551:
2768:
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593:, where high-precision measurements must be made for the relative distance between two widely separated masses. However, the measurements described by quantum metrology are currently not used in this setting, being difficult to implement. Furthermore, there are other sources of noise affecting the detection of gravitational waves which must be overcome first. Nevertheless, plans may call for the use of quantum metrology in LIGO.
832:
entanglement is needed to achieve a better and better accuracy in parameter estimation. Additionally, entanglement in multiple degrees of freedom of quantum systems (known as "hyperentanglement"), can be used to enhance precision, with enhancement arising from entanglement in each degree of freedom.
831:
is needed to outperform classical interferometry in magnetrometry with a fully polarized ensemble of spins. It has been proved that a similar relation is generally valid for any linear interferometer, independent of the details of the scheme. Moreover, higher and higher levels of multipartite
601:
A central question of quantum metrology is how the precision, i.e., the variance of the parameter estimation, scales with the number of particles. Classical interferometers cannot overcome the shot-noise limit. This limit is also frequently called standard quantum limit (SQL)
32:. This field promises to develop measurement techniques that give better precision than the same measurement performed in a classical framework. Together with quantum hypothesis testing, it represents an important theoretical model at the basis of quantum sensing.
154:
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761:
818:
662:
1138:
Braun, Daniel; Adesso, Gerardo; Benatti, Fabio; Floreanini, Roberto; Marzolino, Ugo; Mitchell, Morgan W.; Pirandola, Stefano (September 5, 2018). "Quantum-enhanced measurements without entanglement".
256:
307:
1085:
PezzĂš, Luca; Smerzi, Augusto; Oberthaler, Markus K.; Schmied, Roman; Treutlein, Philipp (September 5, 2018). "Quantum metrology with nonclassical states of atomic ensembles".
24:
is the study of making high-resolution and highly sensitive measurements of physical parameters using quantum theory to describe the physical systems, particularly exploiting
184:
503:
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Escher, B. M.; Filho, R. L. de Matos; Davidovich, L. (May 2011). "General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology".
224:
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337:
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459:
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1443:"Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics"
1997:
1320:
Braunstein, Samuel L.; Caves, Carlton M.; Milburn, G.J. (April 1996). "Generalized
Uncertainty Relations: Theory, Examples, and Lorentz Invariance".
66:
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However, if uncorrelated local noise is present, then for large particle numbers the scaling of the precision returns to shot-noise scaling
1926:
Walborn, S. P.; Pimentel, A. H.; Filho, R. L. de Matos; Davidovich, L. (January 2018). "Quantum-enhanced sensing from hyperentanglement".
529:. In quantum illumination protocols, two-mode squeezed states are widely studied to overcome the limit of classical states represented in
360:
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the number of particles. Shot-noise limit is known to be asymptotically achievable using coherent states and homodyne detection.
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Guha, SaikatĆ; Erkmen, Baris (November 10, 2009). "Gaussian-state quantum-illumination receivers for target detection".
2020:
1983:
1241:
Pirandola, S; Bardhan, B. R.; Gehring, T.; Weedbrook, C.; Lloyd, S. (2018). "Advances in photonic quantum sensing".
885:
Braunstein, Samuel L.; Caves, Carlton M. (May 30, 1994). "Statistical distance and the geometry of quantum states".
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to perform accurate phase measurements. A similar effect can be produced using less exotic states such as
260:
Typically, the system is composed of many particles, and the
Hamiltonian is a sum of single-particle terms
1441:
Kimble, H. J.; Levin, Yuri; Matsko, Andrey B.; Thorne, Kip S.; Vyatchanin, Sergey P. (December 26, 2001).
3036:
2495:
2402:
827:
There are strong links between quantum metrology and quantum information science. It has been shown that
981:
Giovannetti, Vittorio; Lloyd, Seth; Maccone, Lorenzo (March 31, 2011). "Advances in quantum metrology".
