4080:
8065:
6379:
2376:
6162:
3997:
5377:
5538:
6201:
5655:
3415:
5109:
5218:
3616:
2211:
993:
6026:
3787:
3234:
4876:
5224:
1147:
4609:
5388:
6374:{\displaystyle U_{\text{UQSD}}|\varphi \rangle ={\sqrt {1-|\langle \varphi |\psi \rangle |}}|{\text{result Ï}}\rangle +e^{-i\arg(\langle \varphi |\psi \rangle )}{\sqrt {|\langle \varphi |\psi \rangle |}}|{\text{result ?}}\rangle .}
1963:
4386:
4200:
2035:
5874:
Since the POVMs are rank-1, we can use the simple case of the construction above to obtain a projective measurement that physically realises this POVM. Labelling the three possible states of the enlarged
Hilbert space as
2129:
3024:
5546:
1785:
6387:
This POVM has been used to experimentally distinguish non-orthogonal polarisation states of a photon. The realisation of the POVM with a projective measurement was slightly different from the one described here.
1507:
821:
3280:
4725:
2750:
The post-measurement state is not determined by the POVM itself, but rather by the PVM that physically realizes it. Since there are infinitely many different PVMs that realize the same POVM, the operators
347:
4448:, in the sense that there is no measurement, either PVM or POVM, that will distinguish them with certainty. The impossibility of perfectly discriminating between non-orthogonal states is the basis for
5000:
with a single measurement, and the probability of getting a conclusive result is smaller than with POVMs. The POVM that gives the highest probability of a conclusive outcome in this task is given by
552:
1852:
418:
2509:
5006:
5810:
2371:{\displaystyle {\text{Prob}}(i)=\operatorname {tr} \left(V\rho _{A}V^{\dagger }\Pi _{i}\right)=\operatorname {tr} \left(\rho _{A}V^{\dagger }\Pi _{i}V\right)=\operatorname {tr} (\rho _{A}F_{i})}
634:
1655:
1152:
The probability formulas for a PVM are the same as for the POVM. An important difference is that the elements of a POVM are not necessarily orthogonal. As a consequence, the number of elements
5115:
885:
4792:
1586:
1216:
shows how POVMs can be obtained from PVMs acting on a larger space. This result is of critical importance in quantum mechanics, as it gives a way to physically realize POVM measurements.
6157:{\displaystyle U_{\text{UQSD}}|\psi \rangle ={\sqrt {1-|\langle \varphi |\psi \rangle |}}|{\text{result Ï}}\rangle +{\sqrt {|\langle \varphi |\psi \rangle |}}|{\text{result ?}}\rangle ,}
3695:
3480:
2928:
5963:
5933:
5903:
4911:
4126:
2669:
1413:
4644:
3992:{\displaystyle {\text{Prob}}(i_{1}|i_{0})={\operatorname {tr} (M_{i_{1}}M_{i_{0}}\rho M_{i_{0}}^{\dagger }M_{i_{1}}^{\dagger }) \over {\rm {tr}}(M_{i_{0}}\rho M_{i_{0}}^{\dagger })}}
292:
2870:
2740:
1377:
3097:
2800:
2613:
1268:
845:
8183:
8101:
4797:
3272:
3062:
2180:
1299:
6193:
5869:
5751:
5372:{\displaystyle F_{?}=\operatorname {I} -F_{\psi }-F_{\varphi }={\frac {2|\langle \varphi |\psi \rangle |}{1+|\langle \varphi |\psi \rangle |}}|\gamma \rangle \langle \gamma |,}
4998:
4939:
4753:
4525:
4442:
4316:
4260:
5990:
6018:
5841:
5703:
4970:
4672:
4497:
4414:
4288:
4232:
1690:
1045:
877:
685:
488:
262:
107:
1890:
1033:
1445:
4069:
4035:
3472:
464:
5723:
5533:{\displaystyle |\gamma \rangle ={\frac {1}{\sqrt {2(1+|\langle \varphi |\psi \rangle |)}}}(|\psi \rangle +e^{i\arg(\langle \varphi |\psi \rangle )}|\varphi \rangle ).}
4530:
51:(PVM) and, correspondingly, quantum measurements described by POVMs are a generalization of quantum measurement described by PVMs (called projective measurements).
5675:
3779:
3752:
3722:
3445:
3089:
2633:
2559:
1326:
759:
712:
143:
7452:
3640:
2820:
2689:
2532:
2439:
2419:
2399:
2203:
1527:
1190:
1170:
732:
654:
579:
374:
231:
211:
191:
167:
8094:
4321:
7954:
7554:
6543:
I. M. Gelfand and M. A. Neumark, On the embedding of normed rings into the ring of operators in
Hilbert space, Rec. Math. N.S. 12(54) (1943), 197â213.
4527:, at the cost of sometimes having an inconclusive result. It is possible to do this with projective measurements. For example, if you measure the PVM
4079:
1898:
4131:
1968:
2043:
7187:
7790:
5650:{\displaystyle \operatorname {tr} (|\varphi \rangle \langle \varphi |F_{\psi })=\operatorname {tr} (|\psi \rangle \langle \psi |F_{\varphi })=0}
3727:
Another difference from the projective measurements is that a POVM measurement is in general not repeatable. If on the first measurement result
8087:
2936:
1219:
In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional
Hilbert space, Naimark's theorem says that if
7617:
6407:
7209:
4071:
are not orthogonal. In a projective measurement these operators are always orthogonal and therefore the measurement is always repeatable.
2205:
with this PVM, and the state suitably transformed by the isometry, is the same as the probability of obtaining it with the original POVM:
5871:
with the same probability. This result is known as the
IvanoviÄ-Dieks-Peres limit, named after the authors who pioneered UQSD research.
3410:{\displaystyle |\psi '\rangle _{A}={\frac {M_{i_{0}}|\psi \rangle }{\sqrt {\langle \psi |M_{i_{0}}^{\dagger }M_{i_{0}}|\psi \rangle }}}}
1698:
7780:
6866:
R. B. M. Clarke; A. Chefles; S. M. Barnett; E. Riis (2001). "Experimental demonstration of optimal unambiguous state discrimination".
767:
7192:
6965:
7214:
4677:
66:); analogously, POVMs are necessary to describe the effect on a subsystem of a projective measurement performed on a larger system.
7907:
7762:
299:
3029:
in the second construction above will also implement the same POVM. In the case where the state being measured is in a pure state
8408:
8263:
7738:
7539:
7202:
1453:
7432:
1000:
6823:
B. Huttner; A. Muller; J. D. Gautier; H. Zbinden; N. Gisin (1996). "Unambiguous quantum measurement of nonorthogonal states".
500:
8398:
7285:
7083:
6617:
6514:
5104:{\displaystyle F_{\psi }={\frac {1}{1+|\langle \varphi |\psi \rangle |}}|\varphi ^{\perp }\rangle \langle \varphi ^{\perp }|}
379:
237:
8155:
7280:
8188:
5213:{\displaystyle F_{\varphi }={\frac {1}{1+|\langle \varphi |\psi \rangle |}}|\psi ^{\perp }\rangle \langle \psi ^{\perp }|}
8519:
5762:
1172:
of the POVM can be larger than the dimension of the
Hilbert space they act in. On the other hand, the number of elements
988:{\displaystyle {\text{Prob}}(i)=\operatorname {tr} (|\psi \rangle \langle \psi |F_{i})=\langle \psi |F_{i}|\psi \rangle }
595:
4086:
representation of states (in blue) and optimal POVM (in red) for unambiguous quantum state discrimination on the states
1793:
1594:
7630:
5908:
5878:
2447:
8222:
7719:
7610:
7437:
234:
3611:{\displaystyle \rho '_{A}={M_{i_{0}}\rho M_{i_{0}}^{\dagger } \over {\rm {tr}}(M_{i_{0}}\rho M_{i_{0}}^{\dagger })}}
7989:
7255:
4758:
4388:
will form a PVM, and a projective measurement in this basis will determine the state with certainty. If, however,
1535:
8457:
8232:
7634:
7224:
3645:
2878:
5938:
4881:
8452:
8284:
7138:
7022:
6592:
J.A. Bergou; U. Herzog; M. Hillery (2004). "Discrimination of
Quantum States". In M. Paris; J. ĆehĂĄÄek (eds.).
4089:
2381:
This construction can be turned into a recipe for a physical realisation of the POVM by extending the isometry
6938:
7785:
7447:
6958:
4614:
3229:{\displaystyle U_{W}(|\psi \rangle _{A}|0\rangle _{B})=\sum _{i=1}^{n}M_{i}|\psi \rangle _{A}|i\rangle _{B},}
1213:
1201:
267:
63:
2828:
2694:
1331:
8524:
8309:
8068:
7841:
7775:
7603:
7073:
4871:{\displaystyle \{|\varphi \rangle \langle \varphi |,|\varphi ^{\perp }\rangle \langle \varphi ^{\perp }|\}}
4468:
8393:
4206:
Suppose you have a quantum system with a 2-dimensional
Hilbert space that you know is in either the state
8301:
7805:
7584:
7504:
7058:
6534:
M. Nielsen and I. Chuang, Quantum
Computation and Quantum Information, Cambridge University Press, (2000)
2754:
2638:
2567:
1382:
1222:
830:
24:
3242:
3032:
1273:
8376:
8305:
8050:
8004:
7928:
7810:
7559:
7457:
7337:
6170:
5846:
5728:
4975:
4916:
4730:
4502:
4419:
4293:
4237:
2802:
alone do not determine what the post-measurement state will be. To see that, note that for any unitary
1142:{\displaystyle \sum _{i=1}^{N}\Pi _{i}=\operatorname {I} ,\quad \Pi _{i}\Pi _{j}=\delta _{ij}\Pi _{i}.}
592:, the key property of a POVM is that it determines a probability measure on the outcome space, so that
8447:
8360:
8289:
8045:
7861:
7564:
7427:
7260:
7245:
7053:
7017:
6926:, Probabilistic and statistical aspects of quantum theory, North-Holland Publ. Cy., Amsterdam (1982).
6452:
6422:
5968:
558:
48:
5995:
5818:
5680:
4947:
4649:
4474:
4391:
4265:
4209:
1660:
854:
662:
469:
243:
88:
8381:
8268:
8114:
7897:
7795:
7698:
7156:
7146:
7027:
6951:
2137:
428:
40:
1005:
8483:
7994:
7770:
7519:
7494:
7312:
7301:
7012:
6384:
A projective measurement then gives the desired results with the same probabilities as the POVM.
8386:
8215:
8133:
8025:
7969:
7933:
7370:
7360:
7355:
7063:
6601:
4604:{\displaystyle \{|\psi \rangle \langle \psi |,|\psi ^{\perp }\rangle \langle \psi ^{\perp }|\}}
4040:
4006:
3450:
848:
582:
436:
6576:
J. Watrous. The Theory of
Quantum Information. Cambridge University Press, 2018. Chapter 2.3,
5708:
1857:
8275:
8205:
8128:
8110:
7732:
7115:
4457:
69:
POVMs are the most general kind of measurement in quantum mechanics, and can also be used in
36:
8193:
7728:
6593:
1418:
8403:
8317:
8258:
8150:
8008:
7529:
7508:
7422:
7307:
7270:
6885:
6832:
6789:
6746:
6703:
6652:
6471:
5660:
4453:
3757:
3730:
3700:
3423:
3067:
2618:
2537:
1304:
744:
690:
116:
70:
7595:
55:
8:
8439:
8429:
8312:
8227:
7974:
7912:
7626:
7332:
7068:
6920:
K. Kraus, States, Effects, and
Operations, Lecture Notes in Physics 190, Springer (1983).
6402:
4449:
735:
74:
20:
8493:
8079:
6889:
6836:
6793:
6750:
6707:
6656:
6564:
J. Preskill, Lecture Notes for Physics: Quantum Information and Computation, Chapter 3,
6475:
8355:
8210:
7999:
7866:
7462:
7391:
7322:
7166:
7128:
6901:
6875:
6676:
6642:
6565:
6487:
6461:
3625:
2805:
2674:
2517:
2424:
2404:
2384:
2188:
1512:
1175:
1155:
717:
639:
564:
359:
353:
216:
196:
176:
152:
999:
The simplest case of a POVM generalizes the simplest case of a PVM, which is a set of
62:. Mixed states are needed to specify the state of a subsystem of a larger system (see
7979:
7569:
7544:
7229:
7151:
6848:
6805:
6801:
6762:
6758:
6719:
6715:
6680:
6668:
6613:
6594:
6510:
6417:
589:
421:
6905:
8498:
8345:
8335:
8237:
8198:
7984:
7902:
7871:
7851:
7836:
7831:
7826:
7574:
7275:
7123:
7078:
7002:
6923:
6893:
6840:
6797:
6754:
6711:
6660:
6605:
6491:
6479:
4381:{\displaystyle \{|\psi \rangle \langle \psi |,|\varphi \rangle \langle \varphi |\}}
3622:
We see therefore that the post-measurement state depends explicitly on the unitary
1448:
431:
146:
7663:
4471:(UQSD) is the next best thing: to never make a mistake about whether the state is
8473:
8350:
8340:
8165:
8160:
7846:
7800:
7748:
7743:
7714:
7549:
7534:
7442:
7405:
7401:
7365:
7327:
7265:
7250:
7219:
7161:
7120:
7107:
7032:
6974:
6943:
6609:
6555:
A. Peres. Quantum Theory: Concepts and Methods. Kluwer Academic Publishers, 1993.
1036:
491:
7673:
1958:{\displaystyle {\mathcal {H}}_{A'}={\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}}
8488:
8294:
8035:
7887:
7688:
7499:
7478:
7396:
7386:
7197:
7104:
7037:
6997:
6897:
6427:
6412:
4195:{\displaystyle |\varphi \rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )}
2030:{\displaystyle \Pi _{i}=\operatorname {I} _{A}\otimes |i\rangle \langle i|_{B}}
6664:
8513:
8477:
8253:
8145:
8140:
8040:
7964:
7693:
7678:
7668:
6865:
6844:
6809:
6766:
6723:
6694:
Ivanovic, I.D. (1987). "How to differentiate between non-orthogonal states".
6672:
6483:
4461:
2124:{\displaystyle V=\sum _{i=1}^{n}{\sqrt {F_{i}}}_{A}\otimes {|i\rangle }_{B}.}
739:
657:
110:
44:
8424:
8279:
8030:
7683:
7653:
7317:
7171:
7112:
6780:
Peres, Asher (1988). "How to differentiate between non-orthogonal states".
6647:
4083:
6852:
6822:
7959:
7949:
7856:
7658:
7514:
7099:
6880:
6466:
6447:
1895:
In the general case, the isometry and PVM can be constructed by defining
6577:
6450:; Terno, Daniel R. (2004). "Quantum information and relativity theory".
3019:{\displaystyle V_{W}=\sum _{i=1}^{n}{M_{i}}_{A}\otimes {|i\rangle }_{B}}
8175:
7892:
7724:
7007:
4074:
59:
6737:
Dieks, D. (1988). "Overlap and distinguishability of quantum states".
6992:
6978:
4794:, then it is inconclusive. The analogous reasoning holds for the PVM
170:
8327:
7579:
7524:
6397:
4202:. Note that on the Bloch sphere orthogonal states are antiparallel.
1780:{\displaystyle V=\sum _{i=1}^{n}|i\rangle _{A'}\langle f_{i}|_{A}}
3447:. When the state being measured is described by a density matrix
851:
operator. When the quantum state being measured is a pure state
816:{\displaystyle {\text{Prob}}(i)=\operatorname {tr} (\rho F_{i})}
4720:{\displaystyle |\psi ^{\perp }\rangle \langle \psi ^{\perp }|}
3754:
was obtained, the probability of obtaining a different result
342:{\displaystyle E\mapsto \langle F(E)\psi \mid \psi \rangle ,}
6939:
Interactive demonstration about quantum state discrimination
3239:
and the projective measurement on the ancilla will collapse
8184:
Differentiable vectorâvalued functions from Euclidean space
7625:
5756:
The probability of having a conclusive outcome is given by
2691:, and make the projective measurement described by the PVM
1502:{\displaystyle V:{\mathcal {H}}_{A}\to {\mathcal {H}}_{A'}}
636:
can be interpreted as the probability (density) of outcome
1208:
Note: An alternate spelling of this is "Neumark's Theorem"
1192:
of the PVM is at most the dimension of the Hilbert space.
734:, such that the probability of obtaining it when making a
8109:
6633:
Chefles, Anthony (2000). "Quantum state discrimination".
4944:
This is unsatisfactory, though, as you can't detect both
2635:
is then to embed the quantum state in the Hilbert space
547:{\displaystyle \sum _{i=1}^{n}F_{i}=\operatorname {I} .}
3474:, the corresponding post-measurement state is given by
561:
in that, for projection-valued measures, the values of
413:{\displaystyle F(X)=\operatorname {I} _{\mathcal {H}}}
6204:
6173:
6029:
5998:
5971:
5941:
5911:
5881:
5849:
5821:
5765:
5731:
5725:
is obtained we are certain that the quantum state is
5711:
5683:
5677:
is obtained we are certain that the quantum state is
5663:
5549:
5391:
5227:
5118:
5009:
4978:
4950:
4919:
4884:
4800:
4761:
4733:
4680:
4652:
4617:
4533:
4505:
4477:
4422:
4394:
4324:
4296:
4268:
4240:
4212:
4134:
4092:
4043:
4009:
3790:
3760:
3733:
3703:
3648:
3628:
3483:
3453:
3426:
3283:
3245:
3100:
3070:
3035:
2939:
2881:
2831:
2808:
2757:
2697:
2677:
2641:
2621:
2570:
2540:
2520:
2450:
2427:
2407:
2387:
2214:
2191:
2185:
In either case, the probability of obtaining outcome
2140:
2046:
1971:
1901:
1860:
1796:
1701:
1663:
1597:
1591:
For the particular case of a rank-1 POVM, i.e., when
1538:
1515:
1456:
1421:
1385:
1334:
1307:
1276:
1225:
1178:
1158:
1048:
1008:
888:
857:
833:
770:
747:
720:
693:
665:
642:
598:
567:
503:
472:
439:
382:
362:
302:
270:
246:
219:
199:
179:
155:
119:
91:
6566:
http://theory.caltech.edu/~preskill/ph229/index.html
4075:
An example: unambiguous quantum state discrimination
5805:{\displaystyle 1-|\langle \varphi |\psi \rangle |,}
629:{\displaystyle \langle F(E)\psi \mid \psi \rangle }
7955:Spectral theory of ordinary differential equations
7555:Spectral theory of ordinary differential equations
6973:
6373:
6187:
6156:
6012:
5984:
5957:
5927:
5897:
5863:
5835:
5804:
5745:
5717:
5697:
5669:
5649:
5532:
5371:
5212:
5103:
4992:
4964:
4933:
4905:
4870:
4786:
4747:
4727:, then you know with certainty that the state was
4719:
4666:
4638:
4603:
4519:
4491:
4436:
4408:
4380:
4310:
4282:
4254:
4226:
4194:
4120:
4063:
4029:
3991:
3773:
3746:
3716:
3689:
3634:
3610:
3466:
3439:
3409:
3266:
3228:
3083:
3056:
3018:
2922:
2864:
2814:
2794:
2734:
2683:
2663:
2627:
2607:
2553:
2526:
2503:
2433:
2413:
2393:
2370:
2197:
2174:
2123:
2029:
1957:
1884:
1847:{\displaystyle \Pi _{i}=|i\rangle \langle i|_{A'}}
1846:
1779:
1684:
1650:{\displaystyle F_{i}=|f_{i}\rangle \langle f_{i}|}
1649:
1580:
1521:
1501:
1439:
1407:
1371:
1320:
1293:
1262:
1184:
1164:
1141:
1027:
987:
871:
839:
815:
753:
726:
706:
679:
648:
628:
573:
546:
482:
458:
412:
368:
341:
286:
256:
225:
205:
185:
161:
137:
101:
7453:SchröderâBernstein theorems for operator algebras
8511:
4262:, and you want to determine which one it is. If
2504:{\displaystyle V|i\rangle _{A}=U|i\rangle _{A'}}
2564:The recipe for realizing the POVM described by
8095:
7611:
6959:
6558:
6408:Mathematical formulation of quantum mechanics
1195:
6551:
6549:
6365:
6345:
6331:
6316:
6302:
6276:
6256:
6242:
6223:
6182:
6148:
6128:
6114:
6101:
6081:
6067:
6048:
6007:
5952:
5922:
5892:
5858:
5830:
5791:
5777:
5740:
5692:
5617:
5614:
5570:
5567:
5521:
5505:
5491:
5468:
5443:
5429:
5400:
5355:
5352:
5333:
5319:
5298:
5284:
5192:
5189:
5163:
5149:
5083:
5080:
5054:
5040:
4987:
4959:
4928:
4900:
4865:
4847:
4844:
4815:
4812:
4801:
4787:{\displaystyle |\psi \rangle \langle \psi |}
4773:
4770:
4742:
4699:
4696:
4661:
4633:
4598:
4580:
4577:
4548:
4545:
4534:
4514:
4486:
4431:
4403:
4375:
4364:
4361:
4339:
4336:
4325:
4305:
4277:
4249:
4221:
4186:
4172:
4143:
4115:
4101:
3401:
3343:
3338:
3298:
3255:
3214:
3196:
3141:
3123:
3091:takes it together with the ancilla to state
3045:
3006:
2772:
2758:
2712:
2698:
2585:
2571:
2487:
2463:
2108:
2009:
2006:
1821:
1818:
1752:
1738:
1679:
1629:
1626:
1581:{\displaystyle F_{i}=V^{\dagger }\Pi _{i}V.}
1349:
1335:
1240:
1226:
1022:
1009:
982:
953:
926:
923:
866:
674:
623:
599:
453:
440:
333:
309:
73:. They are extensively used in the field of
54:In rough analogy, a POVM is to a PVM what a
16:Generalized measurement in quantum mechanics
4318:are orthogonal, this task is easy: the set
3690:{\displaystyle M_{i}^{\dagger }M_{i}=F_{i}}
2923:{\displaystyle M_{i}^{\dagger }M_{i}=F_{i}}
2182:, so this is a more wasteful construction.
714:is associated with the measurement outcome
8102:
8088:
7618:
7604:
6966:
6952:
6530:
6528:
6526:
6446:
5958:{\displaystyle |{\text{result ?}}\rangle }
5928:{\displaystyle |{\text{result Ï}}\rangle }
5898:{\displaystyle |{\text{result Ï}}\rangle }
4906:{\displaystyle |\varphi ^{\perp }\rangle }
6879:
6646:
6546:
6465:
4121:{\displaystyle |\psi \rangle =|0\rangle }
2745:
427:In the simplest case, a POVM is a set of
8399:No infinite-dimensional Lebesgue measure
7908:Group algebra of a locally compact group
6693:
6587:
6585:
4078:
8409:Structure theorem for Gaussian measures
6632:
6523:
4639:{\displaystyle |\psi ^{\perp }\rangle }
287:{\displaystyle \psi \in {\mathcal {H}}}
8512:
6537:
6504:
2865:{\displaystyle M_{i}=W{\sqrt {F_{i}}}}
2735:{\displaystyle \{\Pi _{i}\}_{i=1}^{n}}
1692:, this isometry can be constructed as
1372:{\displaystyle \{\Pi _{i}\}_{i=1}^{n}}
466:on a finite-dimensional Hilbert space
8285:infinite-dimensional Gaussian measure
8083:
7599:
7286:Spectral theory of normal C*-algebras
7084:Spectral theory of normal C*-algebras
6947:
6779:
6736:
6582:
6578:https://cs.uwaterloo.ca/~watrous/TQI/
8156:Infinite-dimensional vector function
7281:Spectral theory of compact operators
5965:, we see that the resulting unitary
5815:when the quantum system is in state
1270:is a POVM acting on a Hilbert space
6509:. London: Acad. Press. p. 35.
4646:is the quantum state orthogonal to
2795:{\displaystyle \{F_{i}\}_{i=1}^{n}}
2664:{\displaystyle {\mathcal {H}}_{A'}}
2608:{\displaystyle \{F_{i}\}_{i=1}^{n}}
1408:{\displaystyle {\mathcal {H}}_{A'}}
1263:{\displaystyle \{F_{i}\}_{i=1}^{n}}
840:{\displaystyle \operatorname {tr} }
13:
7433:CohenâHewitt factorization theorem
6641:(6). Informa UK Limited: 401â424.
5241:
3933:
3930:
3552:
3549:
3267:{\displaystyle |\psi \rangle _{A}}
3057:{\displaystyle |\psi \rangle _{A}}
2702:
2645:
2316:
2267:
1986:
1973:
1944:
1927:
1905:
1798:
1563:
1483:
1466:
1389:
1339:
1294:{\displaystyle {\mathcal {H}}_{A}}
1280:
1127:
1101:
1091:
1083:
1071:
1013:
538:
475:
404:
399:
279:
249:
94:
14:
8536:
8223:Generalizations of the derivative
8189:Differentiation in Fréchet spaces
7438:Extensions of symmetric operators
6932:
6188:{\displaystyle |\varphi \rangle }
6167:and similarly it takes the state
5864:{\displaystyle |\varphi \rangle }
5746:{\displaystyle |\varphi \rangle }
4993:{\displaystyle |\varphi \rangle }
4934:{\displaystyle |\varphi \rangle }
4748:{\displaystyle |\varphi \rangle }
4520:{\displaystyle |\varphi \rangle }
4444:are not orthogonal, this task is
4437:{\displaystyle |\varphi \rangle }
4311:{\displaystyle |\varphi \rangle }
4255:{\displaystyle |\varphi \rangle }
2875:will also have the property that
8064:
8063:
7990:Topological quantum field theory
7256:Positive operator-valued measure
6591:
3697:is always Hermitian, generally,
1657:for some (unnormalized) vectors
47:. POVMs are a generalization of
41:positive semi-definite operators
29:positive operator-valued measure
8458:Holomorphic functional calculus
7540:RayleighâFaberâKrahn inequality
6859:
6816:
6773:
6730:
6687:
5985:{\displaystyle U_{\text{UQSD}}}
3724:does not have to be Hermitian.
