31:. However, the conventional uncertainty relation like the Robertson-Schrödinger relation cannot give a non-trivial bound for the product of variances of two incompatible observables because the lower bound in the uncertainty inequalities can be null and hence trivial even for observables that are incompatible on the state of the system. The Heisenberg–Robertson–Schrödinger uncertainty relation was proved at the dawn of quantum formalism and is ever-present in the teaching and research on quantum mechanics. After about 85 years of existence of the uncertainty relation this problem was solved recently by Lorenzo Maccone and
1192:
principle captures the measurement disturbance by the apparatus and the impossibility of joint measurements of incompatible observables. The
Maccone–Pati uncertainty relations refer to preparation uncertainty relations. These relations set strong limitations for the nonexistence of common eigenstates for incompatible observables. The Maccone–Pati uncertainty relations have been experimentally tested for qutrit systems. The new uncertainty relations not only capture the incompatibility of observables but also of quantities that are physically measurable (as variances can be measured in the experiment).
75:, and the product can be null even when one of the two variances is different from zero. However, the stronger uncertainty relations due to Maccone and Pati provide different uncertainty relations, based on the sum of variances that are guaranteed to be nontrivial whenever the observables are incompatible on the state of the quantum system. (Earlier works on uncertainty relations formulated as the sum of variances include, e.g., He et al., and Ref. due to Huang.)
1178:
83:
The
Heisenberg–Robertson or Schrödinger uncertainty relations do not fully capture the incompatibility of observables in a given quantum state. The stronger uncertainty relations give non-trivial bounds on the sum of the variances for two incompatible observables. For two non-commuting observables
1191:
In quantum theory, one should distinguish between the uncertainty relation and the uncertainty principle. The former refers solely to the preparation of the system which induces a spread in the measurement outcomes, and does not refer to the disturbance induced by the measurement. The uncertainty
289:
984:
782:
492:
392:
1173:{\displaystyle \Delta A\Delta B\geq {\frac {\pm {\frac {i}{2}}\langle \Psi ||\Psi \rangle }{1-{\frac {1}{2}}|\langle \Psi |({\frac {A}{\Delta A}}\pm i{\frac {B}{\Delta B}})|{\bar {\Psi }}\rangle |^{2}}}.}
130:
578:
1483:
Wang, Kunkun; Zhan, Xiang; Bian, Zhihao; Li, Jian; Zhang, Yongsheng; Xue, Peng (11 May 2016). "Experimental investigation of the stronger uncertainty relations for all incompatible observables".
638:
913:
835:
649:
529:
941:
863:
397:
297:
973:
122:
102:
73:
53:
35:. The standard uncertainty relations are expressed in terms of the product of variances of the measurement results of the observables
23:
is one of the fundamental results in quantum mechanics. Later
Robertson proved the uncertainty relation for two general non-commuting
284:{\displaystyle \Delta A^{2}+\Delta B^{2}\geq \pm i\langle \Psi ||\Psi \rangle +|\langle \Psi |(A\pm iB)|{\bar {\Psi }}\rangle |^{2},}
1377:
He, Qiongyi; Peng, Shi-Guo; Drummond, Peter; Reid, Margaret (10 August 2011). "Planar quantum squeezing and atom interferometry".
534:
1592:
1316:
Maccone, Lorenzo; Pati, Arun K. (31 December 2014). "Stronger
Uncertainty Relations for All Incompatible Observables".
583:
1587:
1302:
E. Schrödinger, "Sitzungsberichte der
Preussischen Akademie der Wissenschaften", Physikalisch-mathematische Klasse
868:
790:
777:{\displaystyle \Delta A^{2}+\Delta B^{2}\geq {\frac {1}{2}}|\langle {\bar {\Psi }}_{A+B}|(A+B)|\Psi \rangle |^{2},}
497:
1209:
Heisenberg, W. (1927). "Ăśber den anschaulichen Inhalt der quantentheoretischen
Kinematik und Mechanik".
