2587:
5081:
2157:
5071:
2582:{\displaystyle \langle \phi |\psi \rangle \langle e|e\rangle \equiv \langle \phi |_{A}\langle e|_{B}|\psi \rangle _{A}|e\rangle _{B}=\langle \phi |_{A}\langle e|_{B}U^{\dagger }U|\psi \rangle _{A}|e\rangle _{B}=e^{-i(\alpha (\phi ,e)-\alpha (\psi ,e))}\langle \phi |_{A}\langle \phi |_{B}|\psi \rangle _{A}|\psi \rangle _{B}\equiv e^{-i(\alpha (\phi ,e)-\alpha (\psi ,e))}\langle \phi |\psi \rangle ^{2}.}
3271:: there is no universal cloning morphism for any non-trivial category of this kind. Although the theorem is inherent in the definition of this category, it is not trivial to see that this is so; the insight is important, as this category includes things that are not finite-dimensional Hilbert spaces, including the
3293:
to the combined system. If the unitary transformation is chosen correctly, several components of the combined system will evolve into approximate copies of the original system. In 1996, V. Buzek and M. Hillery showed that a universal cloning machine can make a clone of an unknown state with the
1029:
had proven the theorem 18 months prior to the published proof by
Wootters and Zurek in his referee report to said proposal (as evidenced by a letter from the editor). However, Juan Ortigoso pointed out in 2018 that a complete proof along with an interpretation in terms of the lack of simple
3052:, which says that it is impossible to convert a quantum state into a sequence of classical bits (even an infinite sequence of bits), copy those bits to some new location, and recreate a copy of the original quantum state in the new location. This should not be confused with
1987:
3063:, which states that quantum entanglement cannot be used to transmit classical information (whether superluminally, or slower). That is, cloning, together with entanglement, would allow such communication to occur. To see this, consider the
3288:
Even though it is impossible to make perfect copies of an unknown quantum state, it is possible to produce imperfect copies. This can be done by coupling a larger auxiliary system to the system that is to be cloned, and applying a
2997:," i.e. treated as a pure state of a larger system. Alternately, a different proof can be given that works directly with mixed states; in this case, the theorem is often known as the no-broadcast theorem. Similarly, an arbitrary
2699:
106:
1319:
1463:
2860:
1263:
3033:
cannot be created and used for correcting subsequent errors. Error correction is vital for practical quantum computing, and for some time it was unclear whether or not it was possible. In 1995,
2965:). In such a case the qubit can be represented by just two real numbers (one polar angle and one radius equal to 1), while the value of the third can be arbitrary in such a representation. Yet a
1863:
906:
authored by James Park, in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist (the same result would be independently derived in 1982 by
938:
without violating the no-cloning theorem as no well-defined state may be defined in terms of a subsystem of an entangled state. The no-cloning theorem (as generally understood) concerns only
2801:
2751:
1725:
2969:
of a qubit (polarisation-encoded photon, for example) is capable of storing the whole qubit information support within its "structure". Thus no single universal unitary evolution
1596:
2128:
2093:
1804:
1397:
1203:
1125:
3232:
3194:
3152:
3114:
2920:
2888:
1839:
1362:
1164:
1090:
2615:
1769:
1659:
2973:
can clone an arbitrary quantum state according to the no-cloning theorem. It would have to depend on the transformed qubit (initial) state and thus would not have been
2053:
The extra phase factor expresses the fact that a quantum-mechanical state defines a normalised vector in
Hilbert space only up to a phase factor i.e. as an element of
1483:
2620:
2007:
3074:
are distributed to Alice and Bob. Alice could send bits to Bob in the following way: If Alice wishes to transmit a "0", she measures the spin of her electron in the
2027:
1625:
2148:
2047:
1859:
3875:
Barnum, Howard; Caves, Carlton M.; Fuchs, Christopher A.; Jozsa, Richard; Schumacher, Benjamin (1996-04-08). "Noncommuting Mixed States Cannot Be
Broadcast".
3158:
direction. Bob will know that Alice has transmitted a "0" if all his measurements produce the same result; otherwise, his measurements will have outcomes
36:
4270:
3154:. To transmit "1", Alice does nothing to her qubit. Bob creates many copies of his electron's state, and measures the spin of each copy in the
1409:
4232:
1528:
372:
4962:
873:
4863:
4524:
2989:
and the proposed copier acts via unitary time evolution. These assumptions cause no loss of generality. If the state to be copied is a
4018:
Ahn, Doyeol; Myers, Casey; Ralph, Timothy; Mann, Robert (2013). "Quantum-state cloning in the presence of a closed timelike curve".
4425:
124:
1268:
4750:
580:
5110:
353:
2810:
1212:
3797:
3706:
3574:
2953:
operation is trivial (up to a finite precision) but the problem manifests if the qubit is unitarily transformed (e.g. by the
536:
5105:
5074:
4260:
459:
3056:, which does allow a quantum state to be destroyed in one location, and an exact copy to be recreated in another location.
5032:
3234:
with equal probability. This would allow Alice and Bob to communicate classical bits between each other (possibly across
2949:), that is three real numbers (two polar angles and one radius). Copying three numbers on a classical computer using any
4608:
1671:
The no-cloning theorem answers the following question in the negative: Is it possible to construct a unitary operator
931:
5084:
4972:
4225:
3770:
Ghirardi, GianCarlo (2013), "Entanglement, Nonlocality, Superluminal
Signaling and Cloning", in Bracken, Paul (ed.),
2756:
2706:
119:
4900:
4558:
1678:
1497:
208:
4895:
4623:
4603:
1520:
866:
315:
295:
163:
4890:
3369:
3272:
555:
285:
4402:
4923:
4745:
4648:
4309:
2994:
2804:
1982:{\displaystyle U(|\phi \rangle _{A}|e\rangle _{B})=e^{i\alpha (\phi ,e)}|\phi \rangle _{A}|\phi \rangle _{B}}
970:
595:
333:
233:
4928:
4796:
4387:
4218:
531:
526:
497:
348:
129:
1542:
4952:
4708:
4568:
4342:
4297:
4241:
3822:
565:
310:
300:
3722:
Herbert, Nick (1982). "FLASH—A superluminal communicator based upon a new kind of quantum measurement".
2098:
2063:
1774:
1367:
1173:
1095:
4824:
4696:
4593:
4469:
4397:
4304:
3261:
1022:
978:
859:
511:
482:
4633:
4598:
4494:
4437:
3199:
3161:
3119:
3081:
3068:
2954:
986:
516:
477:
430:
405:
328:
188:
3527:
3421:
3005:
and performing a suitable unitary evolution. Thus the no-cloning theorem holds in full generality.
2897:
2865:
1809:
1332:
1134:
4806:
4779:
4755:
4718:
4509:
4442:
4377:
4362:
4332:
4255:
4056:
3690:
3334:
3324:
3319:
3060:
3049:
3042:
1049:
1018:
1002:
600:
5115:
4957:
4691:
4583:
4553:
4352:
3637:
3268:
966:
894:
states that it is impossible to create an independent and identical copy of an arbitrary unknown
769:
487:
395:
2592:
1748:
1638:
445:
343:
109:
5027:
4791:
4784:
4531:
3724:
3522:
3416:
3399:
3290:
1501:
954:
774:
492:
320:
290:
253:
3936:
Kalev, Amir; Hen, Itay (2008-05-29). "No-Broadcasting
Theorem and Its Classical Counterpart".