1790:and the PVM is given simply by
1089:
8453:Continuous functional calculus
6626:
6570:
6507:Quantum Theory of Open Systems
6498:
6440:
6356:
6349:
6338:
6327:
6319:
6309:
6299:
6267:
6260:
6249:
6238:
6216:
6175:
6139:
6132:
6121:
6110:
6092:
6085:
6074:
6063:
6041:
6013:{\displaystyle |\psi \rangle }
6000:
5943:
5913:
5883:
5851:
5836:{\displaystyle |\psi \rangle }
5823:
5795:
5784:
5773:
5733:
5698:{\displaystyle |\psi \rangle }
5685:
5638:
5624:
5607:
5603:
5591:
5577:
5560:
5556:
5524:
5514:
5508:
5498:
5488:
5461:
5457:
5451:
5447:
5436:
5425:
5415:
5393:
5362:
5345:
5337:
5326:
5315:
5302:
5291:
5280:
5206:
5175:
5167:
5156:
5145:
5097:
5066:
5058:
5047:
5036:
4980:
4965:{\displaystyle |\psi \rangle }
4952:
4921:
4886:
4861:
4830:
4822:
4805:
4780:
4763:
4735:
4713:
4682:
4667:{\displaystyle |\psi \rangle }
4654:
4619:
4594:
4563:
4555:
4538:
4507:
4492:{\displaystyle |\psi \rangle }
4479:
4424:
4409:{\displaystyle |\psi \rangle }
4396:
4371:
4354:
4346:
4329:
4298:
4283:{\displaystyle |\psi \rangle }
4270:
4242:
4227:{\displaystyle |\psi \rangle }
4214:
4189:
4179:
4165:
4161:
4136:
4108:
4094:
3983:
3938:
3923:
3839:
3824:
3810:
3796:
3602:
3557:
3394:
3350:
3331:
3285:
3247:
3206:
3188:
3150:
3133:
3115:
3111:
3037:
2999:
2479:
2455:
2365:
2342:
2226:
2220:
2101:
2017:
1999:
1829:
1811:
1767:
1730:
1685:{\displaystyle |f_{i}\rangle }
1665:
1643:
1612:
1477:
975:
960:
947:
933:
916:
912:
900:
894:
872:{\displaystyle |\psi \rangle }
859:
810:
794:
782:
776:
680:{\displaystyle |\psi \rangle }
667:
611:
605:
483:{\displaystyle {\mathcal {H}}}
392:
386:
321:
315:
306:
257:{\displaystyle {\mathcal {H}}}
238:bounded self-adjoint operators
132:
120:
102:{\displaystyle {\mathcal {H}}}
1:
7786:Uniform boundedness principle
7448:Limiting absorption principle
6745:(5â6). Elsevier BV: 303â306.
6505:Davies, Edward Brian (1976).
6433:
2930:, so that using the isometry
2671:, evolve it with the unitary
2175:{\displaystyle d_{A'}=nd_{A}}
80:
64:purification of quantum state
7074:Singular value decomposition
6802:10.1016/0375-9601(88)91034-1
6759:10.1016/0375-9601(88)90840-7
6716:10.1016/0375-9601(87)90222-2
6610:10.1007/978-3-540-44481-7_11
4469:quantum state discrimination
1028:{\displaystyle \{\Pi _{i}\}}
687:. That is, the POVM element
7:
7505:Hearing the shape of a drum
7188:Decomposition of a spectrum
6702:(6). Elsevier BV: 257â259.
6391:
4913:is the state orthogonal to
3781:on a second measurement is
2561:. This can always be done.
25:quantum information science
10:
8541:
8520:Quantum information theory
7929:Invariant subspace problem
7093:Special Elements/Operators
6898:10.1103/PhysRevA.63.040305
1379:acting on a Hilbert space
1328:, then there exists a PVM
1214:Naimark's dilation theorem
1202:Naimark's dilation theorem
1199:
1196:Naimark's dilation theorem
49:projection-valued measures
8466:
8448:Borel functional calculus
8438:
8417:
8369:
8326:
8246:
8174:
8121:
8115:topological vector spaces
8059:
8018:
7942:
7921:
7880:
7819:
7761:
7707:
7649:
7642:
7565:Superstrong approximation
7487:
7471:
7428:Banach algebra cohomology
7415:
7379:
7348:
7294:
7261:Projection-valued measure
7246:Borel functional calculus
7238:
7180:
7137:
7092:
7046:
7018:Projection-valued measure
6985:
6665:10.1080/00107510010002599
6453:Reviews of Modern Physics
6423:Projection-valued measure
4064:{\displaystyle M_{i_{1}}}
4030:{\displaystyle M_{i_{0}}}
3467:{\displaystyle \rho _{A}}
879:this formula reduces to
559:projection-valued measure
459:{\displaystyle \{F_{i}\}}
356:measure on the Ï-algebra
8382:Inverse function theorem
8269:Classical Wiener measure
7898:Spectrum of a C*-algebra
7157:Spectrum of a C*-algebra
7028:Spectrum of a C*-algebra
6845:10.1103/PhysRevA.54.3783
6788:(1â2). Elsevier BV: 19.
6596:Quantum State Estimation
6484:10.1103/RevModPhys.76.93
5718:{\displaystyle \varphi }
4467:The task of unambiguous
4003:which can be nonzero if
3064:, the resulting unitary
1885:{\displaystyle d_{A'}=n}
8484:Convenient vector space
7995:Noncommutative geometry
7585:WienerâKhinchin theorem
7520:Kuznetsov trace formula
7495:Almost Mathieu operator
7313:Banach function algebra
7302:Amenable Banach algebra
7059:GelfandâNaimark theorem
7013:Noncommutative topology
193:. A POVM is a function
8377:CameronâMartin theorem
8134:Classical Wiener space
8051:TomitaâTakesaki theory
8026:Approximation property
7970:Calculus of variations
7560:SturmâLiouville theory
7458:ShermanâTakeda theorem
7338:TomitaâTakesaki theory
7113:Hermitian/Self-adjoint
7064:Gelfand representation
6375:
6189:
6158:
6014:
5986:
5959:
5929:
5899:
5865:
5837:
5806:
5747:
5719:
5699:
5671:
5651:
5534:
5373:
5214:
5105:
4994:
4966:
4935:
4907:
4872:
4788:
4749:
4721:
4668:
4640:
4605:
4521:
4493:
4438:
4410:
4382:
4312:
4284:
4256:
4228:
4203:
4196:
4122:
4065:
4031:
3993:
3775:
3748:
3718:
3691:
3636:
3612:
3468:
3441:
3411:
3268:
3230:
3176:
3085:
3058:
3020:
2973:
2924:
2866:
2816:
2796:
2746:Post-measurement state
2736:
2685:
2665:
2629:
2609:
2555:
2528:
2505:
2435:
2415:
2395:
2372:
2199:
2176:
2125:
2073:
2031:
1959:
1886:
1848:
1781:
1728:
1686:
1651:
1582:
1523:
1503:
1441:
1440:{\displaystyle d_{A'}}
1409:
1373:
1322:
1295:
1264:
1186:
1166:
1143:
1069:
1029:
989:
873:
841:
817:
755:
728:
708:
681:
650:
630:
583:orthogonal projections
575:
557:A POVM differs from a
548:
524:
484:
460:
429:positive semi-definite
414:
370:
343:
288:
258:
227:
207:
187:
163:
139:
103:
8394:FeldmanâHĂĄjek theorem
8206:Functional derivative
8129:Abstract Wiener space
8046:BanachâMazur distance
8009:Generalized functions
7054:GelfandâMazur theorem
6874:(4). APS: 040305(R).
6831:(5). APS: 3783â3789.
6600:. Springer. pp.
6376:
6190:
6159:
6015:
5987:
5960:
5930:
5900:
5866:
5838:
5807:
5748:
5720:
5700:
5672:
5670:{\displaystyle \psi }
5652:
5535:
5374:
5215:
5106:
4995:
4967:
4936:
4908:
4873:
4789:
4750:
4722:
4669:
4641:
4606:
4522:
4494:
4458:quantum coin flipping
4439:
4411:
4383:
4313:
4285:
4257:
4229:
4197:
4123:
4082:
4066:
4032:
3994:
3776:
3774:{\displaystyle i_{1}}
3749:
3747:{\displaystyle i_{0}}
3719:
3717:{\displaystyle M_{i}}
3692:
3637:
3613:
3469:
3442:
3440:{\displaystyle i_{0}}
3412:
3269:
3231:
3156:
3086:
3084:{\displaystyle U_{W}}
3059:
3021:
2953:
2925:
2867:
2817:
2797:
2737:
2686:
2666:
2630:
2628:{\displaystyle \rho }
2610:
2556:
2554:{\displaystyle d_{A}}
2529:
2506:
2436:
2416:
2396:
2373:
2200:
2177:
2126:
2053:
2032:
1960:
1887:
1849:
1782:
1708:
1687:
1652:
1583:
1524:
1504:
1442:
1410:
1374:
1323:
1321:{\displaystyle d_{A}}
1296:
1265:
1187:
1167:
1144:
1049:
1030:
1001:orthogonal projectors
990:
874:
842:
818:
756:
754:{\displaystyle \rho }
729:
709:
707:{\displaystyle F_{i}}
682:
651:
631:
576:
549:
504:
485:
461:
415:
371:
344:
289:
259:
228:
208:
188:
164:
140:
138:{\displaystyle (X,M)}
104:
8318:Radonifying function
8259:Cylinder set measure
8151:Cylinder set measure
7791:Kakutani fixed-point
7776:Riesz representation
7530:Proto-value function
7509:Dirichlet eigenvalue
7423:Abstract index group
7308:Approximate identity
7271:Rigged Hilbert space
7147:KreinâRutman theorem
6993:Involution/*-algebra
6635:Contemporary Physics
6202:
6171:
6027:
5996:
5969:
5939:
5909:
5879:
5847:
5819:
5763:
5729:
5709:
5681:
5661:
5547:
5389:
5225:
5116:
5007:
4976:
4948:
4917:
4882:
4798:
4759:
4755:. If the result was
4731:
4678:
4674:, and obtain result
4650:
4615:
4531:
4503:
4475:
4454:quantum cryptography
4420:
4392:
4322:
4294:
4266:
4238:
4210:
4132:
4090:
4041:
4007:
3788:
3758:
3731:
3701:
3646:
3626:
3481:
3451:
3424:
3420:on obtaining result
3281:
3243:
3098:
3068:
3033:
2937:
2879:
2829:
2806:
2755:
2695:
2675:
2639:
2619:
2568:
2538:
2518:
2448:
2425:
2405:
2385:
2212:
2189:
2138:
2044:
1969:
1899:
1858:
1794:
1699:
1661:
1595:
1536:
1513:
1454:
1419:
1383:
1332:
1305:
1274:
1223:
1176:
1156:
1046:
1006:
886:
855:
831:
768:
745:
718:
691:
663:
640:
596:
565:
501:
470:
437:
380:
360:
300:
268:
264:such that for every
244:
217:
197:
177:
153:
117:
89:
71:quantum field theory
8525:Quantum measurement
8440:Functional calculus
8430:Covariance operator
8351:GelfandâPettis/Weak
8313:measurable function
8228:Hadamard derivative
7975:Functional calculus
7934:Mahler's conjecture
7913:Von Neumann algebra
7627:Functional analysis
7333:Von Neumann algebra
7069:Polar decomposition
6890:2001PhRvA..63d0305C
6837:1996PhRvA..54.3783H
6794:1988PhLA..128...19P
6751:1988PhLA..126..303D
6708:1987PhLA..123..257I
6657:2000ConPh..41..401C
6476:2004RvMP...76...93P
6403:Quantum measurement
5705:, and when outcome
4450:quantum information
3982:
3922:
3900:
3663:
3601:
3544:
3496:
3375:
2896:
2791:
2731:
2615:on a quantum state
2604:
2421:, that is, finding
1368:
1259:
736:quantum measurement
581:are required to be
75:quantum information
21:functional analysis
8387:NashâMoser theorem
8264:Canonical Gaussian
8211:Gateaux derivative
8194:Fréchet derivative
8000:Riemann hypothesis
7699:Topological vector
