11253:
3612:
12057:
8677:
transitions from being a pure state, exhibiting superpositions, to a mixed state, an incoherent combination of classical alternatives. This transition is fundamentally reversible, as the combined state of system and environment is still pure, but for all practical purposes irreversible, as the environment is a very large and complex quantum system, and it is not feasible to reverse their interaction. Decoherence is thus very important for explaining the
4718:
5945:
3087:
4495:
1236:
5682:
3520:
4713:{\displaystyle \rho ={\frac {1}{2}}|\mathrm {R} \rangle \langle \mathrm {R} |+{\frac {1}{2}}|\mathrm {L} \rangle \langle \mathrm {L} |={\frac {1}{2}}|\mathrm {H} \rangle \langle \mathrm {H} |+{\frac {1}{2}}|\mathrm {V} \rangle \langle \mathrm {V} |={\frac {1}{2}}{\begin{pmatrix}1&0\\0&1\end{pmatrix}}.}
4826:
A given density operator does not uniquely determine which ensemble of pure states gives rise to it; in general there are infinitely many different ensembles generating the same density matrix. Those cannot be distinguished by any measurement. The equivalent ensembles can be completely characterized:
8676:
theory typically involves non-isolated quantum systems developing entanglement with other systems, including measurement apparatuses. Density matrices make it much easier to describe the process and calculate its consequences. Quantum decoherence explains why a system interacting with an environment
8112:
1066:
5940:{\displaystyle \langle A\rangle =\sum _{j}p_{j}\langle \psi _{j}|A|\psi _{j}\rangle =\sum _{j}p_{j}\operatorname {tr} \left(|\psi _{j}\rangle \langle \psi _{j}|A\right)=\operatorname {tr} \left(\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|A\right)=\operatorname {tr} (\rho A),}
3344:
2220:
1677:
5103:
4727:
with a rough surface, so that slightly different parts of the light beam acquire different polarizations. Another possibility is using entangled states: a radioactive decay can emit two photons traveling in opposite directions, in the quantum state
3074:
3270:
7601:
4806:
8416:
Density matrices are a basic tool of quantum mechanics, and appear at least occasionally in almost any type of quantum-mechanical calculation. Some specific examples where density matrices are especially helpful and common are as follows:
6623:
4036:
2340:
1763:
6847:
2428:
5404:
8981:
The C*-algebraic formulation can be seen to include both classical and quantum systems. When the system is classical, the algebra of observables become an abelian C*-algebra. In that case the states become probability measures.
1840:
4360:(linear, circular, or elliptical polarization). Unlike polarized light, it passes through a polarizer with 50% intensity loss whatever the orientation of the polarizer; and it cannot be made polarized by passing it through any
6322:
2980:
2808:
8246:
7937:
7470:
106:
8909:
It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable. For this reason, observables are identified with elements of an abstract
5663:
4228:
5195:
4358:
3878:
9002:. Von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements. The name density matrix itself relates to its classical correspondence to a
4488:
with probability 1/2. These two ensembles are completely indistinguishable experimentally, and therefore they are considered the same mixed state. For this example of unpolarized light, the density operator equals
2068:
1486:
1231:{\displaystyle (\rho _{ij})=\left({\begin{matrix}\rho _{00}&\rho _{01}\\\rho _{10}&\rho _{11}\end{matrix}}\right)=\left({\begin{matrix}p_{0}&\rho _{01}\\\rho _{01}^{*}&p_{1}\end{matrix}}\right)}
3676:
8898:
3515:{\displaystyle \sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}
7790:
6026:
3743:
7096:
6076:
1985:
6397:
8660:
7903:
7261:
2552:
5001:
2656:
4993:
3597:
8487:
6936:
6729:
5551:
1875:
1765:
a convenient representation for the state of this ensemble. It is easy to check that this operator is positive semi-definite, Hermitian, and has trace one. Conversely, it follows from the
5252:
4879:
3944:
2470:, of other quantum states. There are several equivalent characterizations of pure states in the language of density operators. A density operator represents a pure state if and only if:
8681:
of quantum mechanics, but cannot explain wave function collapse, as all classical alternatives are still present in the mixed state, and wave function collapse selects only one of them.
2456:
2247:
6161:
2287:
1686:
8322:
7648:
6658:
4486:
4456:
4426:
4396:
4292:
4262:
4164:
4134:
4100:
4070:
3822:
3792:
953:). In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement.
8555:
5968:
3332:
8804:
6691:
5287:
2881:
2846:
2733:
2698:
2282:
2063:
1396:
7682:
7315:
2988:
2605:
4935:
5462:
2504:
2372:
1937:
3157:
1338:
7340:
7194:
7496:
4731:
4723:
There are also other ways to generate unpolarized light: one possibility is to introduce uncertainty in the preparation of the photon, for example, passing it through a
1298:
1270:
1057:
1029:
7126:
6966:
6877:
3146:
10108:
Ardila, Luis; Heyl, Markus; Eckardt, André (28 December 2018). "Measuring the Single-Particle
Density Matrix for Fermions and Hard-Core Bosons in an Optical Lattice".
8402:
2032:
1481:
6547:
8507:
8361:
8284:
7266:
which can be interpreted as the state produced by performing the measurement but not recording which outcome occurred, has a von
Neumann entropy larger than that of
3961:
8708:
is a process by which, given a set of data representing the results of quantum measurements, a density matrix consistent with those measurement results is computed.
8595:
7368:
7284:
7169:
7146:
6997:
6757:
6749:
6538:
6518:
6490:
6427:
6226:
6191:
5578:
5482:
5427:
1902:
1423:
7822:
8824:
8769:
8749:
8575:
6470:
6102:
5510:
5307:
4899:
2005:
1451:
2381:
5315:
9730:; Fuchs, Christopher A.; Manne, Kiran K.; Renes, Joseph M. (2004). "Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements".
7924:
8107:{\displaystyle W(x,p)\,\ {\stackrel {\mathrm {def} }{=}}\ \,{\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }\psi ^{*}(x+y)\psi (x-y)e^{2ipy/\hbar }\,dy.}
1772:
942:
when the preparation of the systems lead to numerous pure states in the ensemble, and thus one must deal with the statistics of possible preparations, and
2894:
2741:
11301:
8127:
8424:
uses density matrices, most prominently to express the idea that a system is prepared at a nonzero temperature. Constructing a density matrix using a
7402:
7379:
10911:
6234:
8696:, and other fields where state preparation is noisy and decoherence can occur, density matrices are frequently used. Noise is often modelled via a
36:
9016:
In contrast, the motivation that inspired Landau was the impossibility of describing a subsystem of a composite quantum system by a state vector.
4812:
is pure, but the density matrix for each photon individually, found by taking the partial trace of the joint density matrix, is completely mixed.
11854:
8336:
5586:
4169:
5121:
4311:
3831:
11502:
372:
7128:
do not have orthogonal supports, the sum on the right-hand side is strictly greater than the von
Neumann entropy of the convex combination
1430:
9223:
2509:
873:
3628:
8598:
2616:
12025:
8829:
7824:
is the wavefunction propagator over some interval, then the time evolution of the density matrix over that same interval is given by
7697:
5977:
3695:
11351:
11125:
10344:
9850:
7005:
6034:
124:
1942:
12037:
11216:
949:
with another, without describing their combined state; this case is typical for a system interacting with some environment (e.g.
580:
11721:
11294:
9624:
9041:
2215:{\displaystyle p(m)=\left\langle \Psi \right|\left(\Pi _{m}\otimes I\right)\left|\Psi \right\rangle =\operatorname {tr} \left,}
1672:{\displaystyle p(m)=\sum _{j}p_{j}\left\langle \psi _{j}\right|\Pi _{m}\left|\psi _{j}\right\rangle =\operatorname {tr} \left,}
353:
1911:
Another motivation for the definition of density operators comes from considering local measurements on entangled states. Let
11655:
10275:
10214:
10186:
9970:
9937:
9913:
9887:
9860:
9600:
9345:
9291:
9237:
9193:
9081:
9051:
7928:
5669:
536:
6879:
are states that have support on orthogonal subspaces, then the von
Neumann entropy of a convex combination of these states,
12086:
11578:
11135:
10901:
9282:
8604:
7830:
5098:{\displaystyle {\sqrt {q_{i}}}\left|\varphi _{i}\right\rangle =\sum _{j}U_{ij}{\sqrt {p_{j}}}\left|\psi _{j}\right\rangle }
459:
3098:, each point on the unit sphere stands for a pure state. All other density matrices correspond to points in the interior.
2661:
It is important to emphasize the difference between a probabilistic mixture (i.e. an ensemble) of quantum states and the
6433:
shows that in
Hilbert spaces of dimension 3 or larger the assumption of linearity can be replaced with an assumption of
6337:
7202:
2574:
11324:
11287:
9147:
4944:
3528:
2374:
on subsystem 1. It is easy to check that this operator has all the properties of a density operator. Conversely, the
119:
8431:
6885:
6696:
11434:
9956:
5673:
5519:
1845:
915:
208:
5203:
4830:
3883:
11992:
11701:
11696:
11419:
10936:
4428:
polarization with probability 1/2. The same behavior would occur if each photon had either vertical polarization
866:
315:
295:
163:
8118:
12091:
12004:
11676:
10483:
9379:
8723:
or each having an independent single-particle wavefunction. This is the usual starting point when building the
2433:
555:
285:
4821:
2375:
1905:
957:
Density matrices are thus crucial tools in areas of quantum mechanics that deal with mixed ensembles, such as
11982:
11759:
11681:
11464:
11334:
5112:
2225:
1341:
958:
595:
333:
233:
6110:
12081:
11716:
11650:
11645:
11616:
11329:
10700:
10337:
8697:
8289:
7609:
6628:
4461:
4431:
4401:
4371:
4267:
4237:
4139:
4109:
4075:
4045:
3797:
3767:
531:
526:
497:
348:
129:
3599:
represent pure states, while mixed states are represented by points in the interior. This is known as the
3069:{\displaystyle |\psi \rangle \langle \psi |={\frac {1}{2}}{\begin{pmatrix}1&1\\1&1\end{pmatrix}}.}
1904:
that are non-negative and add up to one. However, this representation will not be unique, as shown by the
11784:
11691:
10775:
9100:
Fano, U. (1957). "Description of States in
Quantum Mechanics by Density Matrix and Operator Techniques".
8918:
8512:
5953:
3278:
2557:
565:
310:
300:
8774:
6663:
6331:
is obtained. In the case where the measurement result is not known the ensemble is instead described by
5257:
2851:
2816:
2703:
2668:
2256:
2037:
1366:
12060:
11822:
11630:
11601:
10931:
10453:
9962:
9229:
9026:
8949:
8689:
7661:
2610:
859:
511:
482:
9583:; Lahti, Pekka (2009), Greenberger, Daniel; Hentschel, Klaus; Weinert, Friedel (eds.), "LĂĽders Rule",
3265:{\displaystyle \rho ={\frac {1}{2}}\left(I+r_{x}\sigma _{x}+r_{y}\sigma _{y}+r_{z}\sigma _{z}\right),}
11844:
11711:
11635:
11596:
11553:
11527:
11484:
11377:
11035:
10906:
10820:
9031:
8701:
8335:
The evolution equation for the Wigner function is then analogous to that of its classical limit, the
6402:
If one assumes that the probabilities of measurement outcomes are linear functions of the projectors
4904:
516:
477:
430:
405:
328:
188:
7486:, even though this equation seems at first look to emulate the Heisenberg equation of motion in the
5432:
2481:
2349:
1914:
11911:
11891:
11881:
11871:
11827:
11402:
11140:
11030:
10738:
10418:
9036:
8663:
7596:{\displaystyle i\hbar {\frac {\operatorname {d} A^{(\mathrm {H} )}}{\operatorname {d} t}}=-\left~,}
4801:{\displaystyle (|\mathrm {R} ,\mathrm {L} \rangle +|\mathrm {L} ,\mathrm {R} \rangle )/{\sqrt {2}}}
1348:
600:
7289:
1307:
11606:
11532:
11175:
11104:
10986:
10846:
10443:
10330:
10047:
Granade, Christopher; Combes, Joshua; Cory, D. G. (2016-01-01). "Practical
Bayesian tomography".
9666:
7914:
4301:
3955:
2342:
a convenient tool to calculate the probabilities of these local measurements. It is known as the
1363:
of the system. This definition can be motivated by considering a situation where each pure state
1304:
of each other (also called coherences); they are restricted in magnitude by the requirement that
769:
487:
395:
12020:
11568:
7385:
6618:{\displaystyle \rho =\textstyle \sum _{i}\lambda _{i}|\varphi _{i}\rangle \langle \varphi _{i}|}
1275:
1247:
1034:
1006:
445:
343:
109:
11931:
11706:
11686:
11611:
11479:
11045:
10628:
10433:
9732:
9661:
9580:
8720:
7691:
If the
Hamiltonian is time-independent, the von Neumann equation can be easily solved to yield
7104:
6944:
6855:
6493:
5971:
5115:, which are pure states that generate the density operator when a partial trace is taken. Let
4031:{\displaystyle |\mathrm {V} \rangle =(|\mathrm {R} \rangle +|\mathrm {L} \rangle )/{\sqrt {2}}}
3125:
2343:
2335:{\displaystyle \rho =\operatorname {tr} _{2}\left|\Psi \right\rangle \left\langle \Psi \right|}
1758:{\displaystyle \rho =\sum _{j}p_{j}\left|\psi _{j}\right\rangle \left\langle \psi _{j}\right|,}
989:
774:
492:
320:
290:
253:
11522:
9877:
8366:
7483:
6842:{\displaystyle S=-\sum _{i}\lambda _{i}\ln \lambda _{i}=-\operatorname {tr} (\rho \ln \rho ).}
5516:
of the system, and suppose the ensemble is in a mixed state such that each of the pure states
2010:
1459:
400:
11956:
11469:
11449:
10991:
10728:
10578:
10573:
10408:
10383:
10378:
9903:
9010:
8421:
6434:
3951:
3825:
3761:
2888:
2662:
2561:
243:
228:
8492:
8346:
8254:
7320:
7174:
5108:
will give rise to the same density operator, and all equivalent ensembles are of this form.
11987:
11916:
11861:
11591:
11414:
11372:
11185:
10543:
10373:
10353:
10127:
10066:
10013:
9804:
9751:
9685:
9664:(2003). "Quantum States and Generalized Observables: A Simple Proof of Gleason's Theorem".
9647:
9546:
9503:
9398:
9109:
8580:
7396:) describes how a density operator evolves in time. The von Neumann equation dictates that
7353:
7269:
7154:
7131:
6975:
6734:
6523:
6503:
6475:
6430:
6405:
6204:
6169:
5556:
5467:
5412:
4724:
3080:
1880:
1401:
1356:
946:
899:
560:
472:
198:
155:
7798:
804:
8:
12096:
11972:
11941:
11886:
11866:
11774:
11731:
11586:
11512:
11439:
11429:
11341:
11206:
11180:
10758:
10563:
10553:
9763:
9619:
9006:
8693:
8685:
8673:
8666:
can be applied, where the states summed over to make the density matrix are drawn from a
6450:
6194:
3947:
3753:
2423:{\displaystyle \operatorname {tr} _{2}\left|\Psi \right\rangle \left\langle \Psi \right|}
1454:
966:
962:
950:
659:
467:
385:
213:
193:
145:
10131:
10078:
10070:
10017:
9808:
9755:
9689:
9550:
9507:
9402:
9113:
5399:{\displaystyle |\Psi \rangle =\sum _{j}{\sqrt {p_{j}}}|\psi _{j}\rangle U|a_{j}\rangle }
12032:
11901:
11799:
11507:
11454:
11346:
11257:
11211:
11201:
11155:
11150:
11079:
11015:
10881:
10618:
10613:
10548:
10538:
10403:
10308:
10267:
10206:
10178:
10151:
10117:
10090:
10056:
10029:
10003:
9828:
9794:
9767:
9741:
9709:
9675:
9562:
9536:
9527:
9494:
9471:
9422:
9388:
9359:
9323:
8809:
8754:
8734:
8724:
8705:
8560:
8425:
7487:
6455:
6087:
5495:
5292:
4884:
4039:
3115:
2467:
2466:
A pure quantum state is a state that can not be written as a probabilistic mixture, or
1990:
1436:
380:
305:
238:
150:
9785:
Andrzej Grudka; Paweł Kurzyński (2008). "Is There
Contextuality for a Single Qubit?".
5111:
A closely related fact is that a given density operator has infinitely many different
3110:
of that set. The simplest case is that of a two-dimensional
Hilbert space, known as a
910:(even if the ensemble contains only one system). It allows for the calculation of the
12042:
11951:
11921:
11849:
11812:
11807:
11789:
11754:
11744:
11459:
11424:
11407:
11310:
11268:
11252:
11055:
11050:
11040:
11020:
10981:
10976:
10805:
10800:
10785:
10780:
10771:
10766:
10713:
10608:
10558:
10503:
10473:
10468:
10448:
10438:
10398:
10312:
10271:
10210:
10182:
10143:
10082:
10033:
9976:
9966:
9933:
9909:
9883:
9856:
9820:
9701:
9615:
9596:
9566:
9475:
9463:
9414:
9351:
9341:
9297:
9287:
9233:
9189:
9153:
9143:
9077:
9046:
8922:
8340:
6437:. This restriction on the dimension can be removed by assuming non-contextuality for
4297:
1835:{\textstyle \sum _{j}p_{j}\left|\psi _{j}\right\rangle \left\langle \psi _{j}\right|}
1301:
887:
814:
789:
729:
724:
624:
590:
570:
168:
27:
10155:
10094:
9832:
9771:
9522:
9489:
9426:
8917:(that is one without a distinguished representation as an algebra of operators) and
11769:
11764:
11621:
11517:
11263:
11231:
11160:
11099:
11094:
11074:
11010:
10916:
10886:
10871:
10851:
10790:
10743:
10718:
10708:
10679:
10598:
10593:
10568:
10498:
10478:
10388:
10368:
10300:
10263:
10229:
10139:
10135:
10074:
10021:
9816:
9812:
9759:
9727:
9713:
9693:
9633:
9588:
9554:
9511:
9455:
9406:
9363:
9333:
9181:
9135:
9117:
8991:
8960:
8930:
8728:
6852:
This definition implies that the von Neumann entropy of any pure state is zero. If
6541:
6497:
4938:
4231:
3148:
2975:{\displaystyle |\psi \rangle =(|\psi _{1}\rangle +|\psi _{2}\rangle )/{\sqrt {2}},}
2803:{\displaystyle \rho ={\frac {1}{2}}{\begin{pmatrix}1&0\\0&1\end{pmatrix}},}
1766:
1352:
819:
809:
799:
699:
679:
664:
634:
502:
390:
10856:
10025:
9697:
9592:
8662:. If the number of particles involved in the system is itself not certain, then a
4102:
polarized light, half of the photons are absorbed in both cases. This may make it
11999:
11926:
11906:
11876:
11839:
11834:
11739:
11563:
10961:
10896:
10876:
10861:
10841:
10825:
10723:
10654:
10644:
10603:
10488:
10458:
9846:
9643:
9185:
8970:) correspond exactly to the density operators, and therefore the pure states of
8678:
8241:{\displaystyle {\frac {\partial W(x,p,t)}{\partial t}}=-\{\{W(x,p,t),H(x,p)\}\},}
6969:
6429:, then they must be given by the trace of the projector with a density operator.