1502:
1449:
1396:
1335:
1269:
1218:
20:
1547:
Research
Highlight, NATURE ASIA, 19 January 2015, "Heisenberg's uncertainty relation gets stronger"
918:
840:
978:
One can prove an improved version of the
Heisenberg–Robertson uncertainty relation which reads as
1526:
1492:
1465:
1439:
1412:
1386:
1325:
1242:
487:{\displaystyle \Delta B^{2}=\langle \Psi |B^{2}|\Psi \rangle -\langle \Psi |B|\Psi \rangle ^{2}}
387:{\displaystyle \Delta A^{2}=\langle \Psi |A^{2}|\Psi \rangle -\langle \Psi |A|\Psi \rangle ^{2}}
1518:
1359:
1351:
1285:
1234:
1564:
1510:
1457:
1404:
1343:
1277:
1226:
1183:
The
Heisenberg–Robertson uncertainty relation follows from the above uncertainty relation.
946:
1506:
1453:
1400:
1339:
1273:
1222:
107:
87:
58:
38:
28:
1581:
1530:
1469:
1246:
915:
implies that the right-hand side of the new uncertainty relation is nonzero unless
1416:
1347:
32:
24:
1514:
1461:
1408:
1522:
1355:
1289:
1238:
1363:
1281:
1568:
1430:
Huang, Yichen (10 August 2012). "Variance-based uncertainty relations".
1230:
1497:
1444:
1391:
1330:
643:
The other non-trivial stronger uncertainty relation is given by
531:
is a vector that is orthogonal to the state of the system, i.e.,
1260:
Robertson, H. P. (1 July 1929). "The
Uncertainty Principle".
1217:(3–4). Springer Science and Business Media LLC: 172–198.
573:{\displaystyle \langle \Psi |{\bar {\Psi }}\rangle =0}
987:
949:
921:
871:
843:
793:
652:
586:
537:
500:
400:
300:
133:
110:
90:
61:
41:
124:
the first stronger uncertainty relation is given by
1559:"Heisenberg's uncertainty relation gets stronger".
1172:
967:
935:
907:
857:
829:
776:
632:
572:
523:
486:
386:
283:
116:
96:
67:
47:
633:{\displaystyle \pm i\langle \Psi ||\Psi \rangle }
1268:(1). American Physical Society (APS): 163–164.
908:{\displaystyle |{\bar {\Psi }}_{A+B}\rangle }
830:{\displaystyle |{\bar {\Psi }}_{A+B}\rangle }
8:
1149:
1079:
1053:
1019:
930:
902:
852:
824:
756:
700:
627:
593:
561:
538:
518:
475:
452:
446:
417:
375:
352:
346:
317:
263:
217:
206:
172:
16:Later developments of Heisenberg's principle
1496:
1443:
1390:
1329:
1158:
1153:
1138:
1137:
1132:
1114:
1093:
1085:
1074:
1064:
1045:
1025:
1009:
1003:
986:
948:
922:
920:
890:
879:
878:
872:
870:
844:
842:
812:
801:
800:
794:
792:
765:
760:
748:
728:
716:
705:
704:
695:
685:
676:
660:
651:
619:
599:
585:
550:
549:
544:
536:
507:
506:
501:
499:
478:
466:
458:
438:
432:
423:
408:
399:
378:
366:
358:
338:
332:
323:
308:
299:
272:
267:
252:
251:
246:
223:
212:
198:
178:
157:
141:
132:
109:
89:
60:
40:
1551:
1201:
524:{\displaystyle |{\bar {\Psi }}\rangle }
79:The Maccone–Pati uncertainty relations
7:
640:so that this is a positive number.
