1290:
941:
1047:
977:, and a referee. The referee asks Alice to fill in one row, and Bob one column, of a 3Ă3 table with plus and minus signs. Their answers must respect the following constraints: Alice's row must contain an even number of minus signs, Bob's column must contain an odd number of minus signs, and they both must assign the same sign to the cell where the row and column intersects. If they manage they win, otherwise they lose.
1285:{\displaystyle \left|\varphi \right\rangle ={\frac {1}{\sqrt {2}}}{\bigg (}\left|+\right\rangle _{a}\otimes \left|+\right\rangle _{b}+\left|-\right\rangle _{a}\otimes \left|-\right\rangle _{b}{\bigg )}\otimes {\frac {1}{\sqrt {2}}}{\bigg (}\left|+\right\rangle _{c}\otimes \left|+\right\rangle _{d}+\left|-\right\rangle _{c}\otimes \left|-\right\rangle _{d}{\bigg )}}
1010:
they must fill, each uses that information to select which measurements they should make to their particles. The result of the measurements will appear to each of them to be random (and the observed partial probability distribution of either particle will be independent of the measurement performed by the other party), so no real "communication" takes place.
2966:
1570:
1871:
The play proceeds by having each player make one measurement on their part of the entangled state per round of play. Each of Alice's measurements will give her the values for a row, and each of Bob's measurements will give him the values for a column. It is possible to do that because all observables
989:
It is easy to see that if Alice and Bob can come up with a classical strategy where they always win, they can represent it as a 3Ă3 table encoding their answers. But this is not possible, as the number of minus signs in this hypothetical table would need to be even and odd at the same time: every row
2260:
is chosen, then Alice receives bit 0, Bob receives bit 1, and Carol receives bit 1 from the referee. Based on the question bit received, Alice, Bob, and Carol each respond with an answer a, b, c also in the form of 0 or 1. The players can formulate a strategy together prior to the start of the game.
994:
Bob's columns. This implies that if the referee asks for a row and column whose intersection is one of the cells where their answers match they win, and otherwise they lose. Under the usual assumption that the referee asks for them uniformly at random, the best classical winning probability is 8/9.
993:
With a bit further analysis one can see that the best possible classical strategy can be represented by a table where each cell now contains both Alice and Bob's answers, that may differ. It is possible to make their answers equal in 8 out of 9 cells, while respecting the parity of Alice's rows and
2970:
Suppose that there is a classical strategy that satisfies all four winning conditions, all four conditions hold true. Through observation, each term appears twice on the left hand side. Hence, the left side sum = 0 mod 2. However, the right side sum = 1 mod 2. The contradiction shows that all four
1009:
This requires Alice and Bob to possess two pairs of particles with entangled states. These particles must have been prepared before the start of the game. One particle of each pair is held by Alice and the other by Bob, so they each have two particles. When Alice and Bob learn which column and row
1032:
entangled states (2N independent Bell pairs, see below) be shared in advance. This is because each round needs 2-bits of information to be measured (the third entry is determined by the first two, so measuring it isn't necessary), which destroys the entanglement. There is no way to reuse old
1579:
set of observables with eigenvalues +1 and â1, and with the product of the observables in each row being the identity operator, and the product of observables in each column equating to minus the identity operator. This is a so-called MerminâPeres magic square. It is shown in below table.
2008:
It has been demonstrated that the above-described game is the simplest two-player game of its type in which quantum pseudo-telepathy allows a win with probability one. Other games in which quantum pseudo-telepathy occurs have been studied, including larger magic square games,
1996:. As long as the table above is used, the measurement results are guaranteed to always multiply out to +1 for Alice along her row, and â1 for Bob down his column. Of course, each completely new round requires a new entangled state, as different rows and columns are
3230:
2726:
2028:
The
GreenbergerâHorneâZeilinger (GHZ) game is another interesting example of quantum pseudo-telepathy. Classically, the game has 75% winning probability. However, with a quantum strategy, the players will always win with winning probability equal to 1.
1863:
Effectively, while it is not possible to construct a 3Ă3 table with entries +1 and â1 such that the product of the elements in each row equals +1 and the product of elements in each column equals â1, it is possible to do so with the richer
1420:
1036:
The trick is for Alice and Bob to share an entangled quantum state and to use specific measurements on their components of the entangled state to derive the table entries. A suitable correlated state consists of a pair of entangled
1017:
of the results of the measurement such that if Alice and Bob choose their actions based on the results of their measurement, then there will exist a set of strategies and measurements allowing the game to be won with probability 1.
2561:
Classically, Alice, Bob, and Carol can employ a deterministic strategy that always end up with odd sum (e.g. Alice always output 1. Bob and Carol always output 0). The players win 75% of the time and only lose if the questions are
912:
refers to the fact that quantum pseudo-telepathy does not involve the exchange of information between any parties. Instead, quantum pseudo-telepathy removes the need for parties to exchange information in some circumstances.
3053:
117:
980:
Alice and Bob are allowed to elaborate a strategy together, but crucially are not allowed to communicate after they know which row and column they will need to fill in (as otherwise the game would be trivial).
944:
When attempting to construct a 3Ă3 table filled with the numbers +1 and â1, such that each row has an even number of negative entries and each column an odd number of negative entries, a conflict is bound to
1021:
Note that Alice and Bob could be light years apart from one another, and the entangled particles will still enable them to coordinate their actions sufficiently well to win the game with certainty.
2731:
2220:
990:
must contain an even number of minus signs, making the total number of minus signs even, and every column must contain an odd number of minus signs, making the total number of minus signs odd.
3113:
2312:
3108:
1349:
1321:
2493:
1872:
in a given row or column commute, so there exists a basis in which they can be measured simultaneously. For Alice's first row she needs to measure both her particles in the
2961:{\displaystyle {\begin{aligned}&a_{0}+b_{0}+c_{0}=0\mod 2\\&a_{1}+b_{1}+c_{0}=1\mod 2\\&a_{1}+b_{0}+c_{1}=1\mod 2\\&a_{0}+b_{1}+c_{1}=1\mod 2\end{aligned}}}
2721:
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2100:
2065:
1856:
1826:
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1764:
1734:
1704:
1672:
1642:
1612:
3232:. In both cases, the players give answer 0 if the result of the measurement is the first state of the pair, and answer 1 if the result is the second state of the pair.
1398:
2598:
2390:
2352:
2258:
2332:
1565:{\displaystyle X={\begin{bmatrix}0&1\\1&0\end{bmatrix}},Y={\begin{bmatrix}0&-i\\i&0\end{bmatrix}},Z={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}
2979:
Now we have come to the interesting part where Alice, Bob, and Carol decided to adopt a quantum strategy. The three of them now share a tripartite entangled state
2460:
2439:
2418:
1990:
1970:
1950:
1930:
1910:
1890:
2020:
In July 2022 a study reported the experimental demonstration of quantum pseudotelepathy via playing the nonlocal version of Mermin-Peres magic square game.
908:
is said to display quantum pseudo-telepathy if players who can use entanglement can win it with certainty while players without it can not. The prefix
4064:
924:. However, quantum pseudo-telepathy is a real-world phenomenon which can be verified experimentally. It is thus an especially striking example of an
2982:
1912:
basis, and for the third row she needs to measure them in an entangled basis. For Bob's first column he needs to measure his first particle in the
47:
3871:
Cameron, Peter J.; Montanaro, Ashley; Newman, Michael W.; Severini, Simone; Winter, Andreas (2007). "On the quantum chromatic number of a graph".
3805:
Kunkri, Samir; Kar, Guruprasad; Ghosh, Sibasish; Roy, Anirban (2006). "Winning strategies for pseudo-telepathy games using single non-local box".
3826:
Avis, D.; Hasegawa, Jun; Kikuchi, Yosuke; Sasaki, Yuuya (2006). "A Quantum
Protocol to Win the Graph Colouring Game on All Hadamard Graphs".
383:
3731:
Brassard, Gilles; Broadbent, Anne; Tapp, Alain (2005-06-16). "Recasting Mermin's multi-player game into the framework of pseudo-telepathy".
884:
3916:
Brassard, Gilles; Broadbent, Anne; Tapp, Alain (2005). "Recasting Mermin's multi-player game into the framework of pseudo-telepathy".
3352:
Brassard, Gilles; Cleve, Richard; Tapp, Alain (1999). "Cost of
Exactly Simulating Quantum Entanglement with Classical Communication".
135:
16:
This article uses quantum theoretical concepts and terminology. For a generally accessible introduction to quantum mechanics, see
3961:
Xu, Jia-Min; Zhen, Yi-Zheng; Yang, Yu-Xiang; Cheng, Zi-Mo; Ren, Zhi-Cheng; Chen, Kai; Wang, Xi-Lin; Wang, Hui-Tian (2022-07-26).
591:
1575:
Products of these Pauli spin operators can be used to fill the 3Ă3 table such that each row and each column contains a mutually
1364:
with eigenvalues +1 and â1, respectively, whilst the subscripts a, b, c, and d identify the components of each Bell state, with
364:
3336:
547:
2603:
In fact, this is the best winning strategy classically. We can only satisfy a maximum of 3 out of 4 winning conditions. Let
4054:
470:
3286:
3225:{\textstyle \left\{{\frac {1}{\sqrt {2}}}(|0\rangle +i|1\rangle ),{\frac {1}{\sqrt {2}}}(|0\rangle -i|1\rangle )\right\}}
2109:
23:
This article is about quantum mechanics' application in game theory. For quantum mechanics-related pseudoscience, see
2267:
130:
17:
4029:
219:
877:
326:
306:
174:
3760:
Gisin, N.; Methot, A. A.; Scarani, V. (2007). "Pseudo-telepathy: Input cardinality and Bell-type inequalities".
3461:
3307:
Brassard, Gilles; Broadbent, Anne; Tapp, Alain (2003). Dehne, Frank; Sack, Jörg-RĂŒdiger; Smid, Michiel (eds.).
1014:
566:
296:
606:
344:
244:
953:
game demonstrating nonclassical correlations was introduced by P.K. Aravind based on a series of papers by
4059:
4049:
3065:
1992:
basis, and for his third column he needs to measure both his particles in a different entangled basis, the
1326:
1298:
542:
537:
508:
359:
140:
3271:
576:
321:
311:
3266:
2723:
be Carol's response to question 0, 1. We can write all constraints that satisfy winning conditions as
2466:
2032:
There are three players, Alice, Bob, and Carol playing against a referee. The referee poses a question
870:
522:
493:
3667:
527:
488:
441:
416:
339:
199:
3602:
3437:
2354:
indicates summation of answers modulo 2. In other words, the sum of three answers has to be even if
611:
2686:
2646:
2606:
2070:
2035:
1832:
1802:
1772:
1740:
1710:
1680:
1648:
1618:
1588:
780:
498:
406:
3563:
456:
354:
120:
3432:
785:
503:
331:
301:
264:
1383:
411:
3250:
2565:
2357:
2337:
2225:
925:
254:
239:
1002:
Use of quantum pseudo-telepathy would enable Alice and Bob to win the game 100% of the time
4069:
3984:
3935:
3845:
3689:
3624:
3575:
3520:
3473:
3424:
3371:
3281:
2317:
1865:
901:
571:
483:
209:
166:
3962:
815:
8:
3245:
3235:
It is easy to check that with this strategy the players win the game with probability 1.
2010:
670:
478:
396:
224:
204:
156:
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3939:
3849:
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3428:
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4008:
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IEICE Transactions on
Fundamentals of Electronics, Communications and Computer Sciences
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3308:
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1895:
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391:
316:
249:
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4012:
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38:
24:
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However, the process of measuring the particles imposes sufficient structure on the
3996:
3992:
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3902:
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3853:
3779:
3697:
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3528:
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2014:
830:
820:
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710:
690:
675:
645:
513:
401:
3857:
3701:
1993:
3603:"Bell's theorem without inequalities and without probabilities for two observers"
3313:. Vol. 2748. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 1â11.
954:
929:
855:
725:
705:
451:
291:
3636:
3532:
3383:
3328:
3048:{\textstyle |{\psi }\rangle ={\frac {1}{\sqrt {2}}}(|000\rangle +|111\rangle )}
1411:
1401:
790:
750:
730:
700:
680:
630:
596:
446:
436:
229:
3783:
4043:
4034:
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974:
905:
850:
845:
775:
745:
715:
586:
532:
259:
234:
112:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
940:
4004:
3709:
3644:
3548:
3110:. If question 1 is received, the player makes a measurement in the Y basis
950:
840:
835:
770:
755:
720:
214:
965:. The game has been reformulated to demonstrate quantum pseudo-telepathy.
3930:
3885:
3840:
3811:
3774:
3745:
3684:
3619:
3419:
3366:
3319:
3261:
3062:
If question 0 is received, the player makes a measurement in the X basis
958:
805:
760:
695:
650:
1952:
basis, for second column he needs to measure his first particle in the
1407:
1352:
1038:
795:
765:
685:
660:
655:
640:
3947:
3446:
2067:
to each of the players. The three players each respond with an answer
3255:
3056:
286:
3485:
3979:
665:
3894:
3509:"Simple unified form for the major no-hidden-variables theorems"
3870:
2017:, and multiplayer games involving more than two participants.
961:
and AdĂĄn
Cabello that developed simplifying demonstrations of
3276:
2261:
However, no communication is allowed during the game itself.
1024:
Each round of this game uses up one entangled state. Playing
3405:
Aravind, P.K. (2004). "Quantum mysteries revisited again".
1892:
basis, for the second row she needs to measure them in the
3825:
3963:"Experimental Demonstration of Quantum Pseudotelepathy"
2971:
winning conditions cannot be simultaneously satisfied.
1410:
for these components can be written as products of the
3116:
3068:
2985:
2643:
be Alice's response to question 0 and 1 respectively,
1528:
1480:
1435:
904:
to eliminate the need for classical communications. A
4030:
Understanding and simulating quantum pseudo-telepathy
3915:
3730:
3668:"All versus nothing inseparability for two observers"
3306:
2729:
2689:
2649:
2609:
2568:
2469:
2448:
2427:
2406:
2360:
2340:
2320:
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2228:
2112:
2073:
2038:
1978:
1958:
1938:
1918:
1898:
1878:
1835:
1805:
1775:
1743:
1713:
1683:
1651:
1621:
1591:
1423:
1386:
1329:
1301:
1050:
50:
2215:{\displaystyle \{(0,0,0),(1,1,0),(1,0,1),(0,1,1)\}}
2023:
3759:
3224:
3102:
3047:
2960:
2715:
2675:
2635:
2592:
2487:
2454:
2433:
2412:
2384:
2346:
2326:
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2252:
2214:
2094:
2059:
1984:
1964:
1944:
1924:
1904:
1884:
1850:
1820:
1790:
1758:
1728:
1698:
1666:
1636:
1606:
1564:
1392:
1343:
1315:
1284:
973:This is a cooperative game featuring two players,
111:
3804:
3351:
1277:
1189:
1167:
1079:
4041:
920:to demonstrate the non-local characteristics of
916:Quantum pseudo-telepathy is generally used as a
3960:
2392:. Otherwise, the sum of answers has to be odd.
2307:{\displaystyle a\oplus b\oplus c=x\lor y\lor z}
997:
3564:"Incompatible results of quantum measurements"
3909:
878:
3864:
3762:International Journal of Quantum Information
3659:
3594:
3345:
3211:
3194:
3162:
3145:
3097:
3094:
3080:
3069:
3039:
3025:
2996:
2209:
2113:
2089:
2077:
2054:
2042:
106:
80:
3798:
3398:
1006:any communication once the game has begun.
3753:
885:
871:
3978:
3929:
3884:
3839:
3819:
3810:
3773:
3744:
3683:
3618:
3436:
3418:
3365:
3318:
2950:
2949:
2894:
2893:
2838:
2837:
2782:
2781:
2222:. As a clarification, if question triple
4065:Thought experiments in quantum mechanics
2683:be Bob's response to question 0, 1, and
939:
935:
3665:
3600:
3404:
2556:
4042:
3506:
3459:
3103:{\textstyle \{|+\rangle ,|-\rangle \}}
3561:
2013:giving rise to the notion of quantum
1344:{\displaystyle \left|-\right\rangle }
1316:{\displaystyle \left|+\right\rangle }
984:
2102:. The referee draws three questions
3918:Quantum Information and Computation
3873:Electronic Journal of Combinatorics
2974:
2003:
13:
417:Sum-over-histories (path integral)
103:
77:
33:Part of a series of articles about
14:
4081:
4023:
2488:{\displaystyle a\oplus b\oplus c}
1033:measurements from earlier games.
54:
18:Introduction to quantum mechanics
3258:â an entangled 3-particle state.
2024:GreenbergerâHorneâZeilinger game
3954:
3507:Mermin, N. David (1990-12-31).
3460:Mermin, N. David (1990-08-01).
3287:WheelerâFeynman absorber theory
2945:
2889:
2833:
2777:
3997:10.1103/PhysRevLett.129.050402
3724:
3562:Peres, Asher (December 1990).
3555:
3500:
3453:
3300:
3214:
3204:
3187:
3183:
3165:
3155:
3138:
3134:
3087:
3073:
3042:
3032:
3018:
3014:
2987:
2587:
2569:
2396:Winning condition of GHZ game
2247:
2229:
2206:
2188:
2182:
2164:
2158:
2140:
2134:
2116:
1015:joint probability distribution
567:Relativistic quantum mechanics
99:
92:
73:
1:
3702:10.1103/PhysRevLett.87.010403
3462:"Quantum mysteries revisited"
2106:uniformly from the 4 options
968:
607:Quantum statistical mechanics
3588:10.1016/0375-9601(90)90172-K
3310:Multi-party Pseudo-Telepathy
2000:compatible with each other.
1972:basis and the second in the
1932:basis and the second in the
998:Pseudo-telepathic strategies
7:
4055:Quantum information science
3858:10.1093/ietfec/e89-a.5.1378
3637:10.1103/PhysRevLett.86.1911
3533:10.1103/PhysRevLett.65.3373
3466:American Journal of Physics
3407:American Journal of Physics
3384:10.1103/PhysRevLett.83.1874
3329:10.1007/978-3-540-45078-8_1
3272:Quantum information science
3238:
2716:{\displaystyle c_{0},c_{1}}
2676:{\displaystyle b_{0},b_{1}}
2636:{\displaystyle a_{0},a_{1}}
2334:indicates OR condition and
2095:{\displaystyle \in \{0,1\}}
2060:{\displaystyle \in \{0,1\}}
1851:{\displaystyle +Y\otimes Y}
1821:{\displaystyle -Z\otimes X}
1791:{\displaystyle -X\otimes Z}
1759:{\displaystyle +X\otimes X}
1729:{\displaystyle +I\otimes X}
1699:{\displaystyle +X\otimes I}
1667:{\displaystyle +Z\otimes Z}
1637:{\displaystyle +Z\otimes I}
1607:{\displaystyle +I\otimes Z}
577:Quantum information science
10:
4086:
22:
15:
3784:10.1142/S021974990700289X
1380:going to Bob. The symbol
926:experimental confirmation
4035:Quantum Pseudo-Telepathy
3293:
1868:based on spin matrices.
1393:{\displaystyle \otimes }
898:Quantum pseudo-telepathy
612:Quantum machine learning
365:Wheeler's delayed-choice
3967:Physical Review Letters
3672:Physical Review Letters
3607:Physical Review Letters
3513:Physical Review Letters
3354:Physical Review Letters
2593:{\displaystyle (0,0,0)}
2385:{\displaystyle x=y=z=0}
2347:{\displaystyle \oplus }
2253:{\displaystyle (0,1,1)}
322:LeggettâGarg inequality
3267:KochenâSpecker theorem
3226:
3104:
3049:
2962:
2717:
2677:
2637:
2594:
2489:
2456:
2435:
2414:
2386:
2348:
2328:
2308:
2254:
2216:
2096:
2061:
1986:
1966:
1946:
1926:
1906:
1886:
1852:
1822:
1792:
1760:
1730:
1700:
1668:
1638:
1608:
1566:
1394:
1355:of the Pauli operator
1345:
1317:
1286:
946:
113:
3251:Quantum refereed game
3227:
3105:
3050:
2963:
2718:
2678:
2638:
2595:
2490:
2457:
2436:
2415:
2387:
2349:
2329:
2327:{\displaystyle \lor }
2309:
2255:
2217:
2097:
2062:
2011:graph colouring games
1987:
1967:
1947:
1927:
1907:
1887:
1853:
1823:
1793:
1761:
1731:
1701:
1669:
1639:
1609:
1567:
1395:
1346:
1318:
1287:
1028:rounds requires that
943:
936:The magic square game
900:describes the use of
307:Elitzur–Vaidman
297:Davisson–Germer
114:
3733:Quantum Info. Comput
3666:Cabello, A. (2001).
3601:Cabello, A. (2001).
3114:
3066:
2983:
2727:
2687:
2647:
2607:
2566:
2467:
2446:
2425:
2404:
2358:
2338:
2318:
2268:
2226:
2110:
2071:
2036:
1976:
1956:
1936:
1916:
1896:
1876:
1833:
1803:
1773:
1741:
1711:
1681:
1649:
1619:
1589:
1421:
1384:
1372:going to Alice, and
1327:
1299:
1048:
902:quantum entanglement
572:Quantum field theory
484:Consistent histories
121:Schrödinger equation
48:
4060:Quantum measurement
4050:Concepts in physics
3989:2022PhRvL.129e0402X
3940:2004quant.ph..8052B
3850:2006IEITF..89.1378A
3694:2001PhRvL..87a0403C
3629:2001PhRvL..86.1911C
3580:1990PhLA..151..107P
3525:1990PhRvL..65.3373M
3478:1990AmJPh..58..731M
3429:2004AmJPh..72.1303A
3376:1999PhRvL..83.1874B
3246:Quantum game theory
2397:
2264:The players win if
1866:algebraic structure
360:Stern–Gerlach
157:Classical mechanics
3222:
3100:
3045:
2958:
2956:
2713:
2673:
2633:
2590:
2557:Classical strategy
2485:
2452:
2431:
2410:
2395:
2382:
2344:
2324:
2304:
2250:
2212:
2092:
2057:
1982:
1962:
1942:
1922:
1902:
1882:
1848:
1818:
1788:
1756:
1726:
1696:
1664:
1634:
1604:
1562:
1556:
1508:
1460:
1390:
1341:
1313:
1282:
985:Classical strategy
947:
918:thought experiment
548:Von NeumannâWigner
528:Objective-collapse
327:Mach–Zehnder
317:Leggett inequality
312:Franck–Hertz
162:Old quantum theory
109:
3948:10.26421/QIC5.7-2
3613:(10): 1911â1914.
3568:Physics Letters A
3519:(27): 3373â3376.
3447:10.1119/1.1773173
3413:(10): 1303â1307.
3338:978-3-540-40545-0
3282:Tsirelson's bound
3181:
3180:
3132:
3131:
3012:
3011:
2554:
2553:
2455:{\displaystyle z}
2434:{\displaystyle y}
2413:{\displaystyle x}
1985:{\displaystyle X}
1965:{\displaystyle Z}
1945:{\displaystyle Z}
1925:{\displaystyle X}
1905:{\displaystyle X}
1885:{\displaystyle Z}
1861:
1860:
1185:
1184:
1075:
1074:
922:quantum mechanics
895:
894:
602:Scattering theory
582:Quantum computing
355:Schrödinger's cat
287:Bell's inequality
95:
70:
39:Quantum mechanics
25:Quantum mysticism
4077:
4017:
4016:
3982:
3958:
3952:
3951:
3933:
3931:quant-ph/0408052
3913:
3907:
3906:
3888:
3886:quant-ph/0608016
3868:
3862:
3861:
3843:
3841:quant-ph/0509047
3834:(5): 1378â1381.
3823:
3817:
3816:
3814:
3812:quant-ph/0602064
3802:
3796:
3795:
3777:
3775:quant-ph/0610175
3757:
3751:
3750:
3748:
3746:quant-ph/0408052
3728:
3722:
3721:
3687:
3685:quant-ph/0101108
3663:
3657:
3656:
3622:
3620:quant-ph/0008085
3598:
3592:
3591:
3574:(3â4): 107â108.
3559:
3553:
3552:
3504:
3498:
3497:
3457:
3451:
3450:
3440:
3422:
3420:quant-ph/0206070
3402:
3396:
3395:
3369:
3367:quant-ph/9901035
3360:(9): 1874â1877.
3349:
3343:
3342:
3322:
3320:quant-ph/0306042
3304:
3231:
3229:
3228:
3223:
3221:
3217:
3207:
3190:
3182:
3176:
3172:
3158:
3141:
3133:
3127:
3123:
3109:
3107:
3106:
3101:
3090:
3076:
3054:
3052:
3051:
3046:
3035:
3021:
3013:
3007:
3003:
2995:
2990:
2975:Quantum strategy
2967:
2965:
2964:
2959:
2957:
2938:
2937:
2925:
2924:
2912:
2911:
2901:
2882:
2881:
2869:
2868:
2856:
2855:
2845:
2826:
2825:
2813:
2812:
2800:
2799:
2789:
2770:
2769:
2757:
2756:
2744:
2743:
2733:
2722:
2720:
2719:
2714:
2712:
2711:
2699:
2698:
2682:
2680:
2679:
2674:
2672:
2671:
2659:
2658:
2642:
2640:
2639:
2634:
2632:
2631:
2619:
2618:
2599:
2597:
2596:
2591:
2494:
2492:
2491:
2486:
2461:
2459:
2458:
2453:
2440:
2438:
2437:
2432:
2419:
2417:
2416:
2411:
2398:
2394:
2391:
2389:
2388:
2383:
2353:
2351:
2350:
2345:
2333:
2331:
2330:
2325:
2313:
2311:
2310:
2305:
2259:
2257:
2256:
2251:
2221:
2219:
2218:
2213:
2101:
2099:
2098:
2093:
2066:
2064:
2063:
2058:
2015:chromatic number
2004:Current research
1991:
1989:
1988:
1983:
1971:
1969:
1968:
1963:
1951:
1949:
1948:
1943:
1931:
1929:
1928:
1923:
1911:
1909:
1908:
1903:
1891:
1889:
1888:
1883:
1857:
1855:
1854:
1849:
1827:
1825:
1824:
1819:
1797:
1795:
1794:
1789:
1765:
1763:
1762:
1757:
1735:
1733:
1732:
1727:
1705:
1703:
1702:
1697:
1673:
1671:
1670:
1665:
1643:
1641:
1640:
1635:
1613:
1611:
1610:
1605:
1583:
1582:
1571:
1569:
1568:
1563:
1561:
1560:
1513:
1512:
1465:
1464:
1399:
1397:
1396:
1391:
1350:
1348:
1347:
1342:
1340:
1322:
1320:
1319:
1314:
1312:
1291:
1289:
1288:
1283:
1281:
1280:
1274:
1273:
1268:
1253:
1252:
1247:
1232:
1231:
1226:
1211:
1210:
1205:
1193:
1192:
1186:
1180:
1176:
1171:
1170:
1164:
1163:
1158:
1143:
1142:
1137:
1122:
1121:
1116:
1101:
1100:
1095:
1083:
1082:
1076:
1070:
1066:
1061:
887:
880:
873:
514:Superdeterminism
167:Braâket notation
118:
116:
115:
110:
102:
97:
96:
88:
76:
71:
69:
58:
30:
29:
4085:
4084:
4080:
4079:
4078:
4076:
4075:
4074:
4040:
4039:
4026:
4021:
4020:
3959:
3955:
3914:
3910:
3869:
3865:
3824:
3820:
3803:
3799:
3758:
3754:
3729:
3725:
3664:
3660:
3599:
3595:
3560:
3556:
3505:
3501:
3486:10.1119/1.16503
3458:
3454:
3438:10.1.1.121.9157
3403:
3399:
3350:
3346:
3339:
3305:
3301:
3296:
3291:
3241:
3203:
3186:
3171:
3154:
3137:
3122:
3121:
3117:
3115:
3112:
3111:
3086:
3072:
3067:
3064:
3063:
3055:, known as the
3031:
3017:
3002:
2991:
2986:
2984:
2981:
2980:
2977:
2955:
2954:
2933:
2929:
2920:
2916:
2907:
2903:
2899:
2898:
2877:
2873:
2864:
2860:
2851:
2847:
2843:
2842:
2821:
2817:
2808:
2804:
2795:
2791:
2787:
2786:
2765:
2761:
2752:
2748:
2739:
2735:
2730:
2728:
2725:
2724:
2707:
2703:
2694:
2690:
2688:
2685:
2684:
2667:
2663:
2654:
2650:
2648:
2645:
2644:
2627:
2623:
2614:
2610:
2608:
2605:
2604:
2567:
2564:
2563:
2559:
2468:
2465:
2464:
2447:
2444:
2443:
2426:
2423:
2422:
2405:
2402:
2401:
2359:
2356:
2355:
2339:
2336:
2335:
2319:
2316:
2315:
2269:
2266:
2265:
2227:
2224:
2223:
2111:
2108:
2107:
2072:
2069:
2068:
2037:
2034:
2033:
2026:
2006:
1977:
1974:
1973:
1957:
1954:
1953:
1937:
1934:
1933:
1917:
1914:
1913:
1897:
1894:
1893:
1877:
1874:
1873:
1834:
1831:
1830:
1804:
1801:
1800:
1774:
1771:
1770:
1742:
1739:
1738:
1712:
1709:
1708:
1682:
1679:
1678:
1650:
1647:
1646:
1620:
1617:
1616:
1590:
1587:
1586:
1555:
1554:
1546:
1540:
1539:
1534:
1524:
1523:
1507:
1506:
1501:
1495:
1494:
1486:
1476:
1475:
1459:
1458:
1453:
1447:
1446:
1441:
1431:
1430:
1422:
1419:
1418:
1385:
1382:
1381:
1363:
1330:
1328:
1325:
1324:
1302:
1300:
1297:
1296:
1276:
1275:
1269:
1258:
1257:
1248:
1237:
1236:
1227:
1216:
1215:
1206:
1195:
1194:
1188:
1187:
1175:
1166:
1165:
1159:
1148:
1147:
1138:
1127:
1126:
1117:
1106:
1105:
1096:
1085:
1084:
1078:
1077:
1065:
1051:
1049:
1046:
1045:
1000:
987:
971:
955:N. David Mermin
938:
930:Bell inequality
891:
862:
861:
860:
625:
617:
616:
562:
561:Advanced topics
554:
553:
552:
504:Hidden-variable
494:de BroglieâBohm
473:
471:Interpretations
463:
462:
461:
431:
423:
422:
421:
379:
371:
370:
369:
336:
292:CHSH inequality
281:
273:
272:
271:
200:Complementarity
194:
186:
185:
184:
152:
123:
98:
87:
86:
72:
62:
57:
49:
46:
45:
28:
21:
12:
11:
5:
4083:
4073:
4072:
4067:
4062:
4057:
4052:
4038:
4037:
4032:
4025:
4024:External links
4022:
4019:
4018:
3953:
3924:(7): 538â550.
3908:
3863:
3818:
3797:
3768:(4): 525â534.
3752:
3739:(7): 538â550.
3723:
3658:
3593:
3554:
3499:
3472:(8): 731â734.
3452:
3397:
3344:
3337:
3298:
3297:
3295:
3292:
3290:
3289:
3284:
3279:
3274:
3269:
3264:
3259:
3253:
3248:
3242:
3240:
3237:
3220:
3216:
3213:
3210:
3206:
3202:
3199:
3196:
3193:
3189:
3185:
3179:
3175:
3170:
3167:
3164:
3161:
3157:
3153:
3150:
3147:
3144:
3140:
3136:
3130:
3126:
3120:
3099:
3096:
3093:
3089:
3085:
3082:
3079:
3075:
3071:
3044:
3041:
3038:
3034:
3030:
3027:
3024:
3020:
3016:
3010:
3006:
3001:
2998:
2994:
2989:
2976:
2973:
2953:
2948:
2944:
2941:
2936:
2932:
2928:
2923:
2919:
2915:
2910:
2906:
2902:
2900:
2897:
2892:
2888:
2885:
2880:
2876:
2872:
2867:
2863:
2859:
2854:
2850:
2846:
2844:
2841:
2836:
2832:
2829:
2824:
2820:
2816:
2811:
2807:
2803:
2798:
2794:
2790:
2788:
2785:
2780:
2776:
2773:
2768:
2764:
2760:
2755:
2751:
2747:
2742:
2738:
2734:
2732:
2710:
2706:
2702:
2697:
2693:
2670:
2666:
2662:
2657:
2653:
2630:
2626:
2622:
2617:
2613:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2558:
2555:
2552:
2551:
2548:
2545:
2542:
2538:
2537:
2534:
2531:
2528:
2524:
2523:
2520:
2517:
2514:
2510:
2509:
2506:
2503:
2500:
2496:
2495:
2484:
2481:
2478:
2475:
2472:
2462:
2451:
2441:
2430:
2420:
2409:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2343:
2323:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2091:
2088:
2085:
2082:
2079:
2076:
2056:
2053:
2050:
2047:
2044:
2041:
2025:
2022:
2005:
2002:
1981:
1961:
1941:
1921:
1901:
1881:
1859:
1858:
1847:
1844:
1841:
1838:
1828:
1817:
1814:
1811:
1808:
1798:
1787:
1784:
1781:
1778:
1767:
1766:
1755:
1752:
1749:
1746:
1736:
1725:
1722:
1719:
1716:
1706:
1695:
1692:
1689:
1686:
1675:
1674:
1663:
1660:
1657:
1654:
1644:
1633:
1630:
1627:
1624:
1614:
1603:
1600:
1597:
1594:
1573:
1572:
1559:
1553:
1550:
1547:
1545:
1542:
1541:
1538:
1535:
1533:
1530:
1529:
1527:
1522:
1519:
1516:
1511:
1505:
1502:
1500:
1497:
1496:
1493:
1490:
1487:
1485:
1482:
1481:
1479:
1474:
1471:
1468:
1463:
1457:
1454:
1452:
1449:
1448:
1445:
1442:
1440:
1437:
1436:
1434:
1429:
1426:
1412:Pauli matrices
1402:tensor product
1389:
1359:
1339:
1336:
1333:
1311:
1308:
1305:
1293:
1292:
1279:
1272:
1267:
1264:
1261:
1256:
1251:
1246:
1243:
1240:
1235:
1230:
1225:
1222:
1219:
1214:
1209:
1204:
1201:
1198:
1191:
1183:
1179:
1174:
1169:
1162:
1157:
1154:
1151:
1146:
1141:
1136:
1133:
1130:
1125:
1120:
1115:
1112:
1109:
1104:
1099:
1094:
1091:
1088:
1081:
1073:
1069:
1064:
1060:
1057:
1054:
999:
996:
986:
983:
970:
967:
963:Bell's theorem
937:
934:
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:
9:
6:
4:
3:
2:
4082:
4071:
4068:
4066:
4063:
4061:
4058:
4056:
4053:
4051:
4048:
4047:
4045:
4036:
4033:
4031:
4028:
4027:
4014:
4010:
4006:
4002:
3998:
3994:
3990:
3986:
3981:
3976:
3973:(5): 050402.
3972:
3968:
3964:
3957:
3949:
3945:
3941:
3937:
3932:
3927:
3923:
3919:
3912:
3904:
3900:
3896:
3892:
3887:
3882:
3878:
3874:
3867:
3859:
3855:
3851:
3847:
3842:
3837:
3833:
3829:
3822:
3813:
3808:
3801:
3793:
3789:
3785:
3781:
3776:
3771:
3767:
3763:
3756:
3747:
3742:
3738:
3734:
3727:
3719:
3715:
3711:
3707:
3703:
3699:
3695:
3691:
3686:
3681:
3678:(1): 010403.
3677:
3673:
3669:
3662:
3654:
3650:
3646:
3642:
3638:
3634:
3630:
3626:
3621:
3616:
3612:
3608:
3604:
3597:
3589:
3585:
3581:
3577:
3573:
3569:
3565:
3558:
3550:
3546:
3542:
3538:
3534:
3530:
3526:
3522:
3518:
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2086:
2083:
2080:
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2051:
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2045:
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2030:
2021:
2018:
2016:
2012:
2001:
1999:
1995:
1979:
1959:
1939:
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1869:
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1400:represents a
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1019:
1016:
1011:
1007:
1005:
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991:
982:
978:
976:
975:Alice and Bob
966:
964:
960:
956:
952:
942:
933:
931:
927:
923:
919:
914:
911:
907:
906:nonlocal game
903:
899:
888:
883:
881:
876:
874:
869:
868:
866:
865:
857:
854:
852:
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714:
712:
709:
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699:
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694:
692:
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679:
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672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
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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:
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443:
440:
438:
435:
434:
427:
426:
418:
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413:
410:
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389:
385:
382:
381:
375:
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366:
363:
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353:
352:
346:
343:
342:
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338:
337:
333:
330:
328:
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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:
37:
36:
32:
31:
26:
19:
3970:
3966:
3956:
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3917:
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3895:10.37236/999
3876:
3872:
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3831:
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3765:
3761:
3755:
3736:
3732:
3726:
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3661:
3610:
3606:
3596:
3571:
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3502:
3469:
3465:
3455:
3410:
3406:
3400:
3357:
3353:
3347:
3309:
3302:
3234:
3061:
2978:
2969:
2602:
2560:
2263:
2103:
2031:
2027:
2019:
2007:
1997:
1870:
1862:
1574:
1406:
1377:
1373:
1369:
1365:
1360:
1356:
1294:
1035:
1029:
1025:
1023:
1020:
1012:
1008:
1003:
1001:
992:
988:
979:
972:
951:magic square
948:
932:violations.
915:
909:
897:
896:
442:KleinâGordon
378:Formulations
215:Energy level
210:Entanglement
193:Fundamentals
180:Interference
131:Introduction
4070:Game theory
3262:EPR paradox
1408:Observables
1353:eigenstates
1039:Bell states
959:Asher Peres
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
4044:Categories
3980:2206.12042
1994:Bell basis
969:Game rules
826:Sommerfeld
741:Heisenberg
736:Gutzwiller
676:de Broglie
624:Scientists
538:Relational
489:Copenhagen
392:Heisenberg
250:Tunnelling
151:Background
4013:250048711
3653:119472501
3541:0031-9007
3494:0002-9505
3433:CiteSeerX
3256:GHZ state
3212:⟩
3198:−
3195:⟩
3163:⟩
3146:⟩
3095:⟩
3092:−
3081:⟩
3057:GHZ state
3040:⟩
3026:⟩
2997:⟩
2993:ψ
2480:⊕
2474:⊕
2342:⊕
2322:∨
2299:∨
2293:∨
2281:⊕
2275:⊕
2075:∈
2040:∈
1843:⊗
1813:⊗
1807:−
1783:⊗
1777:−
1751:⊗
1721:⊗
1691:⊗
1659:⊗
1629:⊗
1599:⊗
1577:commuting
1549:−
1489:−
1388:⊗
1335:−
1263:−
1255:⊗
1242:−
1213:⊗
1173:⊗
1153:−
1145:⊗
1132:−
1103:⊗
1056:φ
949:A simple
856:Zeilinger
701:Ehrenfest
430:Equations
107:⟩
104:Ψ
93:^
81:⟩
78:Ψ
55:ℏ
4005:35960591
3792:11386567
3718:18748483
3710:11461451
3645:11289818
3549:10042855
3239:See also
2550:1 mod 2
2536:1 mod 2
2522:1 mod 2
2508:0 mod 2
2314:, where
1338:⟩
1310:⟩
1266:⟩
1245:⟩
1224:⟩
1203:⟩
1156:⟩
1135:⟩
1114:⟩
1093:⟩
1059:⟩
781:Millikan
706:Einstein
691:Davisson
646:Blackett
631:Aharonov
499:Ensemble
479:Bayesian
384:Overview
265:Collapse
245:Symmetry
136:Glossary
3985:Bibcode
3936:Bibcode
3903:6320177
3846:Bibcode
3690:Bibcode
3625:Bibcode
3576:Bibcode
3521:Bibcode
3474:Bibcode
3425:Bibcode
3392:5837965
3372:Bibcode
2104:x, y, z
1004:without
945:emerge.
821:Simmons
811:Rydberg
776:Moseley
756:Kramers
746:Hilbert
731:Glauber
726:Feynman
711:Everett
681:Compton
452:Rydberg
141:History
4011:
4003:
3901:
3790:
3716:
3708:
3651:
3643:
3547:
3539:
3492:
3435:
3390:
3335:
910:pseudo
851:Zeeman
846:Wigner
796:Planck
766:Landau
751:Jordan
402:Matrix
332:Popper
4009:S2CID
3975:arXiv
3926:arXiv
3899:S2CID
3881:arXiv
3879:(1).
3836:arXiv
3807:arXiv
3788:S2CID
3770:arXiv
3741:arXiv
3714:S2CID
3680:arXiv
3649:S2CID
3615:arXiv
3415:arXiv
3388:S2CID
3362:arXiv
3315:arXiv
3294:Notes
3277:Qubit
1295:here
806:Raman
791:Pauli
786:Onnes
721:Fermi
696:Debye
686:Dirac
651:Bloch
641:Bethe
509:Local
447:Pauli
437:Dirac
235:State
4001:PMID
3706:PMID
3641:PMID
3545:PMID
3537:ISSN
3490:ISSN
3333:ISBN
1376:and
1368:and
1351:are
1323:and
957:and
841:Wien
836:Weyl
801:Rabi
771:Laue
761:Lamb
716:Fock
671:Bose
666:Born
661:Bohr
656:Bohm
636:Bell
3993:doi
3971:129
3944:doi
3891:doi
3854:doi
3780:doi
3698:doi
3633:doi
3584:doi
3572:151
3529:doi
3482:doi
3443:doi
3380:doi
3325:doi
3037:111
3023:000
2947:mod
2891:mod
2835:mod
2779:mod
1998:not
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