of Functional Analysis
2990:Journal of Functional Analysis
2809:Journal of Functional Analysis
2700:
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2657:
2647:
2582:
2572:
2337:{\displaystyle {\mathcal {L}}}
2289:{\displaystyle {\mathcal {L}}}
2209:
2173:
2163:
2138:{\displaystyle {\mathcal {A}}}
2110:{\displaystyle {\mathcal {L}}}
2069:
2033:
2017:
1995:{\displaystyle {\mathcal {T}}}
1945:
1939:
1869:
1863:
1838:
1754:
1744:
1724:{\displaystyle {\mathcal {L}}}
1700:{\displaystyle {\mathcal {T}}}
1679:Infinitesimal generator of QDS
1640:{\displaystyle {\mathcal {T}}}
1577:
1560:
1510:{\displaystyle {\mathcal {T}}}
1362:{\displaystyle {\mathcal {T}}}
1345:A quantum dynamical semigroup
1341:Quantum Markov semigroup (QMS)
1330:{\displaystyle {\mathcal {H}}}
1264:
1255:
1234:
1218:
1187:
1171:
992:
965:{\displaystyle {\mathcal {A}}}
820:{\displaystyle {\mathcal {A}}}
745:
669:{\displaystyle {\mathcal {A}}}
213:{\displaystyle {\mathcal {A}}}
189:{\displaystyle {\mathcal {A}}}
165:{\displaystyle {\mathcal {H}}}
138:{\displaystyle {\mathcal {A}}}
107:
1:
3088:
57:
3124:10.1016/0034-4877(72)90010-9
2907:10.1016/j.anihpb.2004.12.003
2801:Selected recent publications
2272:the infinitesimal generator
2093:the infinitesimal generator
27:describes the dynamics in a
7:
3027:
1656:
1451:
10:
3452:
1971:
84:is replaced by a suitable
3254:10.1142/S1230161217400017
3012:10.1016/j.jfa.2017.05.003
2973:10.1142/S0219025707002762
2933:10.1007/s00440-005-0450-7
2881:Rebolledo, R (May 2005).
2865:10.1007/s00440-003-0268-0
2344:has the characterization
102:quantum Markov semigroups
36:quantum Markov semigroups
3278:Fagnola, Franco (1999).
3175:Lindblad, Goran (1976).
948:of positive elements in
38:was first introduced by
25:quantum Markov semigroup
2309:{\displaystyle \sigma }
2121:on von Neumann algebra
1616:{\displaystyle t\geq 0}
875:{\displaystyle \sigma }
796:{\displaystyle \sigma }
695:{\displaystyle t\geq 0}
645:{\displaystyle \sigma }
592:{\displaystyle t\geq 0}
3384:Rudin, Walter (1991).
3360:10.1006/jfan.1997.3189
2831:10.1006/jfan.1997.3189
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972:, a positive operator
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214:
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139:
3055:Contraction semigroup
2788:
2748:
2746:{\displaystyle \left}
2708:
2665:
2590:
2540:
2507:
2339:
2311:
2291:
2264:
2234:
2191:
2140:
2112:
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1997:
1959:
1917:
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1702:
1649:contraction semigroup
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119:von Neumann algebras
98:one-parameter groups
82:Schrödinger equation
3386:Functional analysis
3246:2017OSID...2440001C
3193:1976CMaPh..48..119L
3151:1976JMP....17..821G
3116:1972RpMP....3..247K
3082:Open quantum system
3040:Von Neumann algebra
3034:Operator topologies
2899:2005AIHPB..41..349R
2626:
2475:
2429:
2316:-weakly continuous.
1311:linear sub-manifold
571:completely positive
32:open quantum system
3201:10.1007/BF01608499
2783:
2743:
2717:. Moreover, above
2703:
2660:
2612:
2611:
2585:
2535:
2502:
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2415:
2409:
2334:
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1992:
1954:
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1132:
1101:
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1002:
962:
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852:
817:
803:-weak topology on
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622:
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526:
493:
458:
356:
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48:E. C. G. Sudarshan
3431:Quantum mechanics
3076:Quantum mechanics
2602:
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2400:
2009:
1901:
1893:
1882:
1830:
1829:
1821:
1459:
1458:
1299:{\displaystyle u}
1199:
1152:
1104:{\displaystyle x}
1090:least upper bound
90:Lindblad equation
44:A. M. Kossakowski
40:A. M. Kossakowski
21:quantum mechanics
3443:
3436:Semigroup theory
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3351:funct-an/9711006
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3023:
3005:
2996:(5): 1810–1869.