580:and one should choose the sign of
1140:
1120:
1099:
1082:
1050:
1022:
994:
988:
927:
881:
849:
803:
753:
707:
669:
653:
624:
596:
552:
541:
509:
471:
455:
443:
420:
401:
371:
355:
343:
320:
301:
254:
220:
203:
175:
150:
134:
14:
837:is a unit vector orthogonal to
1348:10.1103/physrevlett.113.260401
1154:
1143:
1133:
1129:
1090:
1086:
1075:
1046:
1042:
1030:
1026:
962:
950:
936:{\displaystyle |\Psi \rangle }
923:
884:
873:
858:{\displaystyle |\Psi \rangle }
845:
806:
795:
761:
749:
745:
733:
729:
710:
696:
620:
616:
604:
600:
555:
545:
512:
502:
467:
459:
439:
424:
367:
359:
339:
324:
268:
257:
247:
243:
228:
224:
213:
199:
195:
183:
179:
1:
27:, which was strengthened by
1609:
1515:10.1103/physreva.93.052108
1462:10.1103/PhysRevA.86.024101
1409:10.1103/PhysRevA.84.022107
1318:Physical Review Letters
1282:10.1103/physrev.34.163
1211:Zeitschrift fĂĽr Physik
1174:
969:
937:
909:
859:
831:
778:
634:
574:
525:
488:
388:
285:
118:
98:
69:
49:
1569:10.1038/nindia.2015.6
1175:
970:
968:{\displaystyle (A+B)}
938:
910:
860:
832:
779:
635:
575:
526:
489:
389:
286:
119:
99:
70:
50:
1593:Mathematical physics
985:
947:
943:is an eigenstate of
919:
869:
841:
791:
650:
584:
535:
498:
398:
298:
131:
108:
88:
59:
39:
21:uncertainty relation
1507:2016PhRvA..93e2108W
1454:2012PhRvA..86b4101H
1401:2011PhRvA..84b2107H
1340:2014PhRvL.113z0401M
1274:1929PhRv...34..163R
1223:1927ZPhy...43..172H
1231:10.1007/bf01397280
1170:
965:
933:
905:
855:
827:
774:
630:
570:
521:
484:
384:
281:
114:
94:
65:
45:
1588:Quantum mechanics
1485:Physical Review A
1432:Physical Review A
1379:Physical Review A
1165:
1146:
1127:
1106:
1072:
1017:
887:
809:
713:
693:
558:
515:
260:
117:{\displaystyle B}
97:{\displaystyle A}
68:{\displaystyle B}
48:{\displaystyle A}
1600:
1573:
1572:
1556:
1535:
1534:
1500:
1480:
1474:
1473:
1447:
1427:
1421:
1420:
1394:
1374:
1368:
1367:
1333:
1313:
1307:
1300:
1294:
1293:
1257:
1251:
1250:
1206:
1179:
1177:
1176:
1171:
1166:
1164:
1163:
1162:
1157:
1148:
1147:
1139:
1136:
1128:
1126:
1115:
1107:
1105:
1094:
1089:
1078:
1073:
1065:
1056:
1049:
1029:
1018:
1010:
1004:
974:
972:
971:
966:
942:
940:
939:
934:
926:
914:
912:
911:
906:
901:
900:
889:
888:
880:
876:
864:
862:
861:
856:
848:
836:
834:
833:
828:
823:
822:
811:
810:
802:
798:
783:
781:
780:
775:
770:
769:
764:
752:
732:
727:
726:
715:
714:
706:
699:
694:
686:
681:
680:
665:
664:
639:
637:
636:
631:
623:
603:
579:
577:
576:
571:
560:
559:
551:
548:
530:
528:
527:
522:
517:
516:
508:
505:
493:
491:
490:
485:
483:
482:
470:
462:
442:
437:
436:
427:
413:
412:
393:
391:
390:
385:
383:
382:
370:
362:
342:
337:
336:
327:
313:
312:
290:
288:
287:
282:
277:
276:
271:
262:
261:
253:
250:
227:
216:
202:
182:
162:
161:
146:
145:
123:
121:
120:
115:
103:
101:
100:
95:
74:
72:
71:
66:
54:
52:
51:
46:
1608:
1607:
1603:
1602:
1601:
1599:
1598:
1597:
1578:
1577:
1576:
1558:
1557:
1553:
1544:
1539:
1538:
1482:
1481:
1477:
1429:
1428:
1424:
1376:
1375:
1371:
1315:
1314:
1310:
1301:
1297:
1262:Physical Review
1259:
1258:
1254:
1208:
1207:
1203:
1198:
1189:
1152:
1119:
1098:
1057:
1005:
983:
982:
945:
944:
917:
916:
877:
867:
866:
839:
838:
799:
789:
788:
759:
703:
672:
656:
648:
647:
582:
581:
533:
532:
496:
495:
474:
428:
404:
396:
395:
374:
328:
304:
296:
295:
266:
153:
137:
129:
128:
106:
105:
86:
85:
81:
57:
56:
37:
36:
17:
12:
11:
5:
1606:
1604:
1596:
1595:
1590:
1580:
1579:
1575:
1574:
1550:
1549:
1548:
1543:
1540:
1537:
1536:
1475:
1422:
1369:
1324:(26): 260401.