3260:
and the other in the black hole interior. This leads to more radical interpretations, such as
1468:
400:
4947:
4573:
4499:
4464:
3374:
3364:
3249:
3053:
3013:
2990:
2942:
1992:
1727:, under which the state the system B is in always evolves into the state the system A is in,
943:
918:
the same year). The aforementioned theorems do not preclude the state of one system becoming
243:
228:
4737:
4486:
4337:
4159:
4078:
3955:
3894:
3841:
3733:
3656:
3514:
3466:
3408:
3349:
3329:
3306:
3302:
2012:
1603:
1206:
982:
947:
919:
560:
472:
198:
155:
804:
8:
5056:
5009:
4839:
4613:
4367:
4347:
4282:
4277:
3359:
3339:
958:
927:
659:
467:
385:
213:
193:
145:
4163:
4090:
4082:
3959:
3898:
3845:
3737:
3660:
3518:
3470:
3412:
2985:
In the statement of the theorem, two assumptions were made: the state to be copied is a
4772:
4618:
4420:
4183:
4149:
4124:
4102:
4068:
4027:
3998:
3979:
3945:
3918:
3884:
3857:
3831:
3803:
3775:
3749:
3672:
3646:
3617:
3599:
3482:
3434:
3029:
techniques on quantum states. For example, backup copies of a state in the middle of a
2133:
2032:
1844:
1030:
nondisturbing measurements in quantum mechanics was already delivered by Park in 1970.
1026:
380:
305:
238:
150:
2966:
5036:
4681:
4588:
4545:
4476:
4392:
4372:
4327:
4287:
4265:
4175:
4106:
4094:
3971:
3910:
3861:
3807:
3793:
3753:
3702:
3570:
3536:
3505:
3344:
2998:
1489:
1010:
911:
814:
789:
729:
724:
624:
590:
570:
168:
27:
3983:
3922:
3676:
3438:
4703:
4653:
4430:
4187:
4167:
4086:
4037:
3967:
3963:
3902:
3849:
3785:
3741:
3664:
3609:
3540:
3532:
3486:
3474:
3457:
3426:
3030:
3026:
2958:
2054:
1628:
1006:
907:
899:
819:
809:
799:
699:
679:
664:
634:
502:
390:
3621:
4829:
4767:
4457:
4452:
4140:
BuĹľek, V.; Hillery, M. (1996). "Quantum
Copying: Beyond the No-Cloning Theorem".
3354:
3298:
962:
923:
844:
714:
694:
440:
280:
3997:
Noor, Mah; Duda, Jarek (2024). "No-cloning theorem for 2WQC and postselection".
3906:
3455:
Wootters, William; Zurek, Wojciech (1982). "A Single
Quantum Cannot be Cloned".
2962:
2694:{\displaystyle |\langle \phi |\psi \rangle |^{2}=|\langle \phi |\psi \rangle |.}
961:. Together, these underpin the interpretation of quantum mechanics in terms of
4938:
4915:
4882:
4686:
4563:
4041:
3695:
How the
Hippies Saved Physics: Science, Counterculture, and the Quantum Revival
3008:
For extensions of quantum computers, no-cloning theorem remains valid if using
2950:
2938:
1403:
779:
739:
719:
689:
669:
619:
585:
435:
425:
218:
4210:
3613:
922:
with the state of another as cloning specifically refers to the creation of a
5099:
4760:
4578:
4504:
4171:
4098:
3820:
Ortigoso, Juan (2018). "Twelve years before the quantum no-cloning theorem".
3257:
3038:
3009:
2891:
1402:
The state of the initial composite system is then described by the following
903:
895:
839:
834:
764:
734:
704:
575:
521:
248:
223:
101:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
3559:
1663:
the
Hilbert space of the combined system. However, no such unitary operator
4980:
4905:
3975:
3914:
3698:
3668:
1014:
974:
915:
829:
824:
759:
744:
709:
203:
4179:
2946:
4990:
4844:
4382:
4154:
3889:
3651:
3064:
3002:
998:
794:
749:
684:
639:
5051:
4985:
4849:
3745:
3430:
3253:
3235:
3071:
3034:
2986:
1509:
1505:
939:
784:
754:
674:
649:
644:
629:
3853:
3544:
3397:
Park, James (1970). "The concept of transition in quantum mechanics".
3256:
as meaning that there are two copies of information, one lying at the
4834:
3478:
3276:
3239:
973:, allows, in turn, a connection to be made from quantum mechanics to
275:
5019:
4995:
4854:
4819:
4119:
S. Abramsky, "No-Cloning in categorical quantum mechanics", (2008)
4073:
4003:
3836:
654:
4128:
4032:
3950:
3789:
3780:
3604:
5046:
4663:
1005:, the publication of the 1982 proof of the no-cloning theorem by
887:
3635:
Peres, Asher (2003). "How the No-Cloning
Theorem Got its Name".
3067:, and suppose quantum states could be cloned. Assume parts of a
5023:
4519:
898:, a statement which has profound implications in the field of
4292:
3025:
The no-cloning theorem prevents the use of certain classical
1513:
935:
3309:
protocols, among other uses in quantum information science.
2060:
To prove the theorem, we select an arbitrary pair of states
1314:{\displaystyle |\phi \rangle _{A}\otimes |\phi \rangle _{B}}
5041:
4514:
4447:
3560:"Physics, Topology, Logic and Computation: A Rosetta Stone"
4123:, I. Mackie and S. Gay (eds), Cambridge University Press.
4658:
4643:
3248:
The no cloning theorem prevents an interpretation of the
1458:{\displaystyle |\phi \rangle _{A}\otimes |e\rangle _{B}.}
2855:{\displaystyle |\phi \rangle =e^{i\beta }|\psi \rangle }
1258:{\displaystyle |\phi \rangle _{A}\otimes |e\rangle _{B}}
1092:. Suppose we want to have a procedure to copy the state
3041:
showed that it is, by independently devising the first
2937:
Take a qubit for example. It can be represented by two
926:
with identical factors. For example, one might use the
3503:
Dieks, Dennis (1982). "Communication by EPR devices".
902:
among others. The theorem is an evolution of the 1970
4207:, Physics World 14 (11) (2001), pp. 25–29.
3874:
3202:
3164:
3122:
3084:
2900:
2868:
2813:
2759:
2709:
2623:
2595:
2160:
2136:
2101:
2066:
2035:
2015:
1995:
1866:
1847:
1812:
1777:
1751:
1681:
1641:
1606:
1545:
1471:
1412:
1370:
1335:
1271:
1215:
1176:
1137:
1098:
1052:
39:
4485:
4057:"Quantum state discrimination and its applications"
2934:quantum state. This proves the no-cloning theorem.
1492:with which we may manipulate the composite system:
4061:Journal of Physics A: Mathematical and Theoretical
3226:
3188:
3146:
3108:
2914:
2882:
2854:
2795:
2745:
2693:
2609:
2581:
2142:
2122:
2087:
2041:
2021:
2001:
1981:
1853:
1833:
1798:
1763:
1719:
1653:
1619:
1590:
1477:
1457:
1391:
1356:
1313:
1257:
1197:
1158:
1119:
1084:
100:
4017:
3245:Quantum states cannot be discriminated perfectly.
