1123:
906:
442:
962:
598:
839:
60:
398:
368:
242:
645:
562:
506:
262:
793:
213:
999:
141:
618:
486:
466:
345:
164:
702:
542:
760:
729:
673:
184:
86:
296:
115:
739:. Convex linear combinations of rays naturally give rise to density matrix which (still in case of an irreducible representation) correspond to mixed states.
847:
1256:
Cirelli, R; Lanzavecchia, P; Mania, A (1983). "Normal pure states of the von
Neumann algebra of bounded operators as Kahler manifold".
1246:
1125:. In quantum theory, it describes how to make states of the composite system from states of its constituents. It is only an
1349:
407:
933:
992:
is an embedding of the
Cartesian product of two projective spaces into the projective space associated to the
315:
736:
648:
567:
923:
As such, the projectivization of, e.g., two-dimensional complex
Hilbert space (the space describing one
1293:
Kong, Otto C. W.; Liu, Wei-Yin (2021). "Noncommutative
Coordinate Picture of the Quantum Phase Space".
732:
1153:
813:
34:
377:
1148:
800:
445:
350:
400:. The Born rule demands that if the system is physical and measurable, its wave function has unit
1368:
928:
917:
218:
1118:{\displaystyle \mathbf {P} (H)\times \mathbf {P} (H')\to \mathbf {P} (H\otimes H'),(,)\mapsto }
630:
547:
491:
247:
765:
192:
1219:
Ashtekar, Abhay; Schilling, Troy A. (1999). "Geometrical
Formulation of Quantum Mechanics".
120:
1312:
1265:
1135:
603:
471:
451:
330:
149:
144:
678:
518:
8:
1130:
796:
401:
1316:
1269:
1328:
1302:
1224:
745:
714:
658:
169:
71:
1277:
675:). No measurement can recover the phase of a ray; it is not observable. One says that
275:
94:
1345:
1281:
1242:
985:
804:
321:
89:
21:
1332:
272:, applied to a complex Hilbert space. In quantum mechanics, the equivalence classes
1320:
1273:
1234:
1158:
269:
913:
1238:
1324:
993:
973:
969:
509:
63:
1362:
1285:
989:
808:
325:
66:
965:
621:
513:
1129:, not a surjection; most of the tensor product space does not lie in its
705:
17:
652:
1229:
1126:
320:
The physical significance of the projective
Hilbert space is that in
1307:
1196:
901:{\displaystyle \mathbf {P} (H_{n})=\mathbb {C} \mathbf {P} ^{n-1}}
976:
for details of the projectivization construction in this case.
924:
988:
of projective
Hilbert spaces is not a projective space. The
448:. The unit norm constraint does not completely determine
1255:
1202:
1002:
936:
850:
816:
768:
748:
717:
681:
661:
633:
606:
570:
550:
521:
494:
474:
454:
410:
380:
353:
333:
278:
250:
221:
195:
172:
152:
123:
97:
74:
37:
1172:
795:, the Hilbert space reduces to a finite-dimensional
1184:
799:and the set of projective rays may be treated as a
735:of the algebra of observables then the rays induce
1117:
956:
900:
833:
787:
754:
723:
696:
667:
639:
612:
592:
556:
536:
500:
480:
460:
436:
392:
362:
339:
290:
256:
236:
207:
178:
158:
135:
109:
80:
54:
1218:
1360:
1344:. Providence (R.I.): American Mathematical Soc.
1258:Journal of Physics A: Mathematical and General
544:action) and retain its normalization. Such a
437:{\displaystyle \langle \psi |\psi \rangle =1}
957:{\displaystyle \mathbb {C} \mathbf {P} ^{1}}
425:
411:
316:Wigner's theorem § Rays and ray space
1306:
1228:
938:
920:, derived from the Hilbert space's norm.
876:
1339:
1292:
1190:
1178:
655:of observables and a representation on
1361:
1203:Cirelli, Lanzavecchia & Mania 1983
1342:Algebraic Curves and Riemann Surfaces
996:of the two Hilbert spaces, given by
649:quantum state (algebraic definition)
1223:. New York, NY: Springer New York.