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2822:funct-an/9711006
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2144:
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2136:
2134:
2133:
2119:bounded operator
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1758:
1753:
1752:
1730:
1728:
1727:
1722:
1720:
1719:
1707:is the operator
1706:
1704:
1703:
1698:
1696:
1695:
1646:
1644:
1643:
1638:
1636:
1635:
1622:
1620:
1619:
1614:
1596:
1594:
1593:
1588:
1580:
1576:
1575:
1570:
1569:
1551:
1549:
1548:
1543:
1541:
1540:
1535:
1534:
1516:
1514:
1513:
1508:
1506:
1505:
1492:
1490:
1489:
1484:
1482:
1481:
1472:
1453:
1444:
1442:
1441:
1436:
1415:
1407:
1403:
1394:
1393:
1388:
1387:
1372:
1368:
1366:
1365:
1360:
1358:
1357:
1336:
1334:
1333:
1328:
1326:
1325:
1305:
1303:
1302:
1297:
1282:
1280:
1279:
1274:
1233:
1232:
1207:
1186:
1185:
1160:
1141:
1139:
1138:
1133:
1131:
1130:
1125:
1124:
1110:
1108:
1107:
1102:
1087:
1085:
1084:
1079:
1077:
1076:
1071:
1070:
1056:
1054:
1053:
1048:
1046:
1045:
1040:
1036:
1035:
1011:
1009:
1008:
1003:
1001:
1000:
991:
990:
971:
969:
968:
963:
961:
960:
943:
941:
940:
935:
933:
932:
927:
926:
912:
910:
909:
904:
902:
901:
896:
895:
881:
879:
878:
873:
861:
859:
858:
853:
851:
850:
845:
844:
826:
824:
823:
818:
816:
815:
802:
800:
799:
794:
782:
780:
779:
774:
772:
761:
760:
755:
754:
734:
732:
731:
726:
724:
723:
701:
699:
698:
693:
675:
673:
672:
667:
665:
664:
651:
649:
648:
643:
631:
629:
628:
623:
621:
620:
615:
614:
598:
596:
595:
590:
568:
566:
565:
560:
558:
557:
552:
551:
535:
533:
532:
527:
525:
524:
502:
500:
499:
494:
467:
465:
464:
459:
457:
453:
452:
441:
440:
435:
434:
422:
421:
416:
415:
405:
394:
393:
382:
381:
365:
363:
362:
357:
355:
354:
332:
330:
329:
324:
316:
305:
304:
299:
298:
281:
279:
278:
273:
271:
270:
259:
255:
254:
249:
248:
233:
232:
219:
217:
216:
211:
209:
208:
195:
193:
192:
187:
185:
184:
171:
169:
168:
163:
161:
160:
144:
142:
141:
136:
134:
133:
3451:
3450:
3446:
3445:
3444:
3442:
3441:
3440:
3421:
3420:
3419:
3418:
3407:
3403:
3396:
3382:
3375:
3334:
3330:
3323:
3309:
3305:
3276:
3269:
3220:
3216:
3173:
3166:
3135:
3131:
3100:
3096:
3091:
3058:
3049:
3030:
2803:
2795:anti-commutator
2768:
2764:
2762:
2759:
2758:
2728:
2724:
2722:
2719:
2718:
2694:
2693:
2684:
2683:
2675:
2672:
2671:
2651:
2650:
2641:
2640:
2631:
2627:
2621:
2616:
2606:
2600:
2597:
2596:
2576:
2575:
2566:
2565:
2556:
2552:
2550:
2547:
2546:
2529:
2528:
2520:
2517:
2516:
2480:
2476:
2470:
2465:
2460:
2456:
2446:
2437:
2433:
2424:
2419:
2414:
2410:
2404:
2382:
2378:
2361:
2355:
2354:
2352:
2349:
2348:
2328:
2327:
2325:
2322:
2321:
2301:
2298:
2297:
2280:
2279:
2277:
2274:
2273:
2253:
2252:
2244:
2241:
2240:
2220:
2214:
2213:
2212:
2204:
2201:
2200:
2180:
2179:
2167:
2166:
2152:
2150:
2147:
2146:
2129:
2128:
2126:
2123:
2122:
2101:
2100:
2098:
2095:
2094:
2062:
2056:
2055:
2054:
2045:
2039:
2038:
2037:
2036:
2032:
2024:
2020:
2013:
2007:
2004:
2003:
1986:
1985:
1983:
1980:
1979:
1976:
1970:
1933:
1932:
1930:
1927:
1926:
1898:
1890:
1857:
1851:
1850:
1849:
1848:
1846:
1834:
1826:
1822:
1813:
1812:
1805:
1801:
1788:
1787:
1783:
1775:
1772:
1771:
1748:
1747:
1736:
1733:
1732:
1715:
1714:
1712:
1709:
1708:
1691:
1690:
1688:
1685:
1684:
1681:
1673:operator theory
1654:The Condition (
1631:
1630:
1628:
1625:
1624:
1602:
1599:
1598:
1571:
1565:
1564:
1563:
1559:
1557:
1554:
1553:
1536:
1530:
1529:
1528:
1526:
1523:
1522:
1501:
1500:
1498:
1495:
1494:
1477:
1476:
1468:
1466:
1463:
1462:
1411:
1399:
1395:
1389:
1383:
1382:
1381:
1379:
1376:
1375:
1353:
1352:
1350:
1347:
1346:
1343:
1321:
1320:
1318:
1315:
1314:
1291:
1288:
1287:
1228:
1224:
1203:
1181:
1177:
1156:
1150:
1147:
1146:
1126:
1120:
1119:
1118:
1116:
1113:
1112:
1096:
1093:
1092:
1072:
1066:
1065:
1064:
1062:
1059:
1058:
1041:
1031:
1027:
1023:
1022:
1020:
1017:
1016:
996:
995:
986:
985:
977:
974:
973:
956:
955:
953:
950:
949:
928:
922:
921:
920:
918:
915:
914:
897:
891:
890:
889:
887:
884:
883:
867:
864:
863:
846:
840:
839:
838:
836:
833:
832:
811:
810:
808:
805:
804:
788:
785:
784:
762:
756:
750:
749:
748:
740:
737:
736:
719:
718:
710:
707:
706:
681:
678:
677:
660:
659:
657:
654:
653:
637:
634:
633:
616:
610:
609:
608:
606:
603:
602:
578:
575:
574:
553:
547:
546:
545:
543:
540:
539:
520:
519:
508:
505:
504:
473:
470:
469:
442:
436:
430:
429:
428:
427:
423:
417:
411:
410:
409:
395:
383:
377:
376:
375:
373:
370:
369:
350:
349:
338:
335:
334:
306:
300:
294:
293:
292:
290:
287:
286:
260:
250:
244:
243:
242:
238:
237:
228:
227:
225:
222:
221:
204:
203:
201:
198:
197:
180:
179:
177:
174:
173:
156:
155:
153:
150:
149:
129:
128:
126:
123:
122:
115:
110:
86:master equation
60:
17:
12:
11:
5:
3449:
3439:
3438:
3433:
3417:
3416:
3401:
3395:978-0070542365
3394:
3373:
3344:(2): 382–404.
3328:
3321:
3303:
3267:
3230:(3): 1740001.
3214:
3187:(2): 119–130.
3164:
3129:
3110:(4): 247–274.
3093:
3092:
3090:
3087:
3086:
3085:
3079:
3073:
3067:
3061:
3052:
3043:
3037:
3029:
3026:
3025:
3024:
2985:
2957:(3): 335–363.
2946:
2926:(4): 603–623.
2911:
2893:(3): 349–373.
2878:
2858:(2): 289–306.
2843:
2815:(2): 382–404.
