Renormalized quantum tomography

0.22 MB·

by  G. M. D'Ariano

RENORMALIZED QUANTUM TOMOGRAPHY
GIACOMOMAUROD’ARIANOANDMASSIMILIANOFEDERICOSACCHI
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0
0 ABSTRACT. Thecore ofquantum tomography is thepossibility ofwriting agenerally
unbounded complexoperator informofanexpansion overoperators thataregenerally
2
nonlinearfunctionsofagenerallycontinuoussetofspectraldensities—-theso-calledquo-

n rum of observables. The expansion is generally non unique, the non unicity allowing
a furtheroptimizationforgivencriteria. Themathematicalproblemoftomographyisthus
J theclassificationofallsuchoperatorexpansionsforgiven(suitablyclosed)linearspaces

9 ofunboundedoperators—e.g.Banachspacesofoperatorswithanappropriatenorm.Such
1 problemisadifficultone,andremainsstillopen,involvingthetheoryofgeneralbasisin
Banachspaces,astillunfinishedchapterofanalysis. Inthispaperwepresentnewnon-

] trivialoperatorexpansionsforthequorumofquadraturesoftheharmonicoscillator, and
h
introduceafirstverypreliminarygeneralframeworktogeneratenewexpansionsbasedon
p
theKolmogorovconstruction. Thematerialpresentedinthispaperisintendedtobehelp-
-
fulforthesolutionofthegeneralproblemofquantumtomographyininfinitedimensions,
t
n whichcorrespondstoprovideacoherentmathematicalframeworkforoperatorexpansions
a overfunctionsofacontinuoussetofspectraldensities.
u
q
[
2000MathematicsSubjectClassification.47N30,47N40,47N50.PACS:03.65.Wj,02.30.Tb,

1 42.50.-pKeywordsandphrases.Quantumtomography,operatorexpansions.
v
6
6
8
2
.
1
0 CONTENTS
9
1. Introduction 2
0
2. Quantumtomographyandquorumofobservables 3
:
v 2.1. Unbiasingnoise 6
i 3. Thecaseofhomodynetomography 6
X
4. Calculuswithnullfunctions 8
r
a 5. TheKolmogorovconstruction 9
5.1. Homodynetomography 10
5.2. Spintomography 11
5.3. Alternateexpansions 12
6. Expansionofunboundedoperatorsoverthequadratures 12
7. Canonicaldualforhomodynetomography 14
7.1. Alternatedualframes 16
8. Framesofnormal-orderedmoments 16
9. Frameofmomentsversusquadraturedistribution 17
10. Generatingnewframes 18
11. Otherframes 20
Appendix 20
11.1. ProofofLemma4 20
11.2. Alternativederivationofidentity(87) 21
References 23
2 GIACOMOMAUROD’ARIANOANDMASSIMILIANOFEDERICOSACCHI
1. INTRODUCTION
Thestateofaphysicalsystemisthemathematicaldescriptionthatprovidesacompleteinforma-
tiononthesystem.Itsknowledgeisequivalenttoknowtheresultofanypossiblemeasurementonthe
system.Inclassicalmechanicsitisalwayspossible,atleastinprinciple,todeviseaproceduremade
ofmultiplemeasurementswhichfullyrecoversthestateofasinglesystem. Inquantummechanics,
onthecontrary,thereisnoway,noteveninprinciple,toinferthequantumstateofasinglesystem
withouthavingsomepriorknowledgeonit[1]. Itishoweverpossibletoestimatethequantumstate
ofasystemwhenmanyidenticalcopiesareavailablepreparedinthesamestate,sothatadifferent
measurementcanbeperformedoneachcopy. Suchaprocedureiscalledquantumtomography.
Theproblemoffindingastrategyfordeterminingthestateofasystemfrommultiplecopiesdates
backto1957byFano[2],whocalledquorumasetofobservablessufficientforacompletedetermi-
nationofthedensitymatrix.However,quantumtomographyenteredtherealmofexperimentsmore
recently,withthepioneeringexperimentsbyRaymer’sgroup[3]inthedomainofquantumoptics.In
quantumoptics,infact,usingabalancedhomodynedetectoronehastheuniqueopportunityofmea-
suringallpossiblelinearcombinationsofpositionandmomentum—theso-calledquadratures—of
theharmonicoscillatorrepresentingasinglemodeoftheradiationfield.
Thefirsttechniquetoreconstructthedensitymatrixfromhomodynemeasurements—socalled
homodynetomography—originatedfromtheobservationbyVogelandRisken[4]thatthecollec-
tionofprobabilitydistributionsachievedbyhomodynedetectionisjusttheRadontransformofthe
WignerfunctionW. Therefore, similarlytoclassicalimaging, onecanobtainW byinvertingthe
Radontransform,andthenfromW onecanrecoverthematrixelementsofthedensityoperator.This
originalmethod,however,workswellonlyinasemi-classicalregime,whereasgenerallyforsmall
photonnumbersitisaffectedbyanunknownbiascausedbythesmoothingprocedureneededforthe
analyticalinversionoftheRadontransform. Thesolutiontotheproblemistobypasstheevaluation
oftheWignerfunction,andtoevaluatethematrixelementsofthedensityoperatorbysimplyaverag-
ingsuitablefunctionsofhomodynedata:thisisthebasisofthefirstunbiasedtopographictechnique
presentedinRef. [5]. Clearlythestateisperfectlyrecoveredinprincipleonlyinthelimitofinfin-
itelymanymeasurements: however,forfinitelymanymeasurementsonecanestimatethestatistical
error affecting each matrix element. For infinite dimensions there is the further problem that the
propagationofstatisticalerrorsofthedensitymatrixelementsmakethemuselessforestimatingthe
ensembleaverageofsomeoperators(e. g. unbounded), andamethodforestimatingtheensemble
averageisneeded,whichbypassestheevaluationofthedensitymatrixitself,aswasfirstsuggested
in Ref. [6]. For a brief historical excursus on quantum tomography, along with a review on the
generalizationtoanynumber ofradiationmodes, arbitraryquantumsystems, noisedeconvolution,
adaptivemethods,andmaximum-likelihoodstrategiesthereaderisaddressedtoRef.[7].
Themostcomprehensivetheoreticalapproachtoquantumtomographyusestheconceptofframe
of observables, i. e. a set of observables spanning the linear space of operators, from which one
derivescontextuallythequorumofobservablesandtheestimationrule. Theensembleaverage H
ofanyarbitraryoperatorH onaHilbertspaceHisestimatedusingmeasurementoutcomesofhthei
quorum O upon expanding H over a set of functions f (O) of the observables O . What
l n l l
{ } { }
makesthegeneraltheorynontrivialininfinitedimensionsisthecrucialroleofthenonlinearfunc-
tionsf (O)inmakingtheinfiniteexpansionconvergent.Let’sdenotebyP :=f (O),j =(n,l),
n l j n l
suchcompletesetofoperators.OnceyouhavetheP ,thentheproblemisreducedtothelinearprob-
j
lemofexpandinganoperatorasH = jhQ†j,HiPj,forasuitable“dual”setofoperators{Qj}.
(Noticethat generally theindex l iscontinuous, whence also theoperator expansion.) Thescalar
P
productintheexpansionisgenerallynotsimplytheFrobenius’swhenweneedtoexpandoperators
thatarenotHilbert-Schmidt. Themathematicaltheoryofframes[8,9,10,11]istheperfecttoolfor
establishingcompletenessof P andforfindingdualsets Q . Inmostpracticalsituationsthe
j j
{ } { }
set P isover-complete, andtherearemanyalternatedualsets Q ,thenonunicityproviding
j j
{ } { }
roomforoptimization.AgeneraltheoryshouldalsoclassifytheoperatorsHhavingboundedscalar
productwith{Ql}andexpansionH = jhQ†j,HiPj converginginaverageforagivenclassof
quantumstates.Ininfinitedimensions—e.g.forhomodynetomography—theeasyknownapproach
P
RENORMALIZEDQUANTUMTOMOGRAPHY 3
worksonlyforHilbert-Schmidtoperators,whenceincludingtrace-classones—e.g. forestimating
thematrix elements of thedensity operator over an orthonormal basis. For unbounded operators,
however,suchoperatorexpansionbecomesaninfinitesumofunboundedterms. Ontheotherhand,
converging(andevenfinite)alternateexpansionsareknowntoexistforvariousunboundedoperators
[12].Aswewillseeinthispaper,themechanismallowing”renormalization”oftheexpansionrelies
ontheexistenceofnull-estimators—namelyoperator-valuedfunctionsthathavezeromeanoverthe
quorum—theirexistencebeingrelatedtoagroupofsymmetriesofthequorum. Thenotionofnull-
estimatorwasfirstintroducedinRef. [13], inthecontext ofofhomodyne tomography, wherethe
quorumismadeofthequadraturesofthefieldmode—thequantumanalogueoftheRadontransform.
HerethesymmetrygroupofthequorumisthegroupU(1)ofrotationsofthequadraturephase.The
existenceofnull-estimatorsleadstoinfinitelymanyalternateexpansionsofthesameoperatorover
thequorum, allowingcancellationsoftheinfinitiesintheexpansion—a kindof”renormalization”
procedure.
Theproblemofclassifyingalloperatorexpansionsforagivenquorumininfinitedimensionsfor
givenspacesofunboundedoperatorsisadifficultone,andremainsstillopen. Itinvolvesthetheory
offramesor evenmoregeneral notionsofbasisinBanach spaces[8,9,10,11], astillunfinished
chapter of analysis. In this paper we present new nontrivial operator expansions for the quorum
of quadratures of the harmonic oscillator, and introduce a first preliminary general framework to
generate and classify new expansions, based on the Kolmogorov construction. We hope that the
materialpresented herewill open theway tothesolution of theproblem of quantum tomography
ininfinitedimensions, leading toageneral mathematical framework for operator expansions over
functionsofasetofspectraldensities.
2. QUANTUMTOMOGRAPHYANDQUORUMOFOBSERVABLES
Thegeneral ideaof quantum tomography isthatthereisaset of observables Xξ ξ X onthe
HilbertspaceofthesystemH—called”quorum”—bywhichonecanestimateanyd{esir}ed∈ensemble
averageby measuring the observables of thequorum, each at thetime, in ascheme of arepeated
measurements.Theobservablesofthequorummustbeindependenteachother,namelytheyarenot
commuting:
[Xξ′,Xξ′′]=0 ⇐⇒ ξ′=ξ′′. (1)
GenerallythesetXparameterizingthequorumisinfinite,andmostcommonly,isacontinuum. In
thesecases,sinceclearlyonecanmeasureonlyafinitenumberofobservables,thesearerandomly
pickedoutaccordingtoagivenprobabilitymeasureonX, which,therefore, mustbeaprobability
space. Inthefollowing,forsimplicity,wewillalsoassumeaprobabilitydensityoverXanddenote
itwiththesymboldµ(ξ). Itfollowsthattheensembleaverageofa(generallycomplex)operatoris
writtenintheformofdoubleexpectation
X = dµ(ξ) f (X X) , (2)
ξ ξ
h i h | i
ZX
wherethegenerallynonlinearfunctionf (xX)ofthevariablexhasananalyticformwhichdepends
ξ
|
ontheparticularoperatorX. Wewillcallthefunctionf (xX)thetomographicestimatorforX
ξ
|
withquorum Xξ ξ X. Ifwewanttoachievetheestimationof X = Tr[ρX]—theexpectation
{ } ∈ h i
beingsupposedly bounded onthestateρ—byaveragingtheestimator f (xX)over bothquorum
ξ
|
andmeasurementoutcomeswithaboundedvariance,weneedtohavethefunctionf (xX)square-
ξ
|
summableoverxandξ,moreprecisely
dµ(ξ) dE (x) f (xX)2 < , (3)
ξ ξ
h i| | | ∞
ZX ZXξ
where denotes the spectrum of X , and dE (x) its spectral measure. In the following, for
ξ ξ ξ
X
simplicity, wewill consider the spectrum independent on ξ. Clearly, the above square-
ξ
X ≡ X
summabilitywilldependagainonthestateρandontheoperatorX.
Wefirstwanttonoticetwomainfeaturesofestimators:
4 GIACOMOMAUROD’ARIANOANDMASSIMILIANOFEDERICOSACCHI
(1) Theestimatorf (xX)isgenerallynotunique,namelytherecanbemanydifferentestima-
ξ
|
torsforthesameoperatorX.Thisisequivalenttotheexistenceofnullestimators,namely
functionsn (x)suchthat
ξ
dµ(ξ) dE (x)n (x)=0. (4)
ξ ξ
ZX ZX
Accordingly,theestimatorsaregroupedintoequivalenceclasses,eachclasscorresponding
toanoperatorX.Forsuchequivalencewewillusethenotation ,i.e.wewillwritef g
≃ ≃
orf g 0todenotethatthetwoestimatorsareequivalent,namelytheydifferbyanull
− ≃
estimator.
(2) Forfixedxandξtheestimatorf (xX)mustbealinearfunctionalofX,namely
ξ
|
fξ(xaX+bY)=afξ(xX)+bfξ(xY), fξ(xX†)=fξ(xX)∗. (5)
| | | | |
Example1(Homodynetomography). [7]Thequorumisgivenby X withX denotingthe
. { φ}[0,π) φ
quadrature Xφ = 12(a†eiφ +ae−iφ), a,a† being the annihilation and creation operators of the
harmonicoscillatorwithcommutator[a,a†] = 1. Estimatorsforthedyads n m madewiththe
| ih |
orthonormalbasis n ,n=0,..., are
{| i} ∞
fφ(x||nihm|) = +∞ dk4|k|e18−ηηk2−ikxhn+d|eikXφ|ni
Z−∞
= eid(φ+π2)s(n+n!d)! +∞dk|k|e1−2η2ηk2−i2kxkdLdn(k2), (6)
Z−∞
whereLd(x)denotes thegeneralized Laguerre polynomials. For theunbounded operators aand
n
a†aonecancheckthatthefollowingareunbiasedestimators
f (xa) = 2eiφx,
φ
|
fφ(x|a†a) = 2x2− 21. (7)
Theproblemofquantumtomographyistoestablishthegeneralruleforestimation,namely
Definition1(Estimationrule). Giventhequorum Xξ ξ X findthecorrespondence:
{ } ∈
f (xX) X, foreveryoperatorXonH, (8)
ξ
| ⇐⇒
wherewepossiblymeantofindthewholeequivalenceclassofestimatorsf (X X).
ξ ξ
|
Beforesolvingthistask,firstoneneedstoknowthatthesetofobservables Xξ ξ X isactually
{ } ∈
aquorum. Theeasiest thingtodo, however, istoderiveboth thequorum and theestimationrule
contextually,startingfromaspanning-setofobservables—shortlyobservablespanning-set—namely
asetofobservables Fω ω O intermsofwhichwecanlinearlyexpandoperatorsasfollows
{ } ∈
X = dωc (X)F . (9)
ω ω
ZO
Noticethatthenotionofoperatorspanning-setusedheregeneralizesthenotionofframesforBanach
spacestounboundedoperators(seealsothefollowing),andisgenerallynotstrictlyaframeaccord-
ingtothedefinitionofRefs. [8,9]. Inthefollowingthroughout thepaper wewillalwaysassume
probability distributions admitting densities. Generally, the set O isunbounded, and themeasure
dωisnotnormalizable,whence,assuch,theexpansion(9)cannotbeusedforquantumtomography.
However,generallythisfeatureisrelatedtotheredundancyoftheobservablespanning-set,whichin-
cludesmanyobservablesF thatarejustdifferentfunctionsofthesameobservable.Then,collecting
ω
theobservablesofthespanning-setintofunctionalequivalenceclassesK ,eachcorrespondingtoan
ξ .
observableofthequorum Xξ ξ X,onecanrelabeltheobservablespanning-setasFκ,ξ =fκ(Xξ)
withκ K ,writing { } ∈
ξ
∈
X = dµ(ξ) dν(κ)c (X)f (X )= dµ(ξ)f (X X), (10)
κ,ξ κ ξ ξ ξ
|
ZX ZKξ ZX
RENORMALIZEDQUANTUMTOMOGRAPHY 5
wherethefunctionf (xX)istheintegralovertheobservablesequivalenttoX ,namely
ξ ξ
|
.
f (X X)= dν(κ)c (X)f (X ). (11)
ξ ξ κ,ξ κ ξ
|
ZKξ
NoticethatintermsofthespectraldecompositionofX wecanwrite
ξ
X = dµ(ξ) dE (x)f (xX), (12)
ξ ξ
|
ZX ZXξ
andsincethisexpansionislinearinthespectralmeasuredE (x),thelattercanberegardeditselfas
ξ .
anobservablespanning-set. Indeed,uponintroducingthespectraldensitydE (x)=Z dx,and
. . ξ ξ,x
thedensitydµ(ξ)=m(ξ)dξ,andrenamingζ =(ξ,x)andZ= (ξ,x),x ,ξ X Eq.(12)
ξ
{ ∈X ∈ }
canberewritteninthesameformofEq.(9),namely
X = dζc′ζ(X)Zζ, (13)
ZZ
wherethenewexpansioncoefficientsarenowgivenby
c′ξ,x(X)=m(ξ)fξ(x|X)=m(ξ) dν(κ)cκ,ξ(X)fκ(x). (14)
ZKξ
ForhomodynetomographytheabovequantitiesareexplicitlygiveninTable1.
For F an operator frame, the coefficients of the expansion (9) can be rewritten in form of a
ω
pairing( )withadualframeG asfollows
ω
·|·
c (X)=(G X), (15)
ω ω
|
intermsofwhichEq.(14)rewritesinthepairingform
c′ξ,x(X)=m(ξ)(Wξ,x|X), (16)
withdualframe
Wξ,x = dν(κ)Gξ,xfκ∗(x). (17)
ZKξ
Fromthelastequationitfollowsthattheestimatoritselfcanbewrittenusingthepairing
f (xX)=(W X), (18)
ξ ξ,x
| |
or,intermsoftheoriginalobservableframe
f (xX)= dν(κ)(G X)f (x). (19)
ξ ξ,x κ
| |
ZKξ
general homodyne general homodyne general homodyne
Xξ Xφ cω(X) Tr[D†(α)X] dµ(ξ) dπφ
X [0,π) ω α Zξ,x x φφ x
dµξ(ξ) dφφ dνK(ξk) 1dRkk XZξ [0|,iπR) h R|
ω απ cξ,κ(X) Tr[4e−ikX|φ|X] m(ξ) π1×
Fω D(α) fκ(Xξ|X) eikXφ Wξ,x −41π(x−PXφ)2
TABLE1. Tableofcorrespondenceforhomodynetomography.
6 GIACOMOMAUROD’ARIANOANDMASSIMILIANOFEDERICOSACCHI
2.1. Unbiasingnoise. Itispossibletoestimatetheidealensembleaverage X bymeasuringthe
quoruminthepresenceofinstrumental noise, whenthenoisemapN isinvhertiible, or, moregen-
erally,ifthereexiststherightinverseofN . Intermsofobservableframesthisjustcorrespondsto
usingadifferentdualframe.Moreprecisely,onehas:
X =N N −1(X)= dµ(ξ)N [fξ(Xξ N −1(X))]
|
ZX (20)
= dµ(ξ) N (dEξ(x))fξ(xN −1(X)).
|
ZX ZXξ
NoticethatEq.(20)meansthat
X = dµ(ξ) dEξ(x) Nfξ(xN −1(X)), (21)
h i h i |
ZX ZXξ
. .
where = Tr[ρ ] denotes the ideal ensemble average, whereas N = Tr[N τ(ρ) ] de-
notes thhe·iexperiment·al ensemble average, N being the predual mapho·fiN (Schroedinger v·ersus
Heisenberg picture). This also means that fo∗r left invertible map N the noisy spectral measures
N (dE (x))arestillaquorum.IntermsofthepairinginEq.(18)unbiasingthenoiseisequivalent
ξ
tousethenewdualframeN −1†(Wξ,x). WhenN isnotright-invertibleonecanstillestimatethe
ensembleaverageofoperatorsintherangeofthemap. Moreover, ininfinitedimension, whenthe
noiseCPmapN iscompactitsinversemapisunbounded,andonegenerallycannotunbiasthenoise
withoutrestrictingthespaceofreconstructedoperators. Otherwise,onehasaHadamardill-posed
problem,forwhichtherearebiasedcompromises,suchasputtingacutoffonthevanishingsingular
valuesofN .
Example2(PaulitomographyinaPaulichannel). [7]Theoperators[σ /√2]makeanobservable
α
orthonormalbasisforC⊗2.WeconsidernowthenoisedescribedbythedepolarizingPaulichannel
p
N =(1 p)I + T, (22)
− 2
.
where I denotes the identity map and T the trace map T(X) = ITr(X). This noise can be
simplyunbiasedvianoise-mapinversion:
1 p
N −1† = I T. (23)
1 p − 2(1 p)
− −
Example3(Homodynetomographywithη <1). [7]ThesetofdisplacementsoperatorsD(α):=
eαa†−α∗a, α∈CmakeanobservableDirac-orthonormalframeforT1/2,where
Ts = X T, X =:f(a,a†):, s.t. lim f(α,α¯)es|α|2 =0 (24)
{ ∈ α }
→∞
where: :denotesnormalordering. Inthepresenceofnoisefromnonunitquantumefficiencyη,the
unbiased reconstruction ispossible foroperators inTs ifη (2s)−1. Infact, one usesthenew
≥
dual:
D(α) N −1†(D(α))=ηD(η1/2α)e1−2η|α|2. (25)
→
Example4 (Homodyne tomography in Gaussian noise). [7] As forquantum efficiency, Gaussian
noisecanbeunbiasedformeanthermalphotonnumbern¯ s 1.Onehasthenewdual:
≤ − 2
D(α) N −1†(D(α))=D(α)en¯|α|2. (26)
→
3. THECASEOFHOMODYNETOMOGRAPHY
Before addressing the general problem of deriving a general tomographic rule for unbounded
operators,inthissectionwewillre-derivetheknownpatternfunctionofhomodynetomographyin
ordertoillustratethegeneralconceptsintroducedintheprevioussection.
RENORMALIZEDQUANTUMTOMOGRAPHY 7
Thestartingpointistheobservableframe D(α) α C,intermsofwhichthedecomposition(9)
{ } ∈
forHilbert-Schmidtoperatorsrewritesasfollows
d2α
X = Tr[D†(α)X]D(α). (27)
π
ZC
Bychangingtopolarvariablesα=( i/2)keiφ,Eq.(27)becomes
−
X = π dφ +∞ dk|k| Tr[X eikXφ]e−ikXφ. (28)
π 4
Z0 Z−∞
Eq.(28)canbeusedonlyforHilbert-Schmidtoperators,forwhichthetraceundertheintegralinEq.
(28)exists.Intermsofthequadraturespectralmeasure,onehas
X = π dπφ +∞ dk4|k| Tr[X eikXφ]e−ikXφ = π dπφ +∞dEφ(x)Tr[XWφ,x], (29)
Z0 Z−∞ Z0 Z−∞
where
Wφ,x =e−iφa†aD(x)W0,0D†(x)eiφa†a, W0,0 =−12PX12, (30)
0
PdenotingtheCauchyprincipalvalue. Ontheotherhand,inthenextsectionwewillshowthatfor
unboundedoperatorswealsohavetheexpansion
π dφ ∞
X = dtTr[G†(t,φ)X]F(t,φ),
π
Z0 Z−∞ (31)
F(t,φ)= 1 e−21(t−i2Xφ)2, G(t,φ)= d tet22 1dθi(1 θ)teiφ iθteiφ.
√2π dt | − ih− |
Z0
whichintermsofthethequadraturespectralmeasurereads
π dφ ∞
X = π dEφ(x)Tr[XWφ′,x], (32)
Z0 Z−∞
wherenow
Wφ′,x= ∞ dt√12πe−21(t−i2x)2ddttet22 1dθ|i(1−θ)teiφih−iθteiφ|. (33)
Z−∞ Z0
Alternatively,asshowninSec. (9)bymeansoftheframeofnormal-orderedmoments,onehas
theexpansion
X = ∞ a†namTr[gn†,mX]
n,m=0
X (34)
1
=Z0π dπφZ−+∞∞dEφ(x)Tr"Xn,Xm∞=0gn†,m n+nm!− Hn√+m2n(+√m2x)#
where
min(n,m) ( 1)j
gn,m = − n j m j . (35)
j! (n j)!(m j)!| − ih − |
Xj=0 − −
The above expansions in Eqs. (32) apnd (34) are just examples of alternate expansions which are
equivalent for the estimation of the expectation values of (even unbounded) observables, but can
beverydifferentasregardsthestatisticalnoiseaffectingsuchestimation. Asamatteroffact, the
problemofclassifyingallpossibleexpansionshasbeenneversolved,and,hopefully, theresultsof
thepresentpapermaysuggestaunifyingapproachtothesolutionofsuchadifficultproblem.Aswe
willseeinthenextsection,theexistenceofmanyalternateexpansionsisduetothesymmetryofthe
quorumofquadratureoperators,andtheresultingpropertiesofnullestimatorfunctions.
8 GIACOMOMAUROD’ARIANOANDMASSIMILIANOFEDERICOSACCHI
4. CALCULUSWITHNULLFUNCTIONS
Wefirstnoticethattoanullestimatorfunctionn 0itcorrespondsanullexpansionoverthe
ξ
≃
quorum,namely
dµ(ξ) dE (x)n (x)=0 dµ(ξ)n (X )=0. (36)
ξ ξ ξ ξ
⇐⇒
ZX ZX ZX
Letusrecalltheorderingrelation[14]
k,l
{ } k!l! s r j
:a†kal:s = j!(k j)!(l j)! −2 :a†k−jal−j:r , (37)
Xj=0 − − “ ”
where k,l := min(k,l), and s = 1,0, and1 correspond to normal, symmetrical, and anti-
{ } −
normalordering,respectively.WewillalsowritethesymmetricalorderingasS a†kal :=:a†kal:0,
{ }
andthenormalorderingas:a†kal::=:a†kal: 1.
−
Thenwehave:
Lemma1(Mainequivalencerelation). [13]Thefollowingequivalencerelationholds
xke±i(k+2n+2)φ 0, k,n 0. (38)
≃ ∀ ≥
Proof.SinceXφk = 21k kl=0 kl S{a†lak−l}eiφ(2l−k),onehas
P π`dφ´
π e±i(k+2+2n)φXφk =0, ∀k,n≥0, (39)
Z0
whichisequivalentto(38).
Stateddifferently:
Lemma2. Thefollowingequivalencerelationholds
Hk(√2x)e±i(k+2n+2)φ 0, k,n 0, (40)
≃ ∀ ≥
whereH (x)denotethek-thHermitepolynomial.
k
Proof.FromthedefinitionofHermitepolynomialsonehas
√12nHn(√2Xφ)= √12n ∂∂tnn e−t2+√2t(a†eiφ+ae−iφ)
˛t=0
= √12n ∂∂tnn˛˛t=0e√2ta†e˛˛˛iφe√2tae−iφ =Xk=n0 nk!a†kan−keiφ(2k−n) =2n:Xφn:. (41)
Then,itfollowsthat ˛˛
πdφ
π e±i(k+2+2n)φHk(√2Xφ)=0, ∀k,n≥0, (42)
Z0
whichisequivalentto(40).
Moreover,wealsohave
Lemma 3 (Equivalence of truncated Hermite polynomials). The following equivalence relations
hold:
e±inφH2(ll)+n(κx)≃e±inφH2l+n(κx), (43)
whereweintroducedthetruncatedHermitepolynomial
H(l)(z)= l (−)mn!(2z)n−2m. n 2l. (44)
n m!(n 2m)! ≥
m=0 −
X
RENORMALIZEDQUANTUMTOMOGRAPHY 9
Proof. Thetwoidentitiesarejustthecomplexconjugatedofeachother. Therefore,itissufficient
toprovetheidentitywiththeplussign. Thelatterisasimpleconsequenceofidentity(38),namely
xke±i(k+2n+2)φ 0.Byusingtheun-truncatedHermitepolynomial,wehave
≃
einφH2l+n(κx)=einφl+[[n/2]](−)mm(2!(l2+l+n)n!(2κ2)2ml+)n!−2m (45)
m=0 −
X
x2l+n−2mei[n+2(l−m)+2(m−l)]φ,
×
fromwhichitfollowsthatalltermswithm>lareequivalenttozero.
Finally,onecanshowthePoissonidentities(whoseproofcanbefoundintheAppendix)
Lemma4(Poissonidentities). Thefollowingidentitieshold
f(x2)δ (φ) 1 f(x2e2iφ) + f(x2e−2iφ) ,
π ≃ π 1 e 2iφ 1 e2iφ
» − − − – (46)
xf(x2)δ (φ) 1 xeiφf(x2e2iφ) + xe−iφf(x2e−2iφ) ,
π ≃ π 1 e 2iφ 1 e2iφ
» − − − –
Inparticular,wehavetheidentity
1
δ (φ) . (47)
π ≃ π
5. THEKOLMOGOROVCONSTRUCTION
In thisSectionwe present theKolmogorov construction, and itsrelationwiththe fundamental
identityofquantumtomography.
Inthefollowing, byl wedenotetheHilbertspaceofsquaresummablesequencesofcomplex
2
numbers,andbyL (X)wedenotetheHilbertspaceofsquaresummablefunctionsoverthespaceX.
2
Forexample,X=R,andL (X)theHilbertspaceofsquaresummablefunctionsontherealaxis,or
2
X=S1,andL (X)istheHardyspaceofsquare-summablecomplexfunctionsonthecircle. Inthe
2
followingwewillfocusattentiononlytothecaseX R.Considernowacompleteorthonormalset
offunctions[υ (x)]forL (X).Thecompletenessof≡thesetcorrespondstothefollowingdistribution
n 2
identity
υn(x)∗υn(y)=δ(x y), (48)
−
n
whereδ denotestheusualDirac-dXelta. Considernowa(infinite-dimensional)HilbertspaceHand
denoteby[w ]anorthonormalbasisforit.Thefollowingvector
n
υ(x) = υ (x)w , (49)
n n
| i | i
n
X
isDirac-normalizable,inthesensethat
υ(y)υ(x) =δ(x y). (50)
h | i −
Considernowanother(infinite-dimensional)HilbertspaceK H.Totheorthonormalbasis[υ (x)]
n
forL2(X)and [zn]forKweassociate amap fromtheobse≃rvables OX withspectrumX onHto
operatorsinB(H,H K)givenby
⊗
υ(X)= υ (X) z , (51)
n n
⊗| i
n
X
where,asusual,wedefinetheoperatorsυ (X)intermsofthespectralresolutionofXasfollows
n
υ (X)= dE (x)υ (x), (52)
n X n
ZX
wheredE (x)denotesthespectralmeasureofX.
X
ForX,Y OX formallywewrite
∈
υ(X)†υ(Y)= υn(X)†υn(Y), (53)
n
X
10 GIACOMOMAUROD’ARIANOANDMASSIMILIANOFEDERICOSACCHI
ConsidertheintegralkernelK(x,y),x,y XcorrespondingtothepositiveoperatorK B(H)
∈ ∈
K(x,y)= υ(x)K υ(y) . (54)
h | | i
Foranytwoself-adjointoperatorsX,Y onL (R),theexpressionK(X,Y)iswelldefinedinthe
2
followingsense
K(X,Y)=. υ(X)†(I K˜)υ(Y), (55)
⊗
whereK˜ B(K)isgivenby
∈
K˜ = z w K w z , (56)
n n m m
| ih | | ih |
n,m
X
namelywecanalsowrite
K(X,Y)= υn(X)† wnK wm υm(Y)= dEX(x) dEX(y)K(x,y). (57)
h | | i
n,m ZX ZX
X
Thisisalsoequivalent tosaythatfor anyexpansion of K(x,y)inseriesof products offunctions
ofsinglevariable,K(X,Y)isdefinedasthesameexpansion, orderedwiththefunctionsofX on
theleftandthefunctionsofY ontheright. ForcommutingX,Y,thenK(X,Y)simplyrepresents
thesameanalyticexpressionofK(x,y),nowsubstitutingtheoperatorsinplaceofthevariables.As
anexample,theidentityoperatorK = I correspondstotheDirac-deltakernel,andforcommuting
X,Y OX wehave
∈
υ(X)†υ(Y)=δ(X Y). (58)
−
ByreplacingnowH H⊗2,evenfornoncommutingXandY,onehas
→
(υ(X)† I)(I υ(Y))=δ(X I I Y). (59)
⊗ ⊗ ⊗ − ⊗
Moreover,similarlytoEq.(57),onehas
K(X I,I Y) =. (υ(X)† I)(IH⊗2 K˜)(I υ(Y))
⊗ ⊗ ⊗ ⊗ ⊗
= υn(X)† υm(Y) wn K wm
⊗ h | | i
n,m
X
= dE (x) dE (y)K(x,y). (60)
X X
⊗
ZX ZX
Thefundamentalidentitiesofquantumtomographyareobtainedasanexpansionoftheswapoperator
Eoverthequorum,sinceforanystateρandobservableAonehasTr[ρA]=Tr[(ρ A)E],where
⊗
E ψ φ = φ ψ .
| i⊗| i | i⊗| i
5.1. Homodynetomography. FromEq.(28),itiscleartheswapoperatorcanbewrittenas
E = π dφ +∞ dk|k|e−ikXφ eikXφ. (61)
π 4 ⊗
Z0 Z−∞
Then,theusualhomodynetomographicformulacanbeobtainedbytheKolmogorovconstructionin
writingtheswapoperatorasfollows
π dφ
E = π (υ(Xφ)†⊗I)(IH⊗2⊗K˜)I⊗υ(Xφ), (62)
Z0
correspondingtothepositivekernel
K(x,x′)= π x Y x′ = dk keik(x′−x) = P 1 , (63)
2h || || i ZR 4 | | −2 (x−x′)2
whereY isthequadratureconjugatedtoX,with[X,Y] = i. Thekernelisclearlypositive,since
2
onehas
2
dk
K(xi,xj)ξi∗ξj = R 4 |k|˛ eixjξj˛ (64)
Xi,j Z ˛Xj ˛
˛ ˛
˛ ˛
˛ ˛