978:
903:
844:
714:
694:
440:
280:
7465:{\displaystyle i\hbar {\frac {\operatorname {d} \rho }{\operatorname {d} t}}=~,}
2883:
are assumed orthogonal and of dimension 2, for simplicity. On the other hand, a
11977:
11946:
11936:
11558:
11548:
11221:
11165:
11145:
11089:
10966:
10926:
10891:
10815:
10754:
10733:
10674:
10664:
10649:
10583:
10528:
10518:
10513:
10423:
9258:
7658:, and the relative sign ensures that the time derivative of the expected value
3119:
3107:
2665:
of two states. If an ensemble is prepared to have half of its systems in state
779:
739:
719:
689:
669:
619:
435:
425:
218:
9638:
9410:
9337:
6317:{\displaystyle \rho _{i}'={\frac {P_{i}\rho P_{i}}{\operatorname {tr} \left}}}
12075:
11896:
11749:
11640:
11474:
11444:
11397:
11226:
11084:
11025:
10956:
10946:
10941:
10866:
10795:
10669:
10659:
10588:
10508:
10493:
10428:
10170:
10086:
9980:
9952:
9515:
9467:
9418:
9355:
9301:
9157:
9121:
8325:
2475:
2250:
1360:
1060:
985:. The density matrix is obtained from the density operator by a choice of an
911:
907:
839:
834:
764:
734:
704:
575:
521:
248:
223:
101:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
9525:(2006-04-03). "Concerning the state-change due to the measurement process".
11779:
11392:
11387:
11109:
11066:
10971:
10684:
10623:
10533:
10413:
10147:
9824:
9705:
9558:
8719:, an imperfect but useful first approximation is to treat the electrons as
8577:
is the system's Hamiltonian. The normalization condition that the trace of
4368:
be described as a statistical ensemble, e. g. as each photon having either
3600:
3091:
938:). Mixed ensembles arise in quantum mechanics in two different situations:
923:
829:
824:
759:
744:
709:
203:
8990:
The formalism of density operators and matrices was introduced in 1927 by
11817:
10951:
10921:
10689:
10523:
10393:
9746:
9680:
9541:
9393:
9277:
9003:
8999:
8405:
7920:
7482:
This equation only holds when the density operator is taken to be in the
5658:{\displaystyle \rho =\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|.}
4223:{\displaystyle (|\mathrm {R} \rangle +|\mathrm {L} \rangle )/{\sqrt {2}}}
3335:
1241:
986:
794:
749:
684:
639:
5190:{\displaystyle \rho =\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|}
4353:{\displaystyle \alpha |\mathrm {R} \rangle +\beta |\mathrm {L} \rangle }
3873:{\displaystyle \alpha |\mathrm {R} \rangle +\beta |\mathrm {L} \rangle }
3611:
11002:
10463:
10304:
9459:
9446:
8995:
8911:
8667:
8329:
7476:
6198:
5513:
4361:
3103:
2568:
927:
784:
754:
674:
649:
644:
629:
11671:
11367:
11236:
10810:
9441:
9176:
Hall, Brian C. (2013). "Systems and Subsystems, Multiple Particles".
8117:
The equation for the time evolution of the Wigner function, known as
3958:. Consider now a vertically polarized photon, described by the state
919:
275:
11279:
8948:. By properties of the GNS construction these states correspond to
11170:
10203:
Algebraic methods in statistical mechanics and quantum field theory
10122:
10061:
10008:
9377:
Kirkpatrick, K. A. (February 2006). "The Schrödinger-HJW Theorem".
8716:
8404:
reduces to the classical Liouville probability density function in
7380:
Liouville's theorem (Hamiltonian) § Quantum Liouville equation
7171:
and a projective measurement as in the previous section, the state
654:
9799:
9328:
9180:. Graduate Texts in Mathematics. Vol. 267. pp. 419–440.
8121:, is then the Wigner-transform of the above von Neumann equation,
3671:{\displaystyle {\begin{bmatrix}0.5&0\\0&0.5\end{bmatrix}}}
10322:
9009:(probability distribution of position and momentum) in classical
8978:) are exactly the pure states in the sense of quantum mechanics.
6441:
as well, but this has been criticized as physically unmotivated.
3079:
Unlike the probabilistic mixture, this superposition can display
8944:
is a state that is an extreme point of the set of all states on
8893:{\textstyle \sum _{i=1}^{N}|\psi _{i}\rangle \langle \psi _{i}|}
7785:{\displaystyle \rho (t)=e^{-iHt/\hbar }\rho (0)e^{iHt/\hbar }.}
6021:{\displaystyle \langle A\rangle =\langle \psi |A|\psi \rangle }
4234:
there is no absorption whatsoever, but if we pass either state
3757:
3086:
10291:
Fano, Ugo (1995). "Density matrices as polarization vectors".
10233:
7373:
3738:{\displaystyle {\begin{bmatrix}1&0\\0&0\end{bmatrix}}}
3122:, which together with the identity matrix provide a basis for
10234:"Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik"
9784:
7091:{\displaystyle S(\rho )=H(p_{i})+\sum _{i}p_{i}S(\rho _{i}).}
6472:
of a mixture can be expressed in terms of the eigenvalues of
6071:{\displaystyle \langle A\rangle =\operatorname {tr} (\rho A)}
5676:
can be calculated by extending from the case of pure states:
5464:
is an orthogonal basis, and furthermore all purifications of
3111:
3095:
9442:"Some comments on the concept of state in quantum mechanics"
8826:
electrons together can be characterized by a density matrix
5289:
not necessarily orthogonal. Then for all partial isometries
1980:{\displaystyle {\mathcal {H}}_{1}\otimes {\mathcal {H}}_{2}}
1769:
that every operator with these properties can be written as
922:. It is a generalization of the more usual state vectors or
8712:
7908:
7347:
6731:. Then the entropy of a quantum system with density matrix
6438:
3114:. An arbitrary mixed state for a qubit can be written as a
1347:
In operator language, a density operator for a system is a
1244:
that sum to one (also called populations of the two states
16:
Matrix describing a quantum system in a pure or mixed state
9492:(1950). "Über die Zustandsänderung durch den Messprozeß".
4815:
1939:
be a pure entangled state in the composite Hilbert space
8711:
When analyzing a system with many electrons, such as an
1063:, then the density operator is represented by the matrix
3686:(3), the remaining photons are all vertically polarized
8832:
6700:
6557:
5523:
4676:
3704:
3637:
3475:
3419:
3366:
3032:
2766:
1775:
1164:
1097:
9620:"Measures on the closed subspaces of a Hilbert space"
8904:
8812:
8777:
8757:
8737:
8655:{\displaystyle Z(\beta )=\mathrm {tr} \exp(-\beta H)}
8607:
8583:
8563:
8515:
8495:
8434:
8369:
8349:
8292:
8257:
8130:
7940:
7898:{\displaystyle \rho (t)=G(t)\rho (0)G(t)^{\dagger }.}
7833:
7801:
7700:
7664:
7612:
7499:
7405:
7356:
7323:
7292:
7272:
7205:
7177:
7157:
7134:
7107:
7008:
6978:
6947:
6888:
6858:
6760:
6737:
6699:
6666:
6631:
6550:
6526:
6506:
6478:
6458:
6408:
6340:
6237:
6207:
6172:
6113:
6090:
6037:
5980:
5956:
5685:
5589:
5559:
5522:
5498:
5470:
5435:
5415:
5318:
5295:
5260:
5206:
5124:
5004:
4947:
4907:
4887:
4833:
4734:
4498:
4464:
4434:
4404:
4374:
4314:
4270:
4240:
4172:
4142:
4112:
4078:
4048:
3964:
3886:
3834:
3800:
3770:
3698:
3631:
3531:
3347:
3281:
3160:
3128:
2991:
2897:
2854:
2819:
2744:
2706:
2671:
2619:
2577:
2512:
2484:
2436:
2384:
2378:
implies that all density operators can be written as
2352:
2290:
2259:
2228:
2071:
2040:
2013:
1993:
1945:
1917:
1883:
1848:
1689:
1489:
1462:
1439:
1404:
1369:
1310:
1278:
1250:
1069:
1037:
1009:
945:
when one wants to describe a physical system that is
918:
performed upon the systems of the ensemble using the
39:
9726:
7927:, the density matrix transforms into the equivalent
7919:
The density matrix operator may also be realized in
7654:
operator; but in this picture the density matrix is
6941:
is given by the von Neumann entropies of the states
6228:, the post-measurement density operator is given by
9171:
9169:
9167:
2547:{\displaystyle \rho =|\psi \rangle \langle \psi |.}
8892:
8818:
8798:
8763:
8743:
8654:
8589:
8569:
8549:
8501:
8481:
8396:
8355:
8316:
8278:
8240:
8106:
7897:
7816:
7784:
7676:
7642:
7595:
7464:
7362:
7334:
7309:
7278:
7255:
7188:
7163:
7140:
7120:
7090:
6991:
6960:
6930:
6871:
6841:
6743:
6723:
6685:
6652:
6617:
6532:
6512:
6484:
6464:
6421:
6392:{\displaystyle \;\rho '=\sum _{i}P_{i}\rho P_{i}.}
6391:
6316:
6220:
6185:
6155:
6096:
6070:
6020:
5962:
5939:
5657:
5572:
5545:
5504:
5476:
5456:
5421:
5398:
5301:
5281:
5246:
5200:be the density operator generated by the ensemble
5189:
5097:
4987:
4929:
4893:
4873:
4800:
4712:
4480:
4450:
4420:
4390:
4352:
4286:
4256:
4222:
4158:
4128:
4094:
4064:
4030:
3938:
3872:
3816:
3786:
3737:
3670:
3591:
3514:
3326:
3264:
3140:
3068:
2974:
2875:
2840:
2802:
2727:
2692:
2650:
2599:
2546:
2498:
2450:
2422:
2366:
2334:
2276:
2241:
2214:
2057:
2026:
1999:
1987:. The probability of obtaining measurement result
1979:
1931:
1896:
1869:
1834:
1757:
1671:
1475:
1445:
1417:
1390:
1332:
1292:
1264:
1230:
1051:
1023:
100:
10107:
10046:
9875:
5580:. Then the corresponding density operator equals
3102:Geometrically, the set of density operators is a
2651:{\displaystyle \operatorname {tr} (\rho ^{2})=1.}
12073:
10258:"The Damping Problem in Wave Mechanics (1927)".
9994:Schlosshauer, M. (2019). "Quantum Decoherence".
9587:, Springer Berlin Heidelberg, pp. 356–358,
9164:
7256:{\displaystyle \rho '=\sum _{i}P_{i}\rho P_{i},}
992:in the underlying space. In practice, the terms
8994:and independently, but less systematically, by
8343:. In the limit of vanishing Planck's constant
6540:is a positive semi-definite operator, it has a
4988:{\displaystyle \{q_{i},|\varphi _{i}\rangle \}}
3592:{\displaystyle r_{x}^{2}+r_{y}^{2}+r_{z}^{2}=1}
9927:
9876:Breuer, Heinz; Petruccione, Francesco (2002),
8482:{\displaystyle \rho =\exp(-\beta H)/Z(\beta )}
7388:describes how pure states evolve in time, the
6931:{\displaystyle \rho =\sum _{i}p_{i}\rho _{i},}
6724:{\displaystyle \textstyle \sum \lambda _{i}=1}
3682:After passing through vertical plane polarizer
3606:
2735:, it can be described by the density matrix:
11295:
10338:
10253:
10251:
10175:Mathematical Foundations of Quantum Mechanics
9255:Reduced Density Matrices in Quantum Chemistry
9221:
9076:(2. ed., ed.). New York, NY: Springer.
8933:, we can recover Hilbert spaces that realize
5546:{\displaystyle \textstyle |\psi _{j}\rangle }
3764:, described by the orthogonal quantum states
3619:(1) emits completely random polarized photons
1870:{\displaystyle \left|\psi _{j}\right\rangle }
1429:of pure states. The probability of obtaining
867:
9993:
9987:
9845:
9322:(2nd ed.). Cambridge University Press.
9217:
9215:
9213:
9211:
9209:
9207:
9205:
8940:Geometrically, a pure state on a C*-algebra
8872:
8869:
8793:
8311:
8308:
8296:
8293:
8232:
8229:
8181:
8178:
7671:
7665:
6647:
6596:
6593:
6044:
6038:
6015:
5993:
5987:
5981:
5887:
5884:
5809:
5806:
5754:
5718:
5692:
5686:
5634:
5631:
5539:
5451:
5393:
5372:
5327:
5276:
5247:{\displaystyle \{p_{j},|\psi _{j}\rangle \}}
5241:
5238:
5207:
5169:
5166:
4982:
4979:
4948:
4881:be an ensemble. Then for any complex matrix
4874:{\displaystyle \{p_{j},|\psi _{j}\rangle \}}
4868:
4865:
4834:
4780:
4756:
4645:
4642:
4606:
4603:
4567:
4564:
4528:
4525:
4475:
4445:
4415:
4385:
4347:
4328:
4281:
4251:
4202:
4186:
4153:
4123:
4089:
4059:
4010:
3994:
3975:
3939:{\displaystyle |\alpha |^{2}+|\beta |^{2}=1}
3867:
3848:
3811:
3781:
3003:
3000:
2951:
2930:
2906:
2870:
2835:
2722:
2687:
2530:
2527:
2493:
2361:
1926:
1385:
1287:
1259:
1046:
1018:
977:The density matrix is a representation of a
972:
95:
69:
10228:
10040:
9376:
9225:Quantum Computation and Quantum Information
7374:The von Neumann equation for time evolution
7350:, to have a lower von Neumann entropy than
11302:
11288:
10912:Fundamental (linear differential equation)
10345:
10331:
10248:
9579:
9095:
9093:
9013:, which was introduced by Wigner in 1932.
6341:
3334:are the coordinates of a point within the
874:
860:
10121:
10060:
10007:
9798:
9745:
9679:
9637:
9540:
9392:
9327:
9202:
8094:
7990:
7959:
3760:can be described as having right or left
2451:{\displaystyle \left|\Psi \right\rangle }
10177:, Dover Books on Mathematics, New York:
9882:, Oxford University Press, p. 110,
9852:Quantum Computing: A Gentle Introduction
9252:
9222:Nielsen, Michael; Chuang, Isaac (2000),
7909:Wigner functions and classical analogies
3610:
3085:
2461:
11217:Matrix representation of conic sections
9901:
9839:
9614:
9246:
9140:Statistical Structure of Quantum Theory
9090:
9071:
9058:
8411:
3752:An example of pure and mixed states is
3690:(4) and have pure state density matrix:
2242:{\displaystyle \operatorname {tr} _{2}}
12074:
10169:
9951:
9932:, World Scientific, pp. 175–179,
9625:Indiana University Mathematics Journal
9521:
9488:
9142:. Lecture Notes in Physics. Springer.
9134:
7686:the same as in the Schrödinger picture
6156:{\displaystyle A=\sum _{i}a_{i}P_{i},}
4816:Equivalent ensembles and purifications
4166:, but this is not correct: if we pass
4106:like half of the photons are in state
930:, density matrices can also represent
11309:
11283:
10326:
9660:
9317:
9276:
9052:Wigner quasi-probability distribution
8317:{\displaystyle \{\{\cdot ,\cdot \}\}}
7643:{\displaystyle A^{(\mathrm {H} )}(t)}
6653:{\displaystyle |\varphi _{i}\rangle }
4808:. The joint state of the two photons
4481:{\displaystyle |\mathrm {H} \rangle }
4451:{\displaystyle |\mathrm {V} \rangle }
4421:{\displaystyle |\mathrm {L} \rangle }
4391:{\displaystyle |\mathrm {R} \rangle }
4287:{\displaystyle |\mathrm {L} \rangle }
4257:{\displaystyle |\mathrm {R} \rangle }
4159:{\displaystyle |\mathrm {L} \rangle }
4129:{\displaystyle |\mathrm {R} \rangle }
4095:{\displaystyle |\mathrm {L} \rangle }
4065:{\displaystyle |\mathrm {R} \rangle }
3817:{\displaystyle |\mathrm {L} \rangle }
3787:{\displaystyle |\mathrm {R} \rangle }
10290:
10200:
9928:MĂĽller-Kirsten, Harald J.W. (2008),
9439:
9313:
9311:
9283:Quantum Theory: Concepts and Methods
9272:
9270:
9268:
9175:
9099:
3623:(2) with mixed state density matrix:
10284:
9849:; Polak, Wolfgang H. (2011-03-04).
9520:Translated by K. A. Kirkpatrick as
9042:Green's function (many-body theory)
8550:{\displaystyle (k_{\rm {B}}T)^{-1}}
7795:For a more general Hamiltonian, if
5963:{\displaystyle \operatorname {tr} }
3828:of the two: it can be in any state
3327:{\displaystyle (r_{x},r_{y},r_{z})}
13:
10352:
10268:10.1016/B978-0-08-010586-4.50007-9
9945:
9930:Classical Mechanics and Relativity
9879:The theory of open quantum systems
9764:10.1023/B:FOOP.0000019581.00318.a5
8905:C*-algebraic formulation of states
8799:{\displaystyle |\psi _{i}\rangle }
8627:
8624:
8525:
8163:
8134:
8019:
8014:
7979:
7976:
7973:
7622:
7573:
7535:
7524:
7509:
7490:, with a crucial sign difference:
7426:
7415:
7196:defined by the convex combination
6686:{\displaystyle \lambda _{i}\geq 0}
5324:
5282:{\displaystyle |\psi _{j}\rangle }
4776:
4768:
4752:
4744:
4649:
4638:
4610:
4599:
4571:
4560:
4532:
4521:
4471:
4441:
4411:
4381:
4343:
4324:
4294:half of the photons are absorbed.
4277:
4247:
4198:
4182:
4149:
4119:
4085:
4055:
4006:
3990:
3971:
3863:
3844:
3807:
3777:
2876:{\displaystyle |\psi _{2}\rangle }
2841:{\displaystyle |\psi _{1}\rangle }
2728:{\displaystyle |\psi _{2}\rangle }
2693:{\displaystyle |\psi _{1}\rangle }
2441:
2413:
2402:
2358:
2325:
2314:
2277:{\displaystyle {\mathcal {H}}_{2}}
2263:
2192:
2181:
2150:
2128:
2104:
2091:
2058:{\displaystyle {\mathcal {H}}_{1}}
2044:
2015:
1966:
1949:
1923:
1586:
1544:
1464:
1391:{\displaystyle |\psi _{j}\rangle }
1300:). The off-diagonal elements are
406:Sum-over-histories (path integral)
92:
66:
22:Part of a series of articles about
14:
12108:
9308:
9265:
9178:Quantum Theory for Mathematicians
8089:
8000:
7774:
7738:
7677:{\displaystyle \langle A\rangle }
7503:
7409:
7317:. It is however possible for the
926:: while those can only represent
43:
12056:
12055:
11251:
9958:Statistical Physics of Particles
8959:The states of the C*-algebra of
6972:of the probability distribution
5974:. Thus, the familiar expression
2600:{\displaystyle \rho =\rho ^{2}.}
1240:where the diagonal elements are
1000:are often used interchangeably.