7463:Unbounded operator
7392:Essential spectrum
7371:SchurâHorn theorem
7361:BauerâFike theorem
7356:AlonâBoppana bound
7349:Finite-Dimensional
7323:Nuclear C*-algebra
7167:Spectral asymmetry
6648:quant-ph/0010114v1
6371:
6185:
6154:
6010:
5982:
5955:
5925:
5895:
5861:
5833:
5802:
5743:
5715:
5695:
5667:
5657:, so when outcome
5647:
5530:
5369:
5210:
5101:
4990:
4962:
4931:
4903:
4868:
4784:
4745:
4717:
4664:
4636:
4601:
4517:
4489:
4452:protocols such as
4434:
4406:
4378:
4308:
4280:
4252:
4224:
4204:
4192:
4118:
4061:
4027:
3989:
3961:
3901:
3879:
3771:
3744:
3714:
3687:
3649:
3642:. Note that while
3632:
3608:
3580:
3523:
3484:
3464:
3437:
3407:
3354:
3264:
3226:
3081:
3054:
3016:
2920:
2882:
2862:
2812:
2792:
2771:
2732:
2711:
2681:
2661:
2625:
2605:
2584:
2551:
2524:
2501:
2431:
2411:
2391:
2368:
2195:
2172:
2121:
2027:
1955:
1882:
1844:
1777:
1682:
1647:
1578:
1519:
1509:such that for all
1499:
1437:
1405:
1369:
1348:
1318:
1291:
1260:
1239:
1182:
1162:
1139:
1025:
985:
869:
837:
813:
751:
724:
704:
677:
646:
626:
571:
544:
480:
456:
432:Hermitian matrices
410:
366:
354:countably additive
352:is a non-negative
339:
284:
254:
223:
203:
183:
159:
135:
99:
8507:
8506:
8404:Sazonov's theorem
8290:Projection-valued
8077:
8076:
7980:Integral operator
7757:
7756:
7593:
7592:
7570:Transfer operator
7545:Spectral geometry
7230:Spectral abscissa
7210:Approximate point
7152:Normal eigenvalue
6868:Physical Review A
6825:Physical Review A
6782:Physics Letters A
6739:Physics Letters A
6696:Physics Letters A
6619:978-3-540-44481-7
6516:978-0-12-206150-9
6418:Quantum operation
6363:
6353:
6274:
6264:
6212:
6146:
6136:
6099:
6089:
6037:
5979:
5950:
5920:
5890:
5455:
5454:
5342:
5172:
5063:
4159:
4158:
3987:
3794:
3635:{\displaystyle W}
3606:
3405:
3404:
2860:
2815:{\displaystyle W}
2684:{\displaystyle U}
2527:{\displaystyle i}
2434:{\displaystyle U}
2414:{\displaystyle U}
2394:{\displaystyle V}
2218:
2198:{\displaystyle i}
2087:
1854:. Note that here
1522:{\displaystyle i}
1185:{\displaystyle N}
1165:{\displaystyle n}
892:
774:
727:{\displaystyle i}
656:when measuring a
649:{\displaystyle E}
590:quantum mechanics
574:{\displaystyle F}
422:identity operator
369:{\displaystyle M}
233:whose values are
226:{\displaystyle M}
206:{\displaystyle F}
186:{\displaystyle X}
162:{\displaystyle M}
39:whose values are
8532:
8499:Hilbert manifold
8494:Fréchet manifold
8278: like
8238:Quasi-derivative
8104:
8097:
8090:
8081:
8080:
8067:
8066:
7985:Jones polynomial
7903:Operator algebra
7647:
7646:
7620:
7613:
7606:
7597:
7596:
7575:Transform theory
7295:Special algebras
7276:Spectral theorem
7239:Spectral Theorem
7079:Spectral theorem
6968:
6961:
6954:
6945:
6944:
6910:
6909:
6883:
6881:quant-ph/0007063
6863:
6857:
6856:
6820:
6814:
6813:
6777:
6771:
6770:
6734:
6728:
6727:
6691:
6685:
6684:
6650:
6630:
6624:
6623:
6599:
6589:
6580:
6574:
6568:
6562:
6556:
6553:
6544:
6541:
6535:
6532:
6521:
6520:
6502:
6496:
6495:
6469:
6467:quant-ph/0212023
6444:
6380:
6378:
6377:
6372:
6364:
6361:
6359:
6354:
6352:
6341:
6330:
6325:
6323:
6322:
6312:
6275:
6272:
6270:
6265:
6263:
6252:
6241:
6230:
6219:
6214:
6213:
6210:
6194:
6192:
6191:
6186:
6178:
6163:
6161:
6160:
6155:
6147:
6144:
6142:
6137:
6135:
6124:
6113:
6108:
6100:
6097:
6095:
6090:
6088:
6077:
6066:
6055:
6044:
6039:
6038:
6035:
6019:
6017:
6016:
6011:
6003:
5992:takes the state
5991:
5989:
5988:
5983:
5981:
5980:
5977:
5964:
5962:
5961:
5956:
5951:
5948:
5946:
5934:
5932:
5931:
5926:
5921:
5918:
5916:
5904:
5902:
5901:
5896:
5891:
5888:
5886:
5870:
5868:
5867:
5862:
5854:
5842:
5840:
5839:
5834:
5826:
5811:
5809:
5808:
5803:
5798:
5787:
5776:
5752:
5750:
5749:
5744:
5736:
5724:
5722:
5721:
5716:
5704:
5702:
5701:
5696:
5688:
5676:
5674:
5673:
5668:
5656:
5654:
5653:
5648:
5637:
5636:
5627:
5610:
5590:
5589:
5580:
5563:
5539:
5537:
5536:
5531:
5517:
5512:
5511:
5501:
5464:
5456:
5450:
5439:
5428:
5411:
5407:
5396:
5378:
5376:
5375:
5370:
5365:
5348:
5343:
5341:
5340:
5329:
5318:
5306:
5305:
5294:
5283:
5274:
5269:
5268:
5256:
5255:
5237:
5236:
5219:
5217:
5216:
5211:
5209:
5204:
5203:
5188:
5187:
5178:
5173:
5171:
5170:
5159:
5148:
5133:
5128:
5127:
5110:
5108:
5107:
5102:
5100:
5095:
5094:
5079:
5078:
5069:
5064:
5062:
5061:
5050:
5039:
5024:
5019:
5018:
4999:
4997:
4996:
4991:
4983:
4971:
4969:
4968:
4963:
4955:
4940:
4938:
4937:
4932:
4924:
4912:
4910:
4909:
4904:
4899:
4898:
4889:
4877:
4875:
4874:
4869:
4864:
4859:
4858:
4843:
4842:
4833:
4825:
4808:
4793:
4791:
4790:
4785:
4783:
4766:
4754:
4752:
4751:
4746:
4738:
4726:
4724:
4723:
4718:
4716:
4711:
4710:
4695:
4694:
4685:
4673:
4671:
4670:
4665:
4657:
4645:
4643:
4642:
4637:
4632:
4631:
4622:
4610:
4608:
4607:
4602:
4597:
4592:
4591:
4576:
4575:
4566:
4558:
4541:
4526:
4524:
4523:
4518:
4510:
4498:
4496:
4495:
4490:
4482:
4443:
4441:
4440:
4435:
4427:
4415:
4413:
4412:
4407:
4399:
4387:
4385:
4384:
4379:
4374:
4357:
4349:
4332:
4317:
4315:
4314:
4309:
4301:
4289:
4287:
4286:
4281:
4273:
4261:
4259:
4258:
4253:
4245:
4233:
4231:
4230:
4225:
4217:
4201:
4199:
4198:
4193:
4182:
4168:
4160:
4154:
4150:
4139:
4127:
4125:
4124:
4119:
4111:
4097:
4070:
4068:
4067:
4062:
4060:
4059:
4058:
4057:
4036:
4034:
4033:
4028:
4026:
4025:
4024:
4023:
3998:
3996:
3995:
3990:
3988:
3986:
3981:
3976:
3975:
3974:
3957:
3956:
3955:
3954:
3937:
3936:
3926:
3921:
3916:
3915:
3914:
3899:
3894:
3893:
3892:
3875:
3874:
3873:
3872:
3858:
3857:
3856:
3855:
3831:
3823:
3822:
3813:
3808:
3807:
3795:
3792:
3780:
3778:
3777:
3772:
3770:
3769:
3753:
3751:
3750:
3745:
3743:
3742:
3723:
3721:
3720:
3715:
3713:
3712:
3696:
3694:
3693:
3688:
3686:
3685:
3673:
3672:
3662:
3657:
3641:
3639:
3638:
3633:
3617:
3615:
3614:
3609:
3607:
3605:
3600:
3595:
3594:
3593:
3576:
3575:
3574:
3573:
3556:
3555:
3545:
3543:
3538:
3537:
3536:
3519:
3518:
3517:
3516:
3501:
3492:
3473:
3471:
3470:
3465:
3463:
3462:
3446:
3444:
3443:
3438:
3436:
3435:
3416:
3414:
3413:
3408:
3406:
3397:
3392:
3391:
3390:
3389:
3374:
3369:
3368:
3367:
3353:
3342:
3341:
3334:
3329:
3328:
3327:
3326:
3311:
3306:
3305:
3296:
3288:
3273:
3271:
3270:
3265:
3263:
3262:
3250:
3235:
3233:
3232:
3227:
3222:
3221:
3209:
3204:
3203:
3191:
3186:
3185:
3175:
3170:
3149:
3148:
3136:
3131:
3130:
3118:
3110:
3109:
3090:
3088:
3087:
3082:
3080:
3079:
3063:
3061:
3060:
3055:
3053:
3052:
3040:
3025:
3023:
3022:
3017:
3015:
3014:
3009:
3002:
2992:
2991:
2986:
2985:
2984:
2972:
2967:
2949:
2948:
2929:
2927:
2926:
2921:
2919:
2918:
2906:
2905:
2895:
2890:
2871:
2869:
2868:
2863:
2861:
2859:
2858:
2849:
2841:
2840:
2821:
2819:
2818:
2813:
2801:
2799:
2798:
2793:
2790:
2785:
2770:
2769:
2741:
2739:
2738:
2733:
2730:
2725:
2710:
2709:
2690:
2688:
2687:
2682:
2670:
2668:
2667:
2662:
2660:
2659:
2658:
2649:
2648:
2634:
2632:
2631:
2626:
2614:
2612:
2611:
2606:
2603:
2598:
2583:
2582:
2560:
2558:
2557:
2552:
2550:
2549:
2533:
2531:
2530:
2525:
2510:
2508:
2507:
2502:
2500:
2499:
2498:
2482:
2471:
2470:
2458:
2440:
2438:
2437:
2432:
2420:
2418:
2417:
2412:
2400:
2398:
2397:
2392:
2377:
2375:
2374:
2369:
2364:
2363:
2354:
2353:
2332:
2328:
2324:
2323:
2314:
2313:
2304:
2303:
2280:
2276:
2275:
2274:
2265:
2264:
2255:
2254:
2219:
2216:
2204:
2202:
2201:
2196:
2181:
2179:
2178:
2173:
2171:
2170:
2155:
2154:
2153:
2130:
2128:
2127:
2122:
2117:
2116:
2111:
2104:
2094:
2093:
2088:
2086:
2085:
2076:
2072:
2067:
2036:
2034:
2033:
2028:
2026:
2025:
2020:
2002:
1994:
1993:
1981:
1980:
1964:
1962:
1961:
1956:
1954:
1953:
1948:
1947:
1937:
1936:
1931:
1930:
1920:
1919:
1918:
1909:
1908:
1891:
1889:
1888:
1883:
1875:
1874:
1873:
1853:
1851:
1850:
1845:
1843:
1842:
1841:
1832:
1814:
1806:
1805:
1786:
1784:
1783:
1778:
1776:
1775:
1770:
1764:
1763:
1751:
1750:
1749:
1733:
1727:
1722:
1691:
1689:
1688:
1683:
1678:
1677:
1668:
1656:
1654:
1653:
1648:
1646:
1641:
1640:
1625:
1624:
1615:
1607:
1606:
1587:
1585:
1584:
1579:
1571:
1570:
1561:
1560:
1548:
1547:
1528:
1526:
1525:
1520:
1508:
1506:
1505:
1500:
1498:
1497:
1496:
1487:
1486:
1476:
1475:
1470:
1469:
1446:
1444:
1443:
1438:
1436:
1435:
1434:
1414:
1412:
1411:
1406:
1404:
1403:
1402:
1393:
1392:
1378:
1376:
1375:
1370:
1367:
1362:
1347:
1346:
1327:
1325:
1324:
1319:
1317:
1316:
1300:
1298:
1297:
1292:
1290:
1289:
1284:
1283:
1269:
1267:
1266:
1261:
1258:
1253:
1238:
1237:
1191:
1189:
1188:
1183:
1171:
1169:
1168:
1163:
1148:
1146:
1145:
1140:
1135:
1134:
1125:
1124:
1109:
1108:
1099:
1098:
1079:
1078:
1068:
1063:
1035:that sum to the
1034:
1032:
1031:
1026:
1021:
1020:
994:
992:
991:
986:
978:
973:
972:
963:
946:
945:
936:
919:
893:
890:
878:
876:
875:
870:
862:
846:
844:
843:
838:
822:
820:
819:
814:
809:
808:
775:
772:
760:
758:
757:
752:
733:
731:
730:
725:
713:
711:
710:
705:
703:
702:
686:
684:
683:
678:
670:
655:
653:
652:
647:
635:
633:
632:
627:
580:
578:
577:
572:
553:
551:
550:
545:
534:
533:
523:
518:
490:that sum to the
489:
487:
486:
481:
479:
478:
465:
463:
462:
457:
452:
451:
419:
417:
416:
411:
409:
408:
407:
375:
373:
372:
367:
348:
346:
345:
340:
293:
291:
290:
285:
283:
282:
263:
261:
260:
255:
253:
252:
232:
230:
229:
224:
212:
210:
209:
204:
192:
190:
189:
184:
168:
166:
165:
160:
147:measurable space
144:
142:
141:
136:
108:
106:
105:
100:
98:
97:
8540:
8539:
8535:
8534:
8533:
8531:
8530:
8529:
8510:
8509:
8508:
8503:
8474:Banach manifold
8462:
8434:
8413:
8365:
8341:Direct integral
8322:
8242:
8170:
8166:Vector calculus
8161:Matrix calculus
8117:
8108:
8078:
8073:
8055:
8019:Advanced topics
8014:
7938:
7917:
7876:
7842:HilbertâSchmidt
7815:
7806:GelfandâNaimark
7753:
7703:
7638:
7624:
7594:
7589:
7550:Spectral method
7535:Ramanujan graph
7483:
7467:
7443:Fredholm theory
7411:
7406:Shilov boundary
7402:Structure space
7380:Generalizations
7375:
7366:Numerical range
7344:
7328:Uniform algebra
7290:
7266:Riesz projector
7251:Min-max theorem
7234:
7220:Direct integral
7176:
7162:Spectral radius
7133:
7088:
7042:
7033:Spectral radius
6981:
6975:Spectral theory
6972:
6935:
6914:
6913:
6864:
6860:
6821:
6817:
6778:
6774:
6735:
6731:
6692:
6688:
6631:
6627:
6620:
6590:
6583:
6575:
6571:
6563:
6559:
6554:
6547:
6542:
6538:
6533:
6524:
6517:
6503:
6499:
6445:
6441:
6436:
6394:
6360:
6355:
6348:
6337:
6326:
6324:
6308:
6286:
6282:
6271:
6266:
6259:
6248:
6237:
6229:
6215:
6209:
6205:
6203:
6200:
6199:
6174:
6172:
6169:
6168:
6143:
6138:
6131:
6120:
6109:
6107:
6096:
6091:
6084:
6073:
6062:
6054:
6040:
6034:
6030:
6028:
6025:
6024:
5999:
5997:
5994:
5993:
5976:
5972:
5970:
5967:
5966:
5947:
5942:
5940:
5937:
5936:
5917:
5912:
5910:
5907:
5906:
5887:
5882:
5880:
5877:
5876:
5850:
5848:
5845:
5844:
5822:
5820:
5817:
5816:
5794:
5783:
5772:
5764:
5761:
5760:
5732:
5730:
5727:
5726:
5710:
5707:
5706:
5684:
5682:
5679:
5678:
5662:
5659:
5658:
5632:
5628:
5623:
5606:
5585:
5581:
5576:
5559:
5548:
5545:
5544:
5513:
5497:
5478:
5474:
5460:
5446:
5435:
5424:
5406:
5392:
5390:
5387:
5386:
5361:
5344:
5336:
5325:
5314:
5307:
5301:
5290:
5279:
5275:
5273:
5264:
5260:
5251:
5247:
5232:
5228:
5226:
5223:
5222:
5205:
5199:
5195:
5183:
5179:
5174:
5166:
5155:
5144:
5137:
5132:
5123:
5119:
5117:
5114:
5113:
5096:
5090:
5086:
5074:
5070:
5065:
5057:
5046:
5035:
5028:
5023:
5014:
5010:
5008:
5005:
5004:
4979:
4977:
4974:
4973:
4951:
4949:
4946:
4945:
4920:
4918:
4915:
4914:
4894:
4890:
4885:
4883:
4880:
4879:
4860:
4854:
4850:
4838:
4834:
4829:
4821:
4804:
4799:
4796:
4795:
4779:
4762:
4760:
4757:
4756:
4734:
4732:
4729:
4728:
4712:
4706:
4702:
4690:
4686:
4681:
4679:
4676:
4675:
4653:
4651:
4648:
4647:
4627:
4623:
4618:
4616:
4613:
4612:
4593:
4587:
4583:
4571:
4567:
4562:
4554:
4537:
4532:
4529:
4528:
4506:
4504:
4501:
4500:
4478:
4476:
4473:
4472:
4423:
4421:
4418:
4417:
4395:
4393:
4390:
4389:
4370:
4353:
4345:
4328:
4323:
4320:
4319:
4297:
4295:
4292:
4291:
4269:
4267:
4264:
4263:
4241:
4239:
4236:
4235:
4213:
4211:
4208:
4207:
4178:
4164:
4149:
4135:
4133:
4130:
4129:
4107:
4093:
4091:
4088:
4087:
4077:
4053:
4049:
4048:
4044:
4042:
4039:
4038:
4019:
4015:
4014:
4010:
4008:
4005:
4004:
3977:
3970:
3966:
3965:
3950:
3946:
3945:
3941:
3929:
3928:
3927:
3917:
3910:
3906:
3905:
3895:
3888:
3884:
3883:
3868:
3864:
3863:
3859:
3851:
3847:
3846:
3842:
3832:
3830:
3818:
3814:
3809:
3803:
3799:
3791:
3789:
3786:
3785:
3765:
3761:
3759:
3756:
3755:
3738:
3734:
3732:
3729:
3728:
3708:
3704:
3702:
3699:
3698:
3681:
3677:
3668:
3664:
3658:
3653:
3647:
3644:
3643:
3627:
3624:
3623:
3596:
3589:
3585:
3584:
3569:
3565:
3564:
3560:
3548:
3547:
3546:
3539:
3532:
3528:
3527:
3512:
3508:
3507:
3503:
3502:
3500:
3488:
3482:
3479:
3478:
3458:
3454:
3452:
3449:
3448:
3431:
3427:
3425:
3422:
3421:
3393:
3385:
3381:
3380:
3376:
3370:
3363:
3359:
3358:
3349:
3330:
3322:
3318:
3317:
3313:
3312:
3310:
3301:
3297:
3289:
3284:
3282:
3279:
3278:
3258:
3254:
3246:
3244:
3241:
3240:
3217:
3213:
3205:
3199:
3195:
3187:
3181:
3177:
3171:
3160:
3144:
3140:
3132:
3126:
3122:
3114:
3105:
3101:
3099:
3096:
3095:
3075:
3071:
3069:
3066:
3065:
3048:
3044:
3036:
3034:
3031:
3030:
3010:
2998:
2997:
2996:
2987:
2980:
2976:
2975:
2974:
2968:
2957:
2944:
2940:
2938:
2935:
2934:
2914:
2910:
2901:
2897:
2891:
2886:
2880:
2877:
2876:
2854:
2850:
2848:
2836:
2832:
2830:
2827:
2826:
2807:
2804:
2803:
2786:
2775:
2765:
2761:
2756:
2753:
2752:
2748:
2726:
2715:
2705:
2701:
2696:
2693:
2692:
2676:
2673:
2672:
2651:
2650:
2644:
2643:
2642:
2640:
2637:
2636:
2620:
2617:
2616:
2599:
2588:
2578:
2574:
2569:
2566:
2565:
2545:
2541:
2539:
2536:
2535:
2519:
2516:
2515:
2491:
2490:
2486:
2478:
2466:
2462:
2454:
2449:
2446:
2445:
2426:
2423:
2422:
2406:
2403:
2402:
2401:into a unitary
2386:
2383:
2382:
2359:
2355:
2349:
2345:
2319:
2315:
2309:
2305:
2299:
2295:
2294:
2290:
2270:
2266:
2260:
2256:
2250:
2246:
2242:
2238:
2215:
2213:
2210:
2209:
2190:
2187:
2186:
2166:
2162:
2146:
2145:
2141:
2139:
2136:
2135:
2134:Note that here
2112:
2100:
2099:
2098:
2089:
2081:
2077:
2075:
2074:
2068:
2057:
2045:
2042:
2041:
2021:
2016:
2015:
1998:
1989:
1985:
1976:
1972:
1970:
1967:
1966:
1949:
1943:
1942:
1941:
1932:
1926:
1925:
1924:
1911:
1910:
1904:
1903:
1902:
1900:
1897:
1896:
1866:
1865:
1861:
1859:
1856:
1855:
1834:
1833:
1828:
1827:
1810:
1801:
1797:
1795:
1792:
1791:
1771:
1766:
1765:
1759:
1755:
1742:
1741:
1737:
1729:
1723:
1712:
1700:
1697:
1696:
1673:
1669:
1664:
1662:
1659:
1658:
1642:
1636:
1632:
1620:
1616:
1611:
1602:
1598:
1596:
1593:
1592:
1566:
1562:
1556:
1552:
1543:
1539:
1537:
1534:
1533:
1514:
1511:
1510:
1489:
1488:
1482:
1481:
1480:
1471:
1465:
1464:
1463:
1455:
1452:
1451:
1427:
1426:
1422:
1420:
1417:
1416:
1395:
1394:
1388:
1387:
1386:
1384:
1381:
1380:
1363:
1352:
1342:
1338:
1333:
1330:
1329:
1312:
1308:
1306:
1303:
1302:
1285:
1279:
1278:
1277:
1275:
1272:
1271:
1254:
1243:
1233:
1229:
1224:
1221:
1220:
1204:
1198:
1177:
1174:
1173:
1157:
1154:
1153:
1130:
1126:
1117:
1113:
1104:
1100:
1094:
1090:
1074:
1070:
1064:
1053:
1047:
1044:
1043:
1037:identity matrix
1016:
1012:
1007:
1004:
1003:
974:
968:
964:
959:
941:
937:
932:
915:
889:
887:
884:
883:
858:
856:
853:
852:
832:
829:
828:
804:
800:
771:
769:
766:
765:
746:
743:
742:
719:
716:
715:
698:
694:
692:
689:
688:
666:
664:
661:
660:
641:
638:
637:
597:
594:
593:
566:
563:
562:
529:
525:
519:
508:
502:
499:
498:
492:identity matrix
474:
473:
471:
468:
467:
447:
443:
438:
435:
434:
403:
402:
398:
381:
378:
377:
361:
358:
357:
301:
298:
297:
278:
277:
269:
266:
265:
248:
247:
245:
242:
241:
218:
215:
214:
198:
195:
194:
178:
175:
174:
171:Borel Ï-algebra
154:
151:
150:
118:
115:
114:
93:
92:
90:
87:
86:
83:
17:
12:
11:
5:
8538:
8528:
8527:
8522:
8505:
8504:
8502:
8501:
8496:
8491:
8489:Choquet theory
8486:
8481:
8470:
8468:
8464:
8463:
8461:
8460:
8455:
8450:
8444:
8442:
8436:
8435:
8433:
8432:
8427:
8421:
8419:
8415:
8414:
8412:
8411:
8406:
8401:
8396:
8391:
8390:
8389:
8379:
8373:
8371:
8367:
8366:
8364:
8363:
8358:
8353:
8348:
8343:
8338:
8332:
8330:
8324:
8323:
8321:
8320:
8315:
8299:
8298:
8297:
8292:
8287:
8273:
8272:
8271:
8266:
8256:
8250:
8248:
8244:
8243:
8241:
8240:
8235:
8230:
8225:
8220:
8219:
8218:
8208:
8203:
8202:
8201:
8191:
8186:
8180:
8178:
8172:
8171:
8169:
8168:
8163:
8158:
8153:
8148:
8143:
8138:
8137:
8136:
8125:
8123:
8122:Basic concepts
8119:
8118:
8107:
8106:
8099:
8092:
8084:
8075:
8074:
8072:
8071:
8060:
8057:
8056:
8054:
8053:
8048:
8043:
8038:
8036:Choquet theory
8033:
8028:
8022:
8020:
8016:
8015:
8013:
8012:
8002:
7997:
7992:
7987:
7982:
7977:
7972:
7967:
7962:
7957:
7952:
7946:
7944:
7940:
7939:
7937:
7936:
7931:
7925:
7923:
7919:
7918:
7916:
7915:
7910:
7905:
7900:
7895:
7890:
7888:Banach algebra
7884:
7882:
7878:
7877:
7875:
7874:
7869:
7864:
7859:
7854:
7849:
7844:
7839:
7834:
7829:
7823:
7821:
7817:
7816:
7814:
7813:
7811:BanachâAlaoglu
7808:
7803:
7798:
7793:
7788:
7783:
7778:
7773:
7767:
7765:
7759:
7758:
7755:
7754:
7752:
7751:
7746:
7741:
7739:Locally convex
7736:
7722:
7717:
7711:
7709:
7705:
7704:
7702:
7701:
7696:
7691:
7686:
7681:
7676:
7671:
7666:
7661:
7656:
7650:
7644:
7640:
7639:
7623:
7622:
7615:
7608:
7600:
7591:
7590:
7588:
7587:
7582:
7577:
7572:
7567:
7562:
7557:
7552:
7547:
7542:
7537:
7532:
7527:
7522:
7517:
7512:
7502:
7500:Corona theorem
7497:
7491:
7489:
7485:
7484:
7482:
7481:
7479:Wiener algebra
7475:
7473:
7469:
7468:
7466:
7465:
7460:
7455:
7450:
7445:
7440:
7435:
7430:
7425:
7419:
7417:
7413:
7412:
7410:
7409:
7399:
7397:Pseudospectrum
7394:
7389:
7387:Dirac spectrum
7383:
7381:
7377:
7376:
7374:
7373:
7368:
7363:
7358:
7352:
7350:
7346:
7345:
7343:
7342:
7341:
7340:
7330:
7325:
7320:
7315:
7310:
7304:
7298:
7296:
7292:
7291:
7289:
7288:
7283:
7278:
7273:
7268:
7263:
7258:
7253:
7248:
7242:
7240:
7236:
7235:
7233:
7232:
7227:
7222:
7217:
7212:
7207:
7206:
7205:
7200:
7195:
7184:
7182:
7178:
7177:
7175:
7174:
7169:
7164:
7159:
7154:
7149:
7143:
7141:
7135:
7134:
7132:
7131:
7126:
7118:
7110:
7102:
7096:
7094:
7090:
7089:
7087:
7086:
7081:
7076:
7071:
7066:
7061:
7056:
7050:
