Quantum local asymptotic normality and other questions of quantum statistics

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quantum statistics

Kahn, J.

Citation

Kahn, J. (2008, June 17). Quantum local asymptotic normality and other questions of quantum statistics. Retrieved from

https://hdl.handle.net/1887/12956

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License: Licence agreement concerning inclusion of doctoral thesis in the Institutional Repository of the University of Leiden

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Normality

and

SU (d)

^. ^Phys. ^Rev. ^A, ^75:022326,

2007b. arXiv:quant-ph/0603115.

Chapter 4(Clean positive operator valued measures)

Clean positiveoperator valued measures for qubits and similar ases. J. Phys.

A,Math. Theor., 40:48174832,2007a. arXiv:quant-ph/0603117.

Chapter 5(Complementarysubalgebras)

J.KahnandD Petz. Complementary redu tionsfortwoqubits. J. Math. Phy.,

48:012107,2007. arXiv:quant-ph/0608227.

Chapter 6(QLAN forqubits)

M.Guµ andJ.Kahn. Lo alasymptoti normalityforqubit states. Phys. Rev.

A,73:052108,2006. arXiv:quant-ph/0512075.

Chapter 7 (Optimal estimation of qubit states with ontinuous time

measurements)

M. Guµ , B. Janssens, and J. Kahn. Optimal estimation of qubit states with

ontinuous time measurements. Comm. Math. Phy., 277(1):127 160, 2008.

arXiv:quant-ph/0608074.

Chapter 8(QLAN nitedimension)

M.Guµ andJ.Kahn. Lo alasymptoti normalityfornite-dimensionalsystems.

SoumisàComm. Math. Phy.,2008. arXiv:0804.3876

ISBN978-90-9023254-6

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A knowledgments

Ringraziamenti

Köszönetnyilvánítás

This thesis has been written under the s ienti dire tion of Ri hard Gill. I

thank him for introdu ing me to the world of quantum statisti s, and to lo al

asymptoti normality. The periods I spent with himwere alwaysenlightening,

be it onquantum statisti s, oron lassi al statisti s, orthe way astatisti ian

should behave. I also admire his will of introdu ing morepoeple to this eld,

andhisalwayssuggestingproblemsofinterest.

Pas al Massart a été mon deuxième dire teur de thèse. Son aide dans la ba-

taille administrative a été des plus pré ieuses. Il a également su me fournirla

bibliographieidoinequandl'o asions'yprêtait.

Inmymind, M d linGuµ was athird supervisoraswellasmymain ollabo-

rator. Wehaveworkedoutstrongquantumlo alasymptoti normalitytogether.

I havedeeply appre iatedmystaysin Nottingham, andhis wayof ndingnew

quantumproblems outof lassi alones. Thankyouforthepi turethat adorns

thisthesis' over,too.

Nottingham wasalso the pla e to speak with Belavkin, and takeadvantageof

hishugehistori alknowledge.

SpeakingofQLAN,IalsothankBasJanssensforourworkonquantumsto has-

ti dierentialequations. Dis ussions with Anna Jen£ováfurther deepened my

understandingofQLAN,byyieldingtheotherside,namelyweakequivalen e.

Nelle due settimane he ho passato in Pavia, all'inizio della tesi, Professore

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milianoSa hi.Ringraziotuttalasquadralo ale, hehoan oravistoin ongressi

duranteglianniseguenti.

KöszönömPetzDénesfelhívásotBudapestre. Nem sakkészültünkela íkkeim-

mel,desokattanítottis aCCRalgebrák-ról.

I regretnotbeingableto swit hto JapanesetothankHayashiandMatsumoto

formystayinJapaninMar h2007,andthedis ussionswehavehadonquantum

statisti s,espe iallyquantum Cramér-Raobounds.

Cristina Butu ea a été mon interlo uteur en Fran e, la seule ave laquelle je

puissedialoguersurlesujetdesstatistiquesquantiques.

Jeremer ieaussitous euxquiontétéautourdemoi,élèves ommeprofesseurs,

quand j'étais àl'ENS. Les dis ussions s ientiques permanentes, les problèmes

quenousnousposions,restentlaplusbelleformationquej'aiepuavoir.

Mer iYanPautrat,pourlesparolesé hangéessurlaphysiquestatistiquequan-

tique du point de vue mathématique, et pour avoir invité Vladimir Jak²i¢ à

donnerun oursinspiré.

Mer i enn à eux qui ont relu mathèse, Cristina,Patri iaReynaud, Borg,et

surtout mon voisin de bureau Sylvain Arlot, dont je me suis outrageusement

inspirédansl'espoir himérique d'appro hersa larté.

J'aiessayéd'être aussineutrequepossibledansleslignespré édentes, bienque

plusieursdespersonnes évoquéesméritentletitre d'ami.Mais j'espèrequetous

euxqui omptentpourmoi,autrementplusquelesmagniquesstatuesdegla e

quej'étudie,n'ontpasbesoinquejeles ite pourlesavoir.

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Remer iementsA knowledgments Ringraziamenti

Köszönetnyilvánítás i

1 Introdu tion 1

1.1 Statisti s. . . 3

Classi al Statisti s,3.

•

^Quantum^Obje
ts^andOperations, 10.

•

^Quantum

statisti s,19.

1.2 Dis rimination. . . 26

Motivation,26.

•

^Former^results,^27.

•

Contributionsofthethesis,31.

1.3 FastEstimationofUnitaryOperations . . . 31

Motivation,31.

•

1.4 Clean PositiveOperatorValuedMeasures. . . 34

Motivation,34.

•

1.5 Complementarysubalgebras. . . 38

Motivation,38.

•

1.6 Quantumlo alasymptoti normality. . . 40

Classi al lo al asymptoti normality, 40.

•

Motivation, 43.

•

^Former ^and

•

^Outlook,^46.

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I Mis ellaneous Problems in Quantum Statisti s 47

2 Dis rimination 49

2.1 Introdu tion . . . 49

2.2 Optimalminimaxdis riminationoftwoquantumstates. . . 52

2.3 Optimalminimaxdis riminationof

N ≥ 2

^quantum^states^. ^. ^. ^. ^. ⁵⁵

2.4 Optimalminimaxunambiguousdis rimination . . . 58

2.5 Bayesiandis riminationoftwoPauli hannels . . . 59

2.6 Minimaxdis riminationofPauli hannels . . . 61

3 Fastestimationof unitary operations 71 3.1 Introdu tion . . . 71

3.2 Des riptionoftheproblem . . . 74

3.3 Whywe annotexpe tbetterratethan

1/N ²

^. ^. ^. ^. ^. ^. ^. ^. ^. ⁷⁸

3.4 Formulasfortherisk . . . 80

3.5 Choi eofthe oe ients

c(~λ)

ând^proofôf^theirê
ien
y ^. ^. ^. ^. ^. ⁸¹

3.6 Evaluation ofthe onstantinthespeedof onvergen eandnalresult 85 3.7 Con lusion . . . 87

4 Clean positive operatorvalued measures 89 4.1 Introdu tion . . . 89

4.2 Denitions andnotations. . . 91

4.3 AlgorithmandIdeas . . . 92

Algorithm,92.

•

Heuristi s: whatthealgorithmreallytests,93. 4.4 Su ient ondition . . . 95

4.5 Ne essary onditionforquasi-qubitPOVMs . . . 99

4.6 Summaryforquasi-qubitPOVMsandaspe ial ase . . . 110

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5 Complementarysubalgebras 113

5.2 Preliminaries . . . 114

5.3 Complementarysubalgebras. . . 115

II Quantum Lo al Asymptoti Normality 121

6 Quantum lo alasymptoti normalityfor qubits 123

6.2 Lo alasymptoti normalityin statisti sand itsextension to quantum

me hani s . . . 128

6.3 Thebigballpi tureof oherentspinstates . . . 130

6.4 Lo alasymptoti normalityformixedqubit states. . . 133

Blo kde omposition,134.

•

Irredu iblerepresentationsof

SU (2)

^,^136.

6.5 Constru tionofthe hannels

T n

^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ¹³⁷

6.6 Constru tionoftheinverse hannel

S n

^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ¹⁴²

6.7 Appli ations . . . 143

Lo alasymptoti equivalen eoftheoptimalBayesianmeasurementand the

heterodyne measurement, 143.

•

^The ^optimal ^Bayes measurement is also lo ally asymptoti minimax, 146.

•

Dis rimination of states, 152.

•

^Spin

squeezedstatesand ontinuoustimemeasurements,154.

7 Optimalestimationof qubitstates with ontinuoustime

measurements 155

7.2 Stateestimation . . . 160

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7.3 Lo alasymptoti normality . . . 164

Introdu tiontoLANandsomedenitions,165.

•

Convergen etotheGaus- sianmodel,166.

7.4 Timeevolutionoftheintera tingsystem . . . 170

Quantumsto hasti dierentialequations,170.

•

^Solving^the^QSDE^for ^the

os illator,172.

•

^QSDE^for^large^spin,^173.

7.5 These ondstage measurement. . . 175

Theheterodynemeasurement,176.

•

^Energymeasurement,177.

7.6 Asymptoti optimalityoftheestimator. . . 178

7.7 Con lusions. . . 183

7.A Appendix: Proof ofTheorem7.3.1 . . . 185

ProofofTheorem7.3.1;themap

T _n

,185.

•

^Proof^of^Theorem^7.3.1;^the^map

S _n

,189.

7.B Appendix: Proof ofTheorem7.4.1 . . . 190

8 Quantum lo alasymptoti normalityfor

d

-dimensionalstates 197 8.1 Introdu tion . . . 197

8.2 Lo alasymptoti normalityforqubits . . . 199

8.3 Classi alandquantumstatisti alexperiments . . . 201

Classi alandquantumrandomizations,202.

•

^The^Le^Cam^distan
e^and^its

statisti almeaning,205.

8.4 Lo alasymptoti normalityinstatisti s . . . 207

8.5 Lo alasymptoti normalityinquantumstatisti s . . . 209

QuantumGaussianshift experiment,210.

•

^Symmetri^Fo
k^spa
es, ^210.

•

Fo kspa es,211.

•

^Gaussian^states,^212.

•

^Main^theorem,^215.

8.6 Grouptheoryprimer . . . 216

Irredu ible unitary representations, 216.

•

Irredu ible representations of

SU (d)

^,^219.

•

^Tensor^produ
trepresentation,224.

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8.7 Parametrisationofthedensitymatri esand onstru tionofthe

hannels

T n

^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ^. ²³⁰

Thenite-dimensionalexperiment,230.

•

Des riptionof

T _n

,231.

8.8 Mainstepsoftheproof . . . 233

Why

T _n

doesthework,233.

•

^Denition^of

S _n

andproofofitse ien y,239.

8.9 (Evenmore)te hni alproofs . . . 241

A few more tools, 241.

•

^Proof ^of ^Lemma^8.7.1, ^255.

•

^Proof ^of ^Lemmas

8.6.9and8.7.2andworkaroundsfornon-orthogonalityissues,256.

•

^Proof^of

Lemma8.8.4,259.

•

^Proof^of^Lemma^8.8.2,^262.

•

^Proof^of^Lemma^8.8.1^and

Lemma8.8.8,263.

•

Bibliography 287

Samenvatting 289

Résumé 293

Curri ulumVitae 297

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Introdu tion

Statisti sisthes ien e ofpullinginformationoutofdata. Thoughthey anbe

wildlypolymorphi ,anystatisti alproblemmaybesplitintothree omponents:

theobje twestudy,theoperationsweareallowedtouse,andtheexa tmathe-

mati alquestion. Inotherwords,what wehave,what we ando, andwhat we

wantto know.

Quantum statisti sdiverge from lassi al statisti son the rst point, what we

have. Hen e theydieralsoonwhatisallowed,sin ethetwoarelinked.

In lassi al statisti s, we often immediately start from the result of measure-

ments, whi h are modeled by randomvariables with probability laws. Indeed,

ifwe anmeasure quantity AorquantityB,we antheoreti allymeasure both

simultaneously. Experiments often measure every useful and easily a essible

quantity. In theory, what we ando is applyinganymathemati al treatment

onthedata to transformit. Mathemati ally,this meansapplying anyfun tion

onthedata,possiblywitharandomout ome. Inpra ti e, omputationalpower

mightbound su h latitude.

In some ases, however, wemust already onsider the obje tunder study, and

hoose what measurement we arry out. A typi alexamplewould be tryingto

understandwhatabla kboxdoes. Wemustprobeitwithinputs,andea htime

wemust hoosetheinput. Thisthemati is alleddesign ofexperiments. What

we andomaydependhugelyontheproblemathand. Inthebla kbox ase,we

an hoosethe input. The mathemati aldes riptionof this hoi e mightdier

from one bla k box to another, though. Yet, on e the measurement is arried

out,weagainhaveprobabilitylawsandweareba ktothepreviousparagraph.

In quantum statisti s, the design of experiments annot be avoided. Indeed,

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measuring A and B, in general. We must then hoose the measurement that

yields the information we need most. Nevertheless, quantum physi s gives a

framework paralleling that of lassi alprobability,whi h tellsus exa tly what

we ando. Initially,whatwearegiven isaquantumobje t,whi hismodeled

byaquantumstate. Whatwe ando ismeasuringthestate,gettinga lassi al

randomvariableas aresult,ormoregenerallytransformingthequantum state.

The sets of both measurements and transformations have pre ise and general

mathemati aldenitions,allowingtotreatmanyquestionsin auniedway.

Whatwewanttoknowseldomdiersinquantumand lassi alstatisti s. Most

often,wewanteithertosummarizetheinformationinthedata(statisti alinfer-

en e),to disproveahypothesis ortosee whathypothesisinaniteset best ts

thedata (testing),orto guess with pre isionwhat theunderlying phenomenon

wasthatgeneratedthedata(estimation). Allthese anusuallybedes ribedbya

lassi alparameter. Theex eptionwouldbewhenourben hmarkisintrinsi ally

quantum,forexamplewhentryingtoapproximately loneaquantumstate.

Thisthesis,inPartI,studiesanumberofparti ularsystems. Namelywe onsider

in Chapter 2 how to best de ide in whi h state among a nite set a quantum

obje t anbe;in Chapter3,wegiveafast(

1/n

⁾^pro
edure^to ^estimate^a^bla
k

boxunitarytransformation. Chapter4andChapter5dwellmoreonthegeneral

stru ture of quantum experiments: the former deals with an order relation on

measurements, and the latter on ndingmaximally dierent subsystemsof a

quantumsystem,inthesimplest ase.

Now,wemayhaveverydierentquestionsonagivensystem. Forsu hasystem,

orexperiment,whatwehave and whatwe ando will remainthesame. We

maythenwonderaboutwhatwe ansaydire tlyonthesystem,withoutreferen e

to aparti ular question. Thetheory of onvergen e of experimentsin lassi al

statisti sworksouthowwellwe anapproximateanexperimentbyanother. We

an then translate all the pro edures we use in one experiment to the other.

Hen e we get answers to what we want to know in both experiments when

solvingthequestionin one.

PartII,themain ontributionofthisthesis,generalizestothequantumworldthe

mostbasi aseof onvergen eofexperiments,namelylo al asymptoti normal-

ity. We provethat asu iently smoothexperimentwith identi alindependent

(i.i.d.) quantum states onverge to aquantum Gaussian shift experiment. The

pointisthatthisexperimentisverywell-known,andeverythingweknowabout

it anbetranslatedtothelarge lassofsmoothi.i.d. experiments.

Theremainder ofthe introdu tion rst makespre ise the rulesof lassi aland

quantumstatisti s,andthenintrodu eea hofthe haptersofthethesis,andthe

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1.1 Statisti s

1.1.1 Classi al Statisti s

LeCam[1986℄andvanderVaart[1998℄maybe onsultedforfurtherreferen es,

amongmanyotherbooksonstatisti s. Wesummarizein Table1.1,onpage24,

the mostbasi ingredientsof lassi alstatisti s. The sister Table 1.2 givesthe

orrespondingquantumnotions.

What we have

In lassi alstatisti s,wearegivendata,that anbemodeledasarandomvariable

X

^with probability law

p

^. ^What ^we ^know^beforehand^is ^that

p

^is^a probability lawinaset

E = {p ^θ , θ ∈ Θ} ,

^(1.1)

with no onstraint in general on the parameter set

Θ

^. ^The

p θ

^are ^all ^dened

on the sameprobability spa e

(Ω, A)

^. ^This

E

îs âlled ^the êxperiment ôr ^the

statisti al model.

Remarks:

•

^The^dataâreôften^madeôf^manymeasurements,yieldingasmanyrandom variables

X 1 , . . . , X n

^,^withprobabilitylaws

p ¹ , . . . , p ⁿ

^onpotentiallydier- entprobabilityspa es. However,wemaystill onsiderallthedataasasin-

glerandomvariable

X = (X 1 , . . . , X n )

^withprobabilitylaw

p = p ¹ ⊗· · ·⊗p ⁿ

^,

andwestayinthe urrentframework.

•

Âlthough ^there îs ^noônstraint ôn

Θ

^at ^this ^point^of ^the ^theory, ^this ^set

is often either nite orareasonablesubset of

R ^d

. Therst aseleads to dis retestatisti s,andsomefamilies oftestsinparti ular,these ond ase

to parametri statisti s. When theset

Θ

^is innite-dimensional, weenter the omplexrealmofnon-parametri statisti s,themain fo usofresear h

in re entyears.

Examples: Bernoulli experiment,Gaussian shiftexperiment

The mostbasi probabilityspa e wemay nd is the two-elementspa e

{0, 1}

^.

Anexperiment orrespondingtoa ointosswouldbe

E ^Ber = {p ^θ = (θ, 1 − θ), θ ∈ [0, 1]} .

^(1.2)

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Alternatively, we mighttoss the oin

n

^times. ^Denoting

X = (X 1 , . . . , X n )

^the

results,wewouldgetthisexperimenton

{0, 1} ^⊗n

^:

E ^Bin = n

p θ : {X} 7→ θ ^P ^X ⁱ (1 − θ) ^{n−P X} ⁱ , θ ∈ [0, 1] o

.

^(1.3)

Whendealingwith ontinuousfun tions, themostpervading ofthem allis the

Gaussian. Weareespe iallyinterestedinGaussian shiftexperiments,wherethe

varian eoftheGaussianisxed andtheparameteristhemean:

E ^gs =

N (θ, I ⁻¹ ), θ ∈ R ^d

,

^(1.4)

where

N

^means^normal^law,^and

I

^is^any^xed^positive^matrix¹^.

What we an do

On ewehaveourdata

X

^,^how^an^we^pro
ess^them?

Themostgeneralpro edure onsistsin drawinganew randomvariable

Y

^with

probabilitylaw

p X

^depending^only^on

X

^,ând^measurableâsâ^fun
tion ôf

X

^.

We anviewthisproto olintwoways. Therstis onsideringthat

Y

îsânânswer

to what wewantto know. Then

Y

^is ^a(randomized) estimator, typi allyan estimatorof

θ

^,ⁱⁿ^whi
hâse^weâlso^denoteît^by

θ ˆ

^.

Alternatively,we an onsiderthat

Y

îsâ^new^random^variable,ând^that^we^have

transformedourexperiment. Ournewexperiment onsistsof

Y

^withprobability law

q

ⁱⁿ ^the^set

{q ^θ , θ ∈ Θ}

^on^a^spa
e

(Ω 1 , B)

^,^with^density²

q θ (y) = T (p θ )(y) ˆ = Z

Ω

p X (y)dp θ (X).

^(1.5)

Thetransformation

T

^is^a^Markov ^kernel.

Inthe lassi al ase,thetwonotionsarethesame. However,Iinsistonseparating

themsin etheywillbedierentinthequantum ase.

1

Weusethisstrangenotationbe ause thismatrixistheinverseoftheFisherinformation

matrix(1.13).

2

We ouldequivalentlyworkwithnon-dominatedsetsofprobabilitylaws,butthatwould

onlymakenotationsheavier. Wethenassumethatallprobabilitylawshaveadensity,anduse

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Examples

Letusgoba ktoour

n

^-sample^Bernoulli^experiment

E ^Bin

^(1.3). ^Ourprobability spa eis

{0, 1} ^⊗n

^. ^We^may^use^a^Markov^kernel^from^that^spa
e^to

[0, n] ∩ N

^that

simply send

X = (X 1 , . . . , X n )

^to

Y = P

X i

^. ^Here, ^the

p X

^are ^merely ^delta

fun tions. Wethenobtainabinomialprobabilitylawfor

Y

^,^that^is

q θ = B(n, θ)

^.

The orrespondingexperimentis

E = {q ^θ , θ ∈ Θ}

^.

Alternatively, we might want to build an estimator

θ ˆ

^. ^The ^most ^obvious ^one

would be

X 7→ P

X i /n = Y

^. ^The ^law ôf ôur êstimatorîs ^the âbove^binomial

dividedby

n

^.

Wemightalsolookforanestimatorin

E ^gs

^(1.4). ^The^rst^thought^is^yet^simpler:

wejustkeep

X

^. ^The orrespondingMarkovkernelwouldbetheidentity.

What we wantto know

We usually want to haveinformation on the unknown underlying pro ess that

gaverisetoourdata. Inotherwords,wewanttoguesstheparameter 3

θ

^.

We angiveananswereither with a onden einterval,orwithaguess ofour

quantity, maybe with estimates on the varian e of the estimate. This guess

orrespondstogivinganestimator

θ ˆ

^of

θ

^.

Wewanttobuildagoodestimator. Wethereforeneedawaytorateestimators.

Inde isiontheory,we onsidera ostfun tion

c(θ, ˆ θ)

^. ^That^is^the^ost^we^have^to

payifourestimatoryields

θ ˆ

^when^the^true^parameter^is

θ

^. ^Hen
e,^ost^fun
tions

are usually zero on the diagonal, and grow when

θ

^and

θ ˆ

^get ^farther ^apart ⁱⁿ

somesense.

Atypi al ostfun tionwhen

Θ

îs^dis
reteândôuntable^would^be

c(θ, ˆ θ) = δ _{θ, ˆ} _θ

^.

When

Θ

îsânôpen^subsetôf

R ^d

,themostmathemati allytra table ostfun tion is the square of the Eu lidean distan e

c(θ, ˆ θ) = kθ − ˆθk ² 2

^, ^or ^more ^generally

any quadrati ostfun tion

(θ − ˆθ) ^⊤ G(θ − ˆθ)

^for ^a^positive^matrix

G

^, ^possibly

depending on

θ

^.

Sin e

θ ˆ

^is^a^random ^variable, ^we ^want^to ^minimize^the expe tation of the ost, alledtheriskatpoint

θ

^:

r θ (ˆ θ) = Z

Ω 1

c(θ, ˆ θ)dq θ (ˆ θ).

^(1.6)

3

Moregenerally,wemaybeinterestedmerelyinafun tion

f

^of

θ

^.^However,^we^an^always

use

(θ, f (θ))

âs^parameter. ^We^then^hoose^the ôst^fun
tions întrodu
ed ^below^so^that ^they

dependonlyon

f(θ)

^.

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However,we annot dire tly minimizethis expression, sin ethe best guess de-

pendson

θ

^, ^whi
h îs ûnknown. ^We^must ^then^nd â^way ^to^hooseân ê
ient

estimatorforany

θ

^weâre^likely^toên
ounter. ^Thereâre^mainly^twoapproa hes.

A favourite of physi ists is theBayesianparadigm, where we assumethe exis-

ten eof ana priori probabilitylaw ontheparameter

θ

^. Mathemati ians often preferminimax riteria,whereastrategyisratedbytheworst ase.

Bayesian riteria

Wehave onsideredourdatatobe

X

^withprobabilitylaw

p

^. ^We^assumed^that

theonlyinformationwehad wastheexperiment,thesetweknow

p

^belongs^to.

Suppose nowthat we havemore information. Namely, weare told beforehand

that

θ

îs^hosenât^random^withâprobabilitylaw

π

^. ^Then,^on^average,^the^best

estimatorwouldbetheonethat minimizestheaverageoftherisk(1.6),that is:

R π (ˆ θ) = Z

Θ

π(dθ)r θ (ˆ θ)

= Z

Θ

Z

Ω 1

c(θ, ˆ θ)dq θ (ˆ θ)π(dθ).

^(1.7)

Fromthe Bayesriskofaspe i estimator

θ ˆ

^,^we^an^write ^the^Bayes^risk^asso-

iatedtotheprior

π

âs^theînmumôf^the^risks^forâll

θ ˆ

^:

R π = inf

θ ˆ

R π (θ).

^(1.8)

Theweaknessofthisapproa histhatthereisnoreasonwhythereshouldbean

a priori probability lawon

Θ

^,êx
ept â^delta ^fun
tion ôn ^the^real

θ

^... ^whi
h ^is

exa tlywhatwewanttoknow. Wehaveto hooseapriorand onsideritasthe

realone. Theriskofthenal estimatorwillbeunderestimated,however.

Themain strengthofaBayesianestimatoristheoptimaluseoftheinformation

we get from measurements, given the prior. The prior orresponds to a priori

information, whi h is generally wrong. The best priors try then to minimize

theinformation in theprior 4

. Foranite

Θ

^,^we^usually ^hooseequiprobability a priori forea h possible

θ

^. ^Fôr ânôpen ^pre
ompa
t^subset ôf

R ^d

, we hoose Jereys [1946℄prior, proportionalto thesquare root of theFisher information

(1.13)denedbelow. Apointwiseanalysisshowsthattheseestimatorsareoften

verygoodestimators.

4

Subje tiveBayesians onsidertheprobabilitylawsasdegreesofbelief. Hen ethey anuse

(22)

Bayesian estimators an be omputed through the al ulations of a posteriori

distributions. Insomesimple ases, these anbe arriedoutexpli itly andthe

estimatoristhebary enterofthe

θ

^with^weights^thelikelihoods. Inmore omplex situations,we anresorttoMonte-CarloMarkov hains.

Minimax riteria

Themathemati ianiseitherpessimisti ormegalomania ,andassumesheplays

againsttheDevil. Therefore,hewantstodesignastrategythatwillbee ient

whateverthereal

θ

îs. ^Hen
e ^the^ben
hmarkôfânêstimator

θ ˆ

^is^its^valueⁱⁿ^the

worst ase:

R M (ˆ θ) = sup

θ

r θ (ˆ θ).

^(1.9)

Theminimaxriskistheriskofthebest possibleestimator:

R M = inf

θ ˆ

R M (ˆ θ) = inf

θ ˆ

sup

θ

r θ (ˆ θ).

^(1.10)

The weakness of this method is that we might have to worsen mu h an esti-

mator on intuitively many

θ

^for ît ^to ^beê
ientôn ^some^spe
ial âses. ^The

workaroundistorequireadaptiveness,thatis,minimaxe ien yonawhole lass

of subsets of

{p ^θ }

^. ^The ^latter ^te
hnique ^is essentially used for non-parametri statisti s.

Theinterestofthesemethodsisthat theyrequirenoassumption. Theygivean

e ien yweknowweattaininreality,aslongastheexperiment(ormodel)itself

wasright.

Linksbetween Bayesian and minimax riteria

The main link between the two riteria omes from the following remark. Ifa

strategy

θ ˆ

îs ^Bayesôptimal, ând ^su
h ^that ^the ^riskôf

θ ˆ

^does^not^depend ^on

θ

^,

then

θ ˆ

îsâlso^minimaxôptimal.

Indeed,forany

π

^,^the^Bayes^risk^of

θ

^is^more^than^the^minimax^risk:

R π (ˆ θ) ≤ sup

θ

r θ (ˆ θ) = R M (θ),

^(1.11)

withequalityifandonlyiftheriskat

θ

^is^the^same

π

^-almost everywhere.

Under some onditions, a onverse statement is true: a minimax estimator is

optimalfor somepre iseprior, theoneforwhi h theBayesianriskis maximal.

(23)

Example

We ompute the risk of the aforementioned estimator for the Gaussian shift

family(1.4). Thelawof

θ ˆ

îs^the^lawôf^theôriginal^data,^thatîs ^the^normal^law

N (θ, I ⁻¹ )

^. ^So^that

r θ (ˆ θ) = E θ

h (θ − ˆθ) ^⊤ G(θ − ˆθ) i

= Tr(G I ⁻¹ ).

^(1.12)

Thisriskatpoint

θ

^does^not^depend^on

θ

^,^so^that^the^same^value^is^the^minimax

riskandtheBayesianriskforanyprioroftheestimator. Weshallseebelowthat

theestimatorisminimaxforthemodel.

Theremainderofthese tiongivesaqui ksummaryofwhatriskswe anexpe t

inregularenough ases,forquadrati ostfun tions.

Fisherinformation

Theriskswegiveabovedepend onthequestion(the ostfun tion) and onthe

experiment

{p ^θ , θ ∈ Θ}

^,^but^notônâny^parti
ularêstimator. ^We^may^then^read

informationaboutthemdire tlyontheexperiment.

ThemostimportantnotiontothatendistheFisher informationmatrix. Itisa

lo alnotion,that anbeinterpretedasameasureofhowfastwe andistinguish

p θ

^from ^the surrounding

p θ+dθ

^. ^The ^Cramér-Rao ^bound ^des
ribed ⁱⁿ ^the^next

se tionmakesthatexpli it. Noti ethatinthefollowing,weneedsomeregularity

inthemodel. Twi edierentiableismorethanenough.

TheFisherinformationatpoint

θ = (θ α ) α=1...d

^is^given^by

I ^α,β (θ) = Z

Ω

∂ ln(p θ (X))

∂θ α

∂ ln(p θ (X))

∂θ β

dp θ (X).

^(1.13)

The Fisher information matrix is positive denite, and denes a metri on

Θ

^,

whi hisinvariantbyanysmooth hangeofvariables. Thisfa t anbeviewedas

themostbasi onne tionbetweenstatisti sanddierentialgeometry. Dieren-

tialgeometry anbeused to study higher-orderasymptoti s,asexemplied by

Amari[1985℄.

Developingthelogarithmsofprodu ts,itiseasilyseenthat having

n

^samples^of

thedatamultipliestheFisherinformationby

n

^,^that^is

I ⁽ⁿ⁾ (θ) = n I ⁽¹⁾ (θ)

^where

I ⁽ⁿ⁾

^is^the^Fisherinformationmatrixoftheexperiment

E ⁽ⁿ⁾ = {p ^⊗n θ , θ ∈ Θ}

^.

(24)

Cramér-Raobound

We anusetheFisherinformationmatrixtoderivealowerboundonthevarian e

matrixoflo allyestimators:

Z

Ω 1

(θ − ˆθ)(θ − ˆθ) ^⊤ dq θ (ˆ θ) ≥ I ⁻¹ (θ).

^(1.14)

The bound holds 5

for all lo ally unbiased estimators

θ ˆ

^, ^that îs âs ^long âs

R ˆ θdq θ (ˆ θ) = θ

^and

∂/∂θ i R ˆ θ j dq θ (ˆ θ) = δ i,j

^.

Animmediate onsequen eisthat,forlo allyunbiasedestimators,andaquadrati

ostfun tion

(θ − ˆθ) ^⊤ G(θ − ˆθ)

^, ^we^get^this^lower^bound ^on^the^risk^at^point

θ

^:

r θ (ˆ θ) ≥ Tr(GI ⁻¹ ).

^(1.15)

Thisboundisknowntobeasymptoti allysharp. Indeed,a

n

^-sample^experiment

in reasingly resembles aGaussian shift experiment, forwhi h it is sharp. The

pre iseexplanation omesfrom thetheoryof onvergen e ofexperimentsbyLe

Cam,thatwefurthersket hinSe tion1.6.1.

Examples

We ompute the Fisher information for the Bernoulli experiment, at point

θ

dierent from

0

^and

1

^. ^Theêxpression îs^slightlyêasier ^sin
e^we^haveônlyône

parameter.

I(θ) = θ

d ln(θ) dθ

2 + (1 − θ)

d ln(1 − θ) dθ

2 = 1 θ + 1

1 − θ

= 1

θ(1 − θ) .

Fromthatandourpreviousremarkfor

n

^samples,^we^see^that

I(θ) = n/(θ(1−θ))

inthebinomialexperiment

E ^bin

^.

Aslightlymoretedious al ulationwouldshowthat theFisherinformationma-

trixofaGaussianshiftexperimentistheinverseofthevarian eoftheGaussians.

5

Supere ient estimators su h as Stein estimator provethat we annot simplydrop the

unbiasedness ondition. However, addingsomete hni ality(essentially onsideringe ien y

on a whole neighborhood of

θ

^, ^through êither â ^Bayesianôr â ^minimaxâpproa
h), ^weân

(25)

Hen e our hoi e ofnotationin equation (1.4). Moreover,after omparison be-

tweenthebound(1.15)andtherisk(1.12)oftheestimator onsistingin

X

^itself,

weobtainoptimalityofthelatterestimatoramongthe lassoflo ally unbiased

estimators.

Wenowtryto givetheequivalentsofthosenotionsin thequantum world.

1.1.2 Quantum Obje ts and Operations

The books by Helstrom [1976℄ and Holevo [1982℄ are the usual referen es for

quantum statisti s. We also add the more re ent review arti le by Barndor-

Nielsenet al.[2003℄. Asalreadymentionned,wehavesummarized inTable1.2,

onpage25, themostbasi ingredientsof quantum statisti s,withTable1.1for

lassi al orrespondan eonthepagebefore.

States,density operators

Thebasi obje tinquantumprobabilityisthestate. Thestateistheequivalent

ofaprobabilitylaw.

Wedene it overaHilbert spa e

H

^. ^Its mathemati alexpression isgivenbya densityoperator.

Denition 1.1.1. A densityoperator

ρ

^over^a ^Hilbert ^spa
e

H

^is^a tra e- lass operator withthe followingproperties:

•

Self-adjointness:

ρ

^isself-adjoint.

•

Positivity:

ρ

^isnon-negative.

•

Normalization:

Tr(ρ) = 1

^.

Thosearetheequivalentof onditionsforprobabilitymeasures: probabilitymea-

sures are real (

=

self-adjointness), non-negative (

=

positivity) and normalized to

1

⁽

=

normalization).

For nite-dimensional Hilbert spa es, the operators are matri es, and density

matri esalsosatisfytheabove onditions. Therealdimensionofthemanifoldof

statesis

d ² − 1

îf^theômplex^dimensionôf

H

^is

d

^.

(26)

Example: Qubits

Themostelementary situationarises when

dim( H) = 2

^. Physi ally,thesystem ouldbeanele tronspin. Those statesare alled qubit statesand heavilyused

inquantuminformation.

WedenePaulimatri es as

σ x =

0 1 1 0

, σ y =

0 i

−i 0

, σ z =

1 0 0 −1

.

^(1.16)

Self-adjointness implies that adensity matrix must be a linear ombination of

those matri esand theidentity

1

^. ^Positivity ^andnormalization further impose that:

ρ = 1 2

1 + ~ θ · ~σ

, k~θk ≤ 1,

^(1.17)

with

~σ = (σ x , σ y , σ z )

^a^ve
tor^of^matri
es.

We see that we already need three real parameters to des ribe a qubit state,

onfer the one parameterwe need to des ribe aprobability law on a lassi al

two-out omespa e.

Pure states

Thesetof lassi alprobabilitymeasures anbeseenasthe onvexhullof delta

fun tions. Similarly,theset ofstatesisthe onvexhullofpurestates.

Purestatesare hara terizedby beingrank-one operators,with eigenvalueone.

We anwritethem

|ψi hψ|

^,^where

|ψi

îsâ^norm-one^ve
torôf

H

^. ^Pure^states^an

thusberepresentedaspointsoftheproje tivespa easso iatedto

H

^.

Theyareveryimportant: manytreatmentsofquantum me hani sfeature only

purestates. Generalstates anbeseenasa lassi almixingofpurestates.

Unlike for delta fun tions, where we merely draw a random variable with the

unknown law, there is no measurement that an identify unambiguously any

purestate,evenifweknowbeforehandthatthestateispure. Thisfundamental

dieren e with the lassi alworldis a hallmarkof non- ommutativitybetween

dierentstates. Thestudyofpurestatesinthemselvesisalready hallenging.

For qubits with the above parameterization, the pure states orrespond

to

k~θk = 1

^. ^Thisparameterizationby asphere, alledthe Blo h sphere,givesa graphi alintuitionforproblemsonqubits.

Therealdimensionofthepurestatesis

2(d − 1)

^if

dim H = d

^.

(27)

Example: Coherentstates

Qubits aretheparadigmfor nite-dimensionalquantum states. Theother fun-

damentalfamilyofstatesisthat of oherentstates 6

.

ThosestatesliveontheFo kspa e

7

F(C)

^,^that^is^theinnite-dimensionalHilbert spa e

ℓ ² (N)

^. ^We ^denote

{|ki} k∈N

^the ^anoni
al^basis ^on

ℓ ² (N)

^. ^Physi
ists ^all

|ki

^the

k

^-th^Fo
k^state.

StatesonFo kspa esarestatesoftheharmoni os illator,anexampleofwhi h

is the state of mono hromati light (laser). We are thus on the playground of

quantumopti s. Amongthosestates, oherentstatesarein somewaythemost

lassi al: theysaturateHeisenbergun ertaintyrelations.

Theyaregivenbyone omplex,hen etworeal, oe ient

θ

^. ^Sin
e^they^are^pure

states,we andes ribethemwithave torin

F(C)

^,^rather^than^an^operator⁸^:

|θ) = exp(−|θ| ² /2) X n k=0

θ ^k

√ k! |ki .

^(1.18)

Multipartite states,entangled states

Letus onsidertwoquantumobje ts

ρ 1

^and

ρ 2

^on

H ¹

^and

H ²

^. ^They^an^be^seen

asasinglequantumobje ton

H = H ¹ ⊗ H ²

^,^with^state

ρ = ρ 1 ⊗ ρ ²

^.

Any state on su h omposite Hilbert spa e is alled a multipartite state. Now

somemultipartite states annot be writtenas

P c i ρ ⁱ ₁ ⊗ ρ ⁱ 2

^with^positive

c i

^. ^We

mightneedsomenegative

c i

^. Înôther^words,^those^statesâre^notâ^lassi
al^ran-

domizationofa hoi eofapairofstates. They ontainanintrinsi allyquantum

oupling. Theyare alledentangledstates.

Let us provethey do exist. We write

dim H ¹ = d 1

^and

dim H ² = d 2

^. ^Hen
e

dim H = d ¹ d 2

^. ^Puremultipartitestatesarepurestateson

H

^,^so^they^onstitute

a

2(d 1 d 2 − 1)

^manifold. Ôn^theôther^hand,â^pure^stateôf^the^form

P

c i ρ ⁱ 1 ⊗ ρ ⁱ 2

withpositive

c i

ônlyâllowône^termⁱⁿ^the^sum,^with^both

ρ 1

^and

ρ 2

^pure^states.

The orrespondingdimensionis

2(d 1 + d 2 − 2) < 2(d ¹ d 2 − 1)

^. ^Hen
e ^there ^are

manyentangledpurestates.

6

Moregenerally,allpossiblysqueezedGaussianstatesplayanimportantroleinquantum

opti sand,asweshallsee,inquantumstatisti s. Westi kto oherentstatesforsimpli ityof

theexample.

7

Multidimensional oherentstatesaretensorprodu tsof oherentstatesonthetensorized

Fo kspa e

F (C ^d ) = F (C) ^⊗d

^.

8

Weusethe notation

|θ)

înstead ôf ^the ûsual^ket

|θi

^soâs ^to âvoid ônfusion^with^Fo
k

states,inparti ularwhen

θ

^happens^to^be^a^positive^integer.

(28)

A typi al example are maximally entangled states, that is states of the form

|Ψi hΨ|

^, ^with

|Ψi = ^√ ¹ _d P ψ ⁱ

⊗ ψ ⁱ

, where

H ¹ = H ²

^and

{ ψ ⁱ

}

îs ân ôr-

thonormalbasisof

H ¹

^. Âs^their ^nameîmply,^theyârry âs^mu
h entanglement aspossible.

Entanglementmaybethesinglemostbasi and pervasiveresour ein quantum

information. Itliesat theheartof quantum teleportation,mostquantum ryp-

tographyproto olsandthein reasedpro essingpowerofaquantum omputer.

Literatureonthesubje tistoodauntingtobeevens rat hedupon. Inquantum

statisti s, apart from the problems linked to estimating entangled states, they

anbeusedtospeedupestimationofquantumtransformations.

A tions on states

In the lassi al ase, we noti ed that givingan estimator of a parameter

θ

^or

moregenerallyofanyfun tionof

θ

^was^the^same^astransformingourinitialdata togetanewrandomvariable

Y

^with^law

T (p θ )

^.

Inthequantum ase,thetwonotionsaredistin t. Indeed,transformingthedata

meansgettinganewquantumstate,thatisanoperatoronaHilbertspa e. States

undergoatransformationwhentheyaresentthrougha hannel. Anestimatorof

a lassi alparameter,ontheotherhand,isa lassi alquantity. Wethenendup

with a lassi alrandomvariable. We retrievethis lassi aldata from thestate

throughameasurement.

Ifwemerelywantto onsiderestimators,whyarewealsointerestedin hannels?

Indeed,applyingmany hannelsandthenameasurement anbesummedupto

usingonlyamore omplexmeasurement.

Therstreasonisthat wemighttransformourstatestoanewfamilyforwhi h

weknowwhatmeasurementtouse. Infa t,thewholeaimofstronglo alasymp-

toti normality, whosestudy onstitutesmostof this thesis, is to transforman

experimenttoaquasi-equivalentandeasierone.

Se ondly, hannels des ribephysi al transformations. We mightwant to study

thetransformationitselfrather thanthestate. Typi ally,thephysi al transfor-

mation ouldbe generated bya for e we wantto measure. Wedwell on these

mattersinChapter3.

We allinstrument afun tionyielding lassi alandquantumdataoutofaquan-

tum input. Real measurementapparatuses are essentiallyinstruments, even if

we may forget about the out ome state. In parti ular, ontinuous-time mea-

surementsare ommon in pra ti e. Typi ally, we measure the ele tromagneti

(29)

seen asa sequen e of innitesimal instruments, and writing the orresponding

evolutionequationsisthepurposeofquantumltering,pioneeredbyDaviesand

Belavkin[Boutenetal.,2006,foranintrodu tion℄.

Measurements, POVMs

Ifwewanttomake lassi alstatisti alinferen eontheunknownparameters,we

havetotranslateourquantuminformationto lassi alinformation. Tothatend,

we apply ameasurement. Sin e mixed states are lassi almixing of states, we

requirelinearityofthetransformation. Theout omeshouldalwaysbea lassi al

probabilitylaw. Wededu e from that thefollowingform of physi ally allowed

measurements:

Denition1.1.2. Apositiveoperatorvaluedmeasure,or POVM,overa mea-

suredspa e

(Ω, A)

^is^a^set

{M(A)} A∈A

ôf ^bounde^dôperatorsôn

H

^su
h^that:

• M(Ω) = 1 H

^.

• M(A)

^is^p^ositive.

•

^For âny ôuntable ôlle
tion

(A i ) _i∈N

^of ^disjoint

A i

^, ^we ^have

M ( S A i ) = P M (A i )

^.

We noti e that those are exa tly the usual axioms for a probability measure,

ex eptthatweworkwithoperatorsinsteadofrealnumbers. We allea h

M (A)

aPOVMelement.

Applyingameasurement

M

^on^a^state

ρ

^yields ^aprobabilitylaw

P ρ

^on

(Ω, A)

^,

givenbyBorn'srule:

P ρ (A) = Tr(ρM (A)).

^(1.19)

InChapter4,wes rutinize aspe i orderrelationonPOVMs.

Afewremarksareinorder.Firstofall,we anin ludeany lassi alpro essingof

thedatainthePOVM.Indeed,applyingameasurement

M

^and^then^a^Markov

kernel

T

^(dened^by^(1.5))ôn^theôutput^random^variableîs^the^sameâsâpplying

themeasurement

N

^on

(Ω 1 , B)

^with

N (B) = R

Ω p ω (B)M (dω)

^. ^So^that^working

onPOVMsisequivalenttoworking onestimators.

Se ondly, we annotingeneralmeasuresimultaneously

M 1

^and

M 2

^on

(Ω 1 , A ¹ )

and

(Ω 2 , A ² )

^. În ôntrast^to ^the^lassi
alâse,^where ^weôuld^have^simultane-

ously the results of applying

T 1

^and

T 2

^. ^Indeed, ^measuring ^both

M 1

^and

M 2

meansmeasuring

N

^on

(Ω 1 × Ω ² )

^with

N (A 1 × Ω ² ) = M 1 (A 1 )

^and

N (Ω 1 × A ² ) =

(30)

M 2 (A 2 )

^. ^An ^easy ounterexample illustratingthe role of non- ommutativity is givenby

M 1

^and

M 2

^both^dened ^on

{0, 1}

^,^with

M 1 (0) =

1 0 0 0

, M 1 (1) =

0 0 0 1

, M 2 (0) = 1

2 1 1 1 1

, M 2 (1) = 1 2

1 −1

−1 1

.

Allthosematri esarerank-one. Wewouldnowneed

N (0, 0) + N (0, 1) = M 1 (0)

^.

Sin eallPOVMelementsarepositive,wehave

M 1 (0) ≥ N(0, 0)

^. ^Sin
e^moreover

M 1 (0)

^is ^rank-one, ^we ^have

N (0, 0) = c 1 M 1 (0)

^for ^some

0 ≤ c ¹ ≤ 1

^. ^We ^also

know

N (0, 0) + N (1, 0) = M 2 (0)

^. ^So^that

N (0, 0) = c 2 M 2 (0)

^. ^The^only^solution

is

c 1 = c 2 = 0

^and

N (0, 0) = 0

^. ^The^same^holds^for

N (0, 1)

^,

N (1, 0)

^and

N (1, 1)

^.

Ontheotherhandweneed

N ( {0, 1} ² ) = 1 C ²

^. Contradi tion.

Finally,all those measurementsare believedto bephysi ally feasible. However

theymightbeveryhardtoimplementin pra ti e. Inparti ular,ifthestateisa

multipartite state,it anmakesense to restri tourattention tosmaller lasses

ofmeasurements. Notably,ifdierentpeoplehold dierentparti lesindierent

pla es, they annot implement ageneral measurement, even if they ooperate.

The best they an do is: one of them measures his parti le (possibly with a

non-trivialoutput quantum state), tells the resultto the other, who hoosesa

measurement onhis parti le, keeps theoutput state and tellsthe resultto the

rstone,andtheyiterateontheoutputstates. Su h measurements,usingonly

lo alquantumoperationsand lassi al ommuni ation,aredubbedLOCC:Lo al

Operations,Classi alCommuni ation.

Inquantum information whenthe(usuallyentangled)quantum stateisdivided

betweenseveralpeople,wenaturallyrestri tto LOCCmeasurements. Inquan-

tum estimation of a state with

n

ôpies ôf ^theînitial ^state, ^weâre ât ^least în-

terestedin what anbea hievedthroughLOCCmeasurements,mu h easierto

implementthangeneral( olle tive)measurements. We aningeneralreallygain

pre isionwith olle tivemeasurements. Thismightbesurprisingfromthepoint

ofviewofphysi ists,sin ethe

n

^opies^are^totallyindependent. Insome ases,no- tablywhenweknowthattheunknownstateispure[Matsumoto,2002℄, olle tive

measurementsdonotyieldmu h improvementoverLOCCmeasurements. This

mightbesurprisingfromthepointofviewofmathemati ians,sin ethespa eof

olle tivemeasurementsismu hbiggerthanthat ofLOCCmeasurements.

Example: Spin

z

Considerthebinaryout omemeasurementonqubits givenby

M ( ↑) =

1 0 0 0

= 1

2 (1 + σ z ), M ( ↓) =

0 0 0 1

= 1

2 (1 − σ ^z ).

(31)

Thismeasurementappliedto thestate

ρ = ^1+~ ₂ ^θ·~σ

^yields

↑

^with probability

Tr(ρM ( ↑)) = 1 2

Tr(1M ( ↑)) + X

α=x,y,z

θ α Tr(σ α M ( ↑))

= 1

2 (1 + θ z ).

Inparti ular,if

θ z = 1

^,^then^theôut
omeîsâlways

↑

^. Conversely,if

θ z = −1

^,^the

out omeisalways

↓

^. Ôn^theôther^hand,îf

θ x = 1

^,^so^that

θ z = 0

^,^the^out
ome

iseither

↑

^or

↓

^withprobabilityonehalf,eventhoughthestate

ρ

^is^pure.

This kind of measurements, where all the POVM elements are proje tors, are

also alledobservables. Theyonlyyieldinformationonthebasisinwhi hallthe

POVM elementsarediagonal. Noti e that usualaxiomsofquantum me hani s

restri tmeasurementsto observables. However,weget ba kallthe POVMsby

applyinganobservableonamultipartite stateofwhi hourstateis onlyapart

(Naimarktheorem).

Heterodyne measurement

Theheterodynemeasurementgetsitsnamefromthete hniqueusedtoimplement

itin laboratory, withlasersthat are o-phase. This POVM without omein

C

hasamathemati alexpressiongivenby:

M (A) = 1 π

Z

A |z)(z|dz,

^(1.20)

where

|z)

îsâôherent^state^(1.18).

Theprobabilitylawoftheout omewhenmeasuring

ρ

^has^thus ^a^density

(z |ρ|z)

with respe t to Lebesgue at point

z

^. ^Inparti ular, the lawof the result when measuringa oherentstateisaGaussian:

q θ (dz) = 1

π (z |θ)(θ|z) = 1

π exp( −|θ − z| ² ).

^(1.21)

Ifwe onsiderallthe omplex

θ

^,^we^re
ognize^a^lassi
al^Gaussian^shift^experiment

(1.4)in

R ²

.

Moregenerally,theprobabilitydensityfun tionof theout omeof themeasure-

mentonastate

ρ

îsâlled ^the^Husimi ^fun
tion ôf^the^state:

H ρ (dz) = 1

π (z |ρ|z).

^(1.22)

(32)

Channels

We now des ribe how to make anew quantum state out of the original state.

Noti ethattherststateisdestroyedin thepro ess.

A physi al transformationof aquantum obje ttakesastate andyield another

state,possiblyonadierentspa e. It isdes ribedbya hannel, theequivalent

ofaMarkovkernel.

We re all that a positive superoperator

E

îs â^map ^su
h ^that ^for âny ^positive

operator

A

^, ^the^output

E(A)

^is^also^positive.

Denition 1.1.3. A hannel

E

^is ^a^map ^from ^the ^set

T (H ¹ )

^of tra e- lassop- eratorsto

T (H ² )

^,^with^the ^following properties:

•

^Linearity:

E

^is^line^ar.

•

^Completepositiveness: foranyauxiliaryspa e

H ³

^,^thesuperoperator

E⊗Id : T (H ¹ ⊗ H ³ ) → T (H ² ⊗ H ³ )

^given^by

( E ⊗ Id)(ρ ⊗ σ) = E(ρ) ⊗ σ

^is^positive.

•

Tra e-preserving:

Tr( E(A)) = Tr(A)

^.

Noti ethatMarkovkernelssatisfyallthese riteria,whenrepla ingoperatorsby

measures 9

.

Thene essity of linearity an be provedfrom the axiomof unitaryevolution 10

andin ludingtheobserverin thesystem.

Wewanttheimageofastatetobeastate, soapositiveoperatormustbesent

toapositiveoperator. Tounderstandwhyweneed ompletepositivity,wemust

onsider apossiblyentangledstate on

H ¹ ⊗ H ³

^. ^If ^we ^transform^states^on

H ¹

^,

wealsotransformstateson

H ¹ ⊗ H ³

^,^with

E ⊗ Id

^as^the^hannel. ^Therefore^the

lattertransformationmustbepositive. Hen eweneed ompletepositivity.

Finally,the output isastate ifthe inputis astate,and bothare tra e-one,so

tra emustbepreserved.

Weoften onsider the hannelsin the(pre)dualpi ture,thatisasa tingonthe

elementsof

B(H)

^. ^So^that

Tr( E(ρ)A) = Tr(ρE ∗ (A))

^for^all^state

ρ

^and^all^bounded

operator

A

^. ^In^this^ase

E ∗

îsâlsoâômpletely^positive^linear^map,^but^we^must

9

Inthe moregeneral settingof

C ^∗

^-algebras,^the ^spa
es^of^fun
tions ^are ommutative

C ^∗

^-

algebrasandallpositivesuperoperatoronthosespa esis ompletelypositive.

10

Quantumme hani sstatethattheevolutionofasystemisgivenby

ρ(t) = U (t)ρ(0)U ^∗ (t)

^,

where

U(t)

îsâûnitaryôperator^thatân^beômputed^from^theself-adjointoperator

H

^alled

theHamiltonian.IftheHamiltoniandoesnotdependontime,then

U(t) = e ^itH

^.

(33)

repla ethetra e-preserving onditionbytheidentity-preserving ondition,that

is

E ∗ (1) = 1

^.

Notations: Weusuallywrite

E

^or

F

^for^hannels. ^Abusing^notations,^we^usually

drop the star for the pre-dual and also write

E

ⁱⁿ ^that ^ase. ^However, ^those

standardnotationsare alsothe standardnotationsfor experiments. Sothat in

the hapterswhere weuse that notion,weusefor hannels thesamenotations

asforMarkovkernels,thatis

T, T n , S, S n

^.

Kraus representation,Stinespring theorem

Theabovedenitiondoesnotmakeitobvioustodealwith hannels. Fortunately,

tworepresentationtheoremsdes ribe ompletelypositivemapsinamoreusable

way. ThebookbyPaulsen[1987℄isagoodreferen eonthosematters.

Kraus[1983℄representationisthemain toolwhentheHilbert spa esare nite-

dimensional.

Theorem1.1.4. A ompletely positive map

E

^from

M (C ^d ¹ )

^to

M (C ^d ² )

^an ^be

writtenas

E(A) = X

α

R α AR _α ^∗ ,

^(1.23)

with

α

^running^from

1

^to^at^most

d 1 d 2

^,^and

R α ∈ M ^d ² ^,d ¹ (C)

^. ^Star^is^the^adjoint.

Moreover, the hannel istra e-preservingifandonly if

P R ^∗ _α R α = 1 C d1

^.

Thede ompositionisnotunique. Thedual hannelisgivenby

A 7→ P

R ^∗ _α AR α

^.

Ininnitedimension,weratherusethemorepowerfulStinespring[1955℄dilation

theorem 11

.

Theorem 1.1.5. Let

E : B(H ¹ ) → B(H ² )

^be â ômpletely ^pôsitive ^map. ^Then

thereisaHilbertspa e

K

^and^a*-homomorphism(orrepresentation)

π : B(H ¹ ) → B(H ² )

^su
h^that

E(A) = V π(A)V ^∗ ,

^(1.24)

where

V : K → H

îsâ^bounde^dôperator.

Moreover, if

E

^isidentity-preserving, then

V

îsânîsometry, ^thatîs

V V ^∗ = 1 _H

^.

If we further impose that

K

^is ^the ^losed ^linear ^span ^of

π(A)V ^∗ H

^, ^then ^the

dilationisuniqueuptounitarytransformations.

11

Infa t,Stinespringtheoremwasprovedforanyunital

C ^∗

âlgebraâsînitial^spa
e.Îtân^be

showntoimplyKrausrepresentation,butalsotheGNSrepresentation,astapleof

C ^∗

^-algebras.

(34)

Instruments

We give the representation of instruments for nite dimensions 12

. To further

simplify notations, werestri tourselvesto the asewhenthe measurementhas

anitenumberofout omes.

Denition1.1.6. Aninstrument isgiven byaset

{N ^ω,k }

^of^matri
es^from

H ¹

to

H ²

^,^su
h^that

X

ω

X

k

N _ω,k ^∗ N ω,k = 1 _H 1 .

The orresponding measurementisgiven by

M (ω) = X

k

N _ω,k ^∗ N ω,k ,

andthe outputstatewhenthe resultof themeasurement is

ω

^is^given ^by

N (ρ, ω) =

P

k N ω,k ρN _ω,k ^∗ Tr(ρM (ω)) .

The outputstatelives on

H ²

^.

Wenowhave anotherwayto understand whywe annot measure twoPOVMs

simultaneously: aftermeasuring

M

^,^the^quantumôbje
t,^thatîsôur^data,^hasⁱⁿ

generalbeenperturbed. Infa t, ifthemeasurementisri henough, theoutput

statedependsonlyontheout ome

ω

^,ând^notânymoreôn^theînput^state.

Wenowhaveallthetoolsto opythesetupfrom lassi alstatisti stoquantum

statisti s.

1.1.3 Quantum statisti s

Usually,weworkonquantumstates;o asionallywemaywanttogainknowledge

ona hannel. Wetreatthetwo asesseparately.

States: What we have,what we an do,what we wantto know

Inanalogywiththe lassi al ase,weareusually givenaquantum state

ρ

^,^that

weknowtobein aset

E = {ρ ^θ , θ ∈ Θ} .

^(1.25)

12

Ininnitedimension,wehavetousethe

C ^∗

^-algebra^settingândânînstrumentîs^merelyâ

hannelbetween

C ^∗

^-algebras.

(35)

Weagain all thissetanexperiment,oramodel.

With theexamplesof thequbits, the usualmodelswouldbethe3D full mixed

model

E ^m = {ρ ^θ , kθk < 1}

^and ^the ^2D ^pure ^state ^model

E ^p = {ρ ^θ , kθk = 1}

^,

where wehaveused ourformerparameterizationfor thestate

ρ θ

^(1.17). ^When

having

n

^opies^of^the^state,^we^repla
e

ρ θ

^by

ρ ^⊗n _θ

^.

Another typi al experiment would be

E ^t = {ρ ^θ , θ ∈ {θ ¹ , θ 2 }}

^, ^where ^the ^usual

question is to dis riminate between the twopossible

θ

^. ^We ^study ^this ^kind ^of

probleminSe tion 1.2andChapter2.

We anapriori useanysequen eofinstrumentsonthestate. Ifwemerelywant

lassi alinformationon

θ

^,^we^may^restri
t^tomeasurements

M

^,^that^is^POVMs.

We then asso iate to

M

^an ^estimator, ^say

θ ˆ

^, ^with ^law ^depending ^on ^the ^true

parameter

θ

^through

q θ (B) ˆ = P θ

h θ ˆ ∈ B i

= Tr(ρ θ M (B)).

Dependingonthe ir umstan es,wemightallowanyphysi al measurement, or

asmaller lass,su hasseparateorLOCCmeasurement.

Finally, what we want to know is the same as in the lassi al ase. We want

to knowsomefun tion oftheparameter

θ

^. ^So^that^we^want^to^estimate

θ

^, ^and

we rate our estimator

θ ˆ

^through ^a ^ost ^fun
tion

c(θ, ˆ θ)

^. ^As ^before, ^the ^most

ommon ostfun tionsare

(1 − δ θ, ˆ θ )

^,îf^the^parameter^setîs^nite,ând^quadrati

ostfun tions

(ˆ θ − θ) ^⊤ G(ˆ θ − θ)

^for^a^positive^matrix

G

^,^if^the^parameter^lives^on

anopensubsetof

R ^d

. Theweightmatrix

G

^might^depend^on

θ

^.

We anagain write the risk (1.6)of an estimator at point

θ

^. ^Sin
e ^we ^do^not

know

θ

^, ^we^then êither ûse ^the ^Bayesian^risk^(1.7) ^for ânappropriate prior, or the minimax risk (1.9), and optimize (1.8, 1.10)over the available estimators.

Noti ethatthelaststagedependonthesetofallowedestimators.

QuantumFisher informationand Cramér-Raobounds

We antrytomimi thedenitionof lassi alFisherinformationandgetsimilar

bounds onvarian e of estimators. Infa t, we anbuild su h anequivalent for

any hoi eofalogarithmi derivative. We hoosetherightlogarithmi derivative

(RLD),dened forea h

θ

ândêa
h ôordinate

θ α

^as^a^matrix

λ α,θ

^su
h^that:

∂ρ θ

∂θ α = ρ θ λ α,θ

^(1.26)

onthesupport of

ρ θ

^.

Quantum local asymptotic normality and other questions of quantum statistics

quantum statistics

Kahn, J.

Citation

Kahn, J. (2008, June 17). Quantum local asymptotic normality and other questions of quantum statistics. Retrieved from

https://hdl.handle.net/1887/12956

Version: Corrected Publisher’s Version

License: Licence agreement concerning inclusion of doctoral thesis in the Institutional Repository of the University of Leiden

Downloaded from: https://hdl.handle.net/1887/12956

Note: To cite this publication please use the final published version (if

applicable).

SU (d)

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1/n

X

p

p

E = {p θ , θ ∈ Θ} ,

Θ

p θ

(Ω, A)

E

•

X 1 , . . . , X n

p 1 , . . . , p n

X = (X 1 , . . . , X n )

p = p 1 ⊗· · ·⊗p n

1/N ²

T _n

S _n

T _n

T _n

S _n

E = {p ^θ , θ ∈ Θ} ,

p ¹ , . . . , p ⁿ

p = p ¹ ⊗· · ·⊗p ⁿ

R ^d

E ^Ber = {p ^θ = (θ, 1 − θ), θ ∈ [0, 1]} .

{0, 1} ^⊗n

E ^Bin = n

p θ : {X} 7→ θ ^P ^X ⁱ (1 − θ) ^{n−P X} ⁱ , θ ∈ [0, 1] o

E ^gs =

N (θ, I ⁻¹ ), θ ∈ R ^d

{q ^θ , θ ∈ Θ}

E ^Bin

{0, 1} ^⊗n

E = {q ^θ , θ ∈ Θ}

E ^gs

c(θ, ˆ θ) = δ _{θ, ˆ} _θ

R ^d

c(θ, ˆ θ) = kθ − ˆθk ² 2

(θ − ˆθ) ^⊤ G(θ − ˆθ)