11119:Used in science and engineering
10260:Collected Papers of L.D. Landau
10222:
10194:
10162:
10101:
9921:
9895:
9869:
9778:
9720:
9654:
9608:
9573:
9482:
9433:
9074:Principles of quantum mechanics
8328:, the transform of the quantum
6028:for pure states is replaced by
4930:{\displaystyle U^{\dagger }U=I}
3456:
3400:
2887:of these two states with equal
12005:Relativistic quantum mechanics
10362:Explicitly constrained entries
10140:10.1103/PhysRevLett.121.260401
9817:10.1103/PhysRevLett.100.160401
9380:Foundations of Physics Letters
9370:
9128:
9065:
8937:as a subalgebra of operators.
8886:
8855:
8779:
8771:single-particle wavefunctions
8649:
8637:
8617:
8611:
8535:
8516:
8476:
8470:
8459:
8447:
8391:
8373:
8273:
8261:
8226:
8214:
8205:
8187:
8158:
8140:
8064:
8052:
8046:
8034:
7956:
7944:
7883:
7876:
7870:
7864:
7858:
7852:
7843:
7837:
7811:
7805:
7752:
7746:
7710:
7704:
7637:
7631:
7626:
7618:
7577:
7569:
7528:
7520:
7453:
7441:
7394:Liouville–von Neumann equation
7082:
7069:
7040:
7027:
7018:
7012:
6833:
6818:
6633:
6610:
6579:
6065:
6056:
6008:
6000:
5931:
5922:
5901:
5870:
5823:
5792:
5740:
5732:
5648:
5617:
5525:
5487:
5457:{\displaystyle |a_{j}\rangle }
5437:
5379:
5358:
5320:
5262:
5224:
5183:
5152:
4965:
4851:
4783:
4763:
4739:
4735:
4654:
4633:
4615:
4594:
4576:
4555:
4537:
4516:
4466:
4436:
4406:
4376:
4338:
4319:
4272:
4242:
4205:
4193:
4177:
4173:
4144:
4114:
4080:
4050:
4013:
4001:
3985:
3981:
3966:
3920:
3911:
3897:
3888:
3858:
3839:
3802:
3772:
3603:picture of qubit state space.
3321:
3282:
3106:, and the pure states are the
3010:
2993:
2954:
2937:
2916:
2912:
2899:
2856:
2821:
2708:
2673:
2639:
2626:
2537:
2520:
2499:{\displaystyle |\psi \rangle }
2486:
2367:{\displaystyle |\Psi \rangle }
2354:
2081:
2075:
1932:{\displaystyle |\Psi \rangle }
1919:
1499:
1493:
1371:
1327:
1311:
1280:
1252:
1086:
1070:
1039:
1011:
934:(sometimes ambiguously called
902:that describes an ensemble of
556:Relativistic quantum mechanics
88:
81:
62:
1:
11983:Quantum statistical mechanics
11760:Quantum differential calculus
11682:Delayed-choice quantum eraser
11465:Symmetry in quantum mechanics
11136:Fundamental (computer vision)
10079:10.1088/1367-2630/18/3/033024
10026:10.1016/j.physrep.2019.10.001
9698:10.1103/PhysRevLett.91.120403
9593:10.1007/978-3-540-70626-7_110
9585:Compendium of Quantum Physics
9253:Davidson, Ernest Roy (1976).
4364:. However, unpolarized light
1398:is prepared with probability
959:quantum statistical mechanics
596:Quantum statistical mechanics
9440:Ochs, Wilhelm (1981-11-01).
9186:10.1007/978-1-4614-7116-5_19
7475:where the brackets denote a
7310:{\displaystyle \rho =\rho '}
6201:corresponding to eigenvalue
4136:and the other half in state
1333:{\displaystyle (\rho _{ij})}
7:
12087:Quantum information science
11785:Quantum stochastic calculus
11775:Quantum measurement problem
11697:Mach–Zehnder interferometer
10902:Duplication and elimination
10701:eigenvalues or eigenvectors
9072:Shankar, Ramamurti (2014).
9019:
8950:irreducible representations
8509:is the inverse temperature
8428:gives a result of the form
4458:or horizontal polarization
4300:(such as the light from an
3615:The incandescent light bulb
3607:Example: light polarization
2284:. This makes the operator
566:Quantum information science
10:
12113:
10835:With specific applications
10464:Discrete Fourier Transform
9963:Cambridge University Press
9320:Quantum Information Theory
9230:Cambridge University Press
9027:Atomic electron transition
8985:
8690:quantum information theory
8597:be equal to 1 defines the
7912:
7377:
6444:
4819:
4038:. If we pass it through a
2891:results in the pure state
2007:when measuring projectors
1293:{\displaystyle |1\rangle }
1265:{\displaystyle |0\rangle }
1052:{\displaystyle |1\rangle }
1024:{\displaystyle |0\rangle }
12051:
12013:
11965:
11845:Quantum complexity theory
11823:Quantum cellular automata
11798:
11730:
11664:
11577:
11541:
11528:Path integral formulation
11495:
11360:
11317:
11245:
11194:
11126:Cabibbo–Kobayashi–Maskawa
11118:
11064:
11000:
10834:
10753:Satisfying conditions on
10752:
10698:
10637:
10361:
9639:10.1512/iumj.1957.6.56050
9411:10.1007/s10702-006-1852-1
9338:10.1017/9781316809976.001
9102:Reviews of Modern Physics
9032:Density functional theory
8806:, then the collection of
8702:amplitude damping channel
7151:Given a density operator
7121:{\displaystyle \rho _{i}}
6961:{\displaystyle \rho _{i}}
6872:{\displaystyle \rho _{i}}
6660:are orthonormal vectors,
4304:) cannot be described as
4072:polarized light, or only
3141:{\displaystyle 2\times 2}
1003:Pick a basis with states
973:Definition and motivation
11912:Quantum machine learning
11892:Quantum key distribution
11882:Quantum image processing
11872:Quantum error correction
11722:Wheeler's delayed choice
10201:Emch, Gerard G. (1972),
9908:, Springer, p. 16,
9516:10.1002/andp.19504430510
9122:10.1103/RevModPhys.29.74
8929:. However, by using the
8664:grand canonical ensemble
8397:{\displaystyle W(x,p,t)}
8286:is the Hamiltonian, and
6500:of the density operator
6104:has spectral resolution
5553:occurs with probability
4042:that allows either only
2474:it can be written as an
2027:{\displaystyle \Pi _{m}}
1476:{\displaystyle \Pi _{m}}
601:Quantum machine learning
354:Wheeler's delayed-choice
11828:Quantum finite automata
10484:Generalized permutation
10262:. 1965. pp. 8–18.
10171:Mackey, George Whitelaw
10110:Physical Review Letters
9902:Schwabl, Franz (2002),
9787:Physical Review Letters
9667:Physical Review Letters
9318:Wilde, Mark M. (2017).
7915:Phase-space formulation
4822:Schrödinger–HJW theorem
4302:incandescent light bulb
3956:elliptical polarization
3275:where the real numbers
2700:and the other half in
2376:Schrödinger–HJW theorem
2253:over the Hilbert space
1906:Schrödinger–HJW theorem
914:of the outcomes of any
311:Leggett–Garg inequality
11932:Quantum neural network
11258:Mathematics portal
10049:New Journal of Physics
9733:Foundations of Physics
9559:10.1002/andp.200610207
8894:
8853:
8820:
8800:
8765:
8751:electrons filling the
8745:
8656:
8591:
8571:
8551:
8503:
8502:{\displaystyle \beta }
8483:
8398:
8357:
8356:{\displaystyle \hbar }
8318:
8280:
8279:{\displaystyle H(x,p)}
8242:
8108:
7899:
7818:
7786:
7678:
7644:
7597:
7466:
7364:
7336:
7335:{\displaystyle \rho '}
7311:
7280:
7257:
7190:
7189:{\displaystyle \rho '}
7165:
7142:
7122:
7092:
6993:
6962:
6932:
6873:
6843:
6745:
6725:
6687:
6654:
6619:
6542:spectral decomposition
6534:
6514:
6486:
6466:
6423:
6393:
6318:
6222:
6187:
6157:
6098:
6072:
6022:
5964:
5941:
5659:
5574:
5547:
5506:
5478:
5458:
5423:
5400:
5303:
5283:
5248:
5191:
5099:
4989:
4931:
4895:
4875:
4802:
4714:
4482:
4452:
4422:
4392:
4354:
4288:
4258:
4224:
4160:
4130:
4096:
4066:
4032:
3940:
3874:
3818:
3788:
3749:
3739:
3672:
3593:
3516:
3328:
3266:
3142:
3099:
3070:
2976:
2889:probability amplitudes
2877:
2842:
2804:
2729:
2694:
2652:
2601:
2548:
2506:with itself, that is,
2500:
2452:
2424:
2368:
2344:reduced density matrix
2336:
2278:
2243:
2216:
2059:
2028:
2001:
1981:
1933:
1898:
1871:
1836:
1759:
1673:
1477:
1447:
1431:projective measurement
1419:
1392:
1349:positive semi-definite
1342:positive semi-definite
1334:
1294:
1266:
1232:
1053:
1025:
102:
12092:Statistical mechanics
11957:Quantum teleportation
11485:Wave–particle duality
10238:Göttinger Nachrichten
9905:Statistical mechanics
9011:statistical mechanics
8998:and later in 1946 by
8895:
8833:
8821:
8801:
8766:
8746:
8731:method. If there are
8657:
8592:
8590:{\displaystyle \rho }
8572:
8552:
8504:
8484:
8422:Statistical mechanics
8399:
8358:
8319:
8281:
8243:
8109:
7900:
7819:
7787:
7679:
7645:
7598:
7467:
7365:
7363:{\displaystyle \rho }
7337:
7312:
7281:
7279:{\displaystyle \rho }
7258:
7191:
7166:
7164:{\displaystyle \rho }
7143:
7141:{\displaystyle \rho }
7123:
7093:
6994:
6992:{\displaystyle p_{i}}
6963:
6933:
6874:
6844:
6746:
6744:{\displaystyle \rho }
6726:
6688:
6655:
6620:
6535:
6533:{\displaystyle \rho }
6515:
6513:{\displaystyle \rho }
6487:
6485:{\displaystyle \rho }
6467:
6424:
6422:{\displaystyle P_{i}}
6394:
6319:
6223:
6221:{\displaystyle a_{i}}
6188:
6186:{\displaystyle P_{i}}
6158:
6099:
6073:
6023:
5965:
5942:
5660:
5575:
5573:{\displaystyle p_{j}}
5548:
5507:
5479:
5477:{\displaystyle \rho }
5459:
5424:
5422:{\displaystyle \rho }
5409:is a purification of
5401:
5304:
5284:
5249:
5192:
5100:
4990:
4932:
4896:
4876:
4803:
4715:
4483:
4453:
4423:
4393:
4355:
4289:
4259:
4225:
4161:
4131:
4097:
4067:
4033:
3941:
3875:
3819:
3789:
3762:circular polarization
3740:
3673:
3614:
3594:
3517:
3329:
3267:
3149:self-adjoint matrices
3143:
3089:
3071:
2982:with density matrix
2977:
2885:quantum superposition
2878:
2843:
2805:
2730:
2695:
2653:
2602:
2549:
2501:
2462:Pure and mixed states
2453:
2425:
2369:
2337:
2279:
2244:
2217:
2060:
2034:on the Hilbert space
2029:
2002:
1982:
1934:
1899:
1897:{\displaystyle p_{j}}
1872:
1837:
1760:
1674:
1478:
1448:
1420:
1418:{\displaystyle p_{j}}
1393:
1335:
1295:
1267:
1233:
1059:in a two-dimensional
1054:
1026:
296:Elitzur–Vaidman
286:Davisson–Germer
103:
11988:Quantum field theory
11917:Quantum metamaterial
11862:Quantum cryptography
11592:Consistent histories
9136:Holevo, Alexander S.
9059:Notes and references
9037:Green–Kubo relations
8830:
8810:
8775:
8755:
8735:
8698:depolarizing channel
8694:open quantum systems
8605:
8581:
8561:
8513:
8493:
8432:
8412:Example applications
8367:
8347:
8290:
8255:
8128:
7938:
7831:
7817:{\displaystyle G(t)}
7799:
7698:
7662:
7610:
7497:
7403:
7390:von Neumann equation
7386:Schrödinger equation
7354:
7321:
7290:
7270:
7203:
7175:
7155:
7132:
7105:
7006:
6976:
6945:
6886:
6856:
6758:
6735:
6697:
6664:
6629:
6548:
6524:
6504:
6476:
6456:
6406:
6338:
6235:
6205:
6170:
6111:
6088:
6035:
5978:
5954:
5683:
5587:
5557:
5520:
5496:
5468:
5433:
5413:
5316:
5293:
5258:
5204:
5122:
5002:
4945:
4905:
4885:
4831:
4732:
4725:birefringent crystal
4496:
4462:
4432:
4402:
4372:
4312:
4268:
4238:
4170:
4140:
4110:
4076:
4046:
3962:
3946:), corresponding to
3884:
3832:
3798:
3768:
3696:
3629:
3529:
3345:
3279:
3158:
3126:
3094:representation of a
3081:quantum interference
2989:
2895:
2852:
2817:
2742:
2704:
2669:
2617:
2575:
2510:
2482:
2434:
2382:
2350:
2288:
2257:
2226:
2069:
2038:
2011:
1991:
1943:
1915:
1881:
1846:
1773:
1687:
1487:
1460:
1437:
1402:
1367:
1308:
1276:
1248:
1067:
1035:
1007:
963:open quantum systems
561:Quantum field theory
473:Consistent histories
110:Schrödinger equation
37:
12082:Functional analysis
11973:Quantum fluctuation
11942:Quantum programming
11902:Quantum logic gates
11887:Quantum information
11867:Quantum electronics
11342:Classical mechanics
11207:Linear independence
10454:Diagonally dominant
10132:2018PhRvL.121z0401P
10071:2016NJPh...18c3024G
10018:2019PhR...831....1S
9847:Rieffel, Eleanor G.
9809:2008PhRvL.100p0401G
9756:2004FoPh...34..193C
9690:2003PhRvL..91l0403B
9551:2006AnP...518..663L
9508:1950AnP...443..322L
9403:2006FoPhL..19...95K
9114:1957RvMP...29...74F
9007:probability measure
8686:quantum computation
8674:Quantum decoherence
8023:
7484:Schrödinger picture
7392:(also known as the
6492:or in terms of the
6451:von Neumann entropy
6250:
6195:projection operator
3582:
3564:
3546:
2560:, in particular of
1207:
967:quantum information
349:Stern–Gerlach
146:Classical mechanics
12026:in popular culture
11808:Quantum algorithms
11656:Von Neumann–Wigner
11636:Objective collapse
11347:Old quantum theory
11212:Matrix exponential
11202:Jordan normal form
11036:Fisher information
10907:Euclidean distance
10821:Totally unimodular
10305:10.1007/BF03001661
10207:Wiley-Interscience
10179:Dover Publications
9616:Gleason, Andrew M.
9528:Annalen der Physik
9495:Annalen der Physik
9460:10.1007/BF00211375
8923:linear functionals
8890:
8816:
8796:
8761:
8741:
8725:Slater determinant
8706:Quantum tomography
8652:
8599:partition function
8587:
8567:
8547:
8499:
8479:
8426:canonical ensemble
8394:
8353:
8337:Liouville equation
8314:
8276:
8238:
8104:
8006:
7895:
7814:
7782:
7674:
7656:not time-dependent
7652:Heisenberg picture
7640:
7593:
7488:Heisenberg picture
7462:
7360:
7332:
7307:
7276:
7253:
7226:
7186:
7161:
7138:
7118:
7088:
7055:
6989:
6958:
6928:
6904:
6869:
6839:
6779:
6741:
6721:
6720:
6683:
6650:
6615:
6614:
6567:
6530:
6510:
6482:
6462:
6419:
6389:
6362:
6314:
6238:
6218:
6183:
6153:
6129:
6094:
6081:for mixed states.
6068:
6018:
5960:
5937:
5858:
5769:
5707:
5655:
5605:
5570:
5543:
5542:
5502:
5484:are of this form.
5474:
5454:
5419:
5396:
5342:
5299:
5279:
5244:
5187:
5140:
5095:
5049:
4985:
4927:
4891:
4871:
4798:
4710:
4701:
4478:
4448:
4418:
4388:
4350:
4308:state of the form
4284:
4254:
4220:
4156:
4126:
4092:
4062:
4040:circular polarizer
4028:
3936:
3870:
3814:
3784:
3754:light polarization
3750:
3735:
3729:
3668:
3662:
3589:
3568:
3550:
3532:
3512:
3503:
3447:
3391:
3324:
3262:
3138:
3116:linear combination
3100:
3066:
3057:
2972:
2873:
2838:
2800:
2791:
2725:
2690:
2648:
2597:
2544:
2496:
2478:of a state vector
2468:convex combination
2448:
2420:
2364:
2332:
2274:
2239:
2212:
2065:alone is given by
2055:
2024:
1997:
1977:
1929:
1894:
1867:
1832:
1785:
1755:
1705:
1669:
1609:
1514:
1473:
1443:
1415:
1388:
1359:one acting on the
1330:
1302:complex conjugates
1290:
1262:
1228:
1222:
1193:
1150:
1049:
1021:
537:Von Neumann–Wigner
517:Objective-collapse
316:Mach–Zehnder
306:Leggett inequality
301:Franck–Hertz
151:Old quantum theory
98:
12069:
12068:
12043:Quantum mysticism
12021:Schrödinger's cat
11952:Quantum simulator
11922:Quantum metrology
11850:Quantum computing
11813:Quantum amplifier
11790:Quantum spacetime
11755:Quantum cosmology
11745:Quantum chemistry
11460:Scattering theory
11408:Zero-point energy
11403:Degenerate levels
11311:Quantum mechanics
11277:
11276:
11269:Category:Matrices
11141:Fuzzy associative
11031:Doubly stochastic
10739:Positive-definite
10419:Block tridiagonal
10293:Rendiconti Lincei
10277:978-0-08-010586-4
10230:von Neumann, John
10216:978-0-471-23900-0
10188:978-0-486-43517-6
9972:978-0-521-87342-0
9939:978-981-283-251-1
9915:978-3-540-43163-3
9889:978-0-19-852063-4
9862:978-0-262-01506-6
9728:Caves, Carlton M.