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Pezzé, Luca; Smerzi, Augusto (2009). "Entanglement, Nonlinear
Dynamics, and the Heisenberg Limit".
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562:. The reason given is: Advanced LIGO has already deployed measurements with quantum squeezing..
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th particle. In this case, there is no interaction between the particles, and we talk about
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Kapale, Kishor T.; Didomenico, Leo D.; Kok, Pieter; Dowling, Jonathan P. (July 18, 2005).
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Please help update this article to reflect recent events or newly available information.
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Paris, Matteo G. A. (November 21, 2011). "Quantum
Estimation for Quantum Technology".
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Demkowicz-DobrzaĆski, RafaĆ; KoĆodyĆski, Jan; GuĆŁÄ, MÄdÄlin (September 18, 2012).
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1375:"Quantum lithography, entanglement and Heisenberg-limited parameter estimation"
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149:{\displaystyle \varrho (\theta )=\exp(-iH\theta )\varrho _{0}\exp(+iH\theta ),}
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TĂłth, GĂ©za (2012). "Multipartite entanglement and high-precision metrology".
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Hyllus, Philipp (2012). "Fisher information and multiparticle entanglement".
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Kok, Pieter; Braunstein, Samuel L; Dowling, Jonathan P (July 28, 2004).
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SĂžrensen, Anders S. (2001). "Entanglement and
Extreme Spin Squeezing".
1605:
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431:{\displaystyle (\Delta \theta )^{2}\geq {\frac {1}{mF_{\rm {Q}}}},}
1887:
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1036:"Quantum metrology from a quantum information science perspective"
995:
950:
16:
Application of quantum entanglement to high-precision measurement
1287:
1240:
581:
An important application of particular note is the detection of
1925:
756:{\displaystyle (\Delta \theta )^{2}\geq {\tfrac {1}{mN^{2}}}.}
40:
A basic task of quantum metrology is estimating the parameter
1572:"The elusive Heisenberg limit in quantum-enhanced metrology"
1137:
1084:
813:{\displaystyle (\Delta \theta )^{2}\propto {\tfrac {1}{N}}.}
586:
1293:
822:
657:{\displaystyle (\Delta \theta )^{2}\geq {\tfrac {1}{mN}},}
1440:
1218:
Probabilistic and statistical aspects of quantum theory
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635:
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The achievable precision is bounded from below by the
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1040:Journal of Physics A: Mathematical and Theoretical
1034:TĂłth, GĂ©za; Apellaniz, Iagoba (October 24, 2014).
893:(22). American Physical Society (APS): 3439â3443.
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1456:(2). American Physical Society (APS): 022002.
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1991:
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461:is the number of independent repetitions and
938:International Journal of Quantum Information
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949:
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1220:( ed.). Scuola Normale Superiore.
1193:Quantum detection and estimation theory
823:Relation to quantum information science
186:is the initial state of the system and
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1819:
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517:One example of note is the use of the
226:is estimated based on measurements on
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2005:
1979:
935:
1872:
842:Outline of metrology and measurement
544:
537:can be used for phase measurements.
1303:The Old and New Concepts of Physics
13:
776:
707:
615:
474:
401:
367:
251:{\displaystyle \varrho (\theta ).}
206:is the Hamiltonian of the system.
14:
3118:
1296:"Quantum Interferometric Sensors"
302:{\displaystyle H=\sum _{k}H_{k},}
3068:
2767:
2766:
1388:(8). IOP Publishing: S811âS815.
690:Quantum metrology can reach the
549:
1919:
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1813:
1752:
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1563:
1510:
1434:
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597:Scaling and the effect of noise
540:
2716:Relativistic quantum mechanics
1791:10.1103/physrevlett.102.100401
1234:
1209:
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1071:10.1088/1751-8113/47/42/424006
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73:
1:
2694:Quantum statistical mechanics
2471:Quantum differential calculus
2393:Delayed-choice quantum eraser
2161:Symmetry in quantum mechanics
868:
1216:Holevo, Alexander S (1982).