7048:
7044:
7043:
7041:
7040:
7038:Operator space
7035:
7030:
7025:
7020:
7015:
7010:
7005:
7000:
6998:Banach algebra
6995:
6989:
6987:
6986:Basic concepts
6983:
6982:
6971:
6970:
6963:
6956:
6948:
6942:
6941:
6934:
6933:External links
6931:
6930:
6929:
6928:
6927:
6921:
6912:
6911:
6858:
6815:
6772:
6729:
6686:
6625:
6618:
6581:
6569:
6557:
6545:
6536:
6522:
6515:
6497:
6438:
6437:
6435:
6432:
6431:
6430:
6428:Vector measure
6425:
6420:
6415:
6413:Density matrix
6410:
6405:
6400:
6393:
6390:
6382:
6381:
6370:
6367:
6358:
6351:
6347:
6344:
6340:
6336:
6333:
6329:
6321:
6318:
6315:
6311:
6307:
6304:
6301:
6298:
6295:
6292:
6289:
6285:
6281:
6278:
6273:result φ
6269:
6262:
6258:
6255:
6251:
6247:
6244:
6240:
6236:
6233:
6228:
6225:
6222:
6218:
6208:
6184:
6181:
6177:
6165:
6164:
6153:
6150:
6141:
6134:
6130:
6127:
6123:
6119:
6116:
6112:
6106:
6103:
6098:result ψ
6094:
6087:
6083:
6080:
6076:
6072:
6069:
6065:
6061:
6058:
6053:
6050:
6047:
6043:
6033:
6009:
6006:
6002:
5975:
5954:
5945:
5924:
5919:result φ
5915:
5894:
5889:result ψ
5885:
5860:
5857:
5853:
5832:
5829:
5825:
5813:
5812:
5801:
5797:
5793:
5790:
5786:
5782:
5779:
5775:
5771:
5768:
5742:
5739:
5735:
5714:
5694:
5691:
5687:
5666:
5646:
5643:
5640:
5635:
5631:
5626:
5622:
5619:
5616:
5613:
5609:
5605:
5602:
5599:
5596:
5593:
5588:
5584:
5579:
5575:
5572:
5569:
5566:
5562:
5558:
5555:
5552:
5541:
5540:
5529:
5526:
5523:
5520:
5516:
5510:
5507:
5504:
5500:
5496:
5493:
5490:
5487:
5484:
5481:
5477:
5473:
5470:
5467:
5463:
5459:
5453:
5449:
5445:
5442:
5438:
5434:
5431:
5427:
5423:
5420:
5417:
5414:
5410:
5405:
5402:
5399:
5395:
5380:
5379:
5368:
5364:
5360:
5357:
5354:
5351:
5347:
5339:
5335:
5332:
5328:
5324:
5321:
5317:
5313:
5310:
5304:
5300:
5297:
5293:
5289:
5286:
5282:
5278:
5272:
5267:
5263:
5259:
5254:
5250:
5246:
5243:
5240:
5235:
5231:
5220:
5208:
5202:
5198:
5194:
5191:
5186:
5182:
5177:
5169:
5165:
5162:
5158:
5154:
5151:
5147:
5143:
5140:
5136:
5131:
5126:
5122:
5111:
5099:
5093:
5089:
5085:
5082:
5077:
5073:
5068:
5060:
5056:
5053:
5049:
5045:
5042:
5038:
5034:
5031:
5027:
5022:
5017:
5013:
4989:
4986:
4982:
4961:
4958:
4954:
4930:
4927:
4923:
4902:
4897:
4893:
4888:
4867:
4863:
4857:
4853:
4849:
4846:
4841:
4837:
4832:
4828:
4824:
4820:
4817:
4814:
4811:
4807:
4803:
4782:
4778:
4775:
4772:
4769:
4765:
4744:
4741:
4737:
4715:
4709:
4705:
4701:
4698:
4693:
4689:
4684:
4663:
4660:
4656:
4635:
4630:
4626:
4621:
4600:
4596:
4590:
4586:
4582:
4579:
4574:
4570:
4565:
4561:
4557:
4553:
4550:
4547:
4544:
4540:
4536:
4516:
4513:
4509:
4488:
4485:
4481:
4433:
4430:
4426:
4405:
4402:
4398:
4377:
4373:
4369:
4366:
4363:
4360:
4356:
4352:
4348:
4344:
4341:
4338:
4335:
4331:
4327:
4307:
4304:
4300:
4279:
4276:
4272:
4251:
4248:
4244:
4223:
4220:
4216:
4191:
4188:
4185:
4181:
4177:
4174:
4171:
4167:
4163:
4157:
4153:
4148:
4145:
4142:
4138:
4117:
4114:
4110:
4106:
4103:
4100:
4096:
4076:
4073:
4056:
4052:
4047:
4022:
4018:
4013:
4001:
4000:
3985:
3980:
3973:
3969:
3964:
3960:
3953:
3949:
3944:
3940:
3935:
3932:
3925:
3920:
3913:
3909:
3904:
3898:
3891:
3887:
3882:
3878:
3871:
3867:
3862:
3854:
3850:
3845:
3841:
3838:
3835:
3829:
3826:
3821:
3817:
3812:
3806:
3802:
3798:
3768:
3764:
3741:
3737:
3711:
3707:
3684:
3680:
3676:
3671:
3667:
3661:
3656:
3652:
3631:
3620:
3619:
3604:
3599:
3592:
3588:
3583:
3579:
3572:
3568:
3563:
3559:
3554:
3551:
3542:
3535:
3531:
3526:
3522:
3515:
3511:
3506:
3499:
3495:
3491:
3487:
3461:
3457:
3434:
3430:
3418:
3417:
3403:
3400:
3396:
3388:
3384:
3379:
3373:
3366:
3362:
3357:
3352:
3348:
3345:
3340:
3337:
3333:
3325:
3321:
3316:
3309:
3304:
3300:
3295:
3292:
3287:
3261:
3257:
3253:
3249:
3237:
3236:
3225:
3220:
3216:
3212:
3208:
3202:
3198:
3194:
3190:
3184:
3180:
3174:
3169:
3166:
3163:
3159:
3155:
3152:
3147:
3143:
3139:
3135:
3129:
3125:
3121:
3117:
3113:
3108:
3104:
3078:
3074:
3051:
3047:
3043:
3039:
3027:
3026:
3013:
3008:
3005:
3001:
2995:
2990:
2983:
2979:
2971:
2966:
2963:
2960:
2956:
2952:
2947:
2943:
2917:
2913:
2909:
2904:
2900:
2894:
2889:
2885:
2873:
2872:
2857:
2853:
2847:
2844:
2839:
2835:
2822:the operators
2811:
2789:
2784:
2781:
2778:
2774:
2768:
2764:
2760:
2747:
2744:
2729:
2724:
2721:
2718:
2714:
2708:
2704:
2700:
2680:
2657:
2654:
2647:
2624:
2602:
2597:
2594:
2591:
2587:
2581:
2577:
2573:
2548:
2544:
2523:
2512:
2511:
2497:
2494:
2489:
2485:
2481:
2477:
2474:
2469:
2465:
2461:
2457:
2453:
2430:
2410:
2390:
2379:
2378:
2367:
2362:
2358:
2352:
2348:
2344:
2341:
2338:
2335:
2331:
2327:
2322:
2318:
2312:
2308:
2302:
2298:
2293:
2289:
2286:
2283:
2279:
2273:
2269:
2263:
2259:
2253:
2249:
2245:
2241:
2237:
2234:
2231:
2228:
2225:
2222:
2194:
2169:
2165:
2161:
2158:
2152:
2149:
2144:
2132:
2131:
2120:
2115:
2110:
2107:
2103:
2097:
2092:
2084:
2080:
2071:
2066:
2063:
2060:
2056:
2052:
2049:
2024:
2019:
2014:
2011:
2008:
2005:
2001:
1997:
1992:
1988:
1984:
1979:
1975:
1952:
1946:
1940:
1935:
1929:
1923:
1917:
1914:
1907:
1881:
1878:
1872:
1869:
1864:
1840:
1837:
1831:
1826:
1823:
1820:
1817:
1813:
1809:
1804:
1800:
1788:
1787:
1774:
1769:
1762:
1758:
1754:
1748:
1745:
1740:
1736:
1732:
1726:
1721:
1718:
1715:
1711:
1707:
1704:
1681:
1676:
1672:
1667:
1645:
1639:
1635:
1631:
1628:
1623:
1619:
1614:
1610:
1605:
1601:
1589:
1588:
1577:
1574:
1569:
1565:
1559:
1555:
1551:
1546:
1542:
1518:
1495:
1492:
1485:
1479:
1474:
1468:
1462:
1459:
1433:
1430:
1425:
1401:
1398:
1391:
1366:
1361:
1358:
1355:
1351:
1345:
1341:
1337:
1315:
1311:
1288:
1282:
1257:
1252:
1249:
1246:
1242:
1236:
1232:
1228:
1211:
1210:
1200:Main article:
1197:
1194:
1181:
1161:
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1138:
1133:
1129:
1123:
1120:
1116:
1112:
1107:
1103:
1097:
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1088:
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1067:
1062:
1059:
1056:
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1024:
1019:
1015:
1011:
997:
996:
984:
981:
977:
971:
967:
962:
958:
955:
952:
949:
944:
940:
935:
931:
928:
925:
922:
918:
914:
911:
908:
905:
902:
899:
896:
868:
865:
861:
836:
825:
824:
812:
807:
803:
799:
796:
793:
790:
787:
784:
781:
778:
750:
723:
701:
697:
676:
673:
669:
645:
625:
622:
619:
616:
613:
610:
607:
604:
601:
570:
555:
554:
543:
540:
537:
532:
528:
522:
517:
514:
511:
507:
477:
455:
450:
446:
442:
406:
401:
397:
394:
391:
388:
385:
365:
350:
349:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
305:
281:
276:
273:
251:
222:
202:
182:
158:
134:
131:
128:
125:
122:
96:
82:
79:
15:
9:
6:
4:
3:
2:
8537:
8526:
8523:
8521:
8518:
8517:
8515:
8500:
8497:
8495:
8492:
8490:
8487:
8485:
8482:
8479:
8475:
8472:
8471:
8469:
8465:
8459:
8456:
8454:
8451:
8449:
8446:
8445:
8443:
8441:
8437:
8431:
8428:
8426:
8423:
8422:
8420:
8416:
8410:
8407:
8405:
8402:
8400:
8397:
8395:
8392:
8388:
8385:
8384:
8383:
8380:
8378:
8375:
8374:
8372:
8368:
8362:
8359:
8357:
8354:
8352:
8349:
8347:
8344:
8342:
8339:
8337:
8334:
8333:
8331:
8329:
8325:
8319:
8316:
8314:
8311:
8307:
8303:
8300:
8296:
8293:
8291:
8288:
8286:
8283:
8282:
8281:
8280:set functions
8277:
8274:
8270:
8267:
8265:
8262:
8261:
8260:
8257:
8255:
8254:Besov measure
8252:
8251:
8249:
8247:Measurability
8245:
8239:
8236:
8234:
8231:
8229:
8226:
8224:
8221:
8217:
8214:
8213:
8212:
8209:
8207:
8204:
8200:
8197:
8196:
8195:
8192:
8190:
8187:
8185:
8182:
8181:
8179:
8177:
8173:
8167:
8164:
8162:
8159:
8157:
8154:
8152:
8149:
8147:
8146:Convex series
8144:
8142:
8141:Bochner space
8139:
8135:
8132:
8131:
8130:
8127:
8126:
8124:
8120:
8116:
8112:
8105:
8100:
8098:
8093:
8091:
8086:
8085:
8082:
8070:
8062:
8061:
8058:
8052:
8049:
8047:
8044:
8042:
8041:Weak topology
8039:
8037:
8034:
8032:
8029:
8027:
8024:
8023:
8021:
8017:
8010:
8006:
8003:
8001:
7998:
7996:
7993:
7991:
7988:
7986:
7983:
7981:
7978:
7976:
7973:
7971:
7968:
7966:
7965:Index theorem
7963:
7961:
7958:
7956:
7953:
7951:
7948:
7947:
7945:
7941:
7935:
7932:
7930:
7927:
7926:
7924:
7922:Open problems
7920:
7914:
7911:
7909:
7906:
7904:
7901:
7899:
7896:
7894:
7891:
7889:
7886:
7885:
7883:
7879:
7873:
7870:
7868:
7865:
7863:
7860:
7858:
7855:
7853:
7850:
7848:
7845:
7843:
7840:
7838:
7835:
7833:
7830:
7828:
7825:
7824:
7822:
7818:
7812:
7809:
7807:
7804:
7802:
7799:
7797:
7794:
7792:
7789:
7787:
7784:
7782:
7779:
7777:
7774:
7772:
7769:
7768:
7766:
7764:
7760:
7750:
7747:
7745:
7742:
7740:
7737:
7734:
7730:
7726:
7723:
7721:
7718:
7716:
7713:
7712:
7710:
7706:
7700:
7697:
7695:
7692:
7690:
7687:
7685:
7682:
7680:
7677:
7675:
7672:
7670:
7667:
7665:
7662:
7660:
7657:
7655:
7652:
7651:
7648:
7645:
7641:
7636:
7632:
7628:
7621:
7616:
7614:
7609:
7607:
7602:
7601:
7598:
7586:
7583:
7581:
7578:
7576:
7573:
7571:
7568:
7566:
7563:
7561:
7558:
7556:
7553:
7551:
7548:
7546:
7543:
7541:
7538:
7536:
7533:
7531:
7528:
7526:
7523:
7521:
7518:
7516:
7513:
7510:
7506:
7503:
7501:
7498:
7496:
7493:
7492:
7490:
7486:
7480:
7477:
7476:
7474:
7470:
7464:
7461:
7459:
7456:
7454:
7451:
7449:
7446:
7444:
7441:
7439:
7436:
7434:
7431:
7429:
7426:
7424:
7421:
7420:
7418:
7416:Miscellaneous
7414:
7407:
7403:
7400:
7398:
7395:
7393:
7390:
7388:
7385:
7384:
7382:
7378:
7372:
7369:
7367:
7364:
7362:
7359:
7357:
7354:
7353:
7351:
7347:
7339:
7336:
7335:
7334:
7331:
7329:
7326:
7324:
7321:
7319:
7316:
7314:
7311:
7309:
7305:
7303:
7300:
7299:
7297:
7293:
7287:
7284:
7282:
7279:
7277:
7274:
7272:
7269:
7267:
7264:
7262:
7259:
7257:
7254:
7252:
7249:
7247:
7244:
7243:
7241:
7237:
7231:
7228:
7226:
7223:
7221:
7218:
7216:
7213:
7211:
7208:
7204:
7201:
7199:
7196:
7194:
7191:
7190:
7189:
7186:
7185:
7183:
7181:Decomposition
7179:
7173:
7170:
7168:
7165:
7163:
7160:
7158:
7155:
7153:
7150:
7148:
7145:
7144:
7142:
7140:
7136:
7130:
7127:
7125:
7122:
7119:
7117:
7114:
7111:
7109:
7106:
7103:
7101:
7098:
7097:
7095:
7091:
7085:
7082:
7080:
7077:
7075:
7072:
7070:
7067:
7065:
7062:
7060:
7057:
7055:
7052:
7051:
7049:
7045:
7039:
7036:
7034:
7031:
7029:
7026:
7024:
7021:
7019:
7016:
7014:
7011:
7009:
7006:
7004:
7001:
6999:
6996:
6994:
6991:
6990:
6988:
6984:
6980:
6976:
6969:
6964:
6962:
6957:
6955:
6950:
6949:
6946:
6940:
6937:
6936:
6925:
6922:
6919:
6918:
6916:
6915:
6907:
6903:
6899:
6895:
6891:
6887:
6882:
6877:
6873:
6869:
6862:
6854:
6850:
6846:
6842:
6838:
6834:
6830:
6826:
6819:
6811:
6807:
6803:
6799:
6795:
6791:
6787:
6783:
6776:
6768:
6764:
6760:
6756:
6752:
6748:
6744:
6740:
6733:
6725:
6721:
6717:
6713:
6709:
6705:
6701:
6697:
6690:
6682:
6678:
6674:
6670:
6666:
6662:
6658:
6654:
6649:
6644:
6640:
6636:
6629:
6621:
6615:
6611:
6607:
6603:
6598:
6597:
6588:
6586:
6579:
6573:
6567:
6561:
6552:
6550:
6540:
6531:
6529:
6527:
6518:
6512:
6508:
6501:
6493:
6489:
6485:
6481:
6477:
6473:
6468:
6463:
6460:(1): 93â123.
6459:
6455:
6454:
6449:
6443:
6439:
6429:
6426:
6424:
6421:
6419:
6416:
6414:
6411:
6409:
6406:
6404:
6401:
6399:
6396:
6395:
6389:
6385:
6368:
6342:
6334:
6313:
6305:
6296:
6293:
6290:
6287:
6283:
6279:
6253:
6245:
6234:
6231:
6226:
6220:
6206:
6198:
6197:
6196:
6179:
6151:
6125:
6117:
6104:
6078:
6070:
6059:
6056:
6051:
6045:
6031:
6023:
6022:
6021:
6004:
5973:
5872:
5855:
5827:
5799:
5788:
5780:
5769:
5766:
5759:
5758:
5757:
5754:
5737:
5712:
5689:
5664:
5644:
5641:
5633:
5629:
5620:
5611:
5600:
5597:
5594:
5586:
5582:
5573:
5564:
5553:
5550:
5527:
5518:
5502:
5494:
5485:
5482:
5479:
5475:
5471:
5465:
5440:
5432:
5421:
5418:
5412:
5408:
5403:
5397:
5385:
5384:
5383:
5366:
5358:
5349:
5330:
5322:
5311:
5308:
5295:
5287:
5276:
5270:
5265:
5261:
5257:
5252:
5248:
5244:
5238:
5233:
5229:
5221:
5200:
5196:
5184:
5180:
5160:
5152:
5141:
5138:
5134:
5129:
5124:
5120:
5112:
5091:
5087:
5075:
5071:
5051:
5043:
5032:
5029:
5025:
5020:
5015:
5011:
5003:
5002:
5001:
4984:
4956:
4942:
4925:
4895:
4891:
4855:
4851:
4839:
4835:
4826:
4818:
4809:
4776:
4767:
4739:
4707:
4703:
4691:
4687:
4658:
4628:
4624:
4588:
4584:
4572:
4568:
4559:
4551:
4542:
4511:
4483:
4470:
4465:
4463:
4462:quantum money
4459:
4455:
4451:
4447:
4428:
4400:
4367:
4358:
4350:
4342:
4333:
4302:
4274:
4246:
4234:or the state
4218:
4183:
4175:
4169:
4155:
4151:
4146:
4140:
4112:
4104:
4098:
4085:
4081:
4072:
4054:
4050:
4045:
4020:
4016:
4011:
3978:
3971:
3967:
3962:
3958:
3951:
3947:
3942:
3918:
3911:
3907:
3902:
3896:
3889:
3885:
3880:
3876:
3869:
3865:
3860:
3852:
3848:
3843:
3836:
3833:
3827:
3819:
3815:
3804:
3800:
3784:
3783:
3782:
3766:
3762:
3739:
3735:
3725:
3709:
3705:
3682:
3678:
3674:
3669:
3665:
3659:
3654:
3650:
3629:
3597:
3590:
3586:
3581:
3577:
3570:
3566:
3561:
3540:
3533:
3529:
3524:
3520:
3513:
3509:
3504:
3497:
3493:
3489:
3485:
3477:
3476:
3475:
3459:
3455:
3432:
3428:
3398:
3386:
3382:
3377:
3371:
3364:
3360:
3355:
3346:
3335:
3323:
3319:
3314:
3307:
3302:
3293:
3290:
3277:
3276:
3275:
3274:to the state
3259:
3251:
3223:
3218:
3210:
3200:
3192:
3182:
3178:
3172:
3167:
3164:
3161:
3157:
3153:
3145:
3137:
3127:
3119:
3106:
3102:
3094:
3093:
3092:
3076:
3072:
3049:
3041:
3011:
3003:
2993:
2988:
2981:
2977:
2969:
2964:
2961:
2958:
2954:
2950:
2945:
2941:
2933:
2932:
2931:
2915:
2911:
2907:
2902:
2898:
2892:
2887:
2883:
2855:
2851:
2845:
2842:
2837:
2833:
2825:
2824:
2823:
2809:
2787:
2782:
2779:
2776:
2766:
2762:
2743:
2727:
2722:
2719:
2716:
2706:
2678:
2655:
2652:
2622:
2600:
2595:
2592:
2589:
2579:
2575:
2562:
2546:
2542:
2521:
2495:
2492:
2483:
2475:
2472:
2467:
2459:
2451:
2444:
2443:
2442:
2428:
2408:
2388:
2360:
2356:
2350:
2346:
2339:
2336:
2333:
2329:
2325:
2320:
2310:
2306:
2300:
2296:
2291:
2287:
2284:
2281:
2277:
2271:
2261:
2257:
2251:
2247:
2243:
2239:
2235:
2232:
2229:
2223:
2208:
2207:
2206:
2192:
2183:
2167:
2163:
2159:
2156:
2150:
2147:
2142:
2118:
2113:
2105:
2095:
2090:
2082:
2078:
2069:
2064:
2061:
2058:
2054:
2050:
2047:
2040:
2039:
2038:
2022:
2012:
2003:
1995:
1990:
1982:
1977:
1950:
1938:
1933:
1921:
1915:
1912:
1893:
1879:
1876:
1870:
1867:
1862:
1838:
1835:
1824:
1815:
1807:
1802:
1772:
1760:
1756:
1746:
1743:
1734:
1724:
1719:
1716:
1713:
1709:
1705:
1702:
1695:
1694:
1693:
1674:
1670:
1637:
1633:
1621:
1617:
1608:
1603:
1599:
1575:
1572:
1567:
1557:
1553:
1549:
1544:
1540:
1532:
1531:
1530:
1516:
1493:
1490:
1472:
1460:
1457:
1450:
1431:
1428:
1423:
1415:of dimension
1399:
1396:
1364:
1359:
1356:
1353:
1343:
1313:
1309:
1301:of dimension
1286:
1255:
1250:
1247:
1244:
1234:
1230:
1217:
1215:
1209:
1206:
1205:
1203:
1193:
1179:
1159:
1136:
1131:
1121:
1118:
1114:
1110:
1105:
1095:
1086:
1080:
1075:
1065:
1060:
1057:
1054:
1050:
1042:
1041:
1040:
1038:
1017:
1002:
979:
969:
965:
956:
950:
942:
938:
929:
920:
909:
906:
903:
897:
882:
881:
880:
863:
850:
834:
805:
801:
797:
791:
788:
785:
779:
764:
763:
762:
748:
741:
740:quantum state
737:
721:
699:
695:
671:
659:
658:quantum state
643:
620:
617:
614:
608:
602:
591:
586:
584:
568:
560:
541:
535:
530:
526:
520:
515:
512:
509:
505:
497:
496:
495:
493:
448:
444:
433:
430:
425:
423:
395:
389:
383:
363:
355:
336:
330:
327:
324:
318:
312:
303:
296:
295:
294:
274:
271:
239:
236:
220:
200:
180:
172:
156:
148:
129:
126:
123:
112:
111:Hilbert space
78:
76:
72:
67:
65:
61:
57:
52:
50:
46:
45:Hilbert space
42:
38:
34:
30:
26:
22:
8467:Applications
8425:Crinkled arc
8361:PaleyâWiener
8031:Balanced set
8005:Distribution
7943:Applications
7796:KreinâMilman
7781:Closed graph
7488:Applications
7318:Disk algebra
7172:Spectral gap
7047:Main results
6871:
6867:
6861:
6828:
6824:
6818:
6785:
6781:
6775:
6742:
6738:
6732:
6699:
6695:
6689:
6638:
6634:
6628:
6595:
6572:
6560:
6539:
6506:
6500:
6457:
6451:
6448:Peres, Asher
6442:
6386:
6383:
6166:
5873:
5814:
5755:
5542:
5381:
4943:
4466:
4445:
4205:
4084:Bloch sphere
4002:
3726:
3621:
3419:
3238:
3028:
2874:
2749:
2563:
2513:
2380:
2184:
2133:
1894:
1789:
1590:
1218:
1212:
1207:
1151:
998:
826:
761:is given by
587:
556:
426:
351:
84:
68:
53:
32:
28:
18:
8233:Holomorphic
8216:Directional
8176:Derivatives
7960:Heat kernel
7950:Hardy space
7857:Trace class
7771:HahnâBanach
7733:Topological
7515:Heat kernel
7215:Compression
7100:Isospectral
6924:A.S. Holevo
2441:such that
213:defined on
56:mixed state
8514:Categories
7893:C*-algebra
7708:Properties
7193:Continuous
7008:C*-algebra
7003:B*-algebra
6434:References
5543:Note that
4446:impossible
2534:from 1 to
81:Definition
60:pure state
8356:Regulated
8328:Integrals
7867:Unbounded
7862:Transpose
7820:Operators
7749:Separable
7744:Reflexive
7729:Algebraic
7715:Barrelled
6979:-algebras
6810:0375-9601
6767:0375-9601
6724:0375-9601
6681:119340381
6673:0010-7514
6366:⟩
6346:⟩
6343:ψ
6335:φ
6332:⟨
6317:⟩
6314:ψ
6306:φ
6303:⟨
6297:
6288:−
6277:⟩
6257:⟩
6254:ψ
6246:φ
6243:⟨
6235:−
6224:⟩
6221:φ
6183:⟩
6180:φ
6149:⟩
6129:⟩
6126:ψ
6118:φ
6115:⟨
6102:⟩
6082:⟩
6079:ψ
6071:φ
6068:⟨
6060:−
6049:⟩
6046:ψ
6008:⟩
6005:ψ
5953:⟩
5923:⟩
5893:⟩
5859:⟩
5856:φ
5831:⟩
5828:ψ
5792:⟩
5789:ψ
5781:φ
5778:⟨
5770:−
5741:⟩
5738:φ
5713:φ
5693:⟩
5690:ψ
5665:ψ
5634:φ
5621:ψ
5618:⟨
5615:⟩
5612:ψ
5601:
5587:ψ
5574:φ
5571:⟨
5568:⟩
5565:φ
5554:
5522:⟩
5519:φ
5506:⟩
5503:ψ
5495:φ
5492:⟨
5486:
5469:⟩
5466:ψ
5444:⟩
5441:ψ
5433:φ
5430:⟨
5401:⟩
5398:γ
5359:γ
5356:⟨
5353:⟩
5350:γ
5334:⟩
5331:ψ
5323:φ
5320:⟨
5299:⟩
5296:ψ
5288:φ
5285:⟨
5266:φ
5258:−
5253:ψ
5245:−
5201:⊥
5197:ψ
5193:⟨
5190:⟩
5185:⊥
5181:ψ
5164:⟩
5161:ψ
5153:φ
5150:⟨
5125:φ
5092:⊥
5088:φ
5084:⟨
5081:⟩
5076:⊥
5072:φ
5055:⟩
5052:ψ
5044:φ
5041:⟨
5016:ψ
4988:⟩
4985:φ
4960:⟩
4957:ψ
4929:⟩
4926:φ
4901:⟩
4896:⊥
4892:φ
4856:⊥
4852:φ
4848:⟨
4845:⟩
4840:⊥
4836:φ
4819:φ
4816:⟨
4813:⟩
4810:φ
4777:ψ
4774:⟨
4771:⟩
4768:ψ
4743:⟩
4740:φ
4708:⊥
4704:ψ
4700:⟨
4697:⟩
4692:⊥
4688:ψ
4662:⟩
4659:ψ
4634:⟩
4629:⊥
4625:ψ
4611:, where
4589:⊥
4585:ψ
4581:⟨
4578:⟩
4573:⊥
4569:ψ
4552:ψ
4549:⟨
4546:⟩
4543:ψ
4515:⟩
4512:φ
4487:⟩
4484:ψ
4432:⟩
4429:φ
4404:⟩
4401:ψ
4368:φ
4365:⟨
4362:⟩
4359:φ
4343:ψ
4340:⟨
4337:⟩
4334:ψ
4306:⟩
4303:φ
4278:⟩
4275:ψ
4250:⟩
4247:φ
4222:⟩
4219:ψ
4187:⟩
4173:⟩
4144:⟩
4141:φ
4116:⟩
4102:⟩
4099:ψ
3979:†
3959:ρ
3919:†
3897:†
3877:ρ
3837:
3660:†
3598:†
3578:ρ
3541:†
3521:ρ
3486:ρ
3456:ρ
3402:⟩
3399:ψ