9602:978-3-540-70622-9
9347:978-1-107-17616-4
9293:978-0-7923-3632-7
9239:978-0-521-63503-5
9195:978-1-4614-7115-8
9083:978-0-306-44790-7
9047:Lindblad equation
8961:compact operators
8819:{\displaystyle N}
8764:{\displaystyle N}
8744:{\displaystyle N}
8570:{\displaystyle H}
8341:classical physics
8170:
8004:
7989:
7984:
7962:
7589:
7545:
7458:
7436:
7217:
7046:
6895:
6770:
6558:
6465:{\displaystyle S}
6435:non-contextuality
6431:Gleason's theorem
6353:
6312:
6120:
6097:{\displaystyle A}
5849:
5760:
5698:
5670:expectation value
5596:
5505:{\displaystyle A}
5355:
5333:
5302:{\displaystyle U}
5131:
5075:
5040:
5017:
4894:{\displaystyle U}
4796:
4669:
4630:
4591:
4552:
4513:
4298:Unpolarized light
4218:
4026:
3175:
3025:
2967:
2759:
2000:{\displaystyle m}
1877:and coefficients
1776:
1696:
1600:
1505:
1446:{\displaystyle m}
888:quantum mechanics
884:
883:
591:Scattering theory
571:Quantum computing
344:Schrödinger's cat
276:Bell's inequality
84:
59:
28:Quantum mechanics
12104:
12059:
12058:
11770:Quantum geometry
11765:Quantum dynamics
11622:Superdeterminism
11518:Matrix mechanics
11373:Bra–ket notation
11304:
11297:
11290:
11281:
11280:
11264:List of matrices
11256:
11255:
11232:Row echelon form
11176:State transition
11105:Seidel adjacency
10987:Totally positive
10847:Alternating sign
10444:Complex Hadamard
10347:
10340:
10333:
10324:
10323:
10317:
10316:
10288:
10282:
10281:
10255:
10246:
10245:
10226:
10220:
10219:
10198:
10192:
10191:
10166:
10160:
10159:
10125:
10105:
10099:
10098:
10064:
10044:
10038:
10037:
10011:
9991:
9985:
9984:
9949:
9943:
9942:
9925:
9919:
9918:
9899:
9893:
9892:
9873:
9867:
9866:
9843:
9837:
9836:
9802:
9782:
9776:
9775:
9749:
9747:quant-ph/0306179
9724:
9718:
9717:
9683:
9681:quant-ph/9909073
9658:
9652:
9651:
9641:
9612:
9606:
9605:
9577:
9571:
9570:
9544:
9542:quant-ph/0403007
9519:
9486:
9480:
9479:
9437:
9431:
9430:
9396:
9394:quant-ph/0305068
9374:
9368:
9367:
9331:
9315:
9306:
9305:
9274:
9263:
9262:
9250:
9244:
9242:
9219:
9200:
9199:
9173:
9162:
9161:
9132:
9126:
9125:
9097:
9088:
9087:
9069:
8992:John von Neumann
8931:GNS construction
8899:
8897:
8896:
8891:
8889:
8884:
8883:
8868:
8867:
8858:
8852:
8847:
8825:
8823:
8822:
8817:
8805:
8803:
8802:
8797:
8792:
8791:
8782:
8770:
8768:
8767:
8762:
8750:
8748:
8747:
8742:
8661:
8659:
8658:
8653:
8630:
8596:
8594:
8593:
8588:
8576:
8574:
8573:
8568:
8556:
8554:
8553:
8548:
8546:
8545:
8530:
8529:
8528:
8508:
8506:
8505:
8500:
8488:
8486:
8485:
8480:
8466:
8403:
8401:
8400:
8395:
8362:
8360:
8359:
8354:
8323:
8321:
8320:
8315:
8285:
8283:
8282:
8277:
8247:
8245:
8244:
8239:
8171:
8169:
8161:
8132:
8113:
8111:
8110:
8105:
8093:
8092:
8088:
8033:
8032:
8022:
8017:
8005:
8003:
7992:
7987:
7986:
7985:
7983:
7982:
7970:
7965:
7960:
7904:
7902:
7901:
7896:
7891:
7890:
7823:
7821:
7820:
7815:
7791:
7789:
7788:
7783:
7778:
7777:
7773:
7742:
7741:
7737:
7683:
7681:
7680:
7675:
7649:
7647:
7646:
7641:
7630:
7629:
7625:
7602:
7600:
7599:
7594:
7587:
7586:
7582:
7581:
7580:
7576:
7546:
7544:
7533:
7532:
7531:
7527:
7507:
7471:
7469:
7468:
7463:
7456:
7437:
7435:
7424:
7413:
7369:
7367:
7366:
7361:
7346:measurement, or
7341:
7339:
7338:
7333:
7331:
7316:
7314:
7313:
7308:
7306:
7285:
7283:
7282:
7277:
7262:
7260:
7259:
7254:
7249:
7248:
7236:
7235:
7225:
7213:
7195:
7193:
7192:
7187:
7185:
7170:
7168:
7167:
7162:
7147:
7145:
7144:
7139:
7127:
7125:
7124:
7119:
7117:
7116:
7101:When the states
7097:
7095:
7094:
7089:
7081:
7080:
7065:
7064:
7054:
7039:
7038:
6998:
6996:
6995:
6990:
6988:
6987:
6967:
6965:
6964:
6959:
6957:
6956:
6937:
6935:
6934:
6929:
6924:
6923:
6914:
6913:
6903:
6878:
6876:
6875:
6870:
6868:
6867:
6848:
6846:
6845:
6840:
6805:
6804:
6789:
6788:
6778:
6750:
6748:
6747:
6742:
6730:
6728:
6727:
6722:
6713:
6712:
6692:
6690:
6689:
6684:
6676:
6675:
6659:
6657:
6656:
6651:
6646:
6645:
6636:
6624:
6622:
6621:
6616:
6613:
6608:
6607:
6592:
6591:
6582:
6577:
6576:
6566:
6539:
6537:
6536:
6531:
6519:
6517:
6516:
6511:
6491:
6489:
6488:
6483:
6471:
6469:
6468:
6463:
6428:
6426:
6425:
6420:
6418:
6417:
6398:
6396:
6395:
6390:
6385:
6384:
6372:
6371:
6361:
6349:
6323:
6321:
6320:
6315:
6313:
6311:
6310:
6306:
6305:
6304:
6280:
6279:
6278:
6266:
6265:
6255:
6246:
6227:
6225:
6224:
6219:
6217:
6216:
6192:
6190:
6189:
6184:
6182:
6181:
6162:
6160:
6159:
6154:
6149:
6148:
6139:
6138:
6128:
6103:
6101:
6100:
6095:
6077:
6075:
6074:
6069:
6027:
6025:
6024:
6019:
6011:
6003:
5969:
5967:
5966:
5961:
5946:
5944:
5943:
5938:
5912:
5908:
5904:
5899:
5898:
5883:
5882:
5873:
5868:
5867:
5857:
5834:
5830:
5826:
5821:
5820:
5805:
5804:
5795:
5779:
5778:
5768:
5753:
5752:
5743:
5735:
5730:
5729:
5717:
5716:
5706:
5664:
5662:
5661:
5656:
5651:
5646:
5645:
5630:
5629:
5620:
5615:
5614:
5604:
5579:
5577:
5576:
5571:
5569:
5568:
5552:
5550:
5549:
5544:
5538:
5537:
5528:
5511:
5509:
5508:
5503:
5483:
5481:
5480:
5475:
5463:
5461:
5460:
5455:
5450:
5449:
5440:
5428:
5426:
5425:
5420:
5405:
5403:
5402:
5397:
5392:
5391:
5382:
5371:
5370:
5361:
5356:
5354:
5353:
5344:
5341:
5323:
5308:
5306:
5305:
5300:
5288:
5286:
5285:
5280:
5275:
5274:
5265:
5253:
5251:
5250:
5245:
5237:
5236:
5227:
5219:
5218:
5196:
5194:
5193:
5188:
5186:
5181:
5180:
5165:
5164:
5155:
5150:
5149:
5139:
5104:
5102:
5101:
5096:
5094:
5090:
5089:
5076:
5074:
5073:
5064:
5062:
5061:
5048:
5036:
5032:
5031:
5018:
5016:
5015:
5006:
4994:
4992:
4991:
4986:
4978:
4977:
4968:
4960:
4959:
4941:), the ensemble
4939:partial isometry
4936:
4934:
4933:
4928:
4917:
4916:
4900:
4898:
4897:
4892:
4880:
4878:
4877:
4872:
4864:
4863:
4854:
4846:
4845:
4807:
4805:
4804:
4799:
4797:
4792:
4790:
4779:
4771:
4766:
4755:
4747:
4742:
4719:
4717:
4716:
4711:
4706:
4705:
4670:
4662:
4657:
4652:
4641:
4636:
4631:
4623:
4618:
4613:
4602:
4597:
4592:
4584:
4579:
4574:
4563:
4558:
4553:
4545:
4540:
4535:
4524:
4519:
4514:
4506:
4487:
4485:
4484:
4479:
4474:
4469:
4457:
4455:
4454:
4449:
4444:
4439:
4427:
4425:
4424:
4419:
4414:
4409:
4398:polarization or
4397:
4395:
4394:
4389:
4384:
4379:
4359:
4357:
4356:
4351:
4346:
4341:
4327:
4322:
4293:
4291:
4290:
4285:
4280:
4275:
4263:
4261:
4260:
4255:
4250:
4245:
4232:linear polarizer
4229:
4227:
4226:
4221:
4219:
4214:
4212:
4201:
4196:
4185:
4180:
4165:
4163:
4162:
4157:
4152:
4147:
4135:
4133:
4132:
4127:
4122:
4117:
4101:
4099:
4098:
4093:
4088:
4083:
4071:
4069:
4068:
4063:
4058:
4053:
4037:
4035:
4034:
4029:
4027:
4022:
4020:
4009:
4004:
3993:
3988:
3974:
3969:
3945:
3943:
3942:
3937:
3929:
3928:
3923:
3914:
3906:
3905:
3900:
3891:
3879:
3877:
3876:
3871:
3866:
3861:
3847:
3842:
3823:
3821:
3820:
3815:
3810:
3805:
3793:
3791:
3790:
3785:
3780:
3775:
3756:. An individual
3747:
3744:
3742:
3741:
3736:
3734:
3733:
3689:
3685:
3680:
3677:
3675:
3674:
3669:
3667:
3666:
3622:
3618:
3598:
3596:
3595:
3590:
3581:
3576:
3563:
3558:
3545:
3540:
3521:
3519:
3518:
3513:
3508:
3507:
3466:
3465:
3452:
3451:
3410:
3409:
3396:
3395:
3357:
3356:
3333:
3331:
3330:
3325:
3320:
3319:
3307:
3306:
3294:
3293:
3271:
3269:
3268:
3263:
3258:
3254:
3253:
3252:
3243:
3242:
3230:
3229:
3220:
3219:
3207:
3206:
3197:
3196:
3176:
3168:
3147:
3145:
3144:
3139:
3075:
3073:
3072:
3067:
3062:
3061:
3026:
3018:
3013:
2996:
2981:
2979:
2978:
2973:
2968:
2963:
2961:
2950:
2949:
2940:
2929:
2928:
2919:
2902:
2882:
2880:
2879:
2874:
2869:
2868:
2859:
2847:
2845:
2844:
2839:
2834:
2833:
2824:
2809:
2807:
2806:
2801:
2796:
2795:
2760:
2752:
2734:
2732:
2731:
2726:
2721:
2720:
2711:
2699:
2697:
2696:
2691:
2686:
2685:
2676:
2657:
2655:
2654:
2649:
2638:
2637:
2606:
2604:
2603:
2598:
2593:
2592:
2553:
2551:
2550:
2545:
2540:
2523:
2505:
2503:
2502:
2497:
2489:
2457:
2455:
2454:
2449:
2447:
2429:
2427:
2426:
2421:
2419:
2408:
2394:
2393:
2373:
2371:
2370:
2365:
2357:
2341:
2339:
2338:
2333:
2331:
2320:
2306:
2305:
2283:
2281:
2280:
2275:
2273:
2272:
2267:
2266:
2248:
2246:
2245:
2240:
2238:
2237:
2221:
2219:
2218:
2213:
2208:
2204:
2203:
2199:
2198:
2187:
2173:
2172:
2158:
2157:
2134:
2123:
2119:
2112:
2111:
2097:
2064:
2062:
2061:
2056:
2054:
2053:
2048:
2047:
2033:
2031:
2030:
2025:
2023:
2022:
2006:
2004:
2003:
1998:
1986:
1984:
1983:
1978:
1976:
1975:
1970:
1969:
1959:
1958:
1953:
1952:
1938:
1936:
1935:
1930:
1922:
1903:
1901:
1900:
1895:
1893:
1892:
1876:
1874:
1873:
1868:
1866:
1862:
1861:
1842:for some states
1841:
1839:
1838:
1833:
1831:
1827:
1826:
1813:
1809:
1808:
1795:
1794:
1784:
1767:spectral theorem
1764:
1762:
1761:
1756:
1751:
1747:
1746:
1733:
1729:
1728:
1715:
1714:
1704:
1681:density operator
1679:which makes the
1678:
1676:
1675:
1670:
1665:
1661:
1660:
1656:
1655:
1651:
1650:
1637:
1633:
1632:
1619:
1618:
1608:
1594:
1593:
1570:
1566:
1565:
1552:
1551:
1542:
1538:
1537:
1524:
1523:
1513:
1482:
1480:
1479:
1474:
1472:
1471:
1452:
1450:
1449:
1444:
1425:, describing an
1424:
1422:
1421:
1416:
1414:
1413:
1397:
1395:
1394:
1389:
1384:
1383:
1374:
1339:
1337:
1336:
1331:
1326:
1325:
1299:
1297:
1296:
1291:
1283:
1271:
1269:
1268:
1263:
1255:
1237:
1235:
1234:
1229:
1227:
1223:
1219:
1218:
1206:
1201:
1188:
1187:
1176:
1175:
1155:
1151:
1147:
1146:
1135:
1134:
1121:
1120:
1109:
1108:
1085:
1084:
1058:
1056:
1055:
1050:
1042:
1030:
1028:
1027:
1022:
1014:
998:density operator
983:density operator
904:physical systems
896:density operator
876:
869:
862:
503:Superdeterminism
156:Bra–ket notation
107:
105:
104:
99:
91:
86:
85:
77:
65:
60:
58:
47:
19:
18:
12112:
12111:
12107:
12106:
12105:
12103:
12102:
12101:
12072:
12071:
12070:
12065:
12047:
12033:Wigner's friend
12009:
12000:Quantum gravity
11961:
11947:Quantum sensing
11927:Quantum network
11907:Quantum machine
11877:Quantum imaging
11840:Quantum circuit
11835:Quantum channel
11794:
11740:Quantum biology
11726:
11702:Elitzur–Vaidman
11677:Davisson–Germer
11660:
11612:Hidden-variable
11602:de Broglie–Bohm
11579:Interpretations
11573:
11537:
11491:
11378:Complementarity
11356:
11313:
11308:
11278:
11273:
11250:
11241:
11190:
11114:
11060:
10996:
10830:
10748:
10694:
10633:
10434:Centrosymmetric
10357:
10351:
10321:
10320:
10289:
10285:
10278:
10257:
10256:
10249:
10227:
10223:
10217:
10199:
10195:
10189:
10168:See appendix,
10167:
10163:
10106:
10102:
10045:
10041:
9996:Physics Reports
9992:
9988:
9973:
9950:
9946:
9940:
9926:
9922:
9916:
9900:
9896:
9890:
9874:
9870:
9863:
9844:
9840:
9783:
9779:
9725:
9721:
9659:
9655:
9613:
9609:
9603:
9578:
9574:
9523:LĂĽders, Gerhart
9490:LĂĽders, Gerhart
9487:
9483:
9438:
9434:
9375:
9371:
9348:
9316:
9309:
9294:
9275:
9266:
9251:
9247:
9240:
9220:
9203:
9196:
9174:
9165:
9150:
9133:
9129:
9098:
9091:
9084:
9070:
9066:
9061:
9056:
9022:
8988:
8907:
8885:
8879:
8875:
8863:
8859:
8854:
8848:
8837:
8831:
8828:
8827:
8811:
8808:
8807:
8787:
8783:
8778:
8776:
8773:
8772:
8756:
8753:
8752:
8736:
8733:
8732:
8679:classical limit
8623:
8606:
8603:
8602:
8582:
8579:
8578:
8562:
8559:
8558:
8538:
8534:
8524:
8523:
8519:
8514:
8511:
8510:
8494:
8491:
8490:
8462:
8433:
8430:
8429:
8414:
8368:
8365:
8364:
8348:
8345:
8344:
8291:
8288:
8287:
8256:
8253:
8252:
8162:
8133:
8131:
8129:
8126:
8125:
8084:
8071:
8067:
8028:
8024:
8018:
8010:
7996:
7991:
7972:
7971:
7966:
7964:
7963:
7939:
7936:
7935:
7929:Wigner function
7917:
7911:
7886:
7882:
7832:
7829:
7828:
7800:
7797:
7796:
7769:
7759:
7755:
7733:
7720:
7716:
7699:
7696:
7695:
7663:
7660:
7659:
7621:
7617:
7613:
7611:
7608:
7607:
7572:
7568:
7564:
7557:
7553:
7534:
7523:
7519:
7515:
7508:
7506:
7498:
7495:
7494:
7425:
7414:
7412:
7404:
7401:
7400:
7382:
7376:
7355:
7352:
7351:
7324:
7322:
7319:
7318:
7299:
7291:
7288:
7287:
7271:
7268:
7267:
7244:
7240:
7231:
7227:
7221:
7206:
7204:
7201:
7200:
7178:
7176:
7173:
7172:
7156:
7153:
7152:
7133:
7130:
7129:
7112:
7108:
7106:
7103:
7102:
7076:
7072:
7060:
7056:
7050:
7034:
7030:
7007:
7004:
7003:
6983:
6979:
6977:
6974:
6973:
6970:Shannon entropy
6952:
6948:
6946:
6943:
6942:
6919:
6915:
6909:
6905:
6899:
6887:
6884:
6883:
6863:
6859:
6857:
6854:
6853:
6800:
6796:
6784:
6780:
6774:
6759:
6756:
6755:
6736:
6733:
6732:
6708:
6704:
6698:
6695:
6694:
6671:
6667:
6665:
6662:
6661:
6641:
6637:
6632:
6630:
6627:
6626:
6609:
6603:
6599:
6587:
6583:
6578:
6572:
6568:
6562:
6549:
6546:
6545:
6525:
6522:
6521:
6505:
6502:
6501:
6477:
6474:
6473:
6457:
6454:
6453:
6447:
6413:
6409:
6407:
6404:
6403:
6380:
6376:
6367:
6363:
6357:
6342:
6339:
6336:
6335:
6300:
6296:
6292:
6288:
6281:
6274:
6270:
6261:
6257:
6256:
6254:
6242:
6236:
6233:
6232:
6212:
6208:
6206:
6203:
6202:
6177:
6173:
6171:
6168:
6167:
6144:
6140:
6134:
6130:
6124:
6112:
6109:
6108:
6089:
6086:
6085:
6036:
6033:
6032:
6007:
5999:
5979:
5976:
5975:
5955:
5952:
5951:
5900:
5894:
5890:
5878:
5874:
5869:
5863:
5859:
5853:
5848:
5844:
5822:
5816:
5812:
5800:
5796:
5791:
5790:
5786:
5774:
5770:
5764:
5748:
5744:
5739:
5731:
5725:
5721:
5712:
5708:
5702:
5684:
5681:
5680:
5647:
5641:
5637:
5625:
5621:
5616:
5610:
5606:
5600:
5588:
5585:
5584:
5564:
5560:
5558:
5555:
5554:
5533:
5529:
5524:
5521:
5518:
5517:
5497:
5494:
5493:
5490:
5469:
5466:
5465:
5445:
5441:
5436:
5434:
5431:
5430:
5414:
5411:
5410:
5387:
5383:
5378:
5366:
5362:
5357:
5349:
5345:
5343:
5337:
5319:
5317:
5314:
5313:
5294:
5291:
5290:
5270:
5266:
5261:
5259:
5256:
5255:
5232:
5228:
5223:
5214:
5210:
5205:
5202:
5201:
5182:
5176:
5172:
5160:
5156:
5151:
5145:
5141:
5135:
5123:
5120:
5119:
5085:
5081:
5077:
5069:
5065:
5063:
5054:
5050:
5044:
5027:
5023:
5019:
5011:
5007:
5005:
5003:
5000:
4999:
4973:
4969:
4964:
4955:
4951:
4946:
4943:
4942:
4912:
4908:
4906:
4903:
4902:
4886:
4883:
4882:
4859:
4855:
4850:
4841:
4837:
4832:
4829:
4828:
4824:
4818:
4791:
4786:
4775:
4767:
4762:
4751:
4743:
4738:
4733:
4730:
4729:
4700:
4699:
4694:
4688:
4687:
4682:
4672:
4671:
4661:
4653:
4648:
4637:
4632:
4622:
4614:
4609:
4598:
4593:
4583:
4575:
4570:
4559:
4554:
4544:
4536:
4531:
4520:
4515:
4505:
4497:
4494:
4493:
4470:
4465:
4463:
4460:
4459:
4440:
4435:
4433:
4430:
4429:
4410:
4405:
4403:
4400:
4399:
4380:
4375:
4373:
4370:
4369:
4342:
4337:
4323:
4318:
4313:
4310:
4309:
4276:
4271:
4269:
4266:
4265:
4246:
4241:
4239:
4236:
4235:
4213:
4208:
4197:
4192:
4181:
4176:
4171:
4168:
4167:
4148:
4143:
4141:
4138:
4137:
4118:
4113:
4111:
4108:
4107:
4084:
4079:
4077:
4074:
4073:
4054:
4049:
4047:
4044:
4043:
4021:
4016:
4005:
4000:
3989:
3984:
3970:
3965:
3963:
3960:
3959:
3924:
3919:
3918:
3910:
3901:
3896:
3895:
3887:
3885:
3882:
3881:
3862:
3857:
3843:
3838:
3833:
3830:
3829:
3806:
3801:
3799:
3796:
3795:
3776:
3771:
3769:
3766:
3765:
3748:
3745:
3728:
3727:
3722:
3716:
3715:
3710:
3700:
3699:
3697:
3694:
3693:
3691:
3687:
3683:
3681:
3678:
3661:
3660:
3655:
3649:
3648:
3643:
3633:
3632:
3630:
3627:
3626:
3624:
3620:
3616:
3609:
3577:
3572:
3559:
3554:
3541:
3536:
3530:
3527:
3526:
3502:
3501:
3493:
3487:
3486:
3481:
3471:
3470:
3461:
3457:
3446:
3445:
3440:
3434:
3433:
3425:
3415:
3414:
3405:
3401:
3390:
3389:
3384:
3378:
3377:
3372:
3362:
3361:
3352:
3348:
3346:
3343:
3342:
3315:
3311:
3302:
3298:
3289:
3285:
3280:
3277:
3276:
3248:
3244:
3238:
3234:
3225:
3221:
3215:
3211:
3202:
3198:
3192:
3188:
3181:
3177:
3167:
3159:
3156:
3155:
3127:
3124:
3123:
3108:extremal points
3056:
3055:
3050:
3044:
3043:
3038:
3028:
3027:
3017:
3009:
2992:
2990:
2987:
2986:
2962:
2957:
2945:
2941:
2936:
2924:
2920:
2915:
2898:
2896:
2893:
2892:
2864:
2860:
2855:
2853:
2850:
2849:
2829:
2825:
2820:
2818:
2815:
2814:
2790:
2789:
2784:
2778:
2777:
2772:
2762:
2761:
2751:
2743:
2740:
2739:
2716:
2712:
2707:
2705:
2702:
2701:
2681:
2677:
2672:
2670:
2667:
2666:
2633:
2629:
2618:
2615:
2614:
2588:
2584:
2576:
2573:
2572:
2536:
2519:
2511:
2508:
2507:
2485:
2483:
2480:
2479:
2464:
2437:
2435:
2432:
2431:
2430:for some state
2409:
2398:
2389:
2385:
2383:
2380:
2379:
2353:
2351:
2348:
2347:
2321:
2310:
2301:
2297:
2289:
2286:
2285:
2268:
2262:
2261:
2260:
2258:
2255:
2254:
2233:
2229:
2227:
2224:
2223:
2188:
2177:
2168:
2164:
2163:
2159:
2153:
2149:
2148:
2144:
2124:
2107:
2103:
2102:
2098:
2087:
2070:
2067:
2066:
2049:
2043:
2042:
2041:
2039:
2036:
2035:
2018:
2014:
2012:
2009:
2008:
1992:
1989:
1988:
1971:
1965:
1964:
1963:
1954:
1948:
1947:
1946:
1944:
1941:
1940:
1918:
1916:
1913:
1912:
1888:
1884:
1882:
1879:
1878:
1857:
1853:
1849:
1847:
1844:
1843:
1822:
1818:
1814:
1804:
1800:
1796:
1790:
1786:
1780:
1774:
1771:
1770:
1742:
1738:
1734:
1724:
1720:
1716:
1710:
1706:
1700:
1688:
1685:
1684:
1683:, defined as
1646:
1642:
1638:
1628:
1624:
1620:
1614:
1610:
1604:
1599:
1595:
1589:
1585:
1584:
1580:
1561:
1557:
1553:
1547:
1543:
1533:
1529:
1525:
1519:
1515:
1509:
1488:
1485:
1484:
1467:
1463:
1461:
1458:
1457:
1438:
1435:
1434:
1409:
1405:
1403:
1400:
1399:
1379:
1375:
1370:
1368:
1365:
1364:
1318:
1314:
1309:
1306:
1305:
1279:
1277:
1274:
1273:
1251:
1249:
1246:
1245:
1221:
1220:
1214:
1210:
1208:
1202:
1197:
1190:
1189:
1183:
1179:
1177:
1171:
1167:
1163:
1159:
1149:
1148:
1142:
1138:
1136:
1130:
1126:
1123:
1122:
1116:
1112:
1110:
1104:
1100:
1096:
1092:
1077:
1073:
1068:
1065:
1064:
1038:
1036:
1033:
1032:
1010:
1008:
1005:
1004:
979:linear operator
975:
932:mixed ensembles
880:
851:
850:
849:
614:
606:
605:
551:
550:Advanced topics
543:
542:
541:
493:Hidden-variable
483:de Broglie–Bohm
462:
460:Interpretations
452:
451:
450:
420:
412:
411:
410:
368:
360:
359:
358:
325:
281:CHSH inequality
270:
262:
261:
260:
189:Complementarity
183:
175:
174:
173:
141:
112:
87:
76:
75:
61:
51:
46:
38:
35:
34:
17:
12:
11:
5:
12110:
12100:
12099:
12094:
12089:
12084:
12067:
12066:
12064:
12063:
12052:
12049:
12048:
12046:
12045:
12040:
12035:
12030:
12029:
12028:
12017:
12015:
12011:
12010:
12008:
12007:
12002:
11997:
11996:
11995:
11985:
11980:
11978:Casimir effect
11975:
11969:
11967:
11963:
11962:
11960:
11959:
11954:
11949:
11944:
11939:
11937:Quantum optics
11934:
11929:
11924:
11919:
11914:
11909:
11904:
11899:
11894:
11889:
11884:
11879:
11874:
11869:
11864:
11859:
11858:
11857:
11847:
11842:
11837:
11832:
11831:
11830:
11820:
11815:
11810:
11804:
11802:
11796:
11795:
11793:
11792:
11787:
11782:
11777:
11772:
11767:
11762:
11757:
11752:
11747:
11742:
11736:
11734:
11728:
11727:
11725:
11724:
11719:
11714:
11712:Quantum eraser
11709:
11704:
11699:
11694:
11689:
11684:
11679:
11674:
11668:
11666:
11662:
11661:
11659:
11658:
11653:
11648:
11643:
11638:
11633:
11628:
11627:
11626:
11625:
11624:
11609:
11604:
11599:
11594:
11589:
11583:
11581:
11575:
11574:
11572:
11571:
11566:
11561:
11556:
11551:
11545:
11543:
11539:
11538:
11536:
11535:
11530:
11525:
11520:
11515:
11510:
11505:
11499:
11497:
11493:
11492:
11490:
11489:
11488:
11487:
11482:
11472:
11467:
11462:
11457:
11452:
11447:
11442:
11437:
11432:
11427:
11422:
11417:
11412:
11411:
11410:
11405:
11400:
11395:
11385:
11383:Density matrix
11380:
11375:
11370:
11364:
11362:
11358:
11357:
11355:
11354:
11349:
11344:
11339:
11338:
11337:
11327:
11321:
11319:
11315:
11314:
11307:
11306:
11299:
11292:
11284:
11275:
11274:
11272:
11271:
11266:
11261:
11246:
11243:
11242:
11240:
11239:
11234:
11229:
11224:
11222:Perfect matrix
11219:
11214:
11209:
11204:
11198:
11196:
11192:
11191:
11189:
11188:
11183:
11178:
11173:
11168:
11163:
11158:
11153:
11148:
11143:
11138:
11133:
11128:
11122:
11120:
11116:
11115:
11113:
11112:
11107:
11102:
11097:
11092:
11087:
11082:
11077:
11071:
11069:
11062:
11061:
11059:
11058:
11053:
11048:
11043:
11038:
11033:
11028:
11023:
11018:
11013:
11007:
11005:
10998:
10997:
10995:
10994:
10992:Transformation
10989:
10984:
10979:
10974:
10969:
10964:
10959:
10954:
10949:
10944:
10939:
10934:
10929:
10924:
10919:
10914:
10909:
10904:
10899:
10894:
10889:
10884:
10879:
10874:
10869:
10864:
10859:
10854:
10849:
10844:
10838:
10836:
10832:
10831:
10829:
10828:
10823:
10818:
10813:
10808:
10803:
10798:
10793:
10788:
10783:
10778:
10769:
10763:
10761:
10750:
10749:
10747:
10746:
10741:
10736:
10731:
10729:Diagonalizable
10726:
10721:
10716:
10711:
10705:
10703:
10699:Conditions on
10696:
10695:
10693:
10692:
10687:
10682:
10677:
10672:
10667:
10662:
10657:
10652:
10647:
10641:
10639:
10635:
10634:
10632:
10631:
10626:
10621:
10616:
10611:
10606:
10601:
10596:
10591:
10586:
10581:
10579:Skew-symmetric
10576:
10574:Skew-Hermitian
10571:
10566:
10561:
10556:
10551:
10546:
10541:
10536:
10531:
10526:
10521:
10516:
10511:
10506:
10501:
10496:
10491:
10486:
10481:
10476:
10471:
10466:
10461:
10456:
10451:
10446:
10441:
10436:
10431:
10426:
10421:
10416:
10411:
10409:Block-diagonal
10406:
10401:
10396:
10391:
10386:
10384:Anti-symmetric
10381:
10379:Anti-Hermitian
10376:
10371:
10365:
10363:
10359:
10358:
10350:
10349:
10342:
10335:
10327:
10319:
10318:
10299:(2): 123–130.
10283:
10276:
10247:
10221:
10215:
10193:
10187:
10161:
10100:
10039:
9986:
9971:
9953:Kardar, Mehran
9944:
9938:
9920:
9914:
9894:
9888:
9868:
9861:
9838:
9793:(16): 160401.
9777:
9740:(2): 193–209.
9719:
9674:(12): 120403.
9653:
9632:(4): 885–893.
9607:
9601:
9572:
9535:(9): 663–670.
9481:
9454:(3): 339–356.
9432:
9369:
9346:
9307:
9292:
9264:
9259:Academic Press
9245:
9238:
9201:
9194:
9163:
9148:
9127:
9089:
9082:
9063:
9062:
9060:
9057:
9055:
9054:
9049:
9044:
9039:
9034:
9029:
9023:
9021:
9018:
8987:
8984:
8906:
8903:
8902:
8901:
8888:
8882:
8878:
8874:
8871:
8866:
8862:
8857:
8851:
8846:
8843:
8840:
8836:
8815:
8795:
8790:
8786:
8781:
8760:
8740:
8709:
8684:Similarly, in
8682:
8671:
8651:
8648:
8645:
8642:
8639:
8636:
8633:
8629:
8626:
8622:
8619:
8616:
8613:
8610:
8586:
8566:
8544:
8541:
8537:
8533:
8527:
8522:
8518:
8498:
8478:
8475:
8472:
8469:
8465:
8461:
8458:
8455:
8452:
8449:
8446:
8443:
8440:
8437:
8413:
8410:
8393:
8390:
8387:
8384:
8381:
8378:
8375:
8372:
8352:
8313:
8310:
8307:
8304:
8301:
8298:
8295:
8275:
8272:
8269:
8266:
8263:
8260:
8249:
8248:
8237:
8234:
8231:
8228:
8225:
8222:
8219:
8216:
8213:
8210:
8207:
8204:
8201:
8198:
8195:
8192:
8189:
8186:
8183:
8180:
8177:
8174:
8168:
8165:
8160:
8157:
8154:
8151:
8148:
8145:
8142:
8139:
8136:
8119:Moyal equation
8115:
8114:
8103:
8100:
8097:
8091:
8087:
8083:
8080:
8077:
8074:
8070:
8066:
8063:
8060:
8057:
8054:
8051:
8048:
8045:
8042:
8039:
8036:
8031:
8027:
8021:
8016:
8013:
8009:
8002:
7999:
7995:
7981:
7978:
7975:
7969:
7958:
7955:
7952:
7949:
7946:
7943:
7913:Main article:
7910:
7907:
7906:
7905:
7894:
7889:
7885:
7881:
7878:
7875:
7872:
7869:
7866:
7863:
7860:
7857:
7854:
7851:
7848:
7845:
7842:
7839:
7836:
7813:
7810:
7807:
7804:
7793:
7792:
7781:
7776:
7772:
7768:
7765:
7762:
7758:
7754:
7751:
7748:
7745:
7740:
7736:
7732:
7729:
7726:
7723:
7719:
7715:
7712:
7709:
7706:
7703:
7673:
7670:
7667:
7639:
7636:
7633:
7628:
7624:
7620:
7616:
7604:
7603:
7592:
7585:
7579:
7575:
7571:
7567:
7563:
7560:
7556:
7552:
7549:
7543:
7540:
7537:
7530:
7526:
7522:
7518:
7514:
7511:
7505:
7502:
7473:
7472:
7461:
7455:
7452:
7449:
7446:
7443:
7440:
7434:
7431:
7428:
7423:
7420:
7417:
7411:
7408:
7375:
7372:
7359:
7342:produced by a
7330:
7327:
7305:
7302:
7298:
7295:
7275:
7264:
7263:
7252:
7247:
7243:
7239:
7234:
7230:
7224:
7220:
7216:
7212:
7209:
7184:
7181:
7160:
7137:
7115:
7111:
7099:
7098:
7087:
7084:
7079:
7075:
7071:
7068:
7063:
7059:
7053:
7049:
7045:
7042:
7037:
7033:
7029:
7026:
7023:
7020:
7017:
7014:
7011:
6986:
6982:
6955:
6951:
6939:
6938:
6927:
6922:
6918:
6912:
6908:
6902:
6898:
6894:
6891:
6866:
6862:
6850:
6849:
6838:
6835:
6832:
6829:
6826:
6823:
6820:
6817:
6814:
6811:
6808:
6803:
6799:
6795:
6792:
6787:
6783:
6777:
6773:
6769:
6766:
6763:
6740:
6719:
6716:
6711:
6707:
6703:
6682:
6679:
6674:
6670:
6649:
6644:
6640:
6635:
6612:
6606:
6602:
6598:
6595:
6590:
6586:
6581:
6575:
6571:
6565:
6561:
6556:
6553:
6529:
6509:
6481:
6461:
6446:
6443:
6416:
6412:
6400:
6399:
6388:
6383:
6379:
6375:
6370:
6366:
6360:
6356:
6352:
6348:
6345:
6325:
6324:
6309:
6303:
6299:
6295:
6291:
6287:
6284:
6277:
6273:
6269:
6264:
6260:
6253:
6249:
6245:
6241:
6215:
6211:
6180:
6176:
6164:
6163:
6152:
6147:
6143:
6137:
6133:
6127:
6123:
6119:
6116:
6093:
6079:
6078:
6067:
6064:
6061:
6058:
6055:
6052:
6049:
6046:
6043:
6040:
6017:
6014:
6010:
6006:
6002:
5998:
5995:
5992:
5989:
5986:
5983:
5959:
5948:
5947:
5936:
5933:
5930:
5927:
5924:
5921:
5918:
5915:
5911:
5907:
5903:
5897:
5893:
5889:
5886:
5881:
5877:
5872:
5866:
5862:
5856:
5852:
5847:
5843:
5840:
5837:
5833:
5829:
5825:
5819:
5815:
5811:
5808:
5803:
5799:
5794:
5789:
5785:
5782:
5777:
5773:
5767:
5763:
5759:
5756:
5751:
5747:
5742:
5738:
5734:
5728:
5724:
5720:
5715:
5711:
5705:
5701:
5697:
5694:
5691:
5688:
5666:
5665:
5654:
5650:
5644:
5640:
5636:
5633:
5628:
5624:
5619:
5613:
5609:
5603:
5599:
5595:
5592:
5567:
5563:
5541:
5536:
5532:
5527:
5501:
5489:
5486:
5473:
5453:
5448:
5444:
5439:
5418:
5407:
5406:
5395:
5390:
5386:
5381:
5377:
5374:
5369:
5365:
5360:
5352:
5348:
5340:
5336:
5332:
5329:
5326:
5322:
5298:
5278:
5273:
5269:
5264:
5254:, with states
5243:
5240:
5235:
5231:
5226:
5222:
5217:
5213:
5209:
5198:
5197:
5185:
5179:
5175:
5171:
5168:
5163:
5159:
5154:
5148:
5144:
5138:
5134:
5130:
5127:
5106:
5105:
5093:
5088:
5084:
5080:
5072:
5068:
5060:
5057:
5053:
5047:
5043:
5039:
5035:
5030:
5026:
5022:
5014:
5010:
4984:
4981:
4976:
4972:
4967:
4963:
4958:
4954:
4950:
4926:
4923:
4920:
4915:
4911:
4890:
4870:
4867:
4862:
4858:
4853:
4849:
4844:
4840:
4836:
4820:Main article:
4817:
4814:
4795:
4789:
4785:
4782:
4778:
4774:
4770:
4765:
4761:
4758:
4754:
4750:
4746:
4741:
4737:
4721:
4720:
4709:
4704:
4698:
4695:
4693:
4690:
4689:
4686:
4683:
4681:
4678:
4677:
4675:
4668:
4665:
4660:
4656:
4651:
4647:
4644:
4640:
4635:
4629:
4626:
4621:
4617:
4612:
4608:
4605:
4601:
4596:
4590:
4587:
4582:
4578:
4573:
4569:
4566:
4562:
4557:
4551:
4548:
4543:
4539:
4534:
4530:
4527:
4523:
4518:
4512:
4509:
4504:
4501:
4477:
4473:
4468:
4447:
4443:
4438:
4417:
4413:
4408:
4387:
4383:
4378:
4349:
4345:
4340:
4336:
4333:
4330:
4326:
4321:
4317:
4283:
4279:
4274:
4253:
4249:
4244:
4217:
4211:
4207:
4204:
4200:
4195:
4191:
4188:
4184:
4179:
4175:
4155:
4151:
4146:
4125:
4121:
4116:
4091:
4087:
4082:
4061:
4057:
4052:
4025:
4019:
4015:
4012:
4008:
4003:
3999:
3996:
3992:
3987:
3983:
3980:
3977:
3973:
3968:
3935:
3932:
3927:
3922:
3917:
3913:
3909:
3904:
3899:
3894:
3890:
3869:
3865:
3860:
3856:
3853:
3850:
3846:
3841:
3837:
3813:
3809:
3804:
3783:
3779:
3774:
3732:
3726:
3723:
3721:
3718:
3717:
3714:
3711:
3709:
3706:
3705:
3703:
3692:
3665:
3659:
3656:
3654:
3651:
3650:
3647:
3644:
3642:
3639:
3638:
3636:
3625:
3608:
3605:
3588:
3585:
3580:
3575:
3571:
3567:
3562:
3557:
3553:
3549:
3544:
3539:
3535:
3523:
3522:
3511:
3506:
3500:
3497:
3494:
3492:
3489:
3488:
3485:
3482:
3480:
3477:
3476:
3474:
3469:
3464:
3460:
3455:
3450:
3444:
3441:
3439:
3436:
3435:
3432:
3429:
3426:
3424:
3421:
3420:
3418:
3413:
3408:
3404:
3399:
3394:
3388:
3385:
3383:
3380:
3379:
3376:
3373:
3371:
3368:
3367:
3365:
3360:
3355:
3351:
3323:
3318:
3314:
3310:
3305:
3301:
3297:
3292:
3288:
3284:
3273:
3272:
3261:
3257:
3251:
3247:
3241:
3237:
3233:
3228:
3224:
3218:
3214:
3210:
3205:
3201:
3195:
3191:
3187:
3184:
3180:
3174:
3171:
3166:
3163:
3137:
3134:
3131:
3120:Pauli matrices
3077:
3076:
3065:
3060:
3054:
3051:
3049:
3046:
3045:
3042:
3039:
3037:
3034:
3033:
3031:
3024:
3021:
3016:
3012:
3008:
3005:
3002:
2999:
2995:
2971:
2966:
2960:
2956:
2953:
2948:
2944:
2939:
2935:
2932:
2927:
2923:
2918:
2914:
2911:
2908:
2905:
2901:
2872:
2867:
2863:
2858:
2837:
2832:
2828:
2823:
2811:
2810:
2799:
2794:
2788:
2785:
2783:
2780:
2779:
2776:
2773:
2771:
2768:
2767:
2765:
2758:
2755:
2750:
2747:
2724:
2719:
2715:
2710:
2689:
2684:
2680:
2675:
2659:
2658:
2647:
2644:
2641:
2636:
2632:
2628:
2625:
2622:
2613:one, that is,
2607:
2596:
2591:
2587:
2583:
2580:
2565:
2554:
2543:
2539:
2535:
2532:
2529:
2526:
2522:
2518:
2515:
2495:
2492:
2488:
2463:
2460:
2446:
2443:
2440:
2418:
2415:
2412:
2407:
2404:
2401:
2397:
2392:
2388:
2363:
2360:
2356:
2330:
2327:
2324:
2319:
2316:
2313:
2309:
2304:
2300:
2296:
2293:
2271:
2265:
2236:
2232:
2211:
2207:
2202:
2197:
2194:
2191:
2186:
2183:
2180:
2176:
2171:
2167:
2162:
2156:
2152:
2147:
2143:
2140:
2137:
2133:
2130:
2127:
2122:
2118:
2115:
2110:
2106:
2101:
2096:
2093:
2090:
2086:
2083:
2080:
2077:
2074:
2052:
2046:
2021:
2017:
1996:
1974:
1968:
1962:
1957:
1951:
1928:
1925:
1921:
1891:
1887:
1865:
1860:
1856:
1852:
1830:
1825:
1821:
1817:
1812:
1807:
1803:
1799:
1793:
1789:
1783:
1779:
1754:
1750:
1745:
1741:
1737:
1732:
1727:
1723:
1719:
1713:
1709:
1703:
1699:
1695:
1692:
1668:
1664:
1659:
1654:
1649:
1645:
1641:
1636:
1631:
1627:
1623:
1617:
1613:
1607:
1603:
1598:
1592:
1588:
1583:
1579:
1576:
1573:
1569:
1564:
1560:
1556:
1550:
1546:
1541:
1536:
1532:
1528:
1522:
1518:
1512:
1508:
1504:
1501:
1498:
1495:
1492:
1470:
1466:
1442:
1412:
1408:
1387:
1382:
1378:
1373:
1329:
1324:
1321:
1317:
1313:
1289:
1286:
1282:
1261:
1258:
1254:
1226:
1217:
1213:
1209:
1205:
1200:
1196:
1192:
1191:
1186:
1182:
1178:
1174:
1170:
1166:
1165:
1162:
1158:
1154:
1145:
1141:
1137:
1133:
1129:
1125:
1124:
1119:
1115:
1111:
1107:
1103:
1099:
1098:
1095:
1091:
1088:
1083:
1080:
1076:
1072:
1048:
1045:
1041:
1020:
1017:
1013:
994:density matrix
974:
971:
955:
954:
943:
908:quantum states
892:density matrix
882:
881:
879:
878:
871:
864:
856:
853:
852:
848:
847:
842:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
642:
637:
632:
627:
622:
616:
615:
612:
611:
608:
607:
604:
603:
598:
593:
588:
586:Density matrix
583:
578:
573:
568:
563:
558:
552:
549:
548:
545:
544:
540:
539:
534:
529:
524:
519:
514:
509:
508:
507:
506:
505:
490:
485:
480:
475:
470:
464:
463:
458:
457:
454:
453:
449:
448:
443:
438:
433:
428:
422:
421:
418:
417:
414:
413:
409:
408:
403:
398:
393:
388:
383:
377:
376:
375:
369:
366:
365:
362:
361:
357:
356:
351:
346:
340:
339:
338:
337:
336:
334:Delayed-choice
329:Quantum eraser
324:
323:
318:
313:
308:
303:
298:
293:
288:
283:
278:
272:
271:
268:
267:
264:
263:
259:
258:
257:
256:
246:
241:
236:
231:
226:
221:
219:Quantum number
216:
211:
206:
201:
196:
191:
185:
184:
181:
180:
177:
176:
172:
171:
166:
160:
159:
158:
153:
148:
142:
139:
138:
135:
134:
133:
132:
127:
122:
114:
113:
108:
97:
94:
90:
83:
80:
74:
71:
68:
64:
57:
54:
50:
45:
42:
31:
30:
24:
23:
15:
9:
6:
4:
3:
2:
12109:
12098:
12095:
12093:
12090:
12088:
12085:
12083:
12080:
12079:
12077:
12062:
12054:
12053:
12050:
12044:
12041:
12039:
12036:
12034:
12031:
12027:
12024:
12023:
12022:
12019:
12018:
12016:
12012:
12006:
12003:
12001:
11998:
11994:
11991:
11990:
11989:
11986:
11984:
11981:
11979:
11976:
11974:
11971:
11970:
11968:
11964:
11958:
11955:
11953:
11950:
11948:
11945:
11943:
11940:
11938:
11935:
11933:
11930:
11928:
11925:
11923:
11920:
11918:
11915:
11913:
11910:
11908:
11905:
11903:
11900:
11898:
11897:Quantum logic
11895:
11893:
11890:
11888:
11885:
11883:
11880:
11878:
11875:
11873:
11870:
11868:
11865:
11863:
11860:
11856:
11853:
11852:
11851:
11848:
11846:
11843:
11841:
11838:
11836:
11833:
11829:
11826:
11825:
11824:
11821:
11819:
11816:
11814:
11811:
11809:
11806:
11805:
11803:
11801:
11797:
11791:
11788:
11786:
11783:
11781:
11778:
11776:
11773:
11771:
11768:
11766:
11763:
11761:
11758:
11756:
11753:
11751:
11750:Quantum chaos
11748:
11746:
11743:
11741:
11738:
11737:
11735:
11733:
11729:
11723:
11720:
11718:
11717:Stern–Gerlach
11715:
11713:
11710:
11708:
11705:
11703:
11700:
11698:
11695:
11693:
11690:
11688:
11685:
11683:
11680:
11678:
11675:
11673:
11670:
11669:
11667:
11663:
11657:
11654:
11652:
11651:Transactional
11649:
11647:
11644:
11642:
11641:Quantum logic
11639:
11637:
11634:
11632:
11629:
11623:
11620:
11619:
11618:
11615:
11614:
11613:
11610:
11608:
11605:
11603:
11600:
11598:
11595:
11593:
11590:
11588:
11585:
11584:
11582:
11580:
11576:
11570:
11567:
11565:
11562:
11560:
11557:
11555:
11552:
11550:
11547:
11546:
11544:
11540:
11534:
11531:
11529:
11526:
11524:
11521:
11519:
11516:
11514:
11511:
11509:
11506:
11504:
11501:
11500:
11498:
11494:
11486:
11483:
11481:
11478:
11477:
11476:
11475:Wave function
11473:
11471:
11468:
11466:
11463:
11461:
11458:
11456:
11453:
11451:
11450:Superposition
11448:
11446:
11445:Quantum state
11443:
11441:
11438:
11436:
11433:
11431:
11428:
11426:
11423:
11421:
11418:
11416:
11413:
11409:
11406:
11404:
11401:
11399:
11398:Excited state
11396:
11394:
11391:
11390:
11389:
11386:
11384:
11381:
11379:
11376:
11374:
11371:
11369:
11366:
11365:
11363:
11359:
11353:
11350:
11348:
11345:
11343:
11340:
11336:
11333:
11332:
11331:
11328:
11326:
11323:
11322:
11320:
11316:
11312:
11305:
11300:
11298:
11293:
11291:
11286:
11285:
11282:
11270:
11267:
11265:
11262:
11260:
11259:
11254:
11248:
11247:
11244:
11238:
11235:
11233:
11230:
11228:
11227:Pseudoinverse
11225:
11223:
11220:
11218:
11215:
11213:
11210:
11208:
11205:
11203:
11200:
11199:
11197:
11195:Related terms
11193:
11187:
11186:Z (chemistry)
11184:
11182:
11179:
11177:
11174:
11172:
11169:
11167:
11164:
11162:
11159:
11157:
11154:
11152:
11149:
11147:
11144:
11142:
11139:
11137:
11134:
11132:
11129:
11127:
11124:
11123:
11121:
11117:
11111:
11108:
11106:
11103:
11101:
11098:
11096:
11093:
11091:
11088:
11086:
11083:
11081:
11078:
11076:
11073:
11072:
11070:
11068:
11063:
11057:
11054:
11052:
11049:
11047:
11044:
11042:
11039:
11037:
11034:
11032:
11029:
11027:
11024:
11022:
11019:
11017:
11014:
11012:
11009:
11008:
11006:
11004:
10999:
10993:
10990:
10988:
10985:
10983:
10980:
10978:
10975:
10973:
10970:
10968:
10965:
10963:
10960:
10958:
10955:
10953:
10950:
10948:
10945:
10943:
10940:
10938:
10935:
10933:
10930:
10928:
10925:
10923:
10920:
10918:
10915:
10913:
10910:
10908:
10905:
10903:
10900:
10898:
10895:
10893:
10890:
10888:
10885:
10883:
10880:
10878:
10875:
10873:
10870:
10868:
10865:
10863:
10860:
10858:
10855:
10853:
10850:
10848:
10845:
10843:
10840:
10839:
10837:
10833:
10827:
10824:
10822:
10819:
10817:
10814:
10812:
10809:
10807:
10804:
10802:
10799:
10797:
10794:
10792:
10789:
10787:
10784:
10782:
10779:
10777:
10773:
10770:
10768:
10765:
10764:
10762:
10760:
10756:
10751:
10745:
10742:
10740:
10737:
10735:
10732:
10730:
10727:
10725:
10722:
10720:
10717:
10715:
10712:
10710:
10707:
10706:
10704:
10702:
10697:
10691:
10688:
10686:
10683:
10681:
10678:
10676:
10673:
10671:
10668:
10666:
10663:
10661:
10658:
10656:
10653:
10651:
10648:
10646:
10643:
10642:
10640:
10636:
10630:
10627:
10625:
10622:
10620:
10617:
10615:
10612:
10610:
10607:
10605:
10602:
10600:
10597:
10595:
10592:
10590:
10587:
10585:
10582:
10580:
10577:
10575:
10572:
10570:
10567:
10565:
10562:
10560:
10557:
10555:
10552:
10550:
10547:
10545:
10544:Pentadiagonal
10542:
10540:
10537:
10535:
10532:
10530:
10527:
10525:
10522:
10520:
10517:
10515:
10512:
10510:
10507:
10505:
10502:
10500:
10497:
10495:
10492:
10490:
10487:
10485:
10482:
10480:
10477:
10475:
10472:
10470:
10467:
10465:
10462:
10460:
10457:
10455:
10452:
10450:
10447:
10445:
10442:
10440:
10437:
10435:
10432:
10430:
10427:
10425:
10422:
10420:
10417:
10415:
10412:
10410:
10407:
10405:
10402:
10400:
10397:
10395:
10392:
10390:
10387:
10385:
10382:
10380:
10377:
10375:
10374:Anti-diagonal
10372:
10370:
10367:
10366:
10364:
10360:
10355:
10348:
10343:
10341:
10336:
10334:
10329:
10328:
10325:
10314:
10310:
10306:
10302:
10298:
10294:
10287:
10279:
10273:
10269:
10265:
10261:
10254:
10252:
10243:
10239:
10235:
10231:
10225:
10218:
10212:
10208:
10204:
10197:
10190:
10184:
10180:
10176:
10172:
10165:
10157:
10153:
10149:
10145:
10141:
10137:
10133:
10129:
10124:
10119:
10116:(260401): 6.
10115:
10111:
10104:
10096:
10092:
10088:
10084:
10080:
10076:
10072:
10068:
10063:
10058:
10055:(3): 033024.
10054:
10050:
10043:
10035:
10031:
10027:
10023:
10019:
10015:
10010:
10005:
10001:
9997:
9990:
9982:
9978:
9974:
9968:
9964:
9960:
9959:
9954:
9948:
9941:
9935:
9931:
9924:
9917:
9911:
9907:
9906:
9898:
9891:
9885:
9881:
9880:
9872:
9864:
9858:
9855:. MIT Press.
9854:
9853:
9848:
9842:
9834:
9830:
9826:
9822:
9818:
9814:
9810:
9806:
9801:
9796:
9792:
9788:
9781:
9773:
9769:
9765:
9761:
9757:
9753:
9748:
9743:
9739:
9735:
9734:
9729:
9723:
9715:
9711:
9707:
9703:
9699:
9695:
9691:
9687:
9682:
9677:
9673:
9669:
9668:
9663:
9657:
9649:
9645:
9640:
9635:
9631:
9627:
9626:
9621:
9617:
9611:
9604:
9598:
9594:
9590:
9586:
9582:
9576:
9568:
9564:
9560:
9556:
9552:
9548:
9543:
9538:
9534:
9530:
9529:
9524:
9517:
9513:
9509:
9505:
9501:
9497:
9496:
9491:
9485:
9477:
9473:
9469:
9465:
9461:
9457:
9453:
9449:
9448:
9443:
9436:
9428:
9424:
9420:
9416:
9412:
9408:
9404:
9400:
9395:
9390:
9387:(1): 95–102.