593:{\displaystyle \lambda =e^{i\phi }}
13:
818:
647:correspond to the same state (cf.
268:This is the usual construction of
14:
1380:
1264:(16). IOP Publishing: 3829–3835.
244:for some non-zero complex number
1043:
1021:
1004:
944:
882:
852:
39:
834:{\displaystyle \mathrm {U} (n)}
444:, in which case it is called a
55:{\displaystyle \mathbf {P} (H)}
1112:
1100:
1097:
1094:
1091:
1085:
1079:
1073:
1070:
1064:
1047:
1039:
1036:
1025:
1014:
1008:
869:
856:
828:
822:
691:
685:
531:
525:
418:
393:{\displaystyle \lambda \neq 0}
285:
279:
104:
98:
49:
43:
1:
1212:
762:is finite-dimensional, i.e.,
363:{\displaystyle \lambda \psi }
1161:, for the concept in general
7:
1278:10.1088/0305-4470/16/16/020
1239:10.1007/978-1-4612-1422-9_3
1142:
627:Rays that differ by such a
488:could be multiplied by any
309:
237:{\displaystyle v=\lambda w}
10:
1385:
1325:10.1016/j.cjph.2021.03.014
1295:Chinese Journal of Physics
979:
733:irreducible representation
313:
1154:Projective representation
1165:
1149:Complex projective space
801:complex projective space
640:{\displaystyle \lambda }
557:{\displaystyle \lambda }
501:{\displaystyle \lambda }
446:normalized wave function
298:are also referred to as
257:{\displaystyle \lambda }
26:projective Hilbert space
964:. This is known as the
929:complex projective line
788:{\displaystyle H=H_{n}}
208:{\displaystyle w\sim v}
20:and the foundations of
1340:Miranda, Rick (1995).
1119:
968:or, equivalently, the
958:
902:
835:
789:
756:
725:
698:
669:
641:
614:
594:
558:
538:
502:
482:
468:within the ray, since
462:
438:
394:
364:
341:
292:
258:
238:
209:
180:
160:
137:
136:{\displaystyle v\in H}
111:
82:
56:
1120:
959:
903:
836:
790:
757:
726:
699:
670:
642:
615:
613:{\displaystyle \phi }
595:
559:
539:
503:
483:
481:{\displaystyle \psi }
463:
461:{\displaystyle \psi }
439:
395:
365:
342:
340:{\displaystyle \psi }
293:
259:
239:
210:
181:
161:
159:{\displaystyle \sim }
138:
112:
83:
57:
1301:. Elsevier BV: 418.
1000:
934:
848:
814:
766:
746:
715:
697:{\displaystyle U(1)}
679:
659:
631:
604:
568:
548:
537:{\displaystyle U(1)}
519:
492:
472:
452:
408:
378:
351:
331:
276:
248:
219:
193:
170:
150:
145:equivalence relation
121:
117:of non-zero vectors
95:
72:
35:
1317:2021ChJPh..71..418K
1270:1983JPhA...16.3829C
1191:Kong & Liu 2021
918:Fubini–Study metric
797:inner product space
708:of the first kind.