3372:†
3347:ψ
3344:⟨
3339:⟩
3336:ψ
3299:⟩
3291:ψ
3256:⟩
3252:ψ
3215:⟩
3197:⟩
3193:ψ
3158:∑
3142:⟩
3124:⟩
3120:ψ
3046:⟩
3042:ψ
3007:⟩
2994:⊗
2955:∑
2893:†
2703:Π
2623:ρ
2488:⟩
2464:⟩
2347:ρ
2340:
2317:Π
2311:†
2297:ρ
2288:
2268:Π
2262:†
2248:ρ
2236:
2109:⟩
2096:⊗
2055:∑
2010:⟨
2007:⟩
1996:⊗
1974:Π
1939:⊗
1822:⟨
1819:⟩
1799:Π
1753:⟨
1739:⟩
1710:∑
1680:⟩
1630:⟨
1627:⟩
1564:Π
1558:†
1478:→
1340:Π
1128:Π
1115:δ
1102:Π
1092:Π
1072:Π
1051:∑
1014:Π
983:⟩
980:ψ
957:ψ
954:⟨
930:ψ
927:⟨
924:⟩
921:ψ
910:
867:⟩
864:ψ
798:ρ
792:
749:ρ
675:⟩
672:ψ
624:⟩
621:ψ
618:∣
615:ψ
600:⟨
506:∑
334:⟩
331:ψ
328:∣
325:ψ
310:⟨
307:↦
275:∈
272:ψ
109:denote a
8310:Strongly
8111:Analysis
8069:Category
7881:Algebras
7763:Theorems
7720:Complete
7689:Schwartz
7635:glossary
7580:Weyl law
7525:Lax pair
7472:Examples
7306:With an
7225:Discrete
7203:Residual
7139:Spectrum
7124:operator
7116:operator
7108:operator
7023:Spectrum
6906:39481893
6398:SIC-POVM
6392:See also
6362:result ?
6145:result ?
5949:result ?
4878:, where
3494:′
3294:′
2656:′
2496:′
2151:′
1916:′
1871:′
1839:′
1747:′
1494:′
1449:isometry
1432:′
1400:′
235:positive
58:is to a
8476: (
8418:Related
8370:Results
8346:Dunford
8336:Bochner
8302:Bochner
8276:Measure
7872:Unitary
7852:Nuclear
7837:Compact
7832:Bounded
7827:Adjoint
7801:Minâmax
7694:Sobolev
7679:Nuclear
7669:Hilbert
7664:Fréchet
7629: (
7121:Unitary
6886:Bibcode
6853:9913923
6833:Bibcode
6790:Bibcode
6747:Bibcode
6704:Bibcode
6653:Bibcode
6492:7481797
6472:Bibcode
1447:and an
847:is the
738:on the
420:is the
37:measure
35:) is a
8478:bundle
8306:Weakly
8295:Vector
7847:Normal
7684:Orlicz
7674:Hölder
7654:Banach
7643:Spaces
7631:topics
7105:Normal
6917:POVMs
6904:
6851:
6808:
6765:
6722:
6679:
6671:
6616:
6604:â465.
6513:
6490:
5935:, and
5382:where
4460:, and
2037:, and
827:where
8199:Total
7659:Besov
7198:Point
6902:S2CID
6876:arXiv
6677:S2CID
6643:arXiv
6488:S2CID
6462:arXiv
849:trace
376:and
149:with
43:on a
8007:(or
7725:Dual
7129:Unit
6977:and
6849:PMID
6806:ISSN
6763:ISSN
6720:ISSN
6669:ISSN
6614:ISBN
6511:ISBN
6211:UQSD
6036:UQSD
5978:UQSD
4972:and
4416:and
4290:and
4128:and
4037:and
3793:Prob
2514:for
2217:Prob
891:Prob
773:Prob
113:and
85:Let
33:POVM
27:, a
23:and
8113:in
6894:doi
6841:doi
6798:doi
6786:128
6755:doi
6743:126
6712:doi
6700:123
6661:doi
6606:doi
6602:417
6480:doi
6294:arg
6195:to
6020:to
5843:or
5483:arg
4499:or
588:In
240:on
173:on
19:In
8516::
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8304:/
7633:â
6900:.
6892:.
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6870:.
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6839:.
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6710:.
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6675:.
6667:.
6659:.
6651:.
6639:41
6637:.
6612:.
6584:^
6548:^
6525:^
6486:.
6478:.
6470:.
6458:76
6456:.
5905:,
5753:.
5598:tr
5551:tr
4941:.
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2742:.
2337:tr
2285:tr
2233:tr
1965:,
1892:.
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1039::
907:tr
835:tr
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585:.
494:,
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169:a
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7731:/
7727:(
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7507:(
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6967:e
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6953:v
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6888::
6878::
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6843::
6835::
6812:.
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6792::
6769:.
6757::
6749::
6726:.
6714::
6706::
6683:.
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6655::
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6622:.
6608::
6519:.
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6474::
6464::
6369:.
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6310:|
6300:(
6291:i
6284:e
6280:+
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6217:|
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6052:=
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5974:U
5944:|
5914:|
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5800:,
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5630:F
5625:|
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5595:=
5592:)
5583:F
5578:|
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5557:(
5528:.
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5458:(
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5426:|
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5327:|
5316:|
5312:+
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5277:2
5271:=
5262:F
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5242:I
5239:=
5234:?
5230:F
5207:|
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5168:|
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5130:=
5121:F
5098:|
5067:|
5059:|
5048:|
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5033:+
5030:1
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5021:=
5012:F
4981:|
4953:|
4922:|
4887:|
4866:}
4862:|
4831:|
4827:,
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4806:|
4802:{
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4683:|
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4095:|
4055:1
4051:i
4046:M
4021:0
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4012:M
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3972:0
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3890:0
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3840:(
3828:=
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3805:1
3801:i
3797:(
3767:1
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3740:0
3736:i
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3675:=
3670:i
3666:M
3655:i
3651:M
3630:W
3618:.
3603:)
3591:0
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3582:M
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3558:(
3553:r
3550:t
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3525:M
3514:0
3510:i
3505:M
3498:=
3490:A
3460:A
3433:0
3429:i
3395:|
3387:0
3383:i
3378:M
3365:0
3361:i
3356:M
3351:|
3332:|
3324:0
3320:i
3315:M
3308:=
3303:A
3286:|
3260:A
3248:|
3224:,
3219:B
3211:i
3207:|
3201:A
3189:|
3183:i
3179:M
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3165:=
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3154:=
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3146:B
3138:0
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3128:A
3116:|
3112:(
3107:W
3103:U
3077:W
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3004:i
3000:|
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2962:=
2959:i
2951:=
2946:W
2942:V
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2912:F
2908:=
2903:i
2899:M
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2843:=
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2780:=
2777:i
2773:}
2767:i
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2759:{
2728:n
2723:1
2720:=
2717:i
2713:}
2707:i
2699:{
2679:U
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2646:H
2601:n
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2580:i
2576:F
2572:{
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2468:A
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2429:U
2409:U
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2343:(
2334:=
2330:)
2326:V
2321:i
2307:V
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2292:(
2282:=
2278:)
2272:i
2258:V
2252:A
2244:V
2240:(
2230:=
2227:)
2224:i
2221:(
2193:i
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2164:d
2160:n
2157:=
2148:A
2143:d
2119:.
2114:B
2106:i
2102:|
2091:A
2083:i
2079:F
2070:n
2065:1
2062:=
2059:i
2051:=
2048:V
2023:B
2018:|
2013:i
2004:i
2000:|
1991:A
1987:I
1983:=
1978:i
1951:B
1945:H
1934:A
1928:H
1922:=
1913:A
1906:H
1880:n
1877:=
1868:A
1863:d
1836:A
1830:|
1825:i
1816:i
1812:|
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1803:i
1773:A
1768:|
1761:i
1757:f
1744:A
1735:i
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1725:n
1720:1
1717:=
1714:i
1706:=
1703:V
1675:i
1671:f
1666:|
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1609:=
1604:i
1600:F
1576:.
1573:V
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1550:=
1545:i
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1491:A
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1473:A
1467:H
1461::
1458:V
1429:A
1424:d
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1357:=
1354:i
1350:}
1344:i
1336:{
1314:A
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1248:=
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1235:i
1231:F
1227:{
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1023:}
1018:i
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387:(
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313:F
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