1170:10.1103/RevModPhys.90.035006
1117:10.1103/RevModPhys.90.035005
179:{\displaystyle \varrho _{0}}
7:
3097:Quantum information science
2496:Quantum stochastic calculus
2486:Quantum measurement problem
2408:MachâZehnder interferometer
1730:10.1103/physrevlett.86.4431
907:10.1103/physrevlett.72.3439
835:
523:MachâZehnder interferometer
512:
498:{\displaystyle F_{\rm {Q}}}
10:
3123:
3052:Thermoacoustic heat engine
1958:10.1103/PhysRevA.97.010301
1905:10.1103/physreva.85.022322
1852:10.1103/physreva.85.022321
1549:10.1103/PhysRevA.80.052310
1480:10.1103/physrevd.65.022002
839:
507:quantum Fisher information
3064:
3037:Immersive virtual reality
2997:
2827:
2820:
2762:
2724:
2676:
2556:Quantum complexity theory
2534:Quantum cellular automata
2509:
2441:
2375:
2288:
2237:
2224:Path integral formulation
2191:
2056:
2013:
1412:10.1088/1464-4266/6/8/029
1273:10.1038/s41566-018-0301-6
1140:Reviews of Modern Physics
1087:Reviews of Modern Physics
960:10.1142/S0219749909004839
558:This section needs to be
3020:Digital scent technology
2623:Quantum machine learning
2603:Quantum key distribution
2593:Quantum image processing
2583:Quantum error correction
2433:Wheeler's delayed choice
352:quantum Cramér-Rao bound
60:of the unitary dynamics
36:Mathematical foundations
2539:Quantum finite automata
1761:Physical Review Letters
1700:Physical Review Letters
1013:10.1038/nphoton.2011.35
887:Physical Review Letters
583:gravitational radiation
533:. In atomic ensembles,
219:{\displaystyle \theta }
53:{\displaystyle \theta }
3042:Magnetic refrigeration
2643:Quantum neural network
1352:10.1006/aphy.1996.0040
814:
757:
681:
658:
499:
455:
432:
345:linear interferometers
333:
303:
252:
220:
200:
180:
150:
54:
3015:Cloak of invisibility
2814:Emerging technologies
2668:Quantum teleportation
2181:Waveâparticle duality
1576:Nature Communications
848:Dimensional metrology
815:
758:
682:
659:
500:
456:
433:
334:
332:{\displaystyle H_{k}}
304:
253:
221:
201:
181:
151:
55:
2699:Quantum field theory
2628:Quantum metamaterial
2573:Quantum cryptography
2303:Consistent histories
1191:Helstrom, C (1976).
829:quantum entanglement
770:
701:
671:
609:
591:Virgo interferometer
585:in projects such as
535:spin squeezed states
465:
445:
361:
316:
267:
230:
210:
190:
163:
67:
44:
26:quantum entanglement
3047:Phased-array optics
3005:Acoustic levitation
2684:Quantum fluctuation
2653:Quantum programming
2613:Quantum logic gates
2598:Quantum information
2578:Quantum electronics
2038:Classical mechanics
1950:2018PhRvA..97a0301W
1897:2012PhRvA..85b2322T
1844:2012PhRvA..85b2321H
1783:2009PhRvL.102j0401P
1722:2001PhRvL..86.4431S
1661:2011NatPh...7..406E
1598:2012NatCo...3.1063D
1541:2009PhRvA..80e2310G
1472:2001PhRvD..65b2002K
1404:2004JOptB...6S.811K
1344:1996AnPhy.247..135B
1265:2018NaPho..12..724P
1162:2018RvMP...90c5006B
1109:2018RvMP...90c5005P
1062:2014JPhA...47P4006T
1005:2011NaPho...5..222G
944:(supp01): 125â137.
899:1994PhRvL..72.3439B
2737:in popular culture
2519:Quantum algorithms
2367:Von NeumannâWigner
2347:Objective collapse
2043:Old quantum theory
1606:10.1038/ncomms2067
1195:. Academic Press.
853:Forensic metrology
810:
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216:
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146:
50:
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3083:
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2867:complexity theory
2852:cellular automata
2780:
2779:
2754:Quantum mysticism
2732:Schrödinger's cat
2663:Quantum simulator
2633:Quantum metrology
2561:Quantum computing
2524:Quantum amplifier
2501:Quantum spacetime
2466:Quantum cosmology
2456:Quantum chemistry
2156:Scattering theory
2104:Zero-point energy
2099:Degenerate levels
2007:Quantum mechanics
1928:Physical Review A
1875:Physical Review A
1822:Physical Review A
1706:(20): 4431â4434.
1669:10.1038/nphys1958
1519:Physical Review A
1450:Physical Review D
1322:Annals of Physics
1227:978-88-7642-378-9
804:
747:
680:{\displaystyle N}
648:
579:
578:
454:{\displaystyle m}
423:
276:
199:{\displaystyle H}
22:Quantum metrology
3114:
3072:
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2949:machine learning
2924:key distribution
2909:image processing
2899:error correction
2825:
2824:
2807:
2800:
2793:
2784:
2783:
2770:
2769:
2481:Quantum geometry
2476:Quantum dynamics
2333:Superdeterminism
2265:RaritaâSchwinger
2214:Matrix mechanics
2069:Braâket notation
2000:
1993:
1986:
1977:
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1969:
1943:
1934:(1): 010301(R).
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1395:quant-ph/0402083
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1335:quant-ph/9507004
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1243:Nature Photonics
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983:Nature Photonics
978:
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692:Heisenberg limit
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2904:finite automata
2816:
2811:
2781:
2776:
2758:
2744:Wigner's friend
2720:
2711:Quantum gravity
2672:
2658:Quantum sensing
2638:Quantum network
2618:Quantum machine
2588:Quantum imaging
2551:Quantum circuit
2546:Quantum channel
2505:
2451:Quantum biology
2437:
2413:ElitzurâVaidman
2388:DavissonâGermer
2371:
2323:Hidden-variable
2313:de BroglieâBohm
2290:Interpretations
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2074:Complementarity
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1309:(3â4): 225â240.
1298:
1292:
1288:
1249:(12): 724â733.
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858:Smart Metrology
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531:coherent states
527:squeezed states
515:
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466:
463:
462:
446:
443:
442:
400:
399:
395:
391:
386:
377:
373:
362:
359:
358:
323:
319:
317:
314:
313:
290:
286:
280:
268:
265:
264:
231:
228:
227:
211:
208:
207:
191:
188:
187:
170:
166:
164:
161:
160:
113:
109:
68:
65:
64:
45:
42:
41:
38:
17:
12:
11:
5:
3120:
3110:
3109:
3104:
3102:Quantum optics
3099:
3082:
3081:
3079:
3078:
3065:
3062:
3061:
3058:
3057:
3055:
3054:
3049:
3044:
3039:
3034:
3033:
3032:
3022:
3017:
3012:
3007:
3001:
2999:
2995:
2994:
2992:
2991:
2986:
2981:
2976:
2971:
2966:
2964:neural network
2961:
2956:
2951:
2946:
2941:
2936:
2931:
2926:
2921:
2916:
2911:
2906:
2901:
2896:
2891:
2886:
2885:
2884:
2874:
2869:
2864:
2859:
2854:
2849:
2844:
2839:
2833:
2831:
2822:
2818:
2817:
2810:
2809:
2802:
2795:
2787:
2778:
2777:
2775:
2774:
2763:
2760:
2759:
2757:
2756:
2751:
2746:
2741:
2740:
2739:
2728:
2726:
2722:
2721:
2719:
2718:
2713:
2708:
2707:
2706:
2696:
2691:
2689:Casimir effect
2686:
2680:
2678:
2674:
2673:
2671:
2670:
2665:
2660:
2655:
2650:
2648:Quantum optics
2645:
2640:
2635:
2630:
2625:
2620:
2615:
2610:
2605:
2600:
2595:
2590:
2585:
2580:
2575:
2570:
2569:
2568:
2558:
2553:
2548:
2543:
2542:
2541:
2531:
2526:
2521:
2515:
2513:
2507:
2506:
2504:
2503:
2498:
2493:
2488:
2483:
2478:
2473:
2468:
2463:
2458:
2453:
2447:
2445:
2439:
2438:
2436:
2435:
2430:
2425:
2423:Quantum eraser
2420:
2415:
2410:
2405:
2400:
2395:
2390:
2385:
2379:
2377:
2373:
2372:
2370:
2369:
2364:
2359:
2354:
2349:
2344:
2339:
2338:
2337:
2336:
2335:
2320:
2315:
2310:
2305:
2300:
2294:
2292:
2286:
2285:
2283:
2282:
2277:
2272:
2267:
2262:
2257:
2252:
2247:
2241:
2239:
2235:
2234:
2232:
2231:
2226:
2221:
2216:
2211:
2206:
2201:
2195:
2193:
2189:
2188:
2186:
2185:
2184:
2183:
2178:
2168:
2163:
2158:
2153:
2148:
2143:
2138:
2133:
2128:
2123:
2118:
2113:
2108:
2107:
2106:
2101:
2096:
2091:
2081:
2079:Density matrix
2076:
2071:
2066:
2060:
2058:
2054:
2053:
2051:
2050:
2045:
2040:
2035:
2034:
2033:
2023:
2017:
2015:
2011:
2010:
2003:
2002:
1995:
1988:
1980:
1972:
1971:
1918:
1865:
1812:
1767:(10): 100401.
1751:
1690:
1645:(5): 406â411.
1639:Nature Physics
1629:
1562:
1509:
1433:
1365:
1328:(1): 135â173.
1312:
1286:
1233:
1226:
1208:
1201:
1183:
1130:
1077:
1046:(42): 424006.
1026:
989:(4): 222â229.
973:
928:
873:
872:
870:
867:
866:
865:
863:Time metrology
860:
855:
850:
840:Main article:
837:
834:
824:
821:
809:
803:
800:
794:
789:
785:
781:
778:
775:
764:
763:
752:
743:
739:
735:
731:
725:
720:
716:
712:
709:
706:
676:
665:
664:
653:
646:
643:
639:
633:
628:
624:
620:
617:
614:
598:
595:
577:
576:
557:
555:
548:
542:
539:
514:
511:
494:
491:
488:
485:
482:
476:
471:
450:
439:
438:
427:
421:
418:
415:
412:
409:
403:
398:
394:
390:
385:
380:
376:
372:
369:
366:
326:
322:
310:
309:
298:
293:
289:
283:
279:
275:
272:
247:
244:
241:
238:
235:
215:
195:
173:
169:
157:
156:
145:
142:
139:
136:
133:
130:
127:
124:
121:
116:
112:
108:
105:
102:
99:
96:
93:
90:
87:
84:
81:
78:
75:
72:
49:
37:
34:
15:
9:
6:
4:
3:
2:
3119:
3108:
3105:
3103:
3100:
3098:
3095:
3094:
3092:
3077:
3076:
3067:
3066:
3063:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3031:
3030:Plasma window
3028:
3027:
3026:
3023:
3021:
3018:
3016:
3013:
3011:
3008:
3006:
3003:
3002:
3000:
2996:
2990:
2989:teleportation
2987:
2985:
2982:
2980:
2977:
2975:
2972:
2970:
2967:
2965:
2962:
2960:
2957:
2955:
2952:
2950:
2947:
2945:
2942:
2940:
2937:
2935:
2932:
2930:
2927:
2925:
2922:
2920:
2917:
2915:
2912:
2910:
2907:
2905:
2902:
2900:
2897:
2895:
2892:
2890:
2887:
2883:
2880:
2879:
2878:
2875:
2873:
2870:
2868:
2865:
2863:
2860:
2858:
2855:
2853:
2850:
2848:
2845:
2843:
2840:
2838:
2835:
2834:
2832:
2830:
2826:
2823:
2819:
2815:
2808:
2803:
2801:
2796:
2794:
2789:
2788:
2785:
2773:
2765:
2764:
2761:
2755:
2752:
2750:
2747:
2745:
2742:
2738:
2735:
2734:
2733:
2730:
2729:
2727:
2723:
2717:
2714:
2712:
2709:
2705:
2702:
2701:
2700:
2697:
2695:
2692:
2690:
2687:
2685:
2682:
2681:
2679:
2675:
2669:
2666:
2664:
2661:
2659:
2656:
2654:
2651:
2649:
2646:
2644:
2641:
2639:
2636:
2634:
2631:
2629:
2626:
2624:
2621:
2619:
2616:
2614:
2611:
2609:
2608:Quantum logic
2606:
2604:
2601:
2599:
2596:
2594:
2591:
2589:
2586:
2584:
2581:
2579:
2576:
2574:
2571:
2567:
2564:
2563:
2562:
2559:
2557:
2554:
2552:
2549:
2547:
2544:
2540:
2537:
2536:
2535:
2532:
2530:
2527:
2525:
2522:
2520:
2517:
2516:
2514:
2512:
2508:
2502:
2499:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2477:
2474:
2472:
2469:
2467:
2464:
2462:
2461:Quantum chaos
2459:
2457:
2454:
2452:
2449:
2448:
2446:
2444:
2440:
2434:
2431:
2429:
2428:SternâGerlach
2426:
2424:
2421:
2419:
2416:
2414:
2411:
2409:
2406:
2404:
2401:
2399:
2396:
2394:
2391:
2389:
2386:
2384:
2381:
2380:
2378:
2374:
2368:
2365:
2363:
2362:Transactional
2360:
2358:
2355:
2353:
2352:Quantum logic
2350:
2348:
2345:
2343:
2340:
2334:
2331:
2330:
2329:
2326:
2325:
2324:
2321:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2295:
2293:
2291:
2287:
2281:
2278:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2242:
2240:
2236:
2230:
2227:
2225:
2222:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2202:
2200:
2197:
2196:
2194:
2190:
2182:
2179:
2177:
2174:
2173:
2172:
2171:Wave function
2169:
2167:
2164:
2162:
2159:
2157:
2154:
2152:
2149:
2147:
2146:Superposition
2144:
2142:
2141:Quantum state
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2112:
2109:
2105:
2102:
2100:
2097:
2095:
2094:Excited state
2092:
2090:
2087:
2086:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2061:
2059:
2055:
2049:
2046:
2044:
2041:
2039:
2036:
2032:
2029:
2028:
2027:
2024:
2022:
2019:
2018:
2016:
2012:
2008:
2001:
1996:
1994:
1989:
1987:
1982:
1981:
1978:
1967:
1963:
1959:
1955:
1951:
1947:
1942:
1937:
1933:
1929:
1922:
1914:
1910:
1906:
1902:
1898:
1894:
1889:
1884:
1881:(2): 022322.
1880:
1876:
1869:
1861:
1857:
1853:
1849:
1845:
1841:
1836:
1831:
1828:(2): 022321.
1827:
1823:
1816:
1808:
1804:
1800:
1796:
1792:
1788:
1784:
1780:
1775:
1770:
1766:
1762:
1755:
1747:
1743:
1739:
1735:
1731:
1727:
1723:
1719:
1714:
1709:
1705:
1701:
1694:
1686:
1682:
1678:
1674:
1670:
1666:
1662:
1658:
1653:
1648:
1644:
1640:
1633:
1625:
1621:
1616:
1611:
1607:
1603:
1599:
1595:
1590:
1585:
1581:
1577:
1573:
1566:
1558:
1554:
1550:
1546:
1542:
1538:
1533:
1528:
1525:(5): 052310.
1524:
1520:
1513:
1505:
1501:
1497:
1493:
1489:
1488:1969.1/181491
1485:
1481:
1477:
1473:
1469:
1464:
1463:gr-qc/0008026
1459:
1455:
1451:
1444:
1437:
1429:
1425:
1421:
1417:
1413:
1409:
1405:
1401:
1396:
1391:
1387:
1383:
1376:
1369:
1361:
1357:
1353:
1349:
1345:
1341:
1336:
1331:
1327:
1323:
1316:
1308:
1304:
1297:
1290:
1282:
1278:
1274:
1270:
1266:
1262:
1257:
1252:
1248:
1244:
1237:
1229:
1223:
1219:
1212:
1204:
1198:
1194:
1187:
1179:
1175:
1171:
1167:
1163:
1159:
1154:
1149:
1146:(3): 035006.
1145:
1141:
1134:
1126:
1122:
1118:
1114:
1110:
1106:
1101:
1096:
1093:(3): 035005.
1092:
1088:
1081:
1072:
1067:
1063:
1059:
1054:
1049:
1045:
1041:
1037:
1030:
1022:
1018:
1014:
1010:
1006:
1002:
997:
992:
988:
984:
977:
969:
965:
961:
957:
952:
947:
943:
939:
932:
924:
920:
916:
912:
908:
904:
900:
896:
892:
888:
881:
879:
874:
864:
861:
859:
856:
854:
851:
849:
846:
845:
843:
833:
830:
820:
807:
801:
798:
792:
787:
779:
750:
741:
737:
733:
729:
723:
718:
710:
697:
696:
695:
693:
688:
674:
651:
644:
641:
637:
631:
626:
618:
605:
604:
603:
594:
592:
588:
584:
573:
561:
556:
547:
546:
538:
536:
532:
528:
524:
520:
510:
508:
489:
486:
483:
469:
448:
425:
416:
413:
410:
396:
392:
388:
383:
378:
370:
357:
356:
355:
353:
348:
346:
342:
324:
320:
296:
291:
287:
281:
277:
273:
270:
263:
262:
261:
258:
245:
239:
233:
213:
193:
171:
167:
143:
137:
134:
131:
128:
122:
119:
114:
110:
103:
100:
97:
94:
88:
85:
82:
76:
70:
63:
62:
61:
47:
33:
31:
27:
23:
19:
3073:
3010:Anti-gravity
2954:metamaterial
2882:post-quantum
2877:cryptography
2632:
2491:Quantum mind
2403:FranckâHertz
2245:KleinâGordon
2199:Formulations
2192:Formulations
2121:Interference
2111:Entanglement
2089:Ground state
2084:Energy level
2057:Fundamentals
2021:Introduction
1931:
1927:
1921:
1878:
1874:
1868:
1825:
1821:
1815:
1764:
1760:
1754:
1703:
1699:
1693:
1642:
1638:
1632:
1579:
1575:
1565:
1522:
1518:
1512:
1453:
1449:
1436:
1385:
1381:
1368:
1325:
1321:
1315:
1306:
1302:
1289:
1246:
1242:
1236:
1217:
1211:
1192:
1186:
1143:
1139:
1133:
1090:
1086:
1080:
1043:
1039:
1029:
986:
982:
976:
941:
937:
931:
890:
886:
826:
765:
689:
666:
600:
580:
570:October 2022
567:
559:
541:Applications
516:
440:
349:
340:
339:acts on the
311:
259:
158:
39:
28:and quantum
21:
20:
18:
3025:Force field
2974:programming
2934:logic clock
2919:information
2894:electronics
2749:EPR paradox
2529:Quantum bus
2398:Double-slit
2376:Experiments
2342:Many-worlds
2280:Schrödinger
2229:Phase space
2219:Schrödinger
2209:Interaction
2166:Uncertainty
2136:Nonlocality
2131:Measurement
2126:Decoherence
2116:Hamiltonian
3091:Categories
2939:logic gate
2837:algorithms
2677:Extensions
2511:Technology
2357:Relational
2308:Copenhagen
2204:Heisenberg
2151:Tunnelling
2014:Background
1941:1709.04513
1256:1811.01969
1202:0123400503
1153:1701.05152
1100:1609.01609
869:References
519:NOON state
3107:Metrology
2984:simulator
2872:computing
2842:amplifier
2383:Bell test
2238:Equations
2064:Born rule
1913:119110009
1888:1006.4368
1860:118652590
1835:1006.4366
1774:0711.4840
1746:206327094
1677:1745-2481
1652:1201.1693
1589:1201.3940
1557:109058131
1532:0911.0950
1496:0556-2821
1420:1464-4266
1178:119081121
1125:119250709
1053:1405.4878
996:1102.2318
951:0804.2981
915:0031-9007
793:∝
780:θ
777:Δ
724:≥
711:θ
708:Δ
694:given by
667:where is
632:≥
619:θ
616:Δ
484:ϱ
411:ϱ
384:≥
371:θ
368:Δ
278:∑
240:θ
234:ϱ
214:θ
168:ϱ
138:θ
123:
111:ϱ
104:θ
95:−
89:
77:θ
71:ϱ
48:θ
30:squeezing
2889:dynamics
2772:Category
2566:Timeline
2318:Ensemble
2298:Bayesian
2260:Majorana
2176:Collapse
2048:Glossary
2031:Timeline
1966:73689445
1807:13095638
1799:19392092
1738:11384252
1685:12391055
1624:22990859
1582:: 1063.
1504:15339393
1428:15255876
1281:53626745
1021:12591819
923:10056200
836:See also
513:Examples
2979:sensing
2959:network
2944:machine
2914:imaging
2862:circuit
2857:channel
2829:Quantum
2725:Related
2704:History
2443:Science
2275:Rydberg
2026:History
1946:Bibcode
1893:Bibcode
1840:Bibcode
1779:Bibcode
1718:Bibcode
1657:Bibcode
1615:3658100
1594:Bibcode
1537:Bibcode
1468:Bibcode
1400:Bibcode
1340:Bibcode
1261:Bibcode
1158:Bibcode
1105:Bibcode
1058:Bibcode
1001:Bibcode
968:2365312
895:Bibcode
589:or the
560:updated
505:is the
2969:optics
2821:Fields
2418:Popper
1964:
1911:
1858:
1805:
1797:
1744:
1736:
1683:
1675:
1622:
1612:
1555:
1502:
1494:
1426:
1418:
1360:358923
1358:
1279:
1224:
1199:
1176:
1123:
1019:
966:
921:
913:
441:where
312:where
159:where
2998:Other
2929:logic
2328:Local
2270:Pauli
2250:Dirac
1962:S2CID
1936:arXiv
1909:S2CID
1883:arXiv
1856:S2CID
1830:arXiv
1803:S2CID
1769:arXiv
1742:S2CID
1708:arXiv
1681:S2CID
1647:arXiv
1584:arXiv
1553:S2CID
1527:arXiv
1500:S2CID
1458:arXiv
1446:(PDF)
1424:S2CID
1390:arXiv
1378:(PDF)
1356:S2CID
1330:arXiv
1299:(PDF)
1277:S2CID
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