370:represent the same
90:equivalence classes
1221:On Einstein's Path
1115:
954:
898:
831:
785:
752:
721:
694:
665:
637:
620:called the global
610:
590:
564:can be written as
554:
534:
498:
478:
458:
434:
390:
360:
337:
288:
254:
234:
205:
176:
156:
133:
107:
78:
52:
1248:978-1-4612-7137-6
986:Cartesian product
805:homogeneous space
755:{\displaystyle H}
724:{\displaystyle H}
668:{\displaystyle H}
179:{\displaystyle H}
81:{\displaystyle H}
22:quantum mechanics
1376:
1355:
1336:
1310:
1289:
1252:
1232:
1206:
1200:
1194:
1188:
1182:
1176:
1159:Projective space
1136:entangled states
1124:
1122:
1121:
1116:
1063:
1046:
1035:
1024:
1007:
963:
961:
960:
955:
953:
952:
947:
941:
912:which carries a
907:
905:
904:
899:
897:
896:
885:
879:
868:
867:
855:
840:
838:
837:
832:
821:
794:
792:
791:
786:
784:
783:
761:
759:
758:
753:
730:
728:
727:
722:
703:
701:
700:
695:
674:
672:
671:
666:
646:
644:
643:
638:
619:
617:
616:
611:
599:
597:
596:
591:
589:
588:
563:
561:
560:
555:
543:
541:
540:
535:
507:
505:
504:
499:
487:
485:
484:
479:
467:
465:
464:
459:
443:
441:
440:
435:
421:
399:
397:
396:
391:
369:
367:
366:
361:
346:
344:
343:
338:
297:
295:
294:
291:{\displaystyle }
289:
270:projectivization
263:
261:
260:
255:
243:
241:
240:
235:
214:
212:
211:
206:
185:
183:
182:
177:
165:
163:
162:
157:
142:
140:
139:
134:
116:
114:
113:
110:{\displaystyle }
108:
87:
85:
84:
79:
61:
59:
58:
53:
42:
1384:
1383:
1379:
1378:
1377:
1375:
1374:
1373:
1359:
1358:
1352:
1249:
1215:
1210:
1209:
1201:
1197:
1189:
1185:
1177:
1173:
1168:
1145:
1133:and represents
1056:
1042:
1028:
1020:
1003:
1001:
998:
997:
982:
948:
943:
942:
937:
935:
932:
931:
886:
881:
880:
875:
863:
859:
851:
849:
846:
845:
817:
815:
812:
811:
779:
775:
767:
764:
763:
747:
744:
743:
716:
713:
712:
680:
677:
676:
660:
657:
656:
632:
629:
628:
605:
602:
601:
581:
577:
569:
566:
565:
549:
546:
545:
520:
517:
516:
493:
490:
489:
473:
470:
469:
453:
450:
449:
417:
409:
406:
405:
379:
376:
375:
352:
349:
348:
332:
329:
328:
318:
312:
304:projective rays
277:
274:
273:
249:
246:
245:
220:
217:
216:
215:if and only if
194:
191:
190:
171:
168:
167:
151:
148:
147:
122:
119:
118:
96:
93:
92:
73:
70:
69:
38:
36:
33:
32:
12:
11:
5:
1382:
1372:
1371:
1369:Hilbert spaces
1357:
1356:
1350:
1337:
1290:
1253:
1247:
1214:
1211:
1208:
1207:
1195:
1183:
1170:
1169:
1167:
1164:
1163:
1162:
1156:
1151:
1144:
1141:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1062:
1059:
1055:
1052:
1049:
1045:
1041:
1038:
1034:
1031:
1027:
1023:
1019:
1016:
1013:
1010:
1006:
994:tensor product
981:
978:
974:Hopf fibration
970:Riemann sphere
951:
946:
940:
910:
909:
895:
892:
889:
884:
878:
874:
871:
866:
862:
858:
854:
830:
827:
824:
820:
782:
778:
774:
771:
751:
720:
693:
690:
687:
684:
664:
636:
609:
587:
584:
580:
576:
573:
553:
533:
530:
527:
524:
510:absolute value
497:
477:
457:
433:
430:
427:
424:
420:
416:
413:
389:
386:
383:
372:physical state
359:
356:
336:
326:wave functions
322:quantum theory
311:
308:
287:
284:
281:
266:
265:
253:
233:
230:
227:
224:
204:
201:
198:
175:
155:
132:
129:
126:
106:
103:
100:
88:is the set of
77:
51:
48:
45:
41:
9:
6:
4:
3:
2:
1381:
1370:
1367:
1366:
1364:
1353:
1351:0-8218-0268-2
1347:
1343:
1338:
1334:
1330:
1326:
1322:
1318:
1314:
1309:
1304:
1300:
1296:
1291:
1287:
1283:
1279:
1275:
1271:
1267:
1263:
1259:
1254:
1250:
1244:
1240:
1236:
1231:
1230:gr-qc/9706069
1226:
1222:
1217:
1216:
1204:
1199:
1192:
1187:
1181:, p. 94.