2802:
2799:
2781:
2777:
2774:
2771:
2767:
2741:
2737:
2734:
2731:
2727:
2702:
2697:
2692:
2687:
2682:
2679:
2659:
2654:
2649:
2644:
2639:
2634:
2630:
2624:
2619:
2615:
2609:
2605:
2584:
2579:
2574:
2569:
2564:
2559:
2555:
2532:
2527:
2524:
2513:
2512:
2500:
2495:
2491:
2488:
2483:
2479:
2473:
2468:
2464:
2459:
2453:
2450:
2445:
2440:
2436:
2432:
2427:
2422:
2418:
2413:
2407:
2403:
2399:
2395:
2391:
2388:
2385:
2381:
2377:
2374:
2370:
2367:
2364:
2358:
2331:
2318:
2317:
2305:
2283:
2270:
2256:
2251:
2248:
2228:
2223:
2217:
2211:
2208:
2197:
2183:
2178:
2175:
2170:
2165:
2161:
2158:
2155:
2132:
2104:
2078:
2075:
2071:
2065:
2059:
2053:
2048:
2042:
2035:
2027:
2023:
2019:
2016:
2012:
1989:
1972:Main article:
1969:
1966:
1953:
1950:
1947:
1944:
1941:
1936:
1923:
1922:
1910:
1905:
1900:-weak topology
1897:
1892: in
1889:
1886:
1881:
1877:
1874:
1871:
1868:
1865:
1860:
1854:
1843:
1840:
1837:
1833:
1825:
1816:
1811:
1808:
1804:
1800:
1796:
1791:
1786:
1782:
1779:
1756:
1751:
1746:
1743:
1740:
1718:
1694:
1680:
1677:
1634:
1612:
1609:
1606:
1586:
1583:
1579:
1574:
1568:
1562:
1539:
1533:
1504:
1480:
1475:
1471:
1457:
1456:
1447:
1445:
1434:
1431:
1428:
1425:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1392:
1386:
1356:
1342:
1339:
1324:
1295:
1284:
1283:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1231:
1227:
1223:
1220:
1217:
1214:
1211:
1206:
1202:
1198:
1195:
1192:
1189:
1184:
1180:
1176:
1173:
1170:
1167:
1164:
1159:
1155:
1129:
1123:
1100:
1075:
1069:
1044:
1039:
1034:
1030:
1026:
999:
994:
989:
984:
981:
959:
931:
925:
900:
894:
871:
849:
843:
829:
828:
814:
792:
771:
768:
765:
759:
753:
747:
744:
722:
717:
714:
703:
691:
688:
685:
663:
641:
619:
613:
600:
588:
585:
582:
556:
550:
537:
523:
518:
515:
512:
492:
489:
486:
483:
480:
477:
456:
451:
448:
445:
439:
433:
426:
420:
414:
408:
404:
401:
398:
392:
389:
386:
380:
367:
353:
348:
345:
342:
322:
319:
315:
312:
309:
303:
297:
269:
266:
263:
258:
253:
247:
241:
236:
231:
207:
183:
159:
132:
114:
111:
109:
106:
94:quantum noises
64:quantum system
59:
56:
52:Göran Lindblad
15:
9:
6:
4:
3:
2:
3448:
3437:
3434:
3432:
3429:
3428:
3426:
3412:
3405:
3397:
3391:
3387:
3380:
3378:
3369:
3365:
3361:
3357:
3352:
3347:
3343:
3339:
3332:
3324:
3322:3-540-17093-6
3318:
3314:
3307:
3298:
3293:
3289:
3285:
3281:
3274:
3272:
3263:
3259:
3255:
3251:
3247:
3243:
3238:
3233:
3229:
3225:
3218:
3210:
3206:
3202:
3198:
3194:
3190:
3186:
3182:
3178:
3171:
3169:
3160:
3156:
3152:
3148:
3144:
3140:
3133:
3125:
3121:
3117:
3113:
3109:
3105:
3098:
3094:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3056:
3053:
3047:
3044:
3041:
3038:
3035:
3032:
3031:
3021:
3017:
3013:
3009:
3004:
2999:
2995:
2991:
2986:
2982:
2978:
2974:
2970:
2965:
2960:
2956:
2952:
2947:
2943:
2939:
2934:
2929:
2925:
2921:
2917:
2912:
2908:
2904:
2900:
2896:
2892:
2888:
2884:
2879:
2875:
2871:
2866:
2861:
2857:
2853:
2849:
2844:
2840:
2836:
2832:
2828:
2823:
2818:
2814:
2810:
2805:
2804:
2798:
2796:
2779:
2775:
2772:
2769:
2765:
2756:
2739:
2735:
2732:
2729:
2725:
2716:
2680:
2677:
2637:
2632:
2628:
2622:
2617:
2613:
2607:
2603:
2562:
2557:
2553:
2525:
2522:
2498:
2493:
2489:
2486:
2481:
2477:
2471:
2466:
2462:
2457:
2451:
2448:
2443:
2438:
2434:
2430:
2425:
2420:
2416:
2411:
2405:
2401:
2397:
2393:
2389:
2386:
2383:
2379:
2375:
2372:
2368:
2365:
2362:
2347:
2346:
2345:
2303:
2296:will be also
2271:
2249:
2246:
2226:
2221:
2206:
2198:
2176:
2120:
2092:
2091:
2090:
2076:
2073:
2063:
2051:
2046:
2025:
2021:
2014:
1975:
1965:
1951:
1948:
1942:
1908:
1895:
1887:
1884:
1879:
1875:
1872:
1866:
1858:
1841:
1835:
1823:
1809:
1806:
1802:
1798:
1794:
1784:
1780:
1777:
1770:
1769:
1768:
1741:
1738:
1676:
1674:
1670:
1667:
1666:Parthasarathy
1663:
1659:
1658:
1652:
1650:
1610:
1607:
1604:
1584:
1581:
1572:
1537:
1520:
1473:
1455:
1448:
1446:
1432:
1429:
1426:
1423:
1416:
1408:
1404:
1396:
1390:
1374:
1373:
1370:
1338:
1312:
1309:
1293:
1267:
1261:
1258:
1252:
1249:
1243:
1237:
1229:
1225:
1221:
1215:
1212:
1204:
1196:
1190:
1182:
1178:
1174:
1168:
1165:
1157:
1145:
1144:
1143:
1127:
1098:
1091:
1073:
1042:
1037:
1032:
1028:
1024:
1015:
982:
979:
947:
929:
898:
869:
847:
790:
769:
766:
763:
757:
742:
715:
712:
704:
689:
686:
683:
639:
617:
601:
586:
583:
580:
572:
554:
538:
516:
513:
490:
487:
484:
481:
478:
454:
449:
446:
443:
437:
424:
418:
406:
402:
399:
396:
390:
387:
384:
368:
346:
343:
320:
317:
313:
310:
307:
301:
285:
284:
283:
267:
264:
261:
256:
251:
239:
234:
220:, denoted by
148:
147:Hilbert space
120:
105:
103:
99:
95:
91:
87:
83:
78:
73:
69:
65:
55:
53:
49:
45:
41:
37:
33:
30:
26:
22:
3410:
3404:
3385:
3341:
3337:
3331:
3312:
3306:
3290:(3): 1–144.
3287:
3284:Proyecciones
3283:
3227:
3223:
3217:
3184:
3180:
3142:
3138:
3132:
3107:
3103:
3097:
3070:Markov chain
3046:C0 semigroup
2993:
2989:
2954:
2950:
2923:
2919:
2890:
2886:
2855:
2851:
2812:
2808:
2753:denotes the
2715:self-adjoint
2514:
2319:
2145:with domain
1977:
1924:
1731:with domain
1682:
1655:
1653:
1460:
1449:
1344:
1285:
830:
116:
101:
88:, such as a
61:
35:
24:
18:
3064:Lindbladian
1974:Lindbladian
946:convex cone
944:denote the
108:Definitions
3425:Categories
3237:1710.05993
3145:(5): 821.
3089:References
3003:1609.01254
2755:commutator
2117:will be a
1519:positivity
1308:norm-dense
735:, the map
58:Motivation
3020:119734534
2964:0707.2147
2942:119409078
2874:123052568
2776:⋅
2770:⋅
2736:⋅
2730:⋅
2681:∈
2638:∈
2623:†
2604:∑
2563:∈
2526:∈
2472:†
2444:−
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2304:σ
2250:∈
2210:↦
2052:−
2018:→
1896:σ
1873:−
1839:→
1810:∈
1781:
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1608:≥
1474:∈
1427:≥
1421:∀
1286:for each
1271:⟩
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1230:α
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1158:α
1142:one has
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62:An ideal
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1767:, where
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1561:‖
705:For all
676:for all
573:for all
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