9386:
9382:
9381:
9373:
9365:
9361:
9357:
9353:
9349:
9343:
9339:
9335:
9330:
9325:
9321:
9314:
9312:
9303:
9299:
9295:
9289:
9285:
9284:
9279:
9273:
9271:
9269:
9260:
9256:
9249:
9241:
9235:
9231:
9227:
9226:
9218:
9216:
9214:
9212:
9210:
9208:
9206:
9197:
9191:
9187:
9183:
9179:
9172:
9170:
9168:
9159:
9155:
9151:
9149:3-540-42082-7
9145:
9141:
9137:
9131:
9123:
9119:
9115:
9111:
9107:
9103:
9096:
9094:
9085:
9079:
9075:
9068:
9064:
9053:
9050:
9048:
9045:
9043:
9040:
9038:
9035:
9033:
9030:
9028:
9025:
9024:
9017:
9014:
9012:
9008:
9005:
9001:
8997:
8993:
8983:
8979:
8977:
8973:
8969:
8965:
8962:
8957:
8955:
8951:
8947:
8943:
8938:
8936:
8932:
8928:
8924:
8921:are positive
8920:
8916:
8913:
8880:
8876:
8864:
8860:
8849:
8844:
8841:
8838:
8834:
8813:
8788:
8784:
8758:
8738:
8730:
8726:
8722:
8718:
8714:
8710:
8707:
8703:
8699:
8695:
8691:
8687:
8683:
8680:
8675:
8672:
8669:
8665:
8646:
8643:
8640:
8634:
8631:
8620:
8614:
8608:
8600:
8584:
8564:
8542:
8539:
8531:
8520:
8496:
8473:
8467:
8463:
8456:
8453:
8450:
8444:
8441:
8438:
8435:
8427:
8423:
8420:
8419:
8418:
8409:
8407:
8388:
8385:
8382:
8379:
8376:
8370:
8350:
8342:
8338:
8333:
8331:
8327:
8326:Moyal bracket
8305:
8302:
8299:
8270:
8267:
8264:
8258:
8235:
8223:
8220:
8217:
8211:
8208:
8202:
8199:
8196:
8193:
8190:
8184:
8175:
8172:
8166:
8155:
8152:
8149:
8146:
8143:
8137:
8124:
8123:
8122:
8120:
8101:
8098:
8095:
8085:
8081:
8078:
8075:
8072:
8068:
8061:
8058:
8055:
8049:
8043:
8040:
8037:
8029:
8025:
8011:
8007:
7997:
7993:
7967:
7953:
7950:
7947:
7941:
7934:
7933:
7932:
7930:
7926:
7922:
7916:
7892:
7887:
7879:
7873:
7867:
7861:
7855:
7849:
7846:
7840:
7834:
7827:
7826:
7825:
7808:
7802:
7779:
7770:
7766:
7763:
7760:
7756:
7749:
7743:
7734:
7730:
7727:
7724:
7721:
7717:
7713:
7707:
7701:
7694:
7693:
7692:
7689:
7687:
7668:
7657:
7653:
7634:
7614:
7590:
7583:
7565:
7561:
7558:
7554:
7550:
7547:
7541:
7538:
7516:
7512:
7500:
7493:
7492:
7491:
7489:
7485:
7480:
7478:
7459:
7450:
7447:
7444:
7438:
7432:
7429:
7421:
7418:
7406:
7399:
7398:
7397:
7395:
7391:
7387:
7381:
7371:
7357:
7349:
7345:
7328:
7325:
7303:
7300:
7296:
7293:
7273:
7250:
7245:
7241:
7237:
7232:
7228:
7222:
7218:
7214:
7210:
7207:
7199:
7198:
7197:
7182:
7179:
7158:
7149:
7135:
7113:
7109:
7085:
7077:
7073:
7066:
7061:
7057:
7051:
7047:
7043:
7035:
7031:
7024:
7021:
7015:
7009:
7002:
7001:
7000:
6984:
6980:
6971:
6953:
6949:
6925:
6920:
6916:
6910:
6906:
6900:
6896:
6892:
6889:
6882:
6881:
6880:
6864:
6860:
6836:
6830:
6827:
6824:
6821:
6815:
6812:
6809:
6806:
6801:
6797:
6793:
6790:
6785:
6781:
6775:
6771:
6767:
6764:
6761:
6754:
6753:
6752:
6738:
6717:
6714:
6709:
6705:
6701:
6680:
6677:
6672:
6668:
6642:
6638:
6604:
6600:
6588:
6584:
6573:
6569:
6563:
6559:
6554:
6551:
6543:
6527:
6507:
6499:
6495:
6479:
6459:
6452:
6442:
6440:
6436:
6432:
6414:
6410:
6386:
6381:
6377:
6373:
6368:
6364:
6358:
6354:
6350:
6346:
6343:
6334:
6333:
6332:
6330:
6327:when outcome
6307:
6301:
6297:
6293:
6289:
6285:
6282:
6275:
6271:
6267:
6262:
6258:
6251:
6247:
6243:
6239:
6231:
6230:
6229:
6213:
6209:
6200:
6196:
6178:
6174:
6150:
6145:
6141:
6135:
6131:
6125:
6121:
6117:
6114:
6107:
6106:
6105:
6091:
6084:Moreover, if
6082:
6062:
6059:
6053:
6050:
6047:
6041:
6031:
6030:
6029:
6012:
6004:
5996:
5990:
5984:
5973:
5957:
5934:
5928:
5925:
5919:
5916:
5913:
5909:
5905:
5895:
5891:
5879:
5875:
5864:
5860:
5854:
5850:
5845:
5841:
5838:
5835:
5831:
5827:
5817:
5813:
5801:
5797:
5787:
5783:
5780:
5775:
5771:
5765:
5761:
5757:
5749:
5745:
5736:
5726:
5722:
5713:
5709:
5703:
5699:
5695:
5689:
5679:
5678:
5677:
5675:
5671:
5652:
5642:
5638:
5626:
5622:
5611:
5607:
5601:
5597:
5593:
5590:
5583:
5582:
5581:
5565:
5561:
5534:
5530:
5515:
5499:
5485:
5471:
5446:
5442:
5416:
5388:
5384:
5375:
5367:
5363:
5350:
5346:
5338:
5334:
5330:
5312:
5311:
5310:
5309:we have that
5296:
5271:
5267:
5233:
5229:
5220:
5215:
5211:
5177:
5173:
5161:
5157:
5146:
5142:
5136:
5132:
5128:
5125:
5118:
5117:
5116:
5114:
5113:purifications
5109:
5091:
5086:
5082:
5078:
5070:
5066:
5058:
5055:
5051:
5045:
5041:
5037:
5033:
5028:
5024:
5020:
5012:
5008:
4998:
4997:
4996:
4974:
4970:
4961:
4956:
4952:
4940:
4924:
4921:
4918:
4913:
4909:
4888:
4860:
4856:
4847:
4842:
4838:
4823:
4813:
4811:
4793:
4787:
4772:
4759:
4748:
4726:
4707:
4702:
4696:
4691:
4684:
4679:
4673:
4666:
4663:
4658:
4627:
4624:
4619:
4588:
4585:
4580:
4549:
4546:
4541:
4510:
4507:
4502:
4499:
4492:
4491:
4490:
4367:
4363:
4334:
4331:
4315:
4307:
4303:
4299:
4295:
4233:
4215:
4209:
4189:
4105:
4041:
4023:
4017:
3997:
3978:
3957:
3953:
3949:
3933:
3930:
3925:
3915:
3907:
3902:
3892:
3854:
3851:
3835:
3827:
3826:superposition
3763:
3759:
3755:
3730:
3724:
3719:
3712:
3707:
3701:
3663:
3657:
3652:
3645:
3640:
3634:
3613:
3604:
3602:
3586:
3583:
3578:
3573:
3569:
3565:
3560:
3555:
3551:
3547:
3542:
3537:
3533:
3509:
3504:
3498:
3495:
3490:
3483:
3478:
3472:
3467:
3462:
3458:
3453:
3448:
3442:
3437:
3430:
3427:
3422:
3416:
3411:
3406:
3402:
3397:
3392:
3386:
3381:
3374:
3369:
3363:
3358:
3353:
3349:
3341:
3340:
3339:
3337:
3316:
3312:
3308:
3303:
3299:
3295:
3290:
3286:
3259:
3255:
3249:
3245:
3239:
3235:
3231:
3226:
3222:
3216:
3212:
3208:
3203:
3199:
3193:
3189:
3185:
3182:
3178:
3172:
3169:
3164:
3161:
3154:
3153:
3152:
3150:
3135:
3132:
3129:
3121:
3117:
3113:
3109:
3105:
3097:
3093:
3088:
3084:
3082:
3063:
3058:
3052:
3047:
3040:
3035:
3029:
3022:
3019:
3014:
3006:
2997:
2985:
2984:
2983:
2969:
2964:
2958:
2946:
2942:
2933:
2925:
2921:
2909:
2903:
2890:
2886:
2865:
2861:
2830:
2826:
2797:
2792:
2786:
2781:
2774:
2769:
2763:
2756:
2753:
2748:
2745:
2738:
2737:
2736:
2717:
2713:
2682:
2678:
2664:
2663:superposition
2645:
2642:
2634:
2630:
2623:
2620:
2612:
2608:
2594:
2589:
2585:
2581:
2578:
2570:
2566:
2563:
2559:
2555:
2541:
2533:
2524:
2516:
2513:
2490:
2477:
2476:outer product
2473:
2472:
2471:
2469:
2459:
2444:
2438:
2416:
2410:
2405:
2399:
2395:
2390:
2386:
2377:
2345:
2328:
2322:
2317:
2311:
2307:
2302:
2298:
2294:
2291:
2269:
2252:
2251:partial trace
2234:
2230:
2209:
2205:
2200:
2195:
2189:
2184:
2178:
2174:
2169:
2165:
2160:
2154:
2145:
2141:
2138:
2135:
2131:
2125:
2120:
2116:
2113:
2108:
2099:
2094:
2088:
2084:
2078:
2072:
2050:
2019:
1994:
1972:
1960:
1955:
1909:
1907:
1889:
1885:
1863:
1858:
1854:
1850:
1828:
1823:
1819:
1815:
1810:
1805:
1801:
1797:
1791:
1787:
1781:
1777:
1768:
1752:
1748:
1743:
1739:
1735:
1730:
1725:
1721:
1717:
1711:
1707:
1701:
1697:
1693:
1690:
1682:
1666:
1662:
1657:
1652:
1647:
1643:
1639:
1634:
1629:
1625:
1621:
1615:
1611:
1605:
1601:
1596:
1590:
1581:
1577:
1574:
1571:
1567:
1562:
1558:
1554:
1548:
1539:
1534:
1530:
1526:
1520:
1516:
1510:
1506:
1502:
1496:
1490:
1468:
1456:
1440:
1432:
1428:
1410:
1406:
1380:
1376:
1362:
1361:Hilbert space
1358:
1354:
1350:
1345:
1344:, see below.
1343:
1322:
1319:
1315:
1303:
1284:
1256:
1243:
1238:
1224:
1215:
1211:
1203:
1198:
1194:
1184:
1180:
1172:
1168:
1160:
1156:
1152:
1143:
1139:
1131:
1127:
1117:
1113:
1105:
1101:
1093:
1089:
1081:
1078:
1074:
1062:
1061:Hilbert space
1043:
1015:
1001:
999:
995:
991:
988:
984:
980:
970:
968:
964:
960:
952:
948:
944:
941:
940:
939:
937:
933:
929:
925:
924:wavefunctions
921:
917:
913:
912:probabilities
909:
905:
901:
897:
893:
889:
877:
872:
870:
865:
863:
858:
857:
855:
854:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
617:
610:
609:
602:
599:
597:
594:
592:
589:
587:
584:
582:
579:
577:
576:Quantum chaos
574:
572:
569:
567:
564:
562:
559:
557:
554:
553:
547:
546:
538:
535:
533:
532:Transactional
530:
528:
525:
523:
522:Quantum logic
520:
518:
515:
513:
510:
504:
501:
500:
499:
496:
495:
494:
491:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
465:
461:
456:
455:
447:
444:
442:
439:
437:
434:
432:
429:
427:
424:
423:
416:
415:
407:
404:
402:
399:
397:
394:
392:
389:
387:
384:
382:
379:
378:
374:
371:
370:
364:
363:
355:
352:
350:
347:
345:
342:
341:
335:
332:
331:
330:
327:
326:
322:
319:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
292:
289:
287:
284:
282:
279:
277:
274:
273:
266:
265:
255:
252:
251:
250:
249:Wave function
247:
245:
242:
240:
237:
235:
232:
230:
229:Superposition
227:
225:
222:
220:
217:
215:
212:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
186:
179:
178:
170:
167:
165:
162:
161:
157:
154:
152:
149:
147:
144:
143:
137:
136:
131:
128:
126:
123:
121:
118:
117:
116:
115:
111:
78:
72:
55:
52:
48:
40:
33:
32:
29:
26:
25:
21:
20:
11780:Quantum mind
11692:Franck–Hertz
11554:Klein–Gordon
11503:Formulations
11496:Formulations
11425:Interference
11415:Entanglement
11393:Ground state
11388:Energy level
11382:
11361:Fundamentals
11325:Introduction
11249:
11181:Substitution
11130:
11067:graph theory
10564:Quaternionic
10554:Persymmetric
10296:
10292:
10286:
10259:
10241:
10237:
10224:
10202:
10196:
10174:
10164:
10113:
10109:
10103:
10052:
10048:
10042:
9999:
9995:
9989:
9957:
9947:
9929:
9923:
9904:
9897:
9878:
9871:
9851:
9841:
9790:
9786:
9780:
9737:
9731:
9722:
9671:
9665:
9656:
9629:
9623:
9610:
9584:
9575:
9532:
9526:
9502:(5–8): 322.
9499:
9493:
9484:
9451:
9445:
9435:
9384:
9378:
9372:
9319:
9281:
9278:Peres, Asher
9254:
9248:
9224:
9177:
9139:
9130:
9108:(1): 74–93.
9105:
9101:
9073:
9067:
9015:
8989:
8980:
8975:
8971:
8967:
8963:
8958:
8953:
8945:
8941:
8939:
8934:
8926:
8914:
8908:
8729:Hartree–Fock
8721:uncorrelated
8415:
8334:
8250:
8116:
7923:. Under the
7918:
7794:
7690:
7685:
7655:
7651:
7605:
7481:
7474:
7393:
7389:
7384:Just as the
7383:
7343:
7286:, except if
7265:
7150:
7100:
6940:
6851:
6448:
6401:
6328:
6326:
6165:
6083:
6080:
5949:
5667:
5491:
5408:
5199:
5110:
5107:
4825:
4809:
4722:
4365:
4305:
4296:
4103:
3751:
3601:Bloch sphere
3525:Points with
3524:
3274:
3101:
3092:Bloch sphere
3078:
2884:
2812:
2660:
2465:
2249:denotes the
1910:
1680:
1483:is given by
1426:
1355:operator of
1346:
1242:real numbers
1239:
1002:
997:
993:
982:
976:
956:
936:mixed states
935:
931:
916:measurements
895:
891:
885:
585:
431:Klein–Gordon
367:Formulations
204:Energy level
199:Entanglement
182:Fundamentals
169:Interference
120:Introduction
12038:EPR paradox
11818:Quantum bus
11687:Double-slit
11665:Experiments
11631:Many-worlds
11569:Schrödinger
11533:Phase space
11523:Schrödinger
11513:Interaction
11470:Uncertainty
11440:Nonlocality
11435:Measurement
11430:Decoherence
11420:Hamiltonian
11156:Hamiltonian
11080:Biadjacency
11016:Correlation
10932:Householder
10882:Commutation
10619:Vandermonde
10614:Tridiagonal
10549:Permutation
10539:Nonnegative
10524:Matrix unit
10404:Bisymmetric
9662:Busch, Paul
9581:Busch, Paul
9004:phase-space
9000:Felix Bloch
8406:phase space
7921:phase space
7344:generalized
5674:measurement
5488:Measurement
4995:defined by
1453:when using
987:orthonormal
981:called the
951:decoherence
928:pure states
820:von Neumann
805:Schrödinger
581:EPR paradox
512:Many-worlds
446:Schrödinger
401:Schrödinger
396:Phase-space
386:Interaction
291:Double-slit
269:Experiments
244:Uncertainty
214:Nonlocality
209:Measurement
194:Decoherence
164:Hamiltonian
12097:Lev Landau
12076:Categories
11966:Extensions
11800:Technology
11646:Relational
11597:Copenhagen
11508:Heisenberg
11455:Tunnelling
11318:Background
11056:Transition
11051:Stochastic
11021:Covariance
11003:statistics
10982:Symplectic
10977:Similarity
10806:Unimodular
10801:Orthogonal
10786:Involutory
10781:Invertible
10776:Projection
10772:Idempotent
10714:Convergent
10609:Triangular
10559:Polynomial
10504:Hessenberg
10474:Equivalent
10469:Elementary
10449:Copositive
10439:Conference
10399:Bidiagonal
10123:1806.08171
10062:1509.03770
10009:1911.06282
9447:Erkenntnis
9286:. Kluwer.
8996:Lev Landau
8912:C*-algebra
8668:Fock space
8330:commutator
7925:Wigner map
7684:comes out
7477:commutator
7378:See also:
6544:such that
6199:eigenspace
5514:observable
4901:such that
4362:wave plate
4230:through a
3104:convex set
2571:, that is
2569:idempotent
2558:projection
1455:projectors
815:Sommerfeld
730:Heisenberg
725:Gutzwiller
665:de Broglie
613:Scientists
527:Relational
478:Copenhagen
381:Heisenberg
239:Tunnelling
140:Background
11672:Bell test
11542:Equations
11368:Born rule
11237:Wronskian
11161:Irregular
11151:Gell-Mann
11100:Laplacian
11095:Incidence
11075:Adjacency
11046:Precision
11011:Centering
10917:Generator
10887:Confusion
10872:Circulant
10852:Augmented
10811:Unipotent
10791:Nilpotent
10767:Congruent
10744:Stieltjes
10719:Defective
10709:Companion
10680:Redheffer
10599:Symmetric
10594:Sylvester
10569:Signature
10499:Hermitian
10479:Frobenius
10389:Arrowhead
10369:Alternant
10313:128081459
10244:: 245–272
10087:1367-2630
10034:208006050
9981:860391091
9800:0705.0181
9567:119103479
9476:119980948
9468:1572-8420
9419:0894-9875
9356:973404322
9329:1106.1445
9302:901395752
9261:, London.
9158:318268606
8877:ψ
8873:⟨
8870:⟩
8861:ψ
8835:∑
8794:⟩
8785:ψ
8644:β
8641:−
8635:
8615:β
8585:ρ
8540:−
8497:β
8474:β
8454:β
8451:−
8445:
8436:ρ
8351:ℏ
8306:⋅
8300:⋅
8176:−
8164:∂
8135:∂
8090:ℏ
8059:−
8050:ψ
8030:∗
8026:ψ
8020:∞
8015:∞
8012:−
8008:∫
8001:ℏ
7998:π
7888:†
7862:ρ
7835:ρ
7775:ℏ
7744:ρ
7739:ℏ
7722:−
7702:ρ
7672:⟩
7666:⟨
7551:−
7539:
7513:
7504:ℏ
7451:ρ
7430:
7422:ρ
7419:
7410:ℏ
7358:ρ
7326:ρ
7301:ρ
7294:ρ
7274:ρ
7238:ρ
7219:∑
7208:ρ
7180:ρ
7159:ρ
7136:ρ
7110:ρ
7074:ρ
7048:∑
7016:ρ
6950:ρ
6917:ρ
6897:∑
6890:ρ
6861:ρ
6831:ρ
6828:
6822:ρ
6816:
6810:−
6798:λ
6794:
6782:λ
6772:∑
6768:−
6739:ρ
6706:λ
6702:∑
6678:≥
6669:λ
6648:⟩
6639:φ
6601:φ
6597:⟨
6594:⟩
6585:φ
6570:λ
6560:∑
6552:ρ
6528:ρ
6508:ρ
6498:logarithm
6480:ρ
6374:ρ
6355:∑
6344:ρ
6294:ρ
6286:
6268:ρ
6240:ρ
6197:into the
6122:∑
6060:ρ
6054:
6045:⟩
6039:⟨
6016:⟩
6013:ψ
5997:ψ
5994:⟨
5988:⟩
5982:⟨
5926:ρ
5920:
5892:ψ
5888:⟨
5885:⟩
5876:ψ
5851:∑
5842:
5814:ψ
5810:⟨
5807:⟩
5798:ψ
5784:
5762:∑
5755:⟩
5746:ψ
5723:ψ
5719:⟨
5700:∑
5693:⟩
5687:⟨
5639:ψ
5635:⟨
5632:⟩
5623:ψ
5598:∑
5591:ρ
5540:⟩
5531:ψ
5472:ρ
5452:⟩
5417:ρ
5394:⟩
5373:⟩
5364:ψ
5335:∑
5328:⟩
5325:Ψ
5277:⟩
5268:ψ
5239:⟩
5230:ψ
5174:ψ
5170:⟨
5167:⟩
5158:ψ
5133:∑
5126:ρ
5083:ψ
5042:∑
5025:φ
4980:⟩
4971:φ
4914:†
4866:⟩
4857:ψ
4781:⟩
4757:⟩
4646:⟨
4643:⟩
4607:⟨
4604:⟩
4568:⟨
4565:⟩
4529:⟨
4526:⟩
4500:ρ
4476:⟩
4446:⟩
4416:⟩
4386:⟩
4348:⟩
4335:β
4329:⟩
4316:α
4282:⟩
4252:⟩
4203:⟩
4187:⟩
4154:⟩
4124:⟩
4090:⟩
4060:⟩
4011:⟩
3995:⟩
3976:⟩
3916:β
3893:α
3868:⟩
3855:β
3849:⟩
3836:α
3812:⟩
3782:⟩
3496:−
3459:σ
3428:−
3403:σ
3350:σ
3336:unit ball
3246:σ
3223:σ
3200:σ
3162:ρ
3133:×
3007:ψ
3004:⟨
3001:⟩
2998:ψ
2952:⟩
2943:ψ
2931:⟩
2922:ψ
2907:⟩
2904:ψ
2871:⟩
2862:ψ
2836:⟩
2827:ψ
2746:ρ
2723:⟩
2714:ψ
2688:⟩
2679:ψ
2631:ρ
2624:
2586:ρ
2579:ρ
2534:ψ
2531:⟨
2528:⟩
2525:ψ
2514:ρ
2494:⟩
2491:ψ
2442:Ψ
2414:Ψ
2403:Ψ
2396:
2362:⟩
2359:Ψ
2326:Ψ
2315:Ψ
2308:
2292:ρ
2193:Ψ
2182:Ψ
2175:
2151:Π
2142:
2129:Ψ
2114:⊗
2105:Π
2092:Ψ
2016:Π
1961:⊗
1927:⟩
1924:Ψ
1855:ψ
1820:ψ
1802:ψ
1778:∑
1740:ψ
1722:ψ
1698:∑
1691:ρ
1644:ψ
1626:ψ
1602:∑
1587:Π
1578:
1559:ψ
1545:Π
1531:ψ
1507:∑
1465:Π
1386:⟩
1377:ψ
1353:Hermitian
1316:ρ
1288:⟩
1260:⟩
1204:∗
1195:ρ
1181:ρ
1140:ρ
1128:ρ
1114:ρ
1102:ρ
1075:ρ
1047:⟩
1019:⟩
947:entangled
920:Born rule
845:Zeilinger
690:Ehrenfest
419:Equations
96:⟩
93:Ψ
82:^
70:⟩
67:Ψ
44:ℏ
12061:Category
11855:Timeline
11607:Ensemble
11587:Bayesian
11480:Collapse
11352:Glossary
11335:Timeline
11065:Used in
11001:Used in
10962:Rotation
10937:Jacobian
10897:Distance
10877:Cofactor
10862:Carleman
10842:Adjugate
10826:Weighing
10759:inverses
10755:products
10724:Definite
10655:Identity
10645:Exchange
10638:Constant
10604:Toeplitz
10489:Hadamard
10459:Diagonal
10232:(1927),
10173:(1963),
10156:51684413
10148:30636128
10095:88521187
10002:: 1–57.