1180:
1175:
1171:
1160:
1157:
1155:
1152:
1150:
1147:
1146:
1140:
1138:
1137:
1132:
1128:
1109:
1106:
1103:
1088:
1082:
1076:
1067:
1060:
1057:
1053:
1050:
1032:
1029:
1017:
1011:
995:
991:
990:Segre mapping
987:
977:
975:
971:
967:
949:
930:
926:
921:
919:
916:, called the
915:
914:Kähler metric
893:
890:
887:
872:
864:
860:
844:
843:
842:
825:
810:
809:unitary group
806:
802:
798:
780:
776:
772:
769:
749:
740:
738:
734:
718:
709:
707:
688:
682:
662:
654:
650:
634:
625:
623:
607:
585:
582:
578:
574:
571:
551:
528:
522:
515:
511:
495:
475:
455:
447:
431:
428:
422:
414:
403:
387:
384:
381:
373:
357:
354:
334:
327:
323:
317:
307:
305:
301:
282:
271:
251:
231:
228:
225:
222:
202:
199:
196:
189:
188:
187:
173:
153:
146:
130:
127:
124:
101:
91:
75:
68:
67:Hilbert space
65:
46:
31:
27:
23:
19:
1341:
1298:
1294:
1261:
1257:
1220:
1198:
1193:, p. 9.
1186:
1179:Miranda 1995
1174:
1134:
983:
966:Bloch sphere
922:
911:
742:In the case
741:
710:
626:
514:circle group
371:
319:
303:
299:
267:
29:
25:
15:
841:. That is,
737:pure states
706:gauge group
18:mathematics
1308:1903.11962
1213:References
803:; it is a
653:C*-algebra
651:, given a
374:, for any
314:See also:
143:, for the
1286:0305-4470
1127:embedding
1107:⊗
1098:↦
1054:⊗
1040:→
1018:×
927:) is the
891:−
635:λ
608:ϕ
586:ϕ
572:λ
552:λ
496:λ
476:ψ
456:ψ
426:⟩
423:ψ
415:ψ
412:⟨
385:≠
382:λ
358:ψ
355:λ
335:ψ
252:λ
229:λ
200:∼
186:given by
154:∼
128:∈
30:ray space
1363:Category
1333:85543324
1143:See also
1061:′
1033:′
310:Overview
1313:Bibcode
1266:Bibcode
980:Product
512:1 (the
64:complex
1348:
1331:
1284:
1245:
972:. See
807:for a
731:is an
324:, the
24:, the
1329:S2CID
1303:arXiv
1225:arXiv
1166:Notes
1131:range
925:qubit
704:is a
622:phase
600:with
508:with
62:of a
1346:ISBN
1282:ISSN
1243:ISBN
984:The
402:norm
347:and
300:rays
1321:doi
1274:doi
1235:doi
711:If
302:or
166:on
28:or
16:In
1365::
1327:.
1319:.
1311:.
1299:71
1297:.
1280:.
1272:.
1262:16
1260:.
1241:.
1233:.
1139:.
624:.
404:,
306:.
1354:.
1335:.
1323::
1315::
1305::
1288:.
1276::
1268::
1251:.
1237::
1227::
1205:.
1113:]
1110:y
1104:x
1101:[
1095:)
1092:]
1089:y
1086:[
1083:,
1080:]
1077:x
1074:[
1071:(
1068:,
1065:)
1058:H
1051:H
1048:(
1044:P
1037:)
1030:H
1026:(
1022:P
1015:)
1012:H
1009:(
1005:P
950:1
945:P
939:C
908:,
894:1
888:n
883:P
877:C
873:=
870:)
865:n
861:H
857:(
853:P
829:)
826:n
823:(
819:U
781:n
777:H
773:=
770:H
750:H
719:H
692:)
689:1
686:(
683:U
663:H
583:i
579:e
575:=
532:)
529:1
526:(
523:U
432:1
429:=
419:|
388:0
286:]
283:v
280:[
264:.
232:w
226:=
223:v
203:v
197:w
174:H
131:H
125:v
105:]
102:v
99:[
76:H
50:)
47:H
44:(
40:P
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