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1857:) has a determinate (sharp) value. The values of an observable will be obtained non-deterministically in accordance with a probability distribution that is uniquely determined by the system state. Note that the state is destroyed by measurement, so when we refer to a collection of values, each measured value in this collection must be obtained using a freshly prepared state.
1507:
2212:, experiments of chance, such as coin-tossing and dice-throwing, are deterministic, in the sense that, perfect knowledge of the initial conditions would render outcomes perfectly predictable. The ârandomnessâ stems from ignorance of physical information in the initial toss or throw. In diametrical contrast, in the case of
1860:
This indeterminacy might be regarded as a kind of essential incompleteness in our description of a physical system. Notice however, that the indeterminacy as stated above only applies to values of measurements not to the quantum state. For example, in the spin 1/2 example discussed above, the system
1334:
2251:
in a formal system of
Boolean propositions. In experiments measuring photon polarisation, Paterek et al. demonstrate statistics correlating predictable outcomes with logically dependent mathematical propositions, and random outcomes with propositions that are logically independent.
1056:
in measurement may lead to indeterminate outcomes. By the later half of the 18th century, measurement errors were well understood, and it was known that they could either be reduced by better equipment or accounted for by statistical error models. In quantum mechanics, however,
1819:
can be expressed in terms of other variables, for example, a particle with a definitely measured energy has a fundamental limit to how precisely one can specify how long it will have that energy. The magnitude involved in quantum uncertainty is on the order of the
1248:
2255:
In 2020, Steve
Faulkner reported on work following up on the findings of Tomasz Paterek et al.; showing what logical independence in the Paterek Boolean propositions means, in the domain of Matrix Mechanics proper. He showed how indeterminacy's
2063:
2175:
1768:
one accepted the formalism he was proposing. However, according to Bell, von
Neumann's formal proof did not justify his informal conclusion. A definitive but partial negative answer to 1) has been established by experiment: because
2204:
is the statistical manifestation of that indeterminacy, witnessable in results of experiments repeated many times. However, the relationship between quantum indeterminacy and randomness is subtle and can be considered differently.
1181:. It is immediate from this that measurement in general will be non-deterministic. Quantum mechanics, moreover, gives a recipe for computing a probability distribution Pr on the possible outcomes given the initial system state is
117:
1602:
1961:
are a more general kind of state obtained by a statistical mixture of pure states. For mixed states the "quantum recipe" for determining the probability distribution of a measurement is determined as follows:
1814:
Quantum indeterminacy can also be illustrated in terms of a particle with a definitely measured momentum for which there must be a fundamental limit to how precisely its location can be specified. This quantum
1502:{\displaystyle \sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}
1788:, which is distinct from the uncertainty principle). Still, in the most natural interpretation the answer is also no. To see this, consider two sequences of measurements: (A) that measures exclusively
2596:, American Journal of Physics, 1947. Reprinted in Readings in the Philosophy of Science, Ed. H. Feigl and M. Brodbeck, Appleton-Century-Crofts, 1953. Discusses measurement, accuracy and determinism.