1308:
1295:
1252:
1200:
1199:
1197:
1194:
1188:
1185:
1181:
1180:
1169:
1161:
1156:
1151:
1145:
1142:
1135:
1131:
1125:
1122:
1118:
1113:
1110:
1104:
1101:
1097:
1092:
1088:
1084:
1081:
1077:
1071:
1068:
1063:
1060:
1055:
1052:
1048:
1044:
1041:
1038:
1035:
1032:
1028:
1024:
1021:
1016:
1013:
1008:
1002:
999:
996:
993:
990:
964:
961:
958:
955:
952:
932:
929:
925:
904:
899:
896:
893:
886:
883:
875:
865:. The form of
854:
851:
847:
826:
821:
818:
815:
808:
805:
797:
785:
784:
773:
768:
763:
758:
755:
751:
747:
744:
741:
738:
735:
731:
725:
722:
719:
712:
709:
702:
698:
692:
689:
684:
679:
675:
671:
668:
663:
659:
655:
629:
626:
622:
618:
615:
612:
609:
606:
602:
598:
595:
592:
589:
569:
566:
563:
557:
554:
547:
543:
540:
520:
514:
511:
504:
481:
477:
473:
469:
465:
461:
457:
454:
451:
448:
445:
441:
435:
431:
426:
422:
419:
416:
411:
407:
403:
381:
377:
373:
369:
365:
361:
357:
354:
351:
348:
345:
341:
335:
331:
326:
322:
319:
316:
311:
307:
303:
292:
291:
280:
275:
270:
265:
259:
256:
249:
245:
242:
239:
236:
233:
230:
226:
222:
219:
215:
211:
208:
205:
201:
197:
194:
191:
188:
185:
181:
177:
174:
171:
168:
165:
160:
156:
152:
149:
144:
140:
136:
113:
93:
80:
77:
64:
44:
15:
13:
10:
9:
6:
4:
3:
2:
1605:
1594:
1591:
1589:
1586:
1585:
1583:
1570:
1566:
1562:
1555:
1552:
1546:
1545:
1542:Other sources
1541:
1532:
1528:
1524:
1520:
1516:
1512:
1508:
1504:
1499:
1494:
1491:(5): 052108.
1490:
1486:
1479:
1476:
1471:
1467:
1463:
1459:
1455:
1451:
1446:
1441:
1438:(2): 024101.
1437:
1433:
1426:
1423:
1418:
1414:
1410:
1406:
1402:
1398:
1393:
1388:
1385:(2): 022107.
1384:
1380:
1373:
1370:
1365:
1361:
1357:
1353:
1349:
1345:
1341:
1337:
1332:
1327:
1323:
1319:
1312:
1309:
1305:
1299:
1296:
1291:
1287:
1283:
1279:
1275:
1271:
1267:
1263:
1256:
1253:
1248:
1244:
1240:
1236:
1232:
1228:
1224:
1220:
1216:
1213:(in German).
1212:
1205:
1202:
1195:
1193:
1186:
1184:
1167:
1159:
1123:
1116:
1111:
1108:
1102:
1095:
1069:
1066:
1061:
1058:
1039:
1036:
1033:
1014:
1011:
1006:
1000:
997:
991:
981:
980:
979:
976:
959:
956:
953:
897:
894:
891:
819:
816:
813:
771:
766:
742:
739:
736:
723:
720:
717:
690:
687:
682:
677:
673:
666:
661:
657:
646:
645:
644:
641:
613:
610:
607:
590:
587:
567:
564:
479:
463:
449:
433:
429:
414:
409:
405:
379:
363:
349:
333:
329:
314:
309:
305:
278:
273:
240:
237:
234:
231:
209:
192:
189:
186:
169:
166:
163:
158:
154:
147:
142:
138:
127:
126:
125:
111:
91:
78:
76:
62:
42:
34:
30:
26:
22:
19:Heisenberg's
1561:Nature India
1560:
1554:
1488:
1484:
1478:
1435:
1431:
1425:
1382:
1378:
1372:
1321:
1317:
1311:
1306:, 296 (1930)
1303:
1298:
1265:
1261:
1255:
1214:
1210:
1204:
1190:
1182:
977:
786:
642:
293:
82:
33:Arun K. Pati
18:
29:Schrödinger
25:observables
1582:Categories
1498:1604.05901
1196:References
1531:118404774
1523:2469-9926
1470:118507388
1445:1012.3105
1392:1101.0448
1356:0031-9007
1331:1407.0338
1290:0031-899X
1247:122763326
1239:1434-6001
1150:⟩
1144:¯
1141:Ψ
1121:Δ
1109:±
1100:Δ
1083:Ψ
1080:⟨
1062:−
1054:⟩
1051:Ψ
1023:Ψ
1020:⟨
1007:±
1001:≥
995:Δ
989:Δ
931:⟩
928:Ψ
903:⟩
885:¯
882:Ψ
853:⟩
850:Ψ
825:⟩
807:¯
804:Ψ
757:⟩
754:Ψ
711:¯
708:Ψ
701:⟨
683:≥
670:Δ
654:Δ
628:⟩
625:Ψ
597:Ψ
594:⟨
588:±
562:⟩
556:¯
553:Ψ
542:Ψ
539:⟨
519:⟩
513:¯
510:Ψ
476:⟩
472:Ψ
456:Ψ
453:⟨
450:−
447:⟩
444:Ψ
421:Ψ
418:⟨
402:Δ
376:⟩
372:Ψ
356:Ψ
353:⟨
350:−
347:⟩
344:Ψ
321:Ψ
318:⟨
302:Δ
264:⟩
258:¯
255:Ψ
235:±
221:Ψ
218:⟨
207:⟩
204:Ψ
176:Ψ
173:⟨
167:±
164:≥
151:Δ
135:Δ
1563:. 2015.
1364:25615288
1503:Bibcode
1450:Bibcode
1417:7885824
1397:Bibcode
1336:Bibcode
1270:Bibcode
1219:Bibcode
1187:Remarks
1529:
1521:
1468:
1415:
1362:
1354:
1288:
1245:
1237:
787:where
294:where
1527:S2CID
1493:arXiv
1466:S2CID
1440:arXiv
1413:S2CID
1387:arXiv
1326:arXiv
1243:S2CID
1519:ISSN
1360:PMID
1352:ISSN
1286:ISSN
1235:ISSN
104:and
55:and
1565:doi
1511:doi
1458:doi
1405:doi
1344:doi
1322:113
1278:doi
1227:doi
1584::
1525:.
1517:.
1509:.
1501:.
1489:93
1487:.
1464:.
1456:.
1448:.
1436:86
1434:.
1411:.
1403:.
1395:.
1383:84
1381:.
1358:.
1350:.
1342:.
1334:.
1320:.
1304:14
1284:.
1276:.
1266:34
1264:.
1241:.
1233:.
1225:.
1215:43
975:.
494:,
394:,
1571:.
1567::
1533:.
1513::
1505::
1495::
1472:.
1460::
1452::
1442::
1419:.
1407::
1399::
1389::
1366:.
1346::
1338::
1328::
1292:.
1280::
1272::
1249:.
1229::
1221::
1168:.
1160:2
1155:|
1134:|
1130:)
1124:B
1117:B
1112:i
1103:A
1096:A
1091:(
1087:|
1076:|
1070:2
1067:1
1059:1
1047:|
1043:]
1040:B
1037:,
1034:A
1031:[
1027:|
1015:2
1012:i
998:B
992:A
963:)
960:B
957:+
954:A
951:(
924:|
898:B
895:+
892:A
874:|
846:|
820:B
817:+
814:A
796:|
772:,
767:2
762:|
750:|
746:)
743:B
740:+
737:A
734:(
730:|
724:B
721:+
718:A
697:|
691:2
688:1
678:2
674:B
667:+
662:2
658:A
621:|
617:]
614:B
611:,
608:A
605:[
601:|
591:i
568:0
565:=
546:|
503:|
480:2
468:|
464:B
460:|
440:|
434:2
430:B
425:|
415:=
410:2
406:B
380:2
368:|
364:A
360:|
340:|
334:2
330:A
325:|
315:=
310:2
306:A
279:,
274:2
269:|
248:|
244:)
241:B
238:i
232:A
229:(
225:|
214:|
210:+
200:|
196:]
193:B
190:,
187:A
184:[
180:|
170:i
159:2
155:B
148:+
143:2
139:A
112:B
92:A
63:B
43:A
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.