Nonlocality of symmetric states and its applications in quantum information

155 Pages·2017·1.99 MB·English

by  Zizhu Wang

Acknowledgements
First I would liketo thank my supervisors, Damian Markham and Ge¬rard Cohen, for
providingsupport onthisproject. Averyspecial thankstoDamian Markham,whonot
onlyshowedmetheEnglish(i.e. relaxed,informal,cordial,lazy,etc.) waytodoresearch,
but also becomelikea friend and mentor to me. When I started thisthesis, I wasa
hard corecomputer scientist and Damian wasahard corephysicist. I think wemutually
adapted in these3 years, and now I can proudly call myself afakephysicist. I know
sometimesI’m lazyandannoying,thanksfor not kickingmyass!
Lifeinthequantum groupis, for thelack of abetter word,nonchalant. SoI would
liketothank everyoneinour group,Eleni Diamanti,RomainAlle¬aume,IsabelleZaquine,
Tom Lawson, MarcKaplan, Rupesh Kumar, AnneMarin, AnnaPappa, Hao Qin, Adel
Sohbi, Paul Jouguet and Se¬bastien Kunz-Jacquesfor all the memorable momentsof
grouplifefrom thequantum lunchestotheChristmasdinners.
StimulatingdiscussionswithPabloArrighiandElhamKashefiareoftenverybeneficial.
I wouldalsoliketogivePabloArrighi aspecial thanksfor beingthefirst persontoentrust
mewith aresearch problem.
I would also like to thank Samson Abramsky, GillesDowek, Dan Browne, Gilles
Ze¬mor andTerryRudolph for agreeingtobepart of my jury. I alsowant tothank Terry
for a wonderful summer at Imperial. From thesethreemonthsI learned invaluable
lessonsfrom him on how tobeagood scientist.
Aspecial thankstoFengYan and PingYan,for thecountlessafternoon chatsabout
academia,thepast and thefutureandthecountlessdinnerstogether.
Also, I wish togivemymost sincerethankstomyparents. Aspart of thegeneration
who lived through the cultural revolution, they valueeducation aboveall else. If it
weren’t for their dedicationandhard work I would not havehadthechancetocometo
Franceand becomearesearcher.
Lastly, I wish tothank mybetter half, Wei,for her continued support and encourage-
ment. Wehavebeenthroughthetoughest timeof our livestogether,and thefuturecan
onlybebrighter for usand our family!
ii
Abstract
Thisthesisisabout thenonlocal propertiesof permutation symmetricstatesandthepo-
tential usefulnessof such propertiesinquantum information processing. Thenonlocality
of almost all symmetricstates,except Dickestates,isshown byconstructingan n-party
Hardyparadox. With thehelp of theMajoranarepresentation, suitablemeasurement
settingscanbechosenfor thesesymmetricstateswhichsatisfytheparadox. Anextended
CH inequality can bederived from the probabilistic conditionsof theparadox. This
inequality isshown to beviolated by all symmetricstates. Thenonlocality properties
and entanglement properties of symmetric states are also discussed and compared,
notablywithrespect topersistencyand monogamy. It isshown that thedegeneracy of
somesymmetricstatesislinked tothepersistency,which providesawaytousedevice
independent teststoseparatenonlocality classes. It isalsoshown that theinequalities
usedtoshow thenonlocalityof all symmetricstatesarenot strictlymonogamous. Anew
inequalityfor Dickestatesisshown tobemonogamouswhen thenumber of partiesgoes
toinfinity. But all theseinequalitiescan not detect genuinenonlocality. Applicationsof
nonlocalitytocommunicationcomplexityandBayesian gametheoryarealsodiscussed.
iii
iv
Re¬sume¬
Lesujet decettethe¡seest sur lesproprie¬te¬snon-localesdese¬tatssyme¬triquesinvariant
souslespermutationsdessyste¡meset lesusagespotentielsdecese¬tatsdansledomaine
detraitement d’information quantique.Lanon-localite¬depresquetouslese¬tatssyme¬-
triques, horslese¬tatsdeDicke, est e¬tabliepar une version e¬tenduedu paradoxe de
Hardy.Gracea¡ larepre¬sentationdeMajoranapour lese¬tatssyme¬triques,desparame¡tres
demesureaveclesquelstouteslesconditionsdu paradoxesont satisfaitespeuvent e√tre
trouve¬s.Uneversione¬tenduedel’ine¬galite¬deCHpeute√trede¬rive¬ea¡ partirdesconditions
probabilistesdeceparadoxe. Cetteine¬galite¬est viole¬epar touslese¬tatssyme¬triques.
Lesproprie¬te¬sdelanon-localite¬et lesproprie¬te¬sdel’intrication sont aussi discute¬es
et compare¬es,notamment par rapport a¡ lapersistanceet lamonogamie. Desre¬sultats
indiquent quelade¬ge¬ne¬rescencedecertainse¬tatssyme¬triquesest lie¬ea¡ lapersistance,
qui donneunefacÀon d’inventer destestsqui sont inde¬pendantsdesdispositifsvise¬pour
se¬parer lesdiffe¬rentesclassesdenon-localite¬.Il est aussi montre¬quel’ine¬galite¬utilise¬e
pour de¬montrer lanon-localite¬destouslese¬tatssyme¬triquesn’est pasmonogamedans
lesensstrict. Ne¬anmoins, unenouvelleine¬galite¬pour lese¬tatsdeDickeest propose¬e,
qui est monogamequand lenombredeparticipantstendsversl’infinite¬.Malheureuse-
ment,toutescesine¬galite¬ssont incapablesdede¬tecter lanon-localite¬authentique.Des
applicationsdelanon-localite¬a¡ lacomplexite¬decommunication et auxjeuxbaye¬siens
sont discute¬es.
v
vi
Contents
Acknowledgements i
Abstract iii
Re´sume´ v
Publications xii
0 Sommairedelathe¡se 1
0.1 Re¬sume¬deschapitres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.2 Lanon-localite¬dese¬tatssyme¬triques . . . . . . . . . . . . . . . . . . . . . . 5
0.3 Lanon-localite¬et lesclassesd’intrication. . . . . . . . . . . . . . . . . . . . 19
0.4 Uneine¬galite¬monogamepour lese¬tatsdeDicke . . . . . . . . . . . . . . . 23
1 Introduction 27
2 Background 30
2.1 Essential Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 Postulatesof Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 ScratchingtheSurfaceof QuantumEntanglement . . . . . . . . . . . . . . 40
2.4 TheFacetsof Nonlocality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.1 ALittleBit of History . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.2 Bell’sInequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.4.3 Nonlocalityfrom Correlations . . . . . . . . . . . . . . . . . . . . . . 48
2.4.4 Probability-freeNonlocality& Mermin Inequality . . . . . . . . . . 51
2.4.5 Almost Probability-freeNonlocality& HardyParadox . . . . . . . 55
2.4.6 Unification of Different ApproachestoNonlocality . . . . . . . . . 57
2.5 SemidefiniteProgramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
vii
Contents
3 TheMajoranaRepresentation of SymmetricStates 62
3.1 Geometryof ComplexNumbers . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1.1 TheComplex Planeand theRiemann Sphere . . . . . . . . . . . . 63
3.1.2 TheMo(cid:2)biusTransformation . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 From Complex GeometrytotheMajoranaRepresentation . . . . . . . . . 72
3.3 Physical Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.4 TheMajoranaRepresentation and Entanglement . . . . . . . . . . . . . . . 79
4 Nonlocality of SymmetricStates 83
4.1 BipartiteHardyParadoxRevisited . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 MultipartiteHardyParadox andtheInequalityPn . . . . . . . . . . . . . . 88
4.3 Violation of Pn ByAlmost All SymmetricStates . . . . . . . . . . . . . . . 91
4.4 Violation of Pn ByAll SymmetricStates . . . . . . . . . . . . . . . . . . . . 96
5 Degeneracy and itsConsequences 101
5.1 Degeneracyand Persistencyof Nonlocality. . . . . . . . . . . . . . . . . . . 102
5.2 DeviceIndependent Classification of States . . . . . . . . . . . . . . . . . . 104
6 Analysisof Nonlocal Propertiesfor SymmetricStates 109
6.1 LargenResultsfor jW i and jGHZ i . . . . . . . . . . . . . . . . . . . . . . 109
n n
6.2 Monogamyof Entanglement andMonogamyof Correlations . . . . . . . 112
6.3 Monogamyand GenuineNonlocalityof DickeStates . . . . . . . . . . . . 116
7 Applications 120
7.1 Application toCommunication Complexity . . . . . . . . . . . . . . . . . . 120
7.2 Application toBayesian Games. . . . . . . . . . . . . . . . . . . . . . . . . . 124
8 Summary 127
8.1 New Resultsin ThisThesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.2 Recent Progresson Related Topics . . . . . . . . . . . . . . . . . . . . . . . . 128
8.3 Outlooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Bibliography 129
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List of Figures
1 L’e¬tat te¬trae¡dre(a) et l’e¬tat j000+i (b) danslarepre¬sentationdeMajorana. 20
2 L’e¬tat te¬trae¡dre(a) et l’e¬tat GHZa¡ 4qubitsjGHZ i (b) danslarepre¬sen-
4
tation deMajorana. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Comparaison delaviolation deL (75) par jS(n,n)i (n) et jW i (l ) en
2 n
fonction den(lenombredeparties). . . . . . . . . . . . . . . . . . . . . . . 24
3.1 Thecomplexplanewith thenumber a+ bi. . . . . . . . . . . . . . . . . . . 63
3.2 Thestereographicprojection of thecomplexplanetotheunit sphere. . . 64
3.3 TheRiemann spherewithspherical coorinates(cid:2)and (cid:2). . . . . . . . . . . 65
3.4 Onthecomplexplane,apoint Pinsidetheunit circle(blue) getsinverted
to P0outsidetheunit circle(red). . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5 Effectsof complexinversionson generalizedcircles.. . . . . . . . . . . . . 70
3.6 TheBloch sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
∆ ∆
3.7 TheTetrahedron. jTi = 1jS(4,0)i + 2jS(4,3)i. . . . . . . . . . . . . 80
3 3
3.8 TheOctahedron. jOi = p1 (jS(6,1)i + jS(6,5)i). . . . . . . . . . . . . . . . 81
2
p p p
3.9 TheCube. jCi = p1 ( 5jS(8,0)i + 14jS(8,4)i + 5jS(8,8)i). . . . . 81
2 6
p p p
3.10 TheIcosahedron. jIi = 1( 7jS(12,1)i (cid:2) 11jS(12,6)i (cid:2) 7jS(12,11)i). 81
5
p p
3.11 The Dodecahedron. jDi = 1p ( 187jS(20,0)i + 627jS(20,5)i +
p p 25 3 p
247jS(20,10)i (cid:2) 627jS(20,15)i + 187jS(20,20)i). . . . . . . . . . 82
4.1 ThethreesymmetricBell statesin theMajoranarepresentation. . . . . . 87
5.1 The tetrahedron state (a) and the state j000+i (b) in the Majorana
representation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2 Thetetrahedron state(a) and the4-qubit GHZstate(b) in theMajorana
representation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
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