1902:
showed that if quantum mechanics is correct, then the classical view of how the real world works (at least after special relativity) is no longer tenable. This view included the following two ideas:
1753:
Can the apparent indeterminacy be construed as in fact deterministic, but dependent upon quantities not modeled in the current theory, which would therefore be incomplete? More precisely, are there
1717:
2547:
Tomasz
Paterek, Johannes Kofler, Robert Prevedel, Peter Klimek, Markus Aspelmeyer, Anton Zeilinger, and Caslav Brukner, "Logical independence and quantum randomness â with experimental data",
2505:
Tomasz
Paterek, Johannes Kofler, Robert Prevedel, Peter Klimek, Markus Aspelmeyer, Anton Zeilinger, and Caslav Brukner, "Logical independence and quantum randomness â with experimental data",
1941:
direction was determined with certainty, whereas immediately before Alice's measurement Bob's outcome was only statistically determined. From this it follows that either value of spin in the
1188:
1281:
showing eigenvectors for Pauli Spin matrices. The Bloch sphere is a two-dimensional surface the points of which correspond to the state space of a spin 1/2 particle. At the state
1849:
Quantum indeterminacy is the assertion that the state of a system does not determine a unique collection of values for all its measurable properties. Indeed, according to the
1894:
believed that quantum state cannot be a complete description of a physical system and, it is commonly thought, never came to terms with quantum mechanics. In fact, Einstein,
1661:
2531:
Tomasz
Paterek, Johannes Kofler, Robert Prevedel, Peter Klimek, Markus Aspelmeyer, Anton Zeilinger, and Caslav Brukner, "Logical independence and quantum randomness",
2489:
Tomasz
Paterek, Johannes Kofler, Robert Prevedel, Peter Klimek, Markus Aspelmeyer, Anton Zeilinger, and Caslav Brukner, "Logical independence and quantum randomness",
1806:. The measurement outcomes of (A) are all +1, while the statistical distribution of the measurements (B) is still divided between +1, −1 with equal probability.
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continues to be an active research area in both theoretical and experimental physics. Possibly the first systematic attempt at a mathematical theory was developed by
2200:
Quantum indeterminacy is often understood as information (or lack of it) whose existence we infer, occurring in individual quantum systems, prior to measurement.
1906:
A measurable property of a physical system whose value can be predicted with certainty is actually an element of (local) reality (this was the terminology used by
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1991:
2244:. It refers to the null logical connectivity that exists between mathematical propositions (in the same language) that neither prove nor disprove one another.
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1049:. The distribution is uniquely determined by the system state, and moreover quantum mechanics provides a recipe for calculating this probability distribution.
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arises in evolved density operators representing mixed states, where measurement processes encounter irreversible 'lost history' and ingression of ambiguity.
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2405:
Gregg Jaeger, "Quantum randomness and unpredictability" Philosophical
Transactions of the Royal Society of London A doi/10.1002/prop.201600053 (2016)|Online=
884:
1317:(such as an electron) in which we only consider the spin degree of freedom. The corresponding Hilbert space is the two-dimensional complex Hilbert space
1325:(unique up to phase). In this case, the state space can be geometrically represented as the surface of a sphere, as shown in the figure on the right.
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The answer to 2) depends on how disturbance is understood, particularly since measurement entails disturbance (however note that this is the
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Indeterminacy in measurement was not an innovation of quantum mechanics, since it had been established early on by experimentalists that
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An adequate account of quantum indeterminacy requires a theory of measurement. Many theories have been proposed since the beginning of
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1081:. The kinds of measurements he investigated are now called projective measurements. That theory was based in turn on the theory of
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yields +1. By the von
Neumann (so-called) postulates, immediately after the measurement the system is assuredly in the state
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state. It was a conclusion of EPR, using the formal apparatus of quantum theory, that once Alice measured spin in the
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direction is not an element of reality or that the effect of Alice's measurement has infinite speed of propagation.
1929:, perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a
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can produce either +1, −1 each with probability 1/2. In fact, there is no state in which measurement of both
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Experimental realization of
EinsteinâPodolskyâRosenâBohm gedankenexperiment: A new violation of Bellâs inequalities
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2643:, Princeton University Press, 1955. Reprinted in paperback form. Originally published in German in 1932.
1243:{\displaystyle \operatorname {Pr} (\lambda )=\langle \operatorname {E} (\lambda )\psi \mid \psi \rangle }
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In the work of Paterek et al., the researchers demonstrate a link connecting quantum randomness and
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Von Neumann formulated the question 1) and provided an argument why the answer had to be no,
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2058:{\displaystyle \operatorname {E} _{A}(U)=\int _{U}\lambda \,d\operatorname {E} (\lambda ),}
1159:, then immediately after measurement the system will occupy a state that is an eigenvector
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2170:{\displaystyle \operatorname {D} _{A}(U)=\operatorname {Tr} (\operatorname {E} _{A}(U)S).}
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There are various questions that can be asked about the above indeterminacy assertion.