9955:(2007).
9833:13251108
9825:18518167
9772:18132256
9706:14525351
9618:(1957).
9427:15995449
9280:(1995).
9138:(2001).
9020:See also
8717:molecule
8489:, where
7650:is some
7329:′
7304:′
7211:′
7183:′
6968:and the
6625:, where
6520:. Since
6347:′
6248:′
5970:denotes
5429:, where
5092:⟩
5034:⟩
4810:together
3952:circular
2556:it is a
2445:⟩
2411:⟨
2406:⟩
2323:⟨
2318:⟩
2190:⟨
2185:⟩
2132:⟩
2089:⟨
1864:⟩
1816:⟨
1811:⟩
1736:⟨
1731:⟩
1640:⟨
1635:⟩
1568:⟩
1527:⟨
1427:ensemble
770:Millikan
695:Einstein
680:Davisson
635:Blackett
620:Aharonov
488:Ensemble
468:Bayesian
373:Overview
254:Collapse
234:Symmetry
125:Glossary
12014:Related
11993:History
11732:Science
11564:Rydberg
11330:History
11166:Overlap
11131:Density
11090:Edmonds
10967:Seifert
10927:Hessian
10892:Coxeter
10816:Unitary
10734:Hurwitz
10665:Of ones
10650:Hilbert
10584:Skyline
10529:Metzler
10519:Logical
10514:Integer
10424:Boolean
10356:classes
10128:Bibcode
10067:Bibcode
10014:Bibcode
9805:Bibcode
9752:Bibcode
9714:2168715
9686:Bibcode
9648:0096113
9547:Bibcode
9504:Bibcode
9399:Bibcode
9364:2515538
9110:Bibcode
8986:History
8727:in the
8324:is the
6445:Entropy
6193:is the
5672:of the
3118:of the
3090:In the
2609:it has
1433:result
898:) is a
810:Simmons
800:Rydberg
765:Moseley
745:Kramers
735:Hilbert
720:Glauber
715:Feynman
700:Everett
670:Compton
441:Rydberg
130:History
11707:Popper
11085:Degree
11026:Design
10957:Random
10947:Payoff
10942:Moment
10867:Cartan
10857:BĂ©zout
10796:Normal
10670:Pascal
10660:Lehmer
10589:Sparse
10509:Hollow
10494:Hankel
10429:Cauchy
10354:Matrix
10311:
10274:
10213:
10185:
10154:
10146:
10093:
10085:
10032:
9979:
9969:
9936:
9912:
9886:
9859:
9831:
9823:
9770:
9712:
9704:
9646:
9599:
9565:
9474:
9466:
9425:
9417:
9362:
9354:
9344:
9300:
9290:
9236:
9192:
9156:
9146:
9080:
8919:states
8700:or an
8601:to be
8251:where
7988:
7961:
7606:where
7588:
7457:
6693:, and
6166:where
5950:where
5512:be an
3948:linear
3880:(with
3758:photon
3688:
3684:
3621:
3617:
2813:where
2611:purity
2567:it is
2222:where
900:matrix
840:Zeeman
835:Wigner
785:Planck
755:Landau
740:Jordan
391:Matrix
321:Popper
11617:Local
11559:Pauli
11549:Dirac
11146:Gamma
11110:Tutte
10972:Shear
10685:Shift
10675:Pauli
10624:Walsh
10534:Moore
10414:Block
10309:S2CID
10152:S2CID
10118:arXiv
10091:S2CID
10057:arXiv
10030:S2CID
10004:arXiv
9829:S2CID
9795:arXiv
9768:S2CID
9742:arXiv
9710:S2CID
9676:arXiv
9563:S2CID
9537:arXiv
9472:S2CID
9423:S2CID
9389:arXiv
9360:S2CID
9324:arXiv
6494:trace
6439:POVMs
5972:trace
3954:, or
3824:or a
3112:qubit
3096:qubit
1357:trace
1340:be a
990:basis
795:Raman
780:Pauli
775:Onnes
710:Fermi
685:Debye
675:Dirac
640:Bloch
630:Bethe
498:Local
436:Pauli
426:Dirac
224:State
10952:Pick
10922:Gram
10690:Zero
10394:Band
10272:ISBN
10211:ISBN
10183:ISBN
10144:PMID
10083:ISSN
9977:OCLC
9967:ISBN
9934:ISBN
9910:ISBN
9884:ISBN
9857:ISBN
9821:PMID
9702:PMID
9597:ISBN
9464:ISSN
9415:ISSN
9352:OCLC
9342:ISBN
9298:OCLC
9288:ISBN
9234:ISBN
9190:ISBN
9154:OCLC
9144:ISBN
9078:ISBN
8713:atom
8557:and
7348:POVM
6496:and
6449:The
5668:The
5492:Let
4827:let
4104:seem
3794:and
3338:and
2848:and
2564:one.
2562:rank
996:and
965:and
894:(or
890:, a
830:Wien
825:Weyl
790:Rabi
760:Laue
750:Lamb
705:Fock
660:Bose
655:Born
650:Bohr
645:Bohm
625:Bell
11041:Hat
10774:or
10757:or
10301:doi
10264:doi
10136:doi
10114:121
10075:doi
10022:doi
10000:831
9813:doi
9791:100
9760:doi
9694:doi
9634:doi
9589:doi
9555:doi
9512:doi
9500:443
9456:doi
9407:doi
9334:doi
9182:doi
9118:doi
8952:of
8925:on
8715:or
8632:exp
8442:exp
8339:of
6751:is
4937:(a
4366:can
4306:any
4264:or
3658:0.5
3641:0.5
2346:of
906:as
886:In
12078::
10307:.
10295:.
10270:.
10250:^
10240:,
10236:,
10209:,
10205:,
10181:,
10150:.
10142:.
10134:.
10126:.
10112:.
10089:.
10081:.
10073:.
10065:.
10053:18
10051:.
10028:.
10020:.
10012:.
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9827:.
9819:.
9811:.
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9789:.
9766:.
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9750:.
9738:34
9736:.
9708:.
9700:.
9692:.
9684:.
9672:91
9670:.
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9642:.
9628:.
9622:.
9595:,
9561:.
9553:.
9545:.
9533:15
9531:.
9510:.
9498:.
9470:.
9462:.
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9450:.
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9421:.
9413:.
9405:.
9397:.
9385:19
9383:.
9358:.
9350:.
9340:.
9332:.
9310:^
9296:.
9267:^
9257:.
9232:,
9228:,
9204:^
9188:.
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9152:.
9116:.
9106:29
9104:.
9092:^
8956:.
8704:.
8692:,
8688:,
8408:.
8363:,
8332:.
7931:,
7688:.
7479:.
7370:.
7148:.
6999::
6825:ln
6813:tr
6791:ln
6283:tr
6051:tr
5958:tr
5917:tr
5839:tr
5781:tr
3950:,
3151::
3083:.
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2621:tr
2458:.
2387:tr
2299:tr
2231:tr
2166:tr
2139:tr
1908:.
1575:tr
1351:,
1272:,
1199:01
1185:01
1144:11
1132:10
1118:01
1106:00
1031:,
969:.
961:,
11303:e
11296:t
11289:v
11171:S
10629:Z
10346:e
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10332:v
10315:.
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10297:6
10280:.
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10242:1
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10120::
10097:.
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9835:.
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9112::
9086:.
8976:H
8974:(
8972:K
8968:H
8966:(
8964:K
8954:A
8946:A
8942:A
8935:A
8927:A
8915:A
8900:.
8887:|
8881:i
8865:i
8856:|
8850:N
8845:1
8842:=
8839:i
8814:N
8789:i
8780:|
8759:N
8739:N
8670:.
8650:)
8647:H
8638:(
8628:r
8625:t
8621:=
8618:)
8612:(
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8565:H
8543:1
8536:)
8532:T
8526:B
8521:k
8517:(
8477:)
8471:(
8468:Z
8464:/
8460:)
8457:H
8448:(
8439:=
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8389:t
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8383:p
8380:,
8377:x
8374:(
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8312:}
8309:}
8303:,
8297:{
8294:{
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8271:p
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8265:x
8262:(
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8230:}
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8224:p
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8218:x
8215:(
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8182:{
8179:{
8173:=
8167:t
8159:)
8156:t
8153:,
8150:p
8147:,
8144:x
8141:(
8138:W
8102:.
8099:y
8096:d
8086:/
8082:y
8079:p
8076:i
8073:2
8069:e
8065:)
8062:y
8056:x
8053:(
8047:)
8044:y
8041:+
8038:x
8035:(
7994:1
7980:f
7977:e
7974:d
7968:=
7957:)
7954:p
7951:,
7948:x
7945:(
7942:W
7893:.
7884:)
7880:t
7877:(
7874:G
7871:)
7868:0
7865:(
7859:)
7856:t
7853:(
7850:G
7847:=
7844:)
7841:t
7838:(
7812:)
7809:t
7806:(
7803:G
7780:.
7771:/
7767:t
7764:H
7761:i
7757:e
7753:)
7750:0
7747:(
7735:/
7731:t
7728:H
7725:i
7718:e
7714:=
7711:)
7708:t
7705:(
7669:A
7638:)
7635:t
7632:(
7627:)
7623:H
7619:(
7615:A
7591:,
7584:]
7578:)
7574:H
7570:(
7566:A
7562:,
7559:H
7555:[
7548:=
7542:t
7536:d
7529:)
7525:H
7521:(
7517:A
7510:d
7501:i
7460:,
7454:]
7448:,
7445:H
7442:[
7439:=
7433:t
7427:d
7416:d
7407:i
7297:=
7251:,
7246:i
7242:P
7233:i
7229:P
7223:i
7215:=
7114:i
7086:.
7083:)
7078:i
7070:(
7067:S
7062:i
7058:p
7052:i
7044:+
7041:)
7036:i
7032:p
7028:(
7025:H
7022:=
7019:)
7013:(
7010:S
6985:i
6981:p
6954:i
6926:,
6921:i
6911:i
6907:p
6901:i
6893:=
6865:i
6837:.
6834:)
6819:(
6807:=
6802:i
6786:i
6776:i
6765:=
6762:S
6718:1
6715:=
6710:i
6681:0
6673:i
6643:i
6634:|
6611:|
6605:i
6589:i
6580:|
6574:i
6564:i
6555:=
6460:S
6415:i
6411:P
6387:.
6382:i
6378:P
6369:i
6365:P
6359:i
6351:=
6329:i
6308:]
6302:i
6298:P
6290:[
6276:i
6272:P
6263:i
6259:P
6252:=
6244:i
6214:i
6210:a
6179:i
6175:P
6151:,
6146:i
6142:P
6136:i
6132:a
6126:i
6118:=
6115:A
6092:A
6066:)
6063:A
6057:(
6048:=
6042:A
6009:|
6005:A
6001:|
5991:=
5985:A
5935:,
5932:)
5929:A
5923:(
5914:=
5910:)
5906:A
5902:|
5896:j
5880:j
5871:|
5865:j
5861:p
5855:j
5846:(
5836:=
5832:)
5828:A
5824:|
5818:j
5802:j
5793:|
5788:(
5776:j
5772:p
5766:j
5758:=
5750:j
5741:|
5737:A
5733:|
5727:j
5714:j
5710:p
5704:j
5696:=
5690:A
5653:.
5649:|
5643:j
5627:j
5618:|
5612:j
5608:p
5602:j
5594:=
5566:j
5562:p
5535:j
5526:|
5500:A
5447:j
5443:a
5438:|
5389:j
5385:a
5380:|
5376:U
5368:j
5359:|
5351:j
5347:p
5339:j
5331:=
5321:|
5297:U
5272:j
5263:|
5242:}
5234:j
5225:|
5221:,
5216:j
5212:p
5208:{
5184:|
5178:j
5162:j
5153:|
5147:j
5143:p
5137:j
5129:=
5087:j
5079:|
5071:j
5067:p
5059:j
5056:i
5052:U
5046:j
5038:=
5029:i
5021:|
5013:i
5009:q
4983:}
4975:i
4966:|
4962:,
4957:i
4953:q
4949:{
4925:I
4922:=
4919:U
4910:U
4889:U
4869:}
4861:j
4852:|
4848:,
4843:j
4839:p
4835:{
4794:2
4788:/
4784:)
4777:R
4773:,
4769:L
4764:|
4760:+
4753:L
4749:,
4745:R
4740:|
4736:(
4708:.
4703:)
4697:1
4692:0
4685:0
4680:1
4674:(
4667:2
4664:1
4659:=
4655:|
4650:V
4639:V
4634:|
4628:2
4625:1
4620:+
4616:|
4611:H
4600:H
4595:|
4589:2
4586:1
4581:=
4577:|
4572:L
4561:L
4556:|
4550:2
4547:1
4542:+
4538:|
4533:R
4522:R
4517:|
4511:2
4508:1
4503:=
4472:H
4467:|
4442:V
4437:|
4412:L
4407:|
4382:R
4377:|
4344:L
4339:|
4332:+
4325:R
4320:|
4278:L
4273:|
4248:R
4243:|
4216:2
4210:/
4206:)
4199:L
4194:|
4190:+
4183:R
4178:|
4174:(
4150:L
4145:|
4120:R
4115:|
4086:L
4081:|
4056:R
4051:|
4024:2
4018:/
4014:)
4007:L
4002:|
3998:+
3991:R
3986:|
3982:(
3979:=
3972:V
3967:|
3934:1
3931:=
3926:2
3921:|
3912:|
3908:+
3903:2
3898:|
3889:|
3864:L
3859:|
3852:+
3845:R
3840:|
3808:L
3803:|
3778:R
3773:|
3746:.
3731:]
3725:0
3720:0
3713:0
3708:1
3702:[
3679:.
3664:]
3653:0
3646:0
3635:[
3587:1
3584:=
3579:2
3574:z
3570:r
3566:+
3561:2
3556:y
3552:r
3548:+
3543:2
3538:x
3534:r
3510:.
3505:)
3499:1
3491:0
3484:0
3479:1
3473:(
3468:=
3463:z
3454:,
3449:)
3443:0
3438:i
3431:i
3423:0
3417:(
3412:=
3407:y
3398:,
3393:)
3387:0
3382:1
3375:1
3370:0
3364:(
3359:=
3354:x
3322:)
3317:z
3313:r
3309:,
3304:y
3300:r
3296:,
3291:x
3287:r
3283:(
3260:,
3256:)
3250:z
3240:z
3236:r
3232:+
3227:y
3217:y
3213:r
3209:+
3204:x
3194:x
3190:r
3186:+
3183:I
3179:(
3173:2
3170:1
3165:=
3136:2
3130:2
3064:.
3059:)
3053:1
3048:1
3041:1
3036:1
3030:(
3023:2
3020:1
3015:=
3011:|
2994:|
2970:,
2965:2
2959:/
2955:)
2947:2
2938:|
2934:+
2926:1
2917:|
2913:(
2910:=
2900:|
2866:2
2857:|
2831:1
2822:|
2798:,
2793:)
2787:1
2782:0
2775:0
2770:1
2764:(
2757:2
2754:1
2749:=
2718:2
2709:|
2683:1
2674:|
2643:=
2640:)
2635:2
2627:(
2595:.
2590:2
2582:=
2542:.
2538:|
2521:|
2517:=
2487:|
2439:|
2417:|
2400:|
2391:2
2355:|
2329:|
2312:|
2303:2
2295:=
2270:2
2264:H
2235:2
2210:,
2206:]
2201:)
2196:|
2179:|
2170:2
2161:(
2155:m
2146:[
2136:=
2126:|
2121:)
2117:I
2109:m
2100:(
2095:|
2085:=
2082:)
2079:m
2076:(
2073:p
2051:1
2045:H
2020:m
1995:m
1973:2
1967:H
1956:1
1950:H
1920:|
1890:j
1886:p
1859:j
1851:|
1829:|
1824:j
1806:j
1798:|
1792:j
1788:p
1782:j
1753:,
1749:|
1744:j
1726:j
1718:|
1712:j
1708:p
1702:j
1694:=
1667:,
1663:]
1658:)
1653:|
1648:j
1630:j
1622:|
1616:j
1612:p
1606:j
1597:(
1591:m
1582:[
1572:=
1563:j
1555:|
1549:m
1540:|
1535:j
1521:j
1517:p
1511:j
1503:=
1500:)
1497:m
1494:(
1491:p
1469:m
1441:m
1411:j
1407:p
1381:j
1372:|
1328:)
1323:j
1320:i
1312:(
1285:1
1281:|
1257:0
1253:|
1225:)
1216:1
1212:p
1173:0
1169:p
1161:(
1157:=
1153:)
1094:(
1090:=
1087:)
1082:j
1079:i
1071:(
1044:1
1040:|
1016:0
1012:|
875:e
868:t
861:v
89:|
79:H
73:=
63:|
56:t
53:d
49:d
41:i
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