Introduction to relativisticquantum information Daniel R. Terno∗ Perimeter Institute, 31 Caroline St, Waterloo, Ontario, Canada N2L 2Y5 I discuss the role that relativistic considerations play in quantum information processing. First I describe howthecausalityrequirementslimitpossiblemulti-partitemeasurements. ThentheLorentztransformationsof quantumstatesareintroduced,andtheirimplicationsonphysicalqubitsaredescribed. Thisisusedtodescribe relativisticeffectsincommunicationandentanglement. TothememoryofAsherPeres,teacherandfriend 6 I. INTRODUCTION 0 0 2 Informationand physics are closely and fascinatingly intertwined. Their relations become even more interesting when we leave a non-relativisticquantummechanicsformoreexitingvenues. Mynotesare plannedas a guidedtourfor thefirst steps n alongthatroad,withopenquestionsandmoreinvolvedmergerslefttotheremarksandtothelastsection. a J Istartfromabriefintroductiontocausalityrestrictionsonthedistributedmeasurements: thelimitationsthatareimposedby 4 finalpropagationvelocityofthephysicalinteractions. Itisfollowedbytherelativistictransformationsofthestatesofmassive particlesandphotons,fromwhichwecandeducewhathappenstoqubitswhicharerealizedasthediscretedegreesoffreedom. 2 Buildingonthis,Idiscussthedistinguishabilityofquantumsignals,andbrieflytouchcommunicationchannelsandthebipartite v entanglement. 9 I do not follow a historical order or give all of the original references. A review [1] is used as the standard reference on 4 quantuminformationandrelativity.Theresultsofthe“usual”quantuminformationaregivenwithoutanyreference:allofthem 0 canbefoundinatleastoneofthesources[2,3,4]. Finally,awordaboutunits:~=c=1arealwaysassumed. 8 0 5 0 II. CAUSALITYANDDISTRIBUTEDMEASUREMENTS / h p HereIpresentthecausalityconstraintsonquantummeasurements. Forsimplicity,measurementsareconsideredtobepoint- - like interventions. First recall the standard description of the measurementand the induced state transformation. Consider a t n systeminthestateρthatissubjecttomeasurementthatisdescribedbyapositiveoperator-valuedmeasure(POVM) E . The µ { } a probabilityoftheoutcomeµis u q p =trE ρ, (1) µ µ : v whilethestatetransformationisgivenbysomecompletelypositiveevolution i X ρ ρ′ = A ρA† /p , A† A =E . (2) r → µ µm µm µ µm µm µ a Xm Xm Iftheoutcomeisleftunknown,theupdateruleis ρ ρ= A ρA† . (3) → µm µm Xµm Nowconsiderabipartitestateρ . TheoperationsofAliceandBobaregivenbytheoperatorsA andB ,respectively. AB µm νn Itiseasytoseethatiftheseoperatorscommute, [A ,B ]=0, (4) µm νn then the observationstatistics of Bob is independentof Alice’s resultsand vice versa. Indeed,the probabilitythatBob getsa resultν,irrespectiveofwhatAlicefound,is p = tr B A ρA† B† . (5) ν νn µm µm νn Xµ (cid:16)Xm,n (cid:17) ∗Electronicaddress:[email protected] 2 NowmakeuseofEq.(4)toexchangethepositionsofA andB ,andlikewisethoseofA† andB† ,andthenwemove µm νn µm νn A fromthefirstpositiontothelastoneintheproductofoperatorsinthetracedparenthesis. SincetheelementsofaPOVM µm satisfy E =1l,Eq.(5)reducesto µ µ P p =tr B ρB† , (6) ν νn νn (cid:16)Xn (cid:17) whence all the expressions involving Alice’s operators A have totally disappeared. The statistics of Bob’s result are not µm affectedatallbywhatAlicemaysimultaneouslydosomewhereelse. ThisprovesthatEq.(4)indeedisasufficientconditionfor noinstantaneousinformationtransfer. Inparticular,thelocaloperationsA 1l and1l Bareofthisform. B A ⊗ ⊗ Note that any classical communicationbetween distant observers can be considered a kind of long range interaction. The propagationof signals is, of course, bounded by the velocity of light. As a result, there exists a partial time ordering of the variousinterventionsinanexperiment,whichdefinesthenotionsearlierandlater. Theinputparametersofaninterventionare deterministic (or possibly stochastic) functions of the parameters of earlier interventions, but not of the stochastic outcomes resultingfromlaterormutuallyspacelikeinterventions[1]. Eventheseapparentlysimplenotionsleadtonon-trivialresults. ConsideraseparablebipartitesuperoperatorT, T(ρ)= M ρM†, M =A B , (7) k k k k⊗ k Xk wheretheoperatorsA representoperationsofAliceandB thoseofBob. Notallsuchsuperoperatorscanbeimplementedby k k localtransformationsand classical communication(LOCC) [7]. This is the foundationof the “non-localitywithoutentangle- ment”. A classification of bipartite state transformationswas introduced in [8]. It consists of the following categories. There are localizable operationsthat can be implemented locally by Alice and Bob, possibly with the help of prearrangedancillas, but withoutclassicalcomunication. Ideally,localoperationsareinstantaneous,andthewholeprocesscanbeviewedasperformed at a definite time. A finalclassical outputof such distributedinterventionwill be obtainedat some pointof the (joint) causal futureofAlice’sandBob’sinterventions. Forsemilocalizableoperations,therequirementofnocommunicationisrelaxedand one-wayclassicalcommunicationispossible.Itisobviousthatanytensor-productoperationT T islocalizable,butitisnot A B ⊗ anecessarycondition. ForexampletheBellmeasurements,whichdistinguishesbetweenthefourstandardbipartiteentangled qubitstates, 1 1 Ψ± := (0 1 1 0 ), Φ± := (0 0 1 1 ), (8) | i √2 | i| i±| i| i | i √2 | i| i±| i| i arelocalizable. Otherclassesofbipartiteoperatorsaredefinedasfollows:BobperformsalocaloperationT justbeforetheglobaloperation B T. If no local operation of Alice can reveal any information about T , i.e., Bob cannot signal to Alice, the operation T is B semicausal. If the operation is semicausal in both directions, it is causal. In many cases it is easier to prove causality than localizability(seeRemark3).Thereisanecessaryandsufficientconditionforthesemicausality(andtherefore,thecausality)of operations[8]. These definitions of causal and localizable operators appear equivalent. It is easily proved that localizable operators are causal. Itwasshownthatsemicausaloperatorsarealwayssemilocalizable[9]. However,therearecausaloperationsthatarenot localizable[8]. ItiscuriousthatwhileacompleteBellmeasurementiscausal,thetwo-outcomeincompleteBellmeasurementisnot.Indeed, consideratwo-outcomePVM E = Φ+ Φ+ , E =1l E . (9) 1 2 1 | ih | − If the initial state is 01 , then the outcome that is associated with E always occurs and Alice’s reduced density matrix AB 2 | i after the measurementis ρ = 0 0. On the other hand, if before the joint measurementBob performsa unitary operation A | ih | that transformsthe state into 00 , then the two outcomesare equiprobable, the resulting states after the measurementare AB maximallyentangled,andAlic|e’sireduceddensitymatrixisρ = 11l. AsimplecalculationshowsthatafterthisincompleteBell A 2 measurementtwoinputstates 00 and 01 aredistinguishedbyAlicewithaprobabilityof0.75. AB AB | i | i Here is another example of a semicausal and semilocalizable measurement which can be executed with one-way classical communicationfromAlicetoBob. ConsideraPVMmeasurement,whosecompleteorthogonalprojectorsare 0 0 , 0 1 , 1 + , 1 , (10) | i⊗| i | i⊗| i | i| i | i⊗|−i where =(0 1 )/√2. TheKrausmatricesare |±i | i±| i A =E δ , (11) µj µ j0 3 Fromthepropertiesofcompleteorthogonalmeasurements[8],itfollowsthatthisoperationcannotbeperformedwithoutAlice talkingto Bob. A protocolto realize this measurementis the following. Alice measuresher qubitin the basis 0 , 1 , and {| i | i} tellsherresulttoBob. IfAlice’soutcomewas 0 ,Bobmeasureshisqubitinthebasis 0 , 1 ,andifitwas 1 ,inthebasis | i {| i | i} | i + , . {| i |−i} If oneallowsfor morecomplicatedconditionalstate evolution[10], thenmoremeasurementsare localizable. In particular, considera verificationmeasurement,i.e., the measurementyieldsa µ-thresultwith certainty,ifthe state priortothe classical interventionswasgivenbyρ=E ,butwithoutmakinganyspecificdemandontheresultingstateρ′. µ µ Itispossibleto realizea verificationmeasurementsbymeansofa sharedentangledancillaandBell-typemeasurementsby oneoftheparties[11]. VerificationmeasurementofEq.(10)canillustratethisconstruction.Inadditiontothestatetobetested, AliceandBobshareaBellstate Ψ− . Theydonothavetocoordinatetheirmoves.AliceandBobperformtasksindependently | i andconveytheirresultstoacommoncenter,whereafinaldecisionismade. Theprocedureisbasedontheteleportationidentity Ψ Ψ− = 1 Ψ− Ψ + Ψ+ Ψ˜(z) + Φ− Ψ˜(x) + Φ+ Ψ˜(y) , (12) | i1| i23 2(cid:16)| i12| i3 | i12| i3 | i12| i3 | i12| i3(cid:17) where Ψ˜(z) means the state Ψ rotated by π around the z-axis, etc. The first step of this measurementcorrespondsto the | i | i firststepofateleportationofastateofthespinfromB (Bob’ssite)toA(Alice’ssite). BobandAlicedonotperformthefull teleportation(which requiresa classical communicationbetweenthem). Instead, Bob performsonlythe Bell measurementat hissitewhichleadstooneofthebranchesofthesuperpositionintherhsofEq.(12). The second step of the verification measurement is taken by Alice. Instead of completing the teleportation protocol, she measuresthespinofherparticleinthez direction. Accordingtowhetherthatspinisupordown,shemeasuresthespinofher ancillainthezorxdirection,respectively.Thiscompletesthemeasurementanditonlyremainstocombinethelocaloutcomes togettheresultofthenonlocalmeasurement[11]. ThismethodcanbeextendedtoarbitraryHilbertspacedimensions. Remarks 1. Measurementsinquantumfieldtheoryarediscussedin[1,5,6]. 2. Analgebraicfieldtheoryapproachtostatisticalindependenceandtorelatedtopicsispresentedin[12]. 3. TocheckthecausalityofanoperationT whoseoutcomesarethestatesρ = T (ρ)/p withprobabilitiesp = trT (ρ), µ µ µ µ µ p =1itisenoughtoconsiderthecorrespondingsuperoperator µ µ P T′(ρ):= T (ρ) (13) µ Xµ Indeed,assumethatBob’sactionpriortotheglobaloperationleadtooneofthetwodifferentstatesρ andρ . Thenthestates 1 2 T′(ρ )andT′(ρ )aredistinguishableifandonlyifsomeofthepairsofstatesT (ρ )/p andT (ρ )/p aredistinguishable. 1 2 µ 1 µ1 µ 2 µ2 SuchprobabilisticdistinguishabilityshowsthattheoperationT isnotsemicausal. 3. Absenceofthesuperluminalcommunicationmakespossibletoevadethetheoremsontheimpossibilityofabitcommitment. InparticulartheprotocolRBC2allowsabitcommitmenttobeindefinitelymaintainedwithunconditionallysecurityagainstall classicalattacks,andatleastforsomefiniteamountoftimeagainstquantumattacks[13,14]. 4. In these notes I am not going to deal with the relativistic localization POVM. Their properties (and difficulties in their construction)can be foundin [1]. An exhaustivesurvey of the spatial localizationof photonsis given in [15]. Here we only noteinpassingthatifE( )isanoperatorthatcorrespondstothedetectionofaneventinaspacetimeregion ,sincetheyare O O notthoughttobeimplementedbyphysicaloperationsconfinedtothatspacetimearea,thecondition[E( ),E( )]=0isnot 1 2 O O required[16,17]. III. QUANTUMLORENTZTRANSFORMATIONS There is no elementaryparticle that is called “qubit”. Qubitsare realized by particulardegreesof freedomof more or less complicated systems. To decide how qubits transform (e.g., under Lorentz transformations) it may be necessary to consider againthe entiresystem. Inthe followingourqubitwill beeithera spin ofa massiveparticleora polarizationofa photon. A quantumLorentztransformationconnectsthedescriptionofaquantumstate Ψ intworeferenceframesthatareconnectedby | i aLorentztransformationΛ(i.e.,theircoordinateaxesarerotatedwithrespecttoeachotherandtheframeshaveafixedrelative 4 velocity).Then Ψ′ =U(Λ)Ψ ,andtheunitaryU(Λ)isrepresentedonFig.1below. Thepurposeofthissectionistoexplain | i | i theelementsofthisquantumcircuit. From the mathematical point of view the single-particle states belong to some irreducible representation of the Poincare´ group.Anintroductorydiscussionoftheserepresentationsandtheirrelationswithstatesandquantumfieldsmaybefound,e.g., in[18,19]. Withineachparticularirreduciblerepresentationtherearesixcommutingoperators.Theeigenvaluesoftwoofthem areinvariantsthatlabeltherepresentationbydefiningthemassmandtheintrinsicspinj. Thebasisstatesarelabelledbythree componentsofthemomentumpandthespinoperatorΣ . Henceagenericstateisgivenby 3 Ψ = dµ(p)ψ (p)p,σ . (14) σ | i Z | i Xσ Inthisformuladµ(p)istheLorentz-invariantmeasure, 1 d3p dµ(p)= , (15) (2π)32E(p) wheretheenergyE(p)=p0 = p2+m2. Theimpropermomentumandspineigenstatesareδ-normalized, p p,σ q,σ′ =(2π)3(2E(p))δ(3)(p q)δσσ′, (16) h | i − andarecompleteontheone-particlespace,whichis =C2j+1 L2(R3,dµ(p))forspin-jfields. H ⊗ spin D Λ momentum classical info FIG.1: Relativisticstatetransformationasaquantumcircuit:thegateDwhichrepresentsthematrixD [W(Λ,p)]iscontrolledbyboththe ξσ classicalinformationandthemomentump,whichisitselfsubjecttotheclassicalinformationΛ. Tofindthetransformationlawwehavetobemoreconcreteaboutthespinoperator. TheoperatorΣ (p)isafunctionofthe 3 generatorsof the Poincare´ group. One popularoption is helicity, Σ = J P/P, which is applicable for both massive and 3 · | | masslessparticles. Formassiveparticlesweusethez-componentoftherest-frame(orWignerspin,thatwenowdescribeinthe nextsection. A. Massiveparticles Theconstructioninvolvepickinga reference4-momentumk, whichformassiveparticlesis takento bek = (m,0). The R WignerspinS(p)isdefinedtocoincidewiththenon-relativisticspinSinparticle’srestframe. Thestateofaparticleatrestis labelled k ,σ , R | i S2 k ,σ =j(j+1)k ,σ , S k ,σ =σ k ,σ . (17) R R 3 R R | i | i | i | i The spin states of arbitrary momenta are defined as follows. The standard rotation-free boost that brings k to an arbitrary R momentump,pµ =L(p)µkν isgivenby ν E p1 p2 p3 m m m m  p1 1+ p21 p1p2 p1p3  m m(m+E) m(m+E) m(m+E) L(p)= p2 p2p1 1+ p22 p2p3 . (18)  m m(m+E) m(m+E) m(m+E)   p3 p3p1 p3p2 1+ p23   m m(m+E) m(m+E) m(m+E)  5 TheWignerspinS(p)andtheone-particlebasisstatesaredefinedby p,σ U[L(p)]k ,σ , S (p)p,σ =σ p,σ . (19) R 3 | i≡ | i | i | i Inderivingthetransformationruleswebeginwiththemomentumeigenstates. Usingthegrouprepresentationpropertyand Eqs.(19)thetransformationiswrittenas U(Λ)=U[L(Λp)]U[L−1(Λp)Λ.L(p)]U[L−1(p)] (20) TheelementoftheLorentzgroup W(Λ,p) L−1(Λp)ΛL(p), (21) ≡ leavesk invariant,k = Wk . Henceitbelongstothestabilitysubgroup(orWignerlittlegroup)ofk . Fork = (m,0)it R R R R R isarotation.Pressingon U(Λ)p,σ =U[L(Λp)]U[W(Λ,p)]k ,σ , (22) R | i | i andasaresult, U(Λ)p,σ = D [W(Λ,p)]Λp,ξ , (23) ξσ | i | i Xξ whereD arethematrixelementsoftherepresentationoftheWignerrotationW(Λ,p). ξσ Weconsideronlyspin-1 particles,soσ = 1. Any2 2unitarymatrixcanbewrittenasDˆ =exp( iωnˆ σ),whereωisa 2 ±2 × − · rotationangleandnˆ isarotationaxisthatcorrespondstoW(Λ,p). Thewavefunctionstransformaccordingtoψ′(q)= ξ,q U(Λ)Ψ sothesamestateintheLorentz-transformedframeis ξ h | | i ∞ Ψ′ =U(Λ)Ψ = D [W(Λ,Λ−1p)]ψ (Λ−1p)σ,p dµ(p). (24) σξ ξ | i | i Z | i Xσ,ξ −∞ ForpurerotationRthethree-dimensional(moreexactly,3Dblockof4Dmatrix;hereandinthefollowingweusethesame letterfora4Dand3DmatrixforR SO(3))Wignerrotationmatrixistherotationitself, ∈ W(R,p)=R, p=(p0,p). (25) ∀ Asaresult,theactionofWignerspinoperatorson isgivenbythanhalvesofPaulimatricesthataretensoredwiththeidentity 1 H ofL2. B. Photons Thesingle-photonstatesarelabelledbymomentump(the4-momentumvectorisnull,E =p0 = p)andhelicityσ = 1, p | | ± so the state with a definite momentumis givenby α p,σ , where α 2 + α 2 = 1. Polarizationstates are also laalbteerlnleadtivbeyla3b-veellcintogrsofǫσpth,epsa·mǫσpes=tat0e,,tthhaetrecfoorrree,sipsonPdσto=±th1eαtwσp|o,sǫeσnps.iesofpol|ari+za|tion|of−c|lassicalelectromagneticwaves. An σ=±1 σ| pi Helicity is invariant under proper Lorentz transPformation, but the basis states acquire phases. The little group element W(Λ,p) = L−1(Λp)ΛL(p) is defined with respect to the standard four-momentumk = (1,0,0,1). The standard Lorentz R transformationis L(p)=R(pˆ)B (u), (26) z whereB (u)isapureboostalongthez-axiswithavelocityuthattakesk to(p,0,0, p)andR(pˆ)isthestandardrotation z R | | | | thatcarriesthez-axisintothedirectionoftheunitvectorpˆ. Ifpˆ haspolarandazimuthalanglesθ andφ,thestandardrotation R(pˆ)isaccomplishedbyarotationbyθaroundthey-axis,thatisfollowedbyarotationbyφaroundthez-axis.Hence, cosθcosφ sinφ cosφsinθ − R(pˆ)= cosθsinφ cosφ sinφsinθ , (27) sinθ 0 cosθ  −  (hereonlythenon-trivial3Dblockisshown). 6 Anarbitrarylittlegroupelementforamasslessparticleisdecomposedaccordingto W(Λ,p)=S(β,γ)R (ξ), (28) z wheretheelementsS(β,γ)formasubgroupthatisisomorphictothetranslationsoftheEuclideanplaneandR (ξ)isarotation z aroundthez-axis. We areinterestedonlyintheangleξ, sinceβ andγ donotcorrespondto thephysicaldegreesoffreedom. However,theyareimportantforgaugetransformations.Finally,thelittlegroupelementsarerepresentedby Dσ′σ =exp(iξσ)δσ′σ. (29) Itisworthwhiletoderivemoreexplicitexpressionsforξ. Ibeginwithrotations,Λ=R. Sincerotationsformasubgroupofa Lorentzgroup,R−1(Rpˆ)RR(pˆ)isarotationthatleaveszˆinvariantandthusisoftheformR (ω)forsomeω. Aboostin(t,z) z planeandarotationaroundz-axiscommute,[R ,B ]=0,so z z W(R,p)=R−1(Rpˆ)RR(pˆ)=R (ξ). (30) z Any rotation can be described by two angles that give a direction of the axis and the third angle that gives the amount of rotationaroundthataxis. IfRp=q,wedecomposetherotationmatrixas R=R (ω)R(qˆ)R−1(pˆ), (31) qˆ whereR (ω)characterizesarotationaroundqˆ,andR(qˆ)andR(pˆ)arethestandardrotationsthatcarrythez-axistoqˆ andpˆ, qˆ respectively.UsingEq.(30)wefindthatS =1landthetworotationsareofthesameconjugacyclass, Rz(ξ)=R−1(Rpˆ)RRpˆ(ω)R(Rpˆ), (32) soweconcludethatξ =ω. Apracticaldescriptionofpolarizationstatesisgivenbyspatialvectorsthatcorrespondtotheclassicalpolarizationdirections. Takingagaink asthe referencemomentum,twobasisvectorsoflinearpolarizationareǫ1 = (1,0,0)andǫ2 = (0,1,0), R kR kR whiletotherightandleftcircularpolarizationscorrespondǫ± =(ǫ1 iǫ2 )/√2. kR kR ± kR PhasesofthestatesobtainedbythestandardLorentztransformationsL(p)aresetto1. SincethestandardboostB (u)leaves z thefour-vector(0,ǫ± )invariant,wedefineapolarizationbasisforanypas kR ǫ± =ǫ± R(pˆ)ǫ± , (33) p pˆ ≡ kR whilethetransformationofpolarizationvectorsunderanarbitraryrotationRisgivenbytherotationitself.Toseetheagreement betweentransformationsofspatialvectorsandstates,consideragenericstatewithamomentump. Itspolarizationisdescribed bythepolarizationvectorα(p)=α ǫ++α ǫ−,orbythestatevectorα p,+ +α p, . UsingEq.(33)weseethatthe transformationofα(p)isgivenby + p − p +| i −| −i Rα(p)=RRpˆ(ω)R(Rpˆ)R−1(pˆ)α(p)=RRpˆ(ω)R(Rpˆ)α(kR)=RRpˆ(ω)α(Rpˆ). (34) Ifq=Rpthetransformationresultsinα eiωǫ++α e−iωǫ−,andsinceω =ξ,itisequivalenttothestatetransformation + q − q U(R)(α p,+ +α p, )=α eiξ q,+ +α e−iξ q, . (35) + − + − | i | −i | i | −i ForageneralLorentztransformationsthetriad(ǫ1,ǫ2,pˆ)isrigidlyrotated,butinamorecomplicatedfashion.Toobtainthe p p phaseforageneralLorentztransformation,wedecomposethelatterintotworotationsandastandardboostB alongthez-axis: z Λ=R B (u)R . (36) 2 z 1 ItcanbeshownthatB alonedoesnotleadtoaphaserotation.Therefore, z ξ =ω +ω , (37) 1 2 wherebothω andω areduetotherotationsandaregivenbyEq.(31). NotethatalthoughB (u)alonedoesnotleadtoaphase 1 2 z rotation,itcanaffectthevalueofω ,sinceitindirectlyappearsinthedefinitionofR . 2 2 7 Remarks 1. A comprehensivediscussion of the Poincare´ groupin physicscan be foundin [20, 21]. Useful expressionsforWigner rotationsandtheirapplicationsformassiveparticlesaregivenin[22,23,24]. 2. InthistransformationIdonotassumeanyadditionalnormalizationfactors. AconditionofunitarityisUU† = U†U = 1l, buttherealsootherconventionsintheliterature. 3. A doubleinfinity of the positive energysolutionsof the Dirac equation(functionsu(1/2) and u(−1/2)) spansan improper p p basisofthisspace. Thereisaone-to-onecorrespondencebetweenWignerandDiracwavefunctions. BasisvectorsofWigner andDiracHilbertspacesareintheone-to-onecorrespondence[21], u(1/2) 1,p , u(−1/2) 1,p , (38) p ⇔|2 i p ⇔|2 i whilethewavefunctionsarerelatedby Ψα(p)=ψ (p)u(1/2)α+ψ (p)u(−1/2)α (39) 1/2 p −1/2 p 4 2mψ (p)= u(−σ)Ψα(p) (40) σ αp αX=1 4. AnotherapproachtotheconstructionoftheWignerrotationDˆ isbasedonthehomomorphismbetweenLorentzgroupand SL(2)[21]. 5. Whennotrestrictedtoasingle-particlespacetheWignerspinoperatorisgivenby S= 1 σ dµ(p)(aˆ† aˆ +ˆb† ˆb ), (41) 2 ηζZ ηp ζp ηp σp Xη,ζ whereaˆ† createsamodewithamomentumpandspinηalongthez-axis,etc. Acomparisonofdifferentspinoperatorscanbe ηp foundin[25]. 6. If one works with the 4-vectors, then in the helicity gauge the polarization vector is given by ǫ = (0,ǫ ). A formal p p connection between helicity states and polarization vectors is made by first observing that three spin-1 basis states can be constructedfromthe componentsof a symmetric spinorofrank 2. Unitarytransformationsof thisspinor thatare inducedby Rareinone-to-onecorrespondencewithtransformationsbyRofcertainlinearcombinationsofaspatialvector. Inparticular, transformationsofthehelicity 1statesinducedbyrotationsareequivalenttotherotationsof(ǫ1 iǫ2 )/√2(thez-axisis the initialquantizationdirectio±n). While p ǫµ = 0gaugeconditionis Lorentz-invariant,thespatkiSal±orthkoSgonalityis not. The µ p roleofgaugetransformationsin preservingthe helicitygaugeandsomeusefulexpressionsforthephasethatphotonsacquire canbefoundin[26,27,28] IV. IMPLICATIONSOFQUANTUMLORENTZTRANSFORMATIONS A. Reduceddensitymatrices In a relativistic system whatever is outside the past light cone of the observer is unknown to him, but also cannot affect his system, thereforedoes not lead to decoherence(here, I assume that no particle emitted by from the outside the past cone penetratesintothefuturecone). Sincedifferentobservershavedifferentpastlightcones,bytracingouttheyexcludefromtheir descriptions different parts of spacetime. Therefore any transformation law between them must tacitly assume that the part excludedbyoneobserverisirrelevanttothesystemofanother. Another consequence of relativity is that there is a hierarchy of dynamical variables: primary variables have relativistic transformation laws that depend only on the Lorentz transformation matrix Λ that acts on the spacetime coordinates. For example,momentumcomponentsareprimaryvariables. Ontheotherhand,secondaryvariablessuchasspinandpolarization havetransformationlawsthatdependnotonlyonΛ,butalsoonthemomentumoftheparticle. Asaconsequence,thereduced densitymatrixforsecondaryvariables,whichmaybewelldefinedinanycoordinatesystem,hasnotransformationlawrelating itsvaluesindifferentLorentzframes. Moreover, an unambiguous definition of the reduced density matrix is possible only if the secondary degrees of freedom areunconstrained,andphotonsarethesimplestexamplewhenthisdefinitionfails. Intheabsenceofageneralprescription,a case-by-casetreatmentisrequired.Idescribeaparticularconstruction,validwithrespecttoacertainclassoftests. 8 B. Massiveparticles For a massive qubit the usual definition of quantum entropy has no invariantmeaning. The reason is that under a Lorentz boost,thespinundergoesaWignerrotation,thatasshownonFig.1iscontrolledbothbytheclassicaldataandthecorresponding momentum. Eveniftheinitialstateisadirectproductofafunctionofmomentumandafunctionofspin,thetransformedstate isnotadirectproduct.Spinandmomentumbecomeentangled. Letusdefineareduceddensitymatrix, ρ= dµ(p)ψ(p)ψ†(p). (42) Z Itgivesstatisticalpredictionsfortheresultsofmeasurementsofspincomponentsbyanidealapparatuswhichisnotaffectedby themomentumoftheparticle. NotethatItacitlyassumedthattherelevantobservableistheWignerspin. Thespinentropyis S = tr(ρlogρ)= λ logλ , (43) j j − − X whereλ aretheeigenvaluesofρ. j Asusual,ignoringsomedegreesoffreedomleavestheothersinamixedstate. Whatisnotobviousisthatinthepresentcase theamountofmixingdependsontheLorentzframeusedbytheobserver. Indeedconsideranotherobserver(Bob)whomoves with a constantvelocitywith respectto Alice who preparedthat state. In the Lorentzframe where Bob is at rest, the state is givenbyEq.(24). Asanexample,takeaparticlepreparedbyAlicetobe ζ Ψ =χ ψ(p)p dµ(p), χ= (44) | i Z | i (cid:18)η(cid:19) whereψ isconcentratednearzeromomentumandhasacharacteristicspread∆. Spindensitymatricesofallthestatesthatare givenbyEq.(44)are ζ 2 ζη∗ ρ= | | , (45) (cid:18)ζ∗η η 2(cid:19) | | and are independentof the specific form of ψ(p). To make calculationsexplicit (and simpler) I take the wave functionto be Gaussian, ψ(p) = Nexp(p2/2∆2), whereN is a normalizationfactor. Spin and momentumare notentangled, andthe spin entropy is zero. When that particle is described in Bob’s Lorentz frame, moving with velocity v at the angle θ with Alice’s z-axis,adetailedcalculationshowsthatthethespinentropyispositive[1]. ThisphenomenonisillustratedinFig.2. Arelevant parameter,apartfromtheangleθ,isintheleadingorderinmomentumspread, ∆ 1 √1 v2 Γ= − − , (46) m v where∆isthemomentumspreadinAlice’sframe.Theentropyhasnoinvariantmeaning,becausethereduceddensitymatrixτ hasnocovarianttransformationlaw,exceptinthelimitingcaseofsharpmomenta.Onlythecompletedensitymatrixtransforms covariantly. I outline some of the steps in this derivation. First, we calculate the rotation parameters (ω,nˆ) of the orthogonal matrix W(Λ,p)forageneralmomentum.Therotationaxisandanglearegivenby nˆ =vˆ pˆ, cosθ =vˆ pˆ, 0 θ π (47) × · ≤ ≤ wherevˆ isboost’sdirection,whiletheleadingordertermfortheangleis 1 √1 v2 p p2 ω = − − sinθ O (48) v m − (cid:18)m2(cid:19) Withoutalossofgeneralitywecanmakeanothersimplification.Wecanchooseourcoordinateframeinsuchawaythatboth ζ andηarereal. ThematrixD[W(Λ,Λ−1p)]takestheform ω ω D[W(Λ,p′)]=σ cos isin ( sinφσ +cosφσ ), (49) 0 x y 2 − 2 − 9 3 2 θ 1 0 S 0.015 00..0011 00..000055 00 0.1 0.08 0.06 0.04 Γ 0.02 FIG.2:DependenceofthespinentropyS,inBob’sframe,onthevaluesoftheangleθandaparameterΓ=[1−(1−v2)1/2]∆/mv. where(θ,φ)arethesphericalangleofp′(tobeconsistentwithEq.(24)momentuminAliceframecarriesaprime,p′ =Λ−1p). ThereduceddensitymatrixinBob’sframeis ρB = dµ(p)D D∗ ψ (p′)ψ∗(p′). (50) σξ Z σν ξλ ν λ Thesymmetryofψ(Λ−1p)iscylindrical. Hencethepartialtraceistakenbyperformingamomentumintegrationincylindrical coordinates. This simplification is a result of the spherical symmetry of the original ψ. The two remaining integrations are performedbyfirstexpandinginpowersofp/∆andtakingGaussianintegrals.Finally, ζ2(1 Γ2/4)+η2Γ2/4 ζη∗(1 Γ2/4) ρ′ =(cid:18) ζ−∗η(1 Γ2/4) ζ2Γ2/4+η−2(1 Γ2/4)(cid:19). (51) − − Fidelitycanbeusedtoestimatethedifferencebetweenthetwodensitymatrices. Itisdefinedas f =χ†ρ′χ, (52) anditiseasytogetananalyticalresultforthisquantity.Setζ =cosθandη =sinθ. Then Γ2 cos4θ f =1 3+ . (53) − 2 (cid:18) 8 (cid:19) ConsidernowapairoforthogonalstatesthatwerepreparedbyAlice,e.g. theabovestatewithχ = (1,0)andχ = (0,1). 1 2 HowwellcanmovingBobdistinguishthem?Iusethesimplestcriterion,namelytheprobabilityoferrorP ,definedasfollows: E anobserverreceivesasinglecopyofoneofthetwoknownstatesandperformsanyoperationpermittedbyquantumtheoryin ordertodecidewhichstatewassupplied.Theprobabilityofawronganswerforanoptimalmeasurementis P (ρ ,ρ )= 1 1tr (ρ ρ )2. (54) E 1 2 2 − 4 1− 2 p InAlice’sframeP =0. InBob’sframethereduceddensitymatricesare E 1 Γ2/4 0 Γ2/4 0 ρB = − , ρB = (55) 1 (cid:18) 0 Γ2/4(cid:19) 2 (cid:18) 0 1 Γ2/4,(cid:19) − respectively.HencetheprobabilityoferrorisP (ρ ,ρ )=Γ2/4. E 1 2 10 C. Photons Therelativistic effectsin photonsareessentially differentfromthose formassiveparticlesthatwere discussed above. This is becausephotonshaveonlytwo linearly independentpolarizationstates. As we know,polarizationis a secondaryvariable: states that correspond to different momenta belong to distinct Hilbert spaces and cannot be superposed (an expression such as ǫ± + ǫ± is meaningless if k = q). The complete basis p,ǫ± does not violate this superselection rule, owing to the | ki | qi 6 | pi orthogonalityofthemomentumbasis. Thereduceddensitymatrix,accordingtotheusualrules,shouldbe ρ= dµ(p)ψ(p)2 p,α(p) p,α(p). (56) Z | | | ih | However,sinceξ inEq.(29)dependsonthephoton’smomentumevenforordinaryrotations,thisobjectwillhavenotransfor- mationlawatall. Itisstillpossibledefinean“effective”densitymatrixadaptedtoaspecificmethodofmeasuringpolarization [29,30]. Idescribeonesuchscheme. ThelabellingofpolarizationstatesbyEuclideanvectorsǫ±suggeststheuseofa3 3matrixwithentrieslabelledx,yandz. p × Classically,theycorrespondtodifferentdirectionsoftheelectricfield.Forexample,acomponentρ wouldgivetheexpectation xx valuesofoperatorsrepresentingthepolarizationinthexdirection,seeminglyirrespectiveoftheparticle’smomentum. Tohaveamomentum-independentpolarizationistoadmitlongitudinalphotons. Momentum-independentpolarizationstates thusconsistofphysical(transverse)andunphysical(longitudinal)parts,thelattercorrespondingtoapolarizationvectorǫℓ =pˆ. Forexample,ageneralizedpolarizationstatealongthex-axisis xˆ =x (p)ǫ+ +x (p)ǫ− +x (p)ǫℓ , (57) | i + | pi − | pi ℓ | pi wherex (p)=xˆ ǫ±,andx (p)=xˆ pˆ =sinθcosφ. Itfollowsthat x 2+ x 2+ x 2 =1,andwethusdefine ± · p ℓ · | +| | −| | ℓ| x (p)ǫ++x (p)ǫ− e (p)= + p − p, (58) x x2 +x2 + − q as the polarization vector associated with the x direction. It follows from (57) that xˆ xˆ = 1 and xˆ yˆ = xˆ yˆ = 0, and h | i h | i · likewisefortheotherdirections,sothat xˆ xˆ + yˆ yˆ + ˆz ˆz =1l , (59) p | ih | | ih | | ih | where1l istheunitoperatorinmomentumspace. p Tothedirectionxˆ therecorrespondsaprojectionoperator P = xˆ xˆ 1l = xˆ xˆ dµ(k)p p, (60) xx p | ih |⊗ | ih |⊗Z | ih | The actionof P on Ψ followsfromEq. (57) and ǫ± ǫℓ = 0. Onlythe transversepartof xˆ appearsin the expectation xx | i h p| pi | i value: ΨP Ψ = dµ(p)ψ(p)2 x (p)α∗(p)+x (p)α∗(p)2. (61) h | xx| i Z | | | + + − − | Itisconvenienttowritethetransversepartof xˆ as | i b (p) (ǫ+ ǫ+ + ǫ− ǫ− )xˆ =x (p)ǫ+ +x (p)ǫ− . (62) | x i≡ | pih p| | pih p| | i + | pi − | pi Likewisedefine b (p) and b (p) . Thesethreestatevectorsareneitherofunitlengthnormutuallyorthogonal. y z | i | i Finally, a POVM element E which is the physical part of P , namely is equivalentto P for physical states (without xx xx xx longitudinalphotons)is E = dµ(k)p,b (p) p,b (p), (63) xx x x Z | ih | and likewise for the other directions. The operatorsE , E and E indeed form a POVM in the space of physicalstates, xx yy zz owingtoEq.(59). To complete the constructionof the density matrix, we introduce additionaldirections. Following the standard practice of statereconstruction,weconsiderP ,P andsimilarcombinations.Forexample, x+z,x+z x+iz,x+iz P = 1(xˆ + ˆz )( xˆ + ˆz) 1l . (64) x+z,x+z 2 | i | i h | h | ⊗ p