5097:
4240:
3450:
3448:
3045:codes, which circumvent the no-cloning theorem.
2796:{\displaystyle |\langle \phi |\psi \rangle |=0}
2746:{\displaystyle |\langle \phi |\psi \rangle |=1}
4055:Bae, Joonwoo; Kwek, Leong-Chuan (2015-02-27).
3765:
3763:
3012:or two-way quantum computers. However, adding
1512:, corrupting the information contained in the
4226:
3454:
3016:is claimed to allow to clone quantum state.
1720:{\displaystyle H_{A}\otimes H_{B}=H\otimes H}
867:
4139:
3445:
3215:
3177:
3135:
3097:
3078:direction, collapsing Bob's state to either
2909:
2877:
2849:
2822:
2779:
2765:
2729:
2715:
2680:
2666:
2643:
2629:
2604:
2567:
2552:
2482:
2464:
2437:
2419:
2349:
2331:
2291:
2273:
2261:
2243:
2216:
2198:
2192:
2178:
2175:
2161:
2111:
2076:
1970:
1952:
1900:
1882:
1822:
1787:
1538:), e.g. for a time-independent Hamiltonian,
1443:
1422:
1380:
1345:
1302:
1281:
1246:
1225:
1186:
1147:
1108:
942:whereas the generalized statement regarding
95:
69:
4121:Semantic Techniques for Quantum Computation
3760:
3590:Coecke, Bob (2009). "Quantum Picturalism".
2922:. However, this cannot be the case for two
953:The no-cloning theorem has a time-reversed
4233:
4219:
874:
860:
4153:
4072:
4031:
4002:
3949:
3888:
3835:
3779:
3650:
3603:
3526:
3420:
3059:The no-cloning theorem is implied by the
2617:is assumed to be normalized, we thus get
2154:is supposed to be unitary, we would have
1329:in some unknown initial, or blank, state
3996:
3935:
3819:
3774:, IntechOpen (published April 3, 2013),
3769:
3557:
4751:Continuous-variable quantum information
4054:
3721:
3498:
3496:
1399:, of which we have no prior knowledge.
1025:device using quantum entanglement, and
5098:
3689:
3589:
3392:
3390:
3267:The no-cloning theorem applies to all
3001:can be implemented via introducing an
2926:states. Therefore, a single universal
16:Theorem in quantum information science
4214:
3634:
3569:. Berlin: Springer. pp. 95–172.
3502:
3048:Similarly, cloning would violate the
1516:. This is obviously not what we want.
1209:). That is, beginning with the state
3493:
3396:
3283:
1771:such that for all normalised states
1591:{\displaystyle U(t)=e^{-iHt/\hbar }}
1519:Alternatively, we could control the
1038:Suppose we have two quantum systems
1033:
3387:
3294:surprisingly high fidelity of 5/6.
3054:entanglement-assisted teleportation
1465:(in the following we will omit the
1265:, we want to end up with the state
13:
2123:{\displaystyle |\psi \rangle _{A}}
2088:{\displaystyle |\phi \rangle _{A}}
1799:{\displaystyle |\phi \rangle _{A}}
1392:{\displaystyle |\phi \rangle _{A}}
1198:{\displaystyle |\phi \rangle _{A}}
1120:{\displaystyle |\phi \rangle _{A}}
406:Sum-over-histories (path integral)
92:
66:
22:Part of a series of articles about
14:
5127:
2980:
1583:
43:
5080:
5079:
5070:
5069:
4197:
1321:. To make a "copy" of the state
4133:
4113:
4048:
4011:
3990:
3929:
3868:
3813:
3558:Baez, John; Stay, Mike (2010).
3227:{\displaystyle |z-\rangle _{B}}
3189:{\displaystyle |z+\rangle _{B}}
3147:{\displaystyle |z-\rangle _{B}}
3109:{\displaystyle |z+\rangle _{B}}
3019:
1600:Evolving up to some fixed time
1488:There are only two permissible
3968:10.1103/PhysRevLett.100.210502
3715:
3683:
3628:
3583:
3551:
3370:Stronger uncertainty relations
3273:category of sets and relations
3204:
3166:
3124:
3086:
2915:{\displaystyle |\psi \rangle }
2902:
2883:{\displaystyle |\phi \rangle }
2870:
2842:
2815:
2783:
2772:
2761:
2733:
2722:
2711:
2684:
2673:
2662:
2648:
2636:
2625:
2597:
2559:
2547:
2544:
2532:
2523:
2511:
2505:
2474:
2456:
2445:
2427:
2414:
2411:
2399:
2390:
2378:
2372:
2341:
2323:
2299:
2281:
2253:
2235:
2224:
2206:
2185:
2168:
2103:
2068:
1962:
1944:
1938:
1926:
1909:
1892:
1874:
1870:
1834:{\displaystyle |e\rangle _{B}}
1814:
1779:
1555:
1549:
1485:symbol and keep it implicit).
1435:
1414:
1372:
1357:{\displaystyle |e\rangle _{B}}
1337:
1294:
1273:
1238:
1217:
1178:
1159:{\displaystyle |e\rangle _{B}}
1139:
1100:
1017:was prompted by a proposal of
556:Relativistic quantum mechanics
88:
81:
62:
1:
5111:Theorems in quantum mechanics
4746:Adiabatic quantum computation
4091:10.1088/1751-8113/48/8/083001
3772:Advances in Quantum Mechanics
3381:
1741:There is no unitary operator
1731:of the state system A is in?
1085:{\displaystyle H=H_{A}=H_{B}}
971:categorical quantum mechanics
969:. This formulation, known as
596:Quantum statistical mechanics
4797:Topological quantum computer
3537:10.1016/0375-9601(82)90084-6
1325:, we combine it with system
1046:with a common Hilbert space
7:
5106:Quantum information science
5075:Quantum information science
4242:Quantum information science
3907:10.1103/PhysRevLett.76.2818
3823:American Journal of Physics
3312:
2055:projectivised Hilbert space
987:Cartesian closed categories
965:, and, in particular, as a
566:Quantum information science
10:
5132:
4470:quantum gate teleportation
4042:10.1103/PhysRevA.88.022332
3567:New Structures for Physics
3262:black hole complementarity
2610:{\displaystyle |e\rangle }
1764:{\displaystyle H\otimes H}
1654:{\displaystyle H\otimes H}
1023:superluminal communication
992:
979:quantum information theory
5065:
5008:
4971:
4937:
4914:
4881:
4872:
4805:
4734:
4672:
4632:
4599:Quantum Fourier transform
4544:
4495:Post-quantum cryptography
4438:Entanglement distillation
4411:
4320:
4248:
4203:V. Buzek and M. Hillery,
3614:10.1080/00107510903257624
3269:dagger compact categories
2957:) to be polarised (which
2805:Cauchy–Schwarz inequality
2703:This implies that either
2589:Since the quantum state
5085:Quantum mechanics topics
4780:Quantum machine learning
4756:One-way quantum computer
4609:Quantum phase estimation
4510:Quantum key distribution
4443:Monogamy of entanglement
4172:10.1103/PhysRevA.54.1844
3335:No-communication theorem
3325:Monogamy of entanglement
3320:Fundamental Fysiks Group
3061:no-communication theorem
3050:no-teleportation theorem
3043:quantum error correcting
1478:{\displaystyle \otimes }
981:(in the same sense that
601:Quantum machine learning
354:Wheeler's delayed-choice
4692:Randomized benchmarking
4554:Amplitude amplification
3938:Physical Review Letters
3877:Physical Review Letters
3638:Fortschritte der Physik
3238:separations, violating
2002:{\displaystyle \alpha }
1529:time-evolution operator
1170:for any original state
967:dagger compact category
311:Leggett–Garg inequality
4792:Quantum Turing machine
4785:quantum neural network
4532:Quantum secret sharing
3725:Foundations of Physics
3669:10.1002/prop.200310062
3400:Foundations of Physics
3291:unitary transformation
3228:
3190:
3148:
3110:
3065:EPR thought experiment
2959:unitary transformation
2943:probability amplitudes
2916:
2884:
2856:
2797:
2747:
2695:
2611:
2583:
2144:
2124:
2089:
2043:
2023:
2003:
1983:
1855:
1835:
1800:
1765:
1721:
1655:
1621:
1592:
1479:
1459:
1393:
1358:
1315:
1259:
1199:
1160:
1121:
1086:
102:
4864:Entanglement-assisted
4825:quantum convolutional
4500:Quantum coin flipping
4465:Quantum teleportation
4426:entanglement-assisted
4256:DiVincenzo's criteria
3375:Uncertainty principle
3365:Quantum teleportation
3250:holographic principle
3229:
3191:
3149:
3111:
3014:closed timelike curve
2955:Hadamard quantum gate
2917:
2885:
2857:
2798:
2748:
2696:
2612:
2584:
2145:
2130:in the Hilbert space
2125:
2090:
2044:
2024:
2022:{\displaystyle \phi }
2004:
1989:for some real number
1984:
1856:
1836:
1801:
1766:
1722:
1667:can clone all states.
1656:
1622:
1620:{\displaystyle t_{0}}
1593:
1527:system, and thus the
1504:the system into some
1500:, which irreversibly
1480:
1460:
1394:
1359:
1316:
1260:
1200:
1161:
1122:
1087:
296:Elitzur–Vaidman
286:Davisson–Germer
103:
4675:processor benchmarks
4604:Quantum optimization
4487:Quantum cryptography
4298:physical vs. logical
3592:Contemporary Physics
3350:Quantum entanglement
3330:No-broadcast theorem
3307:quantum cryptography
3303:eavesdropping attack
3275:and the category of
3200:
3162:
3120:
3082:
2898:
2866:
2811:
2757:
2707:
2621:
2593:
2158:
2134:
2099:
2064:
2033:
2013:
1993:
1864:
1845:
1810:
1775:
1749:
1679:
1639:
1604:
1543:
1469:
1410:
1368:
1333:
1269:
1213:
1174:
1135:
1096:
1050:
983:intuitionistic logic
948:no-broadcast theorem
561:Quantum field theory
473:Consistent histories
110:Schrödinger equation
37:
4388:Quantum speed limit
4283:Quantum programming
4278:Quantum information
4164:1996PhRvA..54.1844B
4083:2015JPhA...48h3001B
3960:2008PhRvL.100u0502K
3899:1996PhRvL..76.2818B
3846:2018AmJPh..86..201O
3738:1982FoPh...12.1171H
3661:2003ForPh..51..458P
3519:1982PhLA...92..271D
3471:1982Natur.299..802W
3413:1970FoPh....1...23P
3360:Quantum information
3340:No-deleting theorem
3069:maximally entangled
3031:quantum computation
2963:surjective isometry
1739: —
959:no-deleting theorem
932:Walsh–Hadamard gate
928:controlled NOT gate
349:Stern–Gerlach
146:Classical mechanics
5037:Forest/Rigetti QCS
4773:quantum logic gate
4559:Bernstein–Vazirani
4546:Quantum algorithms
4421:Classical capacity
4305:Quantum processors
4288:Quantum simulation
3746:10.1007/BF00729622
3431:10.1007/BF00708652
3301:can be used as an
3224:
3186:
3144:
3106:
2912:
2880:
2852:
2793:
2743:
2691:
2607:
2579:
2140:
2120:
2085:
2039:
2019:
1999:
1979:
1851:
1831:
1796:
1761:
1737:
1717:
1651:
1617:
1588:
1496:We can perform an
1490:quantum operations
1475:
1455:
1389:
1354:
1311:
1255:
1195:
1166:of quantum system
1156:
1127:of quantum system
1117:
1082:
1027:Giancarlo Ghirardi
892:no-cloning theorem
537:Von Neumann–Wigner
517:Objective-collapse
316:Mach–Zehnder
306:Leggett inequality
301:Franck–Hertz
151:Old quantum theory
98:
5093:
5092:
5004:
5003:
4901:Linear optical QC
4682:Quantum supremacy
4636:complexity theory
4589:Quantum annealing
4540:
4539:
4477:Superdense coding
4266:Quantum computing
4020:Physical Review A
3883:(15): 2818–2821.
3854:10.1119/1.5021356
3799:978-953-51-1089-7
3732:(12): 1171–1179.
3708:978-0-393-07636-3
3576:978-3-642-12821-9
3506:Physics Letters A
3465:(5886): 802–803.
3345:No-hiding theorem
3284:Imperfect cloning
2999:quantum operation
2143:{\displaystyle H}
2042:{\displaystyle e}
1854:{\displaystyle H}
1735:
1131:, over the state
1034:Theorem and proof
912:Wojciech H. Zurek
900:quantum computing
884:
883:
591:Scattering theory
571:Quantum computing
344:Schrödinger's cat
276:Bell's inequality
84:
59:
28:Quantum mechanics
5123:
5083:
5082:
5073:
5072:
4879:
4878:
4809:error correction
4738:computing models
4704:Relaxation times
4594:Quantum counting
4483:
4482:
4431:quantum capacity
4378:No-teleportation
4363:No-communication
4235:
4228:
4221:
4212:
4211:
4192:
4191:
4157:
4155:quant-ph/9607018
4148:(3): 1844–1852.
4137:
4131:
4117:
4111:
4110:
4076:
4052:
4046:
4045:
4035:
4015:
4009:
4008:
4006:
3994:
3988:
3987:
3953:
3933:
3927:
3926:
3892:
3890:quant-ph/9511010
3872:
3866:
3865:
3839:
3817:
3811:
3810:
3783:
3767:
3758:
3757:
3719:
3713:
3712:
3687:
3681:
3680:
3654:
3652:quant-ph/0205076
3632:
3626:
3625:
3607:
3587:
3581:
3580:
3564:
3555:
3549:
3548:
3530:
3500:
3491:
3490:
3479:10.1038/299802a0
3452:
3443:
3442:
3424:
3394:
3233:
3231:
3230:
3225:
3223:
3222:
3207:
3195:
3193:
3192:
3187:
3185:
3184:
3169:
3153:
3151:
3150:
3145:
3143:
3142:
3127:
3115:
3113:
3112:
3107:
3105:
3104:
3089:
3027:error correction
2921:
2919:
2918:
2913:
2905:
2889:
2887:
2886:
2881:
2873:
2861:
2859:
2858:
2853:
2845:
2840:
2839:
2818:
2802:
2800:
2799:
2794:
2786:
2775:
2764:
2752:
2750:
2749:
2744:
2736:
2725:
2714:
2700:
2698:
2697:
2692:
2687:
2676:
2665:
2657:
2656:
2651:
2639:
2628:
2616:
2614:
2613:
2608:
2600:
2588:
2586:
2585:
2580:
2575:
2574:
2562:
2551:
2550:
2490:
2489:
2477:
2472:
2471:
2459:
2454:
2453:
2448:
2436:
2435:
2430:
2418:
2417:
2357:
2356:
2344:
2339:
2338:
2326:
2318:
2317:
2308:
2307:
2302:
2290:
2289:
2284:
2269:
2268:
2256:
2251:
2250:
2238:
2233:
2232:
2227:
2215:
2214:
2209:
2188:
2171:
2149:
2147:
2146:
2141:
2129:
2127:
2126:
2121:
2119:
2118:
2106:
2094:
2092:
2091:
2086:
2084:
2083:
2071:
2048:
2046:
2045:
2040:
2028:
2026:
2025:
2020:
2008:
2006:
2005:
2000:
1988:
1986:
1985:
1980:
1978:
1977:
1965:
1960:
1959:
1947:
1942:
1941:
1908:
1907:
1895:
1890:
1889:
1877:
1860:
1858:
1857:
1852:
1840:
1838:
1837:
1832:
1830:
1829:
1817:
1805:
1803:
1802:
1797:
1795:
1794:
1782:
1770:
1768:
1767:
1762:
1740:
1726:
1724:
1723:
1718:
1704:
1703:
1691:
1690:
1662:
1660:
1658:
1657:
1652:
1629:unitary operator
1626:
1624:
1623:
1618:
1616:
1615:
1599:
1597:
1595:
1594:
1589:
1587:
1586:
1582:
1484:
1482:
1481:
1476:
1464:
1462:
1461:
1456:
1451:
1450:
1438:
1430:
1429:
1417:
1398:
1396:
1395:
1390:
1388:
1387:
1375:
1363:
1361:
1360:
1355:
1353:
1352:
1340:
1320:
1318:
1317:
1312:
1310:
1309:
1297:
1289:
1288:
1276:
1264:
1262:
1261:
1256:
1254:
1253:
1241:
1233:
1232:
1220:
1207:bra–ket notation
1204:
1202:
1201:
1196:
1194:
1193:
1181:
1165:
1163:
1162:
1157:
1155:
1154:
1142:
1126:
1124:
1123:
1118:
1116:
1115:
1103:
1091:
1089:
1088:
1083:
1081:
1080:
1068:
1067:
977:as the logic of
946:is known as the
934:to entangle two
908:William Wootters
876:
869:
862:
503:Superdeterminism
156:Bra–ket notation
107:
105:
104:
99:
91:
86:
85:
77:
65:
60:
58:
47:
19:
18:
5131:
5130:
5126:
5125:
5124:
5122:
5121:
5120:
5096:
5095:
5094:
5089:
5061:
5011:
5000:
4973:Superconducting
4967:
4933:
4924:Neutral atom QC
4916:Ultracold atoms
4910:
4875:implementations
4874:
4868:
4808:
4801:
4768:Quantum circuit
4736:
4730:
4724:
4714:
4674:
4668:
4635:
4628:
4584:Hidden subgroup
4536:
4525:other protocols
4481:
4458:quantum network
4453:Quantum channel
4413:
4407:
4353:No-broadcasting
4343:Gottesman–Knill
4316:
4244:
4239:
4205:Quantum cloning
4200:
4195:
4138:
4134:
4118:
4114:
4053:
4049:
4016:
4012:
3995:
3991:
3934:
3930:
3873:
3869:
3818:
3814:
3800:
3768:
3761:
3720:
3716:
3709:
3688:
3684:
3645:(45): 458–461.
3633:
3629:
3588:
3584:
3577:
3562:
3556:
3552:
3528:10.1.1.654.7183
3501:
3494:
3453:
3446:
3422:10.1.1.623.5267
3395:
3388:
3384:
3379:
3355:Quantum cloning
3315:
3299:quantum cloning
3286:
3218:
3214:
3203:
3201:
3198:
3197:
3180:
3176:
3165:
3163:
3160:
3159:
3138:
3134:
3123:
3121:
3118:
3117:
3100:
3096:
3085:
3083:
3080:
3079:
3022:
2983:
2947:normalised to 1
2939:complex numbers
2930:cannot clone a
2901:
2899:
2896:
2895:
2869:
2867:
2864:
2863:
2841:
2832:
2828:
2814:
2812:
2809:
2808:
2803:. Hence by the
2782:
2771:
2760:
2758:
2755:
2754:
2732:
2721:
2710:
2708:
2705:
2704:
2683:
2672:
2661:
2652:
2647:
2646:
2635:
2624:
2622:
2619:
2618:
2596:
2594:
2591:
2590:
2570:
2566:
2558:
2498:
2494:
2485:
2481:
2473:
2467:
2463:
2455:
2449:
2444:
2443:
2431:
2426:
2425:
2365:
2361:
2352:
2348:
2340:
2334:
2330:
2322:
2313:
2309:
2303:
2298:
2297:
2285:
2280:
2279:
2264:
2260:
2252:
2246:
2242:
2234:
2228:
2223:
2222:
2210:
2205:
2204:
2184:
2167:
2159:
2156:
2155:
2135:
2132:
2131:
2114:
2110:
2102:
2100:
2097:
2096:
2079:
2075:
2067:
2065:
2062:
2061:
2051:
2034:
2031:
2030:
2014:
2011:
2010:
1994:
1991:
1990:
1973:
1969:
1961:
1955:
1951:
1943:
1919:
1915:
1903:
1899:
1891:
1885:
1881:
1873:
1865:
1862:
1861:
1846:
1843:
1842:
1825:
1821:
1813:
1811:
1808:
1807:
1790:
1786:
1778:
1776:
1773:
1772:
1750:
1747:
1746:
1738:
1699:
1695:
1686:
1682:
1680:
1677:
1676:
1640:
1637:
1636:
1635:
1611:
1607:
1605:
1602:
1601:
1578:
1565:
1561:
1544:
1541:
1540:
1539:
1470:
1467:
1466:
1446:
1442:
1434:
1425:
1421:
1413:
1411:
1408:
1407:
1383:
1379:
1371:
1369:
1366:
1365:
1364:independent of
1348:
1344:
1336:
1334:
1331:
1330:
1305:
1301:
1293:
1284:
1280:
1272:
1270:
1267:
1266:
1249:
1245:
1237:
1228:
1224:
1216:
1214:
1211:
1210:
1189:
1185:
1177:
1175:
1172:
1171:
1150:
1146:
1138:
1136:
1133:
1132:
1111:
1107:
1099:
1097:
1094:
1093:
1076:
1072:
1063:
1059:
1051:
1048:
1047:
1036:
995:
963:category theory
924:separable state
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:
5129:
5119:
5118:
5116:No-go theorems
5113:
5108:
5091:
5090:
5088:
5087:
5077:
5066:
5063:
5062:
5060:
5059:
5057:many others...
5054:
5049:
5044:
5039:
5030:
5016:
5014:
5006:
5005:
5002:
5001:
4999:
4998:
4993:
4988:
4983:
4977:
4975:
4969:
4968:
4966:
4965:
4960:
4955:
4950:
4944:
4942:
4935:
4934:
4932:
4931:
4929:Trapped-ion QC
4926:
4920:
4918:
4912:
4911:
4909:
4908:
4903:
4898:
4893:
4887:
4885:
4883:Quantum optics
4876:
4870:
4869:
4867:
4866:
4861:
4860:
4859:
4852:
4847:
4842:
4837:
4832:
4827:
4822:
4813:
4811:
4803:
4802:
4800:
4799:
4794:
4789:
4788:
4787:
4777:
4776:
4775:
4765:
4764:
4763:
4753:
4748:
4742:
4740:
4732:
4731:
4729:
4728:
4727:
4726:
4722:
4716:
4712:
4701:
4700:
4699:
4689:
4687:Quantum volume
4684:
4678:
4676:
4670:
4669:
4667:
4666:
4661:
4656:
4651:
4646:
4640:
4638:
4630:
4629:
4627:
4626:
4621:
4616:
4611:
4606:
4601:
4596:
4591:
4586:
4581:
4576:
4571:
4566:
4564:Boson sampling
4561:
4556:
4550:
4548:
4542:
4541:
4538:
4537:
4535:
4534:
4529:
4528:
4527:
4522:
4517:
4507:
4502:
4497:
4491:
4489:
4480:
4479:
4474:
4473:
4472:
4462:
4461:
4460:
4450:
4445:
4440:
4435:
4434:
4433:
4428:
4417:
4415:
4409:
4408:
4406:
4405:
4400:
4398:Solovay–Kitaev
4395:
4390:
4385:
4380:
4375:
4370:
4365:
4360:
4355:
4350:
4345:
4340:
4335:
4330:
4324:
4322:
4318:
4317:
4315:
4314:
4313:
4312:
4302:
4301:
4300:
4290:
4285:
4280:
4275:
4274:
4273:
4263:
4258:
4252:
4250:
4246:
4245:
4238:
4237:
4230:
4223:
4215:
4209:
4208:
4199:
4196:
4194:
4193:
4132:
4112:
4047:
4010:
3989:
3944:(21): 210502.
3928:
3867:
3830:(3): 201–205.
3812:
3798:
3759:
3714:
3707:
3682:
3627:
3582:
3575:
3550:
3513:(6): 271–272.
3492:
3444:
3385:
3383:
3380:
3378:
3377:
3372:
3367:
3362:
3357:
3352:
3347:
3342:
3337:
3332:
3327:
3322:
3316:
3314:
3311:
3285:
3282:
3281:
3280:
3265:
3246:
3243:
3221:
3217:
3213:
3210:
3206:
3183:
3179:
3175:
3172:
3168:
3141:
3137:
3133:
3130:
3126:
3103:
3099:
3095:
3092:
3088:
3057:
3046:
3021:
3018:
2982:
2981:Generalization
2979:
2951:copy and paste
2911:
2908:
2904:
2879:
2876:
2872:
2851:
2848:
2844:
2838:
2835:
2831:
2827:
2824:
2821:
2817:
2792:
2789:
2785:
2781:
2778:
2774:
2770:
2767:
2763:
2742:
2739:
2735:
2731:
2728:
2724:
2720:
2717:
2713:
2690:
2686:
2682:
2679:
2675:
2671:
2668:
2664:
2660:
2655:
2650:
2645:
2642:
2638:
2634:
2631:
2627:
2606:
2603:
2599:
2578:
2573:
2569:
2565:
2561:
2557:
2554:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2497:
2493:
2488:
2484:
2480:
2476:
2470:
2466:
2462:
2458:
2452:
2447:
2442:
2439:
2434:
2429:
2424:
2421:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2364:
2360:
2355:
2351:
2347:
2343:
2337:
2333:
2329:
2325:
2321:
2316:
2312:
2306:
2301:
2296:
2293:
2288:
2283:
2278:
2275:
2272:
2267:
2263:
2259:
2255:
2249:
2245:
2241:
2237:
2231:
2226:
2221:
2218:
2213:
2208:
2203:
2200:
2197:
2194:
2191:
2187:
2183:
2180:
2177:
2174:
2170:
2166:
2163:
2139:
2117:
2113:
2109:
2105:
2082:
2078:
2074:
2070:
2038:
2018:
1998:
1976:
1972:
1968:
1964:
1958:
1954:
1950:
1946:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1918:
1914:
1911:
1906:
1902:
1898:
1894:
1888:
1884:
1880:
1876:
1872:
1869:
1850:
1828:
1824:
1820:
1816:
1793:
1789:
1785:
1781:
1760:
1757:
1754:
1733:
1716:
1713:
1710:
1707:
1702:
1698:
1694:
1689:
1685:
1669:
1668:
1650:
1647:
1644:
1614:
1610:
1585:
1581:
1577:
1574:
1571:
1568:
1564:
1560:
1557:
1554:
1551:
1548:
1517:
1474:
1454:
1449:
1445:
1441:
1437:
1433:
1428:
1424:
1420:
1416:
1404:tensor product
1386:
1382:
1378:
1374:
1351:
1347:
1343:
1339:
1308:
1304:
1300:
1296:
1292:
1287:
1283:
1279:
1275:
1252:
1248:
1244:
1240:
1236:
1231:
1227:
1223:
1219:
1192:
1188:
1184:
1180:
1153:
1149:
1145:
1141:
1114:
1110:
1106:
1102:
1079:
1075:
1071:
1066:
1062:
1058:
1055:
1035:
1032:
994:
991:
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:
5128:
5117:
5114:
5112:
5109:
5107:
5104:
5103:
5101:
5086:
5078:
5076:
5068:
5067:
5064:
5058:
5055:
5053:
5050:
5048:
5045:
5043:
5040:
5038:
5034:
5031:
5029:
5025:
5021:
5018:
5017:
5015:
5013:
5007:
4997:
4994:
4992:
4989:
4987:
4984:
4982:
4979:
4978:
4976:
4974:
4970:
4964:
4961:
4959:
4956:
4954:
4953:Spin qubit QC
4951:
4949:
4946:
4945:
4943:
4940:
4936:
4930:
4927:
4925:
4922:
4921:
4919:
4917:
4913:
4907:
4904:
4902:
4899:
4897:
4894:
4892:
4889:
4888:
4886:
4884:
4880:
4877:
4871:
4865:
4862:
4858:
4857:
4853:
4851:
4848:
4846:
4843:
4841:
4838:
4836:
4833:
4831:
4828:
4826:
4823:
4821:
4818:
4817:
4815:
4814:
4812:
4810:
4804:
4798:
4795:
4793:
4790:
4786:
4783:
4782:
4781:
4778:
4774:
4771:
4770:
4769:
4766:
4762:
4761:cluster state
4759:
4758:
4757:
4754:
4752:
4749:
4747:
4744:
4743:
4741:
4739:
4733:
4725:
4721:
4717:
4715:
4711:
4707:
4706:
4705:
4702:
4698:
4695:
4694:
4693:
4690:
4688:
4685:
4683:
4680:
4679:
4677:
4671:
4665:
4662:
4660:
4657:
4655:
4652:
4650:
4647:
4645:
4642:
4641:
4639:
4637:
4631:
4625:
4622:
4620:
4617:
4615:
4612:
4610:
4607:
4605:
4602:
4600:
4597:
4595:
4592:
4590:
4587:
4585:
4582:
4580:
4577:
4575:
4572:
4570:
4569:Deutsch–Jozsa
4567:
4565:
4562:
4560:
4557:
4555:
4552:
4551:
4549:
4547:
4543:
4533:
4530:
4526:
4523:
4521:
4518:
4516:
4513:
4512:
4511:
4508:
4506:
4505:Quantum money
4503:
4501:
4498:
4496:
4493:
4492:
4490:
4488:
4484:
4478:
4475:
4471:
4468:
4467:
4466:
4463:
4459:
4456:
4455:
4454:
4451:
4449:
4446:
4444:
4441:
4439:
4436:
4432:
4429:
4427:
4424:
4423:
4422:
4419:
4418:
4416:
4414:communication
4410:
4404:
4401:
4399:
4396:
4394:
4391:
4389:
4386:
4384:
4381:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4351:
4349:
4346:
4344:
4341:
4339:
4336:
4334:
4331:
4329:
4326:
4325:
4323:
4319:
4311:
4308:
4307:
4306:
4303:
4299:
4296:
4295:
4294:
4291:
4289:
4286:
4284:
4281:
4279:
4276:
4272:
4269:
4268:
4267:
4264:
4262:
4259:
4257:
4254:
4253:
4251:
4247:
4243:
4236:
4231:
4229:
4224:
4222:
4217:
4216:
4213:
4206:
4202:
4201:
4198:Other sources
4189:
4185:
4181:
4177:
4173:
4169:
4165:
4161:
4156:
4151:
4147:
4143:
4136:
4130:
4126:
4122:
4116:
4108:
4104:
4100:
4096:
4092:
4088:
4084:
4080:
4075:
4070:
4067:(8): 083001.
4066:
4062:
4058:
4051:
4043:
4039:
4034:
4029:
4025:
4021:
4014:
4005:
4000:
3993:
3985:
3981:
3977:
3973:
3969:
3965:
3961:
3957:
3952:
3947:
3943:
3939:
3932:
3924:
3920:
3916:
3912:
3908:
3904:
3900:
3896:
3891:
3886:
3882:
3878:
3871:
3863:
3859:
3855:
3851:
3847:
3843:
3838:
3833:
3829:
3825:
3824:
3816:
3809:
3805:
3801:
3795:
3791:
3790:10.5772/56429
3787:
3782:
3777:
3773:
3766:
3764:
3755:
3751:
3747:
3743:
3739:
3735:
3731:
3727:
3726:
3718:
3710:
3704:
3700:
3696:
3692:
3691:Kaiser, David
3686:
3678:
3674:
3670:
3666:
3662:
3658:
3653:
3648:
3644:
3640:
3639:
3631:
3623:
3619:
3615:
3611:
3606:
3601:
3597:
3593:
3586:
3578:
3572:
3568:
3561:
3554:
3546:
3542:
3538:
3534:
3529:
3524:
3520:
3516:
3512:
3508:
3507:
3499:
3497:
3488:
3484:
3480:
3476:
3472:
3468:
3464:
3460:
3459:
3451:
3449:
3440:
3436:
3432:
3428:
3423:
3418:
3414:
3410:
3406:
3402:
3401:
3393:
3391:
3386:
3376:
3373:
3371:
3368:
3366:
3363:
3361:
3358:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3338:
3336:
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3317:
3310:
3308:
3304:
3300:
3295:
3292:
3278:
3274:
3270:
3266:
3263:
3259:
3258:event horizon
3255:
3251:
3247:
3244:
3241:
3237:
3219:
3211:
3208:
3181:
3173:
3170:
3157:
3139:
3131:
3128:
3101:
3093:
3090:
3077:
3073:
3070:
3066:
3062:
3058:
3055:
3051:
3047:
3044:
3040:
3036:
3032:
3028:
3024:
3023:
3017:
3015:
3011:
3010:postselection
3006:
3004:
3000:
2996:
2992:
2988:
2978:
2976:
2972:
2968:
2964:
2960:
2956:
2952:
2948:
2944:
2940:
2935:
2933:
2929:
2925:
2906:
2893:
2874:
2846:
2836:
2833:
2829:
2825:
2819:
2806:
2790:
2787:
2776:
2768:
2740:
2737:
2726:
2718:
2701:
2688:
2677:
2669:
2658:
2653:
2640:
2632:
2601:
2576:
2571:
2563:
2555:
2541:
2538:
2535:
2529:
2526:
2520:
2517:
2514:
2508:
2502:
2499:
2495:
2491:
2486:
2478:
2468:
2460:
2450:
2440:
2432:
2422:
2408:
2405:
2402:
2396:
2393:
2387:
2384:
2381:
2375:
2369:
2366:
2362:
2358:
2353:
2345:
2335:
2327:
2319:
2314:
2310:
2304:
2294:
2286:
2276:
2270:
2265:
2257:
2247:
2239:
2229:
2219:
2211:
2201:
2195:
2189:
2181:
2172:
2164:
2153:
2137:
2115:
2107:
2080:
2072:
2058:
2056:
2050:
2036:
2016:
2009:depending on
1996:
1974:
1966:
1956:
1948:
1935:
1932:
1929:
1923:
1920:
1916:
1912:
1904:
1896:
1886:
1878:
1867:
1848:
1826:
1818:
1791:
1783:
1758:
1755:
1752:
1744:
1732:
1730:
1714:
1711:
1708:
1705:
1700:
1696:
1692:
1687:
1683:
1674:
1666:
1648:
1645:
1642:
1633:
1630:
1612:
1608:
1579:
1575:
1572:
1569:
1566:
1562:
1558:
1552:
1546:
1537:
1533:
1530:
1526:
1522:
1518:
1515:
1511:
1507:
1503:
1499:
1495:
1494:
1493:
1491:
1486:
1472:
1452:
1447:
1439:
1431:
1426:
1418:
1405:
1400:
1384:
1376:
1349:
1341:
1328:
1324:
1306:
1298:
1290:
1285:
1277:
1250:
1242:
1234:
1229:
1221:
1208:
1190:
1182:
1169:
1151:
1143:
1130:
1112:
1104:
1077:
1073:
1069:
1064:
1060:
1056:
1053:
1045:
1041:
1031:
1028:
1024:
1020:
1016:
1012:
1008:
1004:
1000:
997:According to
990:
988:
984:
980:
976:
972:
968:
964:
960:
956:
951:
949:
945:
941:
937:
933:
929:
925:
921:
917:
913:
909:
905:
904:no-go theorem
901:
897:
896:quantum state
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:
4981:Charge qubit
4906:KLM protocol
4855:
4719:
4709:
4403:Purification
4357:
4333:Eastin–Knill
4204:
4145:
4142:Phys. Rev. A
4141:
4135:
4120:
4115:
4064:
4060:
4050:
4023:
4019:
4013:
3992:
3941:
3937:
3931:
3880:
3876:
3870:
3827:
3821:
3815:
3771:
3729:
3723:
3717:
3699:W. W. Norton
3694:
3685:
3642:
3636:
3630:
3595:
3591:
3585:
3566:
3553:
3510:
3504:
3462:
3456:
3407:(1): 23–33.
3404:
3398:
3296:
3287:
3155:
3075:
3020:Consequences
3007:
2993:, it can be
2984:
2974:
2970:
2936:
2931:
2927:
2923:
2702:
2151:
2059:
2052:
1742:
1734:
1728:
1675:, acting on
1672:
1670:
1664:
1631:
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1531:
1524:
1487:
1401:
1326:
1322:
1167:
1128:
1043:
1039:
1037:
1019:Nick Herbert
1003:David Kaiser
996:
985:arises from
975:linear logic
952:
944:mixed states
916:Dennis Dieks
891:
885:
431:Klein–Gordon
367:Formulations
204:Energy level
199:Entanglement
182:Fundamentals
169:Interference
120:Introduction
5012:programming
4991:Phase qubit
4896:Circuit QED
4368:No-deleting
4310:cloud-based
3254:black holes
2991:mixed state
2967:realisation
1521:Hamiltonian
1498:observation
999:Asher Peres
940:pure states
914:as well as
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
5100:Categories
5052:libquantum
4986:Flux qubit
4891:Cavity QED
4840:Bacon–Shor
4830:stabilizer
4358:No-cloning
4074:1707.02571
4004:2407.15623
3837:1707.06910
3545:1874/16932
3382:References
3297:Imperfect
3277:cobordisms
3236:space-like
3072:Bell state
2987:pure state
2892:orthogonal
2150:. Because
1729:regardless
1510:observable
1506:eigenstate
815:Sommerfeld
730:Heisenberg
725:Gutzwiller
665:de Broglie
613:Scientists
527:Relational
478:Copenhagen
381:Heisenberg
239:Tunnelling
140:Background
4958:NV center
4393:Threshold
4373:No-hiding
4338:Gleason's
4129:0910.2401
4107:119199057
4099:1751-8113
4033:1207.6062
3951:0704.1754
3862:119192142
3808:118778014
3781:1305.2305
3754:123118337
3605:0908.1787
3598:: 59–83.
3523:CiteSeerX
3417:CiteSeerX
3240:causality
3216:⟩
3212:−
3178:⟩
3136:⟩
3132:−
3098:⟩
2995:"purified
2975:universal
2941:, called
2924:arbitrary
2910:⟩
2907:ψ
2878:⟩
2875:ϕ
2850:⟩
2847:ψ
2837:β
2823:⟩
2820:ϕ
2780:⟩
2777:ψ
2769:ϕ
2766:⟨
2730:⟩
2727:ψ
2719:ϕ
2716:⟨
2681:⟩
2678:ψ
2670:ϕ
2667:⟨
2644:⟩
2641:ψ
2633:ϕ
2630:⟨
2605:⟩
2568:⟩
2564:ψ
2556:ϕ
2553:⟨
2536:ψ
2530:α
2527:−
2515:ϕ
2509:α
2500:−
2492:≡
2483:⟩
2479:ψ
2465:⟩
2461:ψ
2441:ϕ
2438:⟨
2423:ϕ
2420:⟨
2403:ψ
2397:α
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2382:ϕ
2376:α
2367:−
2350:⟩
2332:⟩
2328:ψ
2315:†
2292:⟨
2277:ϕ
2274:⟨
2262:⟩
2244:⟩
2240:ψ
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2202:ϕ
2199:⟨
2196:≡
2193:⟩
2179:⟨
2176:⟩
2173:ψ
2165:ϕ
2162:⟨
2112:⟩
2108:ψ
2077:⟩
2073:ϕ
2017:ϕ
1997:α
1971:⟩
1967:ϕ
1953:⟩
1949:ϕ
1930:ϕ
1924:α
1901:⟩
1883:⟩
1879:ϕ
1823:⟩
1788:⟩
1784:ϕ
1756:⊗
1712:⊗
1693:⊗
1646:⊗
1627:yields a
1584:ℏ
1567:−
1502:collapses
1473:⊗
1444:⟩
1432:⊗
1423:⟩
1419:ϕ
1381:⟩
1377:ϕ
1346:⟩
1303:⟩
1299:ϕ
1291:⊗
1282:⟩
1278:ϕ
1247:⟩
1235:⊗
1226:⟩
1222:ϕ
1187:⟩
1183:ϕ
1148:⟩
1109:⟩
1105:ϕ
920:entangled
845:Zeilinger
690:Ehrenfest
419:Equations
96:⟩
93:Ψ
82:^
70:⟩
67:Ψ
44:ℏ
5020:OpenQASM
4996:Transmon
4873:Physical
4673:Quantum
4574:Grover's
4348:Holevo's
4321:Theorems
4271:timeline
4261:NISQ era
3984:40349990
3976:18518590
3923:11724387
3915:10060796
3693:(2011).
3677:16588882
3439:55890485
3313:See also
1525:combined
1514:qubit(s)
1007:Wootters
930:and the
770:Millikan
695:Einstein
680:Davisson
635:Blackett
620:Aharonov
488:Ensemble
468:Bayesian
373:Overview
254:Collapse
234:Symmetry
125:Glossary
5010:Quantum
4948:Kane QC
4807:Quantum
4735:Quantum
4664:PostBQP
4634:Quantum
4619:Simon's
4412:Quantum
4249:General
4188:1446565
4180:9913670
4160:Bibcode
4079:Bibcode
3956:Bibcode
3895:Bibcode
3842:Bibcode
3734:Bibcode
3657:Bibcode
3515:Bibcode
3487:4339227
3467:Bibcode
3409:Bibcode
3003:ancilla
2932:general
2807:either
1736:Theorem
1523:of the
1013:and by
993:History
888:physics
810:Simmons
800:Rydberg
765:Moseley
745:Kramers
735:Hilbert
720:Glauber
715:Feynman
700:Everett
670:Compton
441:Rydberg
130:History
5028:IBM QX
5024:Qiskit
4963:NMR QC
4941:-based
4845:Steane
4816:Codes
4614:Shor's
4520:SARG04
4328:Bell's
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3458:Nature
3437:
3419:
3039:Steane
1508:of an
1021:for a
957:, the
936:qubits
890:, the
840:Zeeman
835:Wigner
785:Planck
755:Landau
740:Jordan
391:Matrix
321:Popper
4850:Toric
4293:Qubit
4184:S2CID
4150:arXiv
4125:arXiv
4103:S2CID
4069:arXiv
4028:arXiv
4026:(2).
3999:arXiv
3980:S2CID
3946:arXiv
3919:S2CID
3885:arXiv
3858:S2CID
3832:arXiv
3804:S2CID
3776:arXiv
3750:S2CID
3673:S2CID
3647:arXiv
3618:S2CID
3600:arXiv
3563:(PDF)
3483:S2CID
3435:S2CID
2961:is a
1205:(see
1015:Dieks
1011:Zurek
910:and
795:Raman
780:Pauli
775:Onnes
710:Fermi
685:Debye
675:Dirac
640:Bloch
630:Bethe
498:Local
436:Pauli
426:Dirac
224:State
5042:Cirq
5033:Quil
4939:Spin
4835:Shor
4515:BB84
4448:LOCC
4176:PMID
4095:ISSN
3972:PMID
3911:PMID
3794:ISBN
3703:ISBN
3571:ISBN
3305:on
3252:for
3037:and
3035:Shor
2095:and
2029:and
1806:and
1042:and
1009:and
1001:and
955:dual
830:Wien
825:Weyl
790:Rabi
760:Laue
750:Lamb
705:Fock
660:Bose
655:Born
650:Bohr
645:Bohm
625:Bell
4856:gnu
4820:CSS
4697:XEB
4659:QMA
4654:QIP
4649:EQP
4644:BQP
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4383:PBR
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4087:doi
4038:doi
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