Volume130, number4,5 PHYSICSLETTERSA 11 July 1988

IS QUANTUM MECHANICS WITH CPNONCONSERVATION INCOMPATIBLE WITHEINSTEIN’S LOCALITY CONDITION AT THE STATISTICAL LEVEL?

AmitavaDATTA’Institutfür Physik, (JnjversitätDortmund,4600Dortmund50, FRG

DipankarHOMEDepartmentofPhysics,BoseInstitute,Calcutta 700009, India

and

AmitavaRAYCHAUDHURIDepartmentofPhysics,Calcutta University,Calcutta700 009, India

Received23October1987;revisedmanuscriptreceived30 March1988; acceptedfor publication9 May 1988Communicatedby J.P.Vigier

As a sequelto our earlierwork, wepresenta generalanalysis,in termsofdensityoperators,oftheEPR-typegedankensituation(in thepresenceofCPnoninvariance)involvingbasisstateswhicharemutuallynon-orthogonalbutpartially distinguishable.Wealsocommenton thepublishedcriticismsof ourearlierwork.

In an earlierpaper [1] we hadpointedout a cu- principle,impliesthepossibilityof a non-localeffectrious gedankenexampleof the Einstein—Podolsky— manifestingat the statisticallevel. This result wasRosen (EPR) paradoxusingCF nonconservation. obtainedby consideringa certain transitionof theThe exampleinvolves a pair of correlatedneutral pure stateinto a mixed statecomposedof non-or-pseudo-scalarmesons(M°—M°)originatingfrom the thogonalcomponents.The collapseto sucha mixeddecayof a J~=1 — vectormeson.The exponen- statewasfirst assumedto be “total” andthenthe er-tially decayingstateswith definite massesandlife- ror involved (duetooverlapbetweentheprobabilitytimes are denotedby I ML> and IM~>(which are distributions of the invariant massesof the decaycertainlinearcombinationsof I M°>and I M°>)and productscorrespondingtothe non-orthogonalstates)they are usedto describethe quantummechanical wasestimated.It wasarguedthat the errorcouldbe,time-evolutionof the system.In thepresenceof CF in principle,madesmallcomparedto themeasureofnoninvariance,I ML> and I M5> arenon-orthogonal. the non-localeffect.This non-orthogonalityof the physically relevant In a subsequentnote, Squiresand Siegwart [21states (unique characteristicof the quantumme- havequestionedthe resultof ref. [1]. However,it ischanicaltreatmentof CF nonconservation)wasex- importantto note that they consideran inherentlyploredin ref. [1]. Thereit was indicatedthat in an different scenariowhere the collapseof the waveEPR-typesituationinvolving thesestateswhich are function (eq. (9) of their paper) takesplaceto apartially distinguishable(through certain physical mixed statewith mutually orthogonalcomponentsattributes),the quantummechanicaltreatment,in (eq. (10) of their paper).This amountsto identi-

fying anddistinguishingthemutuallyorthogonalin-

Permanentaddress:Departmentof Physics,JadavpurUniver- dividualdecay-productcomponents(in the context

sity,Calcutta700032, India. of the exampledealtwith in ref. [11). Theissue of

0375-9601/887$03.50 © ElsevierSciencePublishersB.V. 187(North-HollandPhysicsPublishingDivision)

Volume 130,number4,5 PHYSICSLETTERSA 11 July 1988

partialdistinctionbetweenthenon-orthogonalstates productstates I ØL>, I øs> (commondecayproductsof the total decayproducts(the crucial elementin from ML and M~)on the right, which is non-van-our treatment)is not addressedto by Squiresand ishingin the presenceof CF violation [1]. TheverySiegwartandwhat they have consideredis essen- fact that the statisticalpropertyof the particlesontially thestandardcasewhereit isalreadywell known oneside hassomeformal dependencepertainingtothatno non-localeffectexistsat the statisticallevel, the interferencebetweenthe physical statesof theThe argumentby FinkelsteinandStapp[31is in ef- particleson the othersideis the key featureof thisfectsimilarto thatof ref. [2]. It wasalsoclaimedby example.Whetherthisinterferencecanbephysicallythe authorsof refs. [2] and [31that the error anal- tamperedby suitablyselectingthe particleson theysis in ref. [lJ, referredto earlier, wasambiguous. right is the point at issue.It needsto be notedthatTheysuggesteda changein theparameterusedin ref. thesedecayproduct stateshavephysicalattributes[11 as a measureof the error, which swampedthe (e.g. invariantmasses,lifetimes of their parentpar-non-local effect. Absenceof a rigorousschemefor tides,etc.) associatedwith them. In the absenceofestimating the error, therefore, makes the issue CF violation, the states I ø~>~I øs> are orthogonalunclear, and they can be distinguishedunambiguouslyac-

In this paperwe show that the result of ref. [1] cordingto the ideasof the standardquantummea-canbe corroboratedby a mode of analysisdifferent surementtheory. However, if in the presenceof afrom thatadoptedin ourearlierpaper.Herewe avoid smallbutnon-vanishingCF violatinginteraction,onethe “ambiguity” mentionedaboveanddirectly in- canpartially discriminatebetweenthe states I ~L>

corporatethe notionof whatwecall the“partial col- and I øs> by exploitingthe differencesin their phys-lapse” (or “partial information”)type measurement ical attributes,therearisesa possibility, in principle,which implies transitionsuchthat the coherenceof to affectthecoherenceofthewavefunctiongivenbythe original purestate is only partially destroyed. eq. (2).

Takingthecuefromref. [1], thetwo-particlewave Sucha schemeenvisagesnon-orthodoxmeasure-functionat thetimeof production(t = 0) of the pair mentspartially destroyingthe coherenceof theorig-is givenby inal pure state and leading to mixtures of non-

orthogonalstates.It shouldbe emphasisedthat the9’(t=O)>=(IMsML>—IMLMs>)/N, (1)

conceptof suchmeasurements( partial collapse )whereN is a normalizationfactorandthe first (sec- is notprimafacie inadmissibleandcanbedealtwith,ond) memberof eachpair refersto the left (right) at leastin principle,by suitablegeneralisationof thehemisphere.Notethatweshallfollow closelytheno- standardquantumtheory of measurement,as hastation of ref. [1]. beenshownby variousauthors [4]. In this context

Followingthediscussiongivenin ref. [1], thesub- it may be notedthat recently Ivanoviá [5] hasan-sequenttime evolvedwave function canbe written alysedthe viability of possiblenon-standardschemesin the form to differentiatebetweennon-orthogonalstates.There

are variousexamplesof realisticmeasurements[4]I~P(t)>=C1IMLØs>+C2IMsØL>+C31x>, (2)

such as approximatemeasurementsand/or mea-whereC1, C2, C3 aretime-dependentconstants,and surementswith imperfectapparatuswhich cannotbe

Ix> —‘ I MSML> — I MLMS> representsthe unde- describedby the standardquantummeasurementcayed piece with <xlx> = 1. Iø,~>(Iø~>)corre- theory basedon orthogonalprojectionsonly. Thisspondsto thedecayproductson therightfrom I ML> aspecthasrecentlybeendiscussedby Ghirardