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then be flawlessly identified from the measured value. Moreover, it can be easily shown that if the individual states are not orthogonal to each other, there is no way to tell them apart with certainty. Therefore, in such a case, it is always necessary to take into account the possibility of incorrect or inconclusive determination of the state of the system. However, there are techniques that try to alleviate this deficiency. With exceptions, these techniques can be divided into two groups, namely those based on error minimization and then those that allow the state to be determined unambiguously in exchange for lower efficiency.
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requires the use of generalized measurement, in which the elements of the POVM set are taken as measurement operators. Both groups of techniques are currently the subject of active, primarily theoretical, research, and apart from a number of special cases, there is no general solution that would allow choosing measurement operators in the form of expressible analytical formula.
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from the 60s and 70s of the 20th century and in its basic form consists in the implementation of projective quantum measurement, where the measurement operators are projective representations. The second group is based on the conclusions of a scientific article published by ID Ivanovich in 1987 and
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collectively refers to quantum-informatics techniques, with the help of which, by performing a small number of measurements on a physical system, its specific quantum state can be identified . And this is provided that the set of states in which the system can be is known in advance, and we only need
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If the set of states in which the investigated system can be is represented by orthogonal vectors, the situation is particularly simple. To unambiguously determine the state of the system, it is enough to perform a quantum measurement in the basis formed by these vectors. The given quantum state can
1180:{\displaystyle {\begin{aligned}P_{\text{success}}&=p_{0}\operatorname {tr} (\sigma _{0}E_{0})+p_{1}\operatorname {tr} (\sigma _{1}E_{1})\\&=p_{0}\operatorname {tr} (\sigma _{0}E_{0})+p_{1}\operatorname {tr} (\sigma _{1}I-\sigma _{1}E_{0})\\&=p_{1}+\operatorname {tr} \end{aligned}}}
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If the task is to discriminate between more than two quantum states, there is no general formula for the optimal POVM and success probability. Nonetheless, the optimal success probability, for the task of discriminating between the elements of a given ensemble
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62:, is a useful starting point for translations, but translators must revise errors as necessary and confirm that the translation is accurate, rather than simply copy-pasting machine-translated text into the English Knowledge (XXG).
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to determine which one it is. This assumption distinguishes such techniques from quantum tomography, which does not impose additional requirements on the state of the system, but requires many times more measurements.
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1812:. An alternative approach to discriminate between a given ensemble of states is to the use the so-called Pretty Good Measurement (PGM), also known as the
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Do not translate text that appears unreliable or low-quality. If possible, verify the text with references provided in the foreign-language article.
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Content in this edit is translated from the existing Czech
Knowledge (XXG) article at ]; see its history for attribution.
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While this expression cannot be given an explicit form in the general case, it can be solved numerically via
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1397:{\displaystyle P_{\text{success}}={\frac {1}{2}}+{\frac {1}{2}}\|p_{0}\sigma _{0}-p_{1}\sigma _{1}\|_{1},}
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1644:{\displaystyle P_{\rm {success}}=\max _{\{E_{i}\}}\sum _{i}p_{i}\operatorname {tr} (E_{i}\sigma _{i}).}
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Bae, Joonwoo; Kwek, Leong-Chuan (2015). "Quantum state discrimination and its applications".
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correctly guessing which state was received. Since the probability of the POVM returning the
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Ivanovic, I.D. (August 1987). "How to differentiate between non-orthogonal states".
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More precisely, in its standard formulation, the problem involves performing some
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Montanaro, Ashley (2007). "On the distinguishability of random quantum states".
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Barnett, Stephen M.; Croke, Sarah (2009). "Quantum state discrimination".
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to the source of your translation. A model attribution edit summary is
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The discrimination of two states can be solved optimally using the
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Interactive demonstration about quantum state discrimination
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is the probability of (correctly) guessing the input to be
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The first group of techniques is based on the works of
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1973:. Cambridge University Press.
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2655:Adiabatic quantum computation
1941:10.1088/1751-8113/48/8/083001
841:{\displaystyle E_{1}=I-E_{0}}
2706:Topological quantum computer
1967:Watrous, John (2018-04-26).
1888:10.1016/0375-9601(87)90222-2
1837:. New York: Academic Press.
118:quantum state discrimination
3015:Quantum information science
2984:Quantum information science
2151:Quantum information science
1798:{\displaystyle \sigma _{i}}
1771:{\displaystyle \sigma _{i}}
1682:probability of getting the
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379:{\displaystyle (E_{i})_{i}}
266:occurring with probability
259:{\displaystyle \sigma _{i}}
172:{\displaystyle (E_{i})_{i}}
103:Knowledge (XXG):Translation
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3020:Quantum information theory
2379:quantum gate teleportation
1833:Helstrom, Carl W. (1976).
1523:, can always be written as
54:Machine translation, like
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2508:Quantum Fourier transform
2404:Post-quantum cryptography
2347:Entanglement distillation
2090:10.1007/s00220-007-0221-7
179:on a given unknown state
35:the corresponding article
2994:Quantum mechanics topics
2689:Quantum machine learning
2665:One-way quantum computer
2518:Quantum phase estimation
2419:Quantum key distribution
2352:Monogamy of entanglement
1810:Semidefinite programming
667:comes two probabilities
2601:Randomized benchmarking
2463:Amplitude amplification
1814:square root measurement
101:For more guidance, see
2701:Quantum Turing machine
2694:quantum neural network
2441:Quantum secret sharing
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1933:2015JPhA...48h3001B
1880:1987PhLA..123..257I
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621:. With two states
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1988:978-1-316-84814-2
1868:Physics Letters A
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12:
11:
5:
3043:
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3017:
3007:
3006:
3000:
2999:
2997:
2996:
2986:
2975:
2972:
2971:
2969:
2968:
2966:many others...
2963:
2958:
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2875:
2874:
2869:
2864:
2859:
2853:
2851:
2844:
2843:
2841:
2840:
2838:Trapped-ion QC
2835:
2829:
2827:
2821:
2820:
2818:
2817:
2812:
2807:
2802:
2796:
2794:
2792:Quantum optics
2785:
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2596:Quantum volume
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2515:
2510:
2505:
2500:
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2490:
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2475:
2473:Boson sampling
2470:
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2457:
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2309:
2307:Solovay–Kitaev
2304:
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2279:
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2118:
2117:
2110:
2109:External links
2107:
2104:
2103:
2066:(3): 619–636.
2050:
2013:(8): 238–278.
1994:
1987:
1954:
1901:
1874:(6): 257–259.
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2863:
2862:Spin qubit QC
2860:
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2670:cluster state
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2478:Deutsch–Jozsa
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2442:
2439:
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2425:
2422:
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2417:
2415:
2414:Quantum money
2412:
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2016:
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1971:
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1961:
1959:
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1917:(8): 083001.
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1105:
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850:
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814:
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620:
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587:
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562:
558:
552:
548:
544:
515:
489:
485:
479:
475:
468:
465:
462:
456:
448:
433:has the form
418:
414:
393:
371:
361:
357:
331:
327:
321:
317:
311:
307:
303:
300:
278:
274:
251:
247:
224:
214:
210:
186:
164:
154:
150:
139:
134:
131:
126:
122:
119:
104:
100:
97:
89:
83:
79:
75:
71:
67:
64:
61:
57:
53:
51:
50:
44:
38:
36:
31:You can help
27:
18:
17:
2890:Charge qubit
2815:KLM protocol
2764:
2628:
2618:
2312:Purification
2242:Eastin–Knill
2063:
2059:
2053:
2010:
2006:
1969:
1914:
1910:
1904:
1871:
1867:
1861:
1834:
1828:
1813:
1807:
1679:
1451:
1440:denotes the
1406:
1189:
616:
135:
127:
123:
117:
115:
78:edit summary
69:
40:
32:
2921:programming
2900:Phase qubit
2805:Circuit QED
2277:No-deleting
2219:cloud-based
293:, that is,
3009:Categories
2961:libquantum
2895:Flux qubit
2800:Cavity QED
2749:Bacon–Shor
2739:stabilizer
2267:No-cloning
1924:1707.02571
1820:References
1442:trace norm
1227:eigenspace
713:and POVMs
2867:NV center
2302:Threshold
2282:No-hiding
2247:Gleason's
2020:0810.1970
1949:119199057
1896:0375-9601
1853:316552953
1787:σ
1760:σ
1730:σ
1713:
1627:σ
1610:
1588:∑
1481:σ
1422:‖
1418:⋅
1415:‖
1383:‖
1373:σ
1359:−
1350:σ
1336:‖
1271:σ
1257:−
1248:σ
1223:projector
1149:σ
1135:−
1126:σ
1106:
1058:σ
1054:−
1042:σ
1035:
997:σ
990:
945:σ
938:
900:σ
893:
826:−
768:∑
759:. Since
646:σ
633:σ
578:σ
571:
549:∑
486:σ
469:
415:σ
328:σ
308:∑
301:ρ
248:σ
211:σ
187:ρ
116:The term
96:talk page
2929:OpenQASM
2905:Transmon
2782:Physical
2582:Quantum
2483:Grover's
2257:Holevo's
2230:Theorems
2180:timeline
2170:NISQ era
2098:12516161
2045:15398601
1680:a priori
72:provide
37:in Czech
2919:Quantum
2857:Kane QC
2716:Quantum
2644:Quantum
2573:PostBQP
2543:Quantum
2528:Simon's
2321:Quantum
2158:General
2078:Bibcode
2025:Bibcode
1929:Bibcode
1876:Bibcode
1678:is the
1305:success
868:success
239:, with
94:to the
76:in the
39:.
2937:IBM QX
2933:Qiskit
2872:NMR QC
2850:-based
2754:Steane
2725:Codes
2523:Shor's
2429:SARG04
2237:Bell's
2096:
2043:
1985:
1947:
1894:
1851:
1841:
1407:where
2759:Toric
2202:Qubit
2094:S2CID
2068:arXiv
2041:S2CID
2015:arXiv
1945:S2CID
1919:arXiv
1221:is a
1192:trace
56:DeepL
2951:Cirq
2942:Quil
2848:Spin
2744:Shor
2424:BB84
2357:LOCC
1983:ISBN
1892:ISSN
1849:OCLC
1839:ISBN
442:Prob
138:POVM
70:must
68:You
2765:gnu
2729:CSS
2606:XEB
2568:QMA
2563:QIP
2558:EQP
2553:BQP
2533:VQE
2488:HHL
2292:PBR
2086:doi
2064:273
2033:doi
1975:doi
1937:doi
1884:doi
1872:123
1565:max
1229:of
58:or
3011::
2956:Q#
2092:.
2084:.
2076:.
2062:.
2039:.
2031:.
2023:.
2009:.
1997:^
1981:.
1957:^
1943:.
1935:.
1927:.
1915:48
1913:.
1890:.
1882:.
1870:.
1847:.
1805:.
1710:tr
1607:tr
1444:.
1103:tr
1032:tr
987:tr
935:tr
890:tr
609:.
568:tr
466:tr
2944:–
2935:–
2931:–
2632:2
2629:T
2622:1
2619:T
2143:e
2136:t
2129:v
2100:.
2088::
2080::
2070::
2047:.
2035::
2027::
2017::
2011:1
1991:.
1977::
1951:.
1939::
1931::
1921::
1898:.
1886::
1878::
1855:.
1791:i
1764:i
1739:)
1734:i
1724:i
1720:E
1716:(
1690:i
1664:i
1660:p
1639:.
1636:)
1631:i
1621:i
1617:E
1613:(
1602:i
1598:p
1592:i
1582:}
1577:i
1573:E
1569:{
1561:=
1555:s
1552:s
1549:e
1546:c
1543:c
1540:u
1537:s
1532:P
1509:N
1504:1
1501:=
1498:i
1494:}
1490:)
1485:i
1477:,
1472:i
1468:p
1464:(
1461:{
1426:1
1392:,
1387:1
1377:1
1367:1
1363:p
1354:0
1344:0
1340:p
1331:2
1328:1
1323:+
1318:2
1315:1
1310:=
1301:P
1275:1
1265:1
1261:p
1252:0
1242:0
1238:p
1207:0
1203:E
1171:]
1166:0
1162:E
1158:)
1153:1
1143:1
1139:p
1130:0
1120:0
1116:p
1112:(
1109:[
1100:+
1095:1
1091:p
1087:=
1077:)
1072:0
1068:E
1062:1
1051:I
1046:1
1038:(
1027:1
1023:p
1019:+
1016:)
1011:0
1007:E
1001:0
993:(
982:0
978:p
974:=
964:)
959:1
955:E
949:1
941:(
930:1
926:p
922:+
919:)
914:0
910:E
904:0
896:(
885:0
881:p
877:=
864:P
834:0
830:E
823:I
820:=
815:1
811:E
790:I
787:=
782:i
778:E
772:i
747:}
742:1
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721:{
701:}
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629:{
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592:i
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582:i
574:(
563:i
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545:=
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536:s
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524:u
521:s
516:P
495:)
490:j
480:i
476:E
472:(
463:=
460:)
457:j
453:|
449:i
446:(
419:j
394:i
372:i
368:)
362:i
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354:(
332:i
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304:=
279:i
275:p
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225:i
221:}
215:i
207:{
165:i
161:)
155:i
151:E
147:(
105:.
98:.
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