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is of a much more fundamental nature, having nothing to do with errors or disturbance.
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that could account for the statistical indeterminacy in a completely classical way?
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Can the indeterminacy be understood as a disturbance of the system being measured?
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Can quantum-mechanical description of physical reality be considered complete?
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112:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
1010:, that has become one of the characteristics of the standard description of
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2522:, Addison-Wesley Publishing Company Inc., Reading Massachusetts USA, 1948.
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This failure of the classical view was one of the conclusions of the EPR
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that uniquely determined all the values of its measurable properties, and
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1597:{\displaystyle {\frac {1}{\sqrt {2}}}(1,1),{\frac {1}{\sqrt {2}}}(1,-1)}
1029:, the values of its measurable properties uniquely determined the state.
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that had been recently developed (by von Neumann and independently by
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Experimental test of Bellâs inequalities using time-varying analyzers
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In this formulation, the state of a physical system corresponds to a
27:
Apparent lack of definite state before measurement of quantum systems
2220:, all indicate that quantum randomness does not stem from any such
1953:
We have described indeterminacy for a quantum system that is in a
1273:
1513:
and correspond to spin-measurements along the 3 coordinate axes.
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See especially part III "Misconceptions regarding measurement".
2407:
http://onlinelibrary.wiley.com/doi/10.1002/prop.201600053/epdf
2180:
This is a probability measure defined on the Borel subsets of
896:
1321:, with each quantum state corresponding to a unit vector in
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that is the probability distribution obtained by measuring
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In 2008, Tomasz Paterek et al. provided an explanation in
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The Pauli matrices all have the eigenvalues +1, −1.
2636:, W. A. Benjamin, 1963 (paperback reprint by Dover 2004).
1913:
Effects of local actions have a finite propagation speed.
1985:
is a projection-valued measure defined by the condition
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The NIST Reference on Constants, Units, and Uncertainty
1973:
is given by a densely defined self-adjoint operator on
1119:. An observable is represented by a self-adjoint (i.e.
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1258:) is the projection onto the space of eigenvectors of
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1994:
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will be the corresponding eigenvalue of the equation
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The problem of hidden variables in quantum mechanics
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are violated, any such hidden variable(s) cannot be
2472:Alain Aspect, Philippe Grangier, and GĂ©rard Roger,
2240:Logical independence is a well-known phenomenon in
1725:has the determinate value +1, while measurement of
1527:, these eigenvalues correspond to the eigenvectors
927:. Unsourced material may be challenged and removed.
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1712:{\displaystyle \psi ={\frac {1}{\sqrt {2}}}(1,1),}
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2661:Common Misconceptions Regarding Quantum Mechanics
2562:The Underlying Machinery of Quantum Indeterminacy
1969:be an observable of a quantum mechanical system.
1014:. Prior to quantum physics, it was thought that
2452:Alain Aspect, Jean Dalibard, and GĂ©rard Roger,
2365:Speakable and Unspeakable in Quantum Mechanics
1095:Hilbert space formulation of quantum mechanics
2641:Mathematical Foundations of Quantum Mechanics
2634:Mathematical Foundations of Quantum Mechanics
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8:
2581:Bell's inequality test: more ideal than ever
1876:that retains only those particles such that
1306:take the values +1, â1 with probability 1/2.
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2196:Logical independence and quantum randomness
2367:, Cambridge University Press, 2004, pg. 5.
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2610:A. Einstein, B. Podolsky, and N. Rosen,
2601:On the EinsteinâPoldolskyâRosen paradox
2520:An introduction to mathematical thought
2422:, Journal of Mathematics and Mechanics
2343:
1006:incompleteness in the description of a
54:
37:
2437:On the Einstein Podolsky Rosen paradox
1613:, they correspond to the eigenvectors
1310:In this example, we consider a single
7:
2378:"2022 CODATA Value: Planck constant"
2302:Interpretations of quantum mechanics
1937:direction, Bob's measurement in the
1019:a physical system had a determinate
925:adding citations to reliable sources
2650:, Princeton University Press, 1999.
2549:https://arxiv.org/pdf/0811.4542.pdf
2507:https://arxiv.org/pdf/0811.4542.pdf
2354:, Cambridge University Press, 1992.
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1996:
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417:Sum-over-histories (path integral)
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33:Part of a series of articles about
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1845:Indeterminacy and incompleteness
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2648:Understanding Quantum Mechanics
1810:Other examples of indeterminacy
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912:needs additional citations for
2538:(2010), no. 013019, 1367â2630.
2496:(2010), no. 013019, 1367â2630.
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1949:Indeterminacy for mixed states
1925:, now commonly referred to as
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1802:of a spin system in the state
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2350:V. Braginski and F. Khalili,
1861:can be prepared in the state
1292:are +1 whereas the values of
607:Quantum statistical mechanics
1155:. If the system is in state
2292:Counterfactual definiteness
1795:and (B) that measures only
1656:{\displaystyle (1,0),(0,1)}
577:Quantum information science
2698:
2476:, Physical Review Letters
2463:(1982), no. 25, 1804â1807.
2418:S Kochen and E P Specker,
1083:projection-valued measures
1041:on the set of outcomes of
2287:Complementarity (physics)
1746:have determinate values.
2518:Edward Russell Stabler,
2229:mathematical information
1865:by using measurement of
1167:and the observed value
1039:probability distribution
612:Quantum machine learning
365:Wheeler's delayed-choice
2076:. Given a mixed state
2068:for every Borel subset
936:"Quantum indeterminacy"
322:LeggettâGarg inequality
2533:New Journal of Physics
2491:New Journal of Physics
2458:Physical Revue Letters
2237:into quantum systems.
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1851:KochenâSpecker theorem
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1087:self-adjoint operators
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2480:(1982), no. 2, 91â94.
2331:Uncertainty principle
2311:Quantum contextuality
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1817:uncertainty principle
1779:Bell test experiments
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1185:. The probability is
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1000:Quantum indeterminacy
307:Elitzur–Vaidman
297:Davisson–Germer
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18:Quantum indeterminism
2352:Quantum Measurements
2316:Quantum entanglement
2249:logical independence
2234:logical independence
2222:physical information
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921:improve this article
572:Quantum field theory
484:Consistent histories
121:Schrödinger equation
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2594:The Logic of Quanta
2321:Quantum measurement
2080:, we introduce the
1771:Bell's inequalities
1330:Pauli spin matrices
1075:quantum measurement
360:Stern–Gerlach
157:Classical mechanics
2626:2006-02-08 at the
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2278:Physics portal
2265:
2262:
2258:indefiniteness
2197:
2194:
2178:
2177:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2143:
2139:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2109:
2105:
2066:
2065:
2054:
2051:
2048:
2045:
2042:
2039:
2036:
2032:
2027:
2023:
2019:
2016:
2013:
2010:
2007:
2002:
1998:
1950:
1947:
1915:
1914:
1911:
1896:Boris Podolsky
1880:
1869:
1846:
1843:
1811:
1808:
1799:
1792:
1762:
1761:
1758:
1743:
1736:
1729:
1722:
1708:
1705:
1702:
1699:
1696:
1693:
1687:
1683:
1678:
1675:
1664:
1663:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1610:
1604:
1593:
1590:
1587:
1584:
1581:
1578:
1572:
1568:
1563:
1560:
1557:
1554:
1551:
1548:
1542:
1538:
1524:
1496:
1490:
1487:
1484:
1482:
1479:
1478:
1475:
1472:
1470:
1467:
1466:
1464:
1459:
1454:
1450:
1445:
1440:
1434:
1431:
1429:
1426:
1425:
1422:
1419:
1416:
1414:
1411:
1410:
1408:
1403:
1398:
1394:
1389:
1384:
1378:
1375:
1373:
1370:
1369:
1366:
1363:
1361:
1358:
1357:
1355:
1350:
1345:
1341:
1303:
1296:
1289:
1285:the values of
1271:
1268:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1091:Marshall Stone
1066:
1063:
1031:
1030:
1024:
1016:
995:
994:
909:
907:
900:
893:
892:
890:
889:
882:
875:
867:
864:
863:
859:
858:
853:
848:
843:
838:
833:
828:
823:
818:
813:
808:
803:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
748:
743:
738:
733:
728:
723:
718:
713:
708:
703:
698:
693:
688:
683:
678:
673:
668:
663:
658:
653:
648:
643:
638:
633:
627:
626:
623:
622:
619:
618:
615:
614:
609:
604:
599:
597:Density matrix
594:
589:
584:
579:
574:
569:
563:
560:
559:
556:
555:
551:
550:
545:
540:
535:
530:
525:
520:
519:
518:
517:
516:
501:
496:
491:
486:
481:
475:
474:
469:
468:
465:
464:
460:
459:
454:
449:
444:
439:
433:
432:
429:
428:
425:
424:
420:
419:
414:
409:
404:
399:
394:
388:
387:
386:
380:
377:
376:
373:
372:
368:
367:
362:
357:
351:
350:
349:
348:
347:
345:Delayed-choice
340:Quantum eraser
335:
334:
329:
324:
319:
314:
309:
304:
299:
294:
289:
283:
282:
279:
278:
275:
274:
270:
269:
268:
267:
257:
252:
247:
242:
237:
232:
230:Quantum number
227:
222:
217:
212:
207:
202:
196:
195:
192:
191:
188:
187:
183:
182:
177:
171:
170:
169:
164:
159:
153:
150:
149:
146:
145:
144:
143:
138:
133:
125:
124:
119:
108:
105:
101:
94:
91:
85:
82:
79:
75:
68:
65:
61:
56:
53:
42:
41:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2694:
2683:
2680:
2678:
2675:
2674:
2672:
2662:
2659:
2658:
2654:
2649:
2645:
2642:
2638:
2635:
2631:
2629:
2625:
2622:
2619:
2615:
2614:
2609:
2606:
2602:
2598:
2595:
2592:G. Bergmann,
2591:
2589:
2586:
2582:
2578:
2577:
2573:
2566:
2563:
2557:
2554:
2550:
2544:
2541:
2537:
2534:
2528:
2525:
2521:
2515:
2512:
2508:
2502:
2499:
2495:
2492:
2486:
2483:
2479:
2475:
2469:
2466:
2462:
2459:
2455:
2449:
2446:
2442:
2438:
2432:
2429:
2425:
2421:
2415:
2412:
2408:
2402:
2399:
2387:
2383:
2379:
2373:
2370:
2366:
2360:
2357:
2353:
2347:
2344:
2337:
2332:
2329:
2327:
2324:
2322:
2319:
2317:
2314:
2312:
2309:
2307:
2303:
2300:
2298:
2295:
2293:
2290:
2288:
2285:
2284:
2279:
2273:
2268:
2263:
2261:
2259:
2253:
2250:
2245:
2243:
2238:
2236:
2235:
2230:
2225:
2223:
2219:
2215:
2211:
2206:
2203:
2195:
2193:
2191:
2187:
2183:
2164:
2158:
2152:
2146:
2141:
2130:
2127:
2124:
2118:
2112:
2107:
2095:
2094:
2093:
2091:
2087:
2083:
2079:
2075:
2071:
2052:
2046:
2040:
2034:
2030:
2025:
2021:
2017:
2011:
2005:
2000:
1988:
1987:
1986:
1984:
1980:
1976:
1972:
1968:
1963:
1960:
1956:
1948:
1946:
1944:
1940:
1936:
1932:
1928:
1927:Alice and Bob
1924:
1920:
1912:
1909:
1905:
1904:
1903:
1901:
1897:
1893:
1888:
1886:
1879:
1875:
1868:
1864:
1858:
1856:
1852:
1844:
1842:
1823:
1818:
1809:
1807:
1805:
1798:
1791:
1787:
1782:
1780:
1776:
1772:
1767:
1759:
1756:
1752:
1751:
1750:
1747:
1742:
1735:
1728:
1721:
1706:
1700:
1697:
1694:
1685:
1681:
1676:
1673:
1647:
1644:
1641:
1635:
1629:
1626:
1623:
1609:
1605:
1588:
1585:
1582:
1579:
1570:
1566:
1561:
1555:
1552:
1549:
1540:
1536:
1523:
1519:
1518:
1517:
1514:
1512:
1494:
1488:
1485:
1480:
1473:
1468:
1462:
1457:
1452:
1448:
1443:
1438:
1432:
1427:
1420:
1417:
1412:
1406:
1401:
1396:
1392:
1387:
1382:
1376:
1371:
1364:
1359:
1353:
1348:
1343:
1339:
1331:
1326:
1324:
1320:
1316:
1313:
1302:
1295:
1288:
1284:
1280:
1276:
1269:
1267:
1265:
1261:
1257:
1253:
1234:
1231:
1228:
1222:
1216:
1207:
1201:
1195:
1192:
1184:
1179:
1175:
1170:
1166:
1162:
1158:
1154:
1150:
1146:
1142:
1138:
1134:
1130:
1126:
1122:
1118:
1114:
1111:
1110:Hilbert space
1107:
1102:
1100:
1096:
1092:
1088:
1084:
1080:
1076:
1072:
1064:
1062:
1060:
1059:indeterminacy
1055:
1050:
1048:
1044:
1040:
1036:
1028:
1025:
1022:
1018:
1017:
1015:
1013:
1009:
1005:
1001:
991:
988:
980:
977:December 2008
969:
966:
962:
959:
955:
952:
948:
945:
941:
938: â
937:
933:
932:Find sources:
926:
922:
916:
915:
910:This article
908:
904:
899:
898:
888:
883:
881:
876:
874:
869:
868:
866:
865:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
629:
628:
621:
620:
613:
610:
608:
605:
603:
600:
598:
595:
593:
590:
588:
587:Quantum chaos
585:
583:
580:
578:
575:
573:
570:
568:
565:
564:
558:
557:
549:
546:
544:
543:Transactional
541:
539:
536:
534:
533:Quantum logic
531:
529:
526:
524:
521:
515:
512:
511:
510:
507:
506:
505:
502:
500:
497:
495:
492:
490:
487:
485:
482:
480:
477:
476:
472:
467:
466:
458:
455:
453:
450:
448:
445:
443:
440:
438:
435:
434:
427:
426:
418:
415:
413:
410:
408:
405:
403:
400:
398:
395:
393:
390:
389:
385:
382:
381:
375:
374:
366:
363:
361:
358:
356:
353:
352:
346:
343:
342:
341:
338:
337:
333:
330:
328:
325:
323:
320:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
285:
284:
277:
276:
266:
263:
262:
261:
260:Wave function
258:
256:
253:
251:
248:
246:
243:
241:
240:Superposition
238:
236:
233:
231:
228:
226:
223:
221:
218:
216:
213:
211:
208:
206:
203:
201:
198:
197:
190:
189:
181:
178:
176:
173:
172:
168:
165:
163:
160:
158:
155:
154:
148:
147:
142:
139:
137:
134:
132:
129:
128:
127:
126:
122:
89:
83:
66:
63:
59:
51:
44:
43:
40:
36:
32:
31:
19:
2647:
2640:
2633:
2620:777 (1935).
2617:
2612:
2604:
2600:
2593:
2587:189 (1999).
2584:
2580:
2561:
2556:
2543:
2535:
2532:
2527:
2519:
2514:
2501:
2493:
2490:
2485:
2477:
2473:
2468:
2460:
2457:
2453:
2448:
2440:
2436:
2431:
2423:
2419:
2414:
2401:
2390:. Retrieved
2381:
2372:
2364:
2359:
2351:
2346:
2257:
2254:
2248:
2246:
2239:
2232:
2228:
2226:
2221:
2218:Alain Aspect
2213:
2209:
2207:
2201:
2199:
2189:
2185:
2181:
2179:
2092:as follows:
2089:
2085:
2082:distribution
2081:
2077:
2073:
2069:
2067:
1982:
1974:
1970:
1966:
1964:
1959:Mixed states
1952:
1942:
1938:
1934:
1931:spin singlet
1916:
1900:Nathan Rosen
1889:
1884:
1877:
1873:
1866:
1862:
1859:
1848:
1836:10 Jâ
Hz
1813:
1803:
1796:
1789:
1783:
1774:
1765:
1763:
1754:
1748:
1740:
1733:
1726:
1719:
1665:
1607:
1521:
1515:
1511:self-adjoint
1327:
1322:
1318:
1309:
1300:
1293:
1286:
1282:
1279:Bloch sphere
1263:
1259:
1255:
1251:
1182:
1177:
1173:
1168:
1164:
1160:
1156:
1153:eigenvectors
1144:
1132:
1128:
1124:
1112:
1103:
1068:
1051:
1043:measurements
1034:
1033:
1003:
999:
998:
983:
974:
964:
957:
950:
943:
931:
919:Please help
914:verification
911:
442:KleinâGordon
378:Formulations
215:Energy level
210:Entanglement
193:Fundamentals
180:Interference
131:Introduction
2682:Determinism
2632:G. Mackey,
2616:Phys. Rev.
2607:195 (1964).
2599:J.S. Bell,
2579:A. Aspect,
2435:John Bell,
2363:J.S. Bell,
2297:EPR paradox
1855:observables
1137:dimensional
1123:) operator
1065:Measurement
831:von Neumann
816:Schrödinger
592:EPR paradox
523:Many-worlds
457:Schrödinger
412:Schrödinger
407:Phase-space
397:Interaction
302:Double-slit
280:Experiments
255:Uncertainty
225:Nonlocality
220:Measurement
205:Decoherence
175:Hamiltonian
2671:Categories
2646:R. OmnĂšs,
2603:, Physics
2574:References
2439:, Physics
2392:2024-05-18
2388:. May 2024
1955:pure state
1135:is finite
1099:Paul Dirac
1093:) and the
1047:observable
1027:conversely
947:newspapers
826:Sommerfeld
741:Heisenberg
736:Gutzwiller
676:de Broglie
624:Scientists
538:Relational
489:Copenhagen
392:Heisenberg
250:Tunnelling
151:Background
2583:, Nature
2147:
2131:
2113:
2047:λ
2041:
2031:λ
2022:∫
2006:
1923:observers
1890:However,
1674:ψ
1586:−
1486:−
1449:σ
1418:−
1393:σ
1340:σ
1238:⟩
1235:ψ
1232:∣
1229:ψ
1223:λ
1217:
1211:⟨
1202:λ
1196:
1139:, by the
1121:Hermitian
1115:over the
1004:necessary
856:Zeilinger
701:Ehrenfest
430:Equations
107:⟩
104:Ψ
93:^
81:⟩
78:Ψ
55:ℏ
2624:Archived
2564:(2020).
2264:See also
1315:particle
1312:spin 1/2
781:Millikan
706:Einstein
691:Davisson
646:Blackett
631:Aharonov
499:Ensemble
479:Bayesian
384:Overview
265:Collapse
245:Symmetry
136:Glossary
2551:(2010).
2509:(2010).
1977:. The
1839:
1270:Example
1147:has an
961:scholar
821:Simmons
811:Rydberg
776:Moseley
756:Kramers
746:Hilbert
731:Glauber
726:Feynman
711:Everett
681:Compton
452:Rydberg
141:History
2088:under
1874:filter
1250:where
1106:vector
1054:errors
1045:of an
963:
956:
949:
942:
934:
851:Zeeman
846:Wigner
796:Planck
766:Landau
751:Jordan
402:Matrix
332:Popper
2338:Notes
1872:as a
1826:6.626
1777:(see
1775:local
1131:. If
1021:state
968:JSTOR
954:books
806:Raman
791:Pauli
786:Onnes
721:Fermi
696:Debye
686:Dirac
651:Bloch
641:Bethe
509:Local
447:Pauli
437:Dirac
235:State
2386:NIST
1965:Let
1898:and
1739:and
1606:For
1520:For
1509:are
1328:The
1299:and
1085:for
1073:and
940:news
841:Wien
836:Weyl
801:Rabi
771:Laue
761:Lamb
716:Fock
671:Bose
666:Born
661:Bohr
656:Bohm
636:Bell
2585:398
2409:PDF
2208:In
2188:in
2084:of
2072:of
1981:of
1908:EPR
1841:).
1828:070
1781:).
1163:of
1151:of
1127:on
1101:).
923:by
2673::
2618:47
2536:12
2494:12
2478:49
2461:49
2456:,
2424:17
2384:.
2380:.
2304::
2224:.
2192:.
2128:Tr
1957:.
1910:).
1887:.
1831:15
1766:if
1266:.
1193:Pr
1178:λe
1176:=
1174:Ae
1143:,
2605:1
2441:1
2395:.
2190:S
2186:A
2182:R
2165:.
2162:)
2159:S
2156:)
2153:U
2150:(
2142:A
2138:E
2134:(
2125:=
2122:)
2119:U
2116:(
2108:A
2104:D
2090:S
2086:A
2078:S
2074:R
2070:U
2053:,
2050:)
2044:(
2038:E
2035:d
2026:U
2018:=
2015:)
2012:U
2009:(
2001:A
1997:E
1983:A
1975:H
1971:A
1967:A
1943:x
1939:x
1935:x
1885:Ï
1881:1
1878:Ï
1870:1
1867:Ï
1863:Ï
1834:Ă
1824:(
1804:Ï
1800:3
1797:Ï
1793:1
1790:Ï
1744:3
1741:Ï
1737:1
1734:Ï
1730:3
1727:Ï
1723:1
1720:Ï
1707:,
1704:)
1701:1
1698:,
1695:1
1692:(
1686:2
1682:1
1677:=
1651:)
1648:1
1645:,
1642:0
1639:(
1636:,
1633:)
1630:0
1627:,
1624:1
1621:(
1611:3
1608:Ï
1592:)
1589:1
1583:,
1580:1
1577:(
1571:2
1567:1
1562:,
1559:)
1556:1
1553:,
1550:1
1547:(
1541:2
1537:1
1525:1
1522:Ï
1495:)
1489:1
1481:0
1474:0
1469:1
1463:(
1458:=
1453:3
1444:,
1439:)
1433:0
1428:i
1421:i
1413:0
1407:(
1402:=
1397:2
1388:,
1383:)
1377:0
1372:1
1365:1
1360:0
1354:(
1349:=
1344:1
1323:C
1319:C
1304:3
1301:Ï
1297:2
1294:Ï
1290:1
1287:Ï
1283:Ï
1264:λ
1260:A
1256:λ
1254:(
1252:E
1226:)
1220:(
1214:E
1208:=
1205:)
1199:(
1183:Ï
1169:λ
1165:A
1161:e
1157:Ï
1145:A
1133:H
1129:H
1125:A
1113:H
990:)
984:(
979:)
975:(
965:·
958:·
951:·
944:·
917:.
886:e
879:t
872:v
100:|
90:H
84:=
74:|
67:t
64:d
60:d
52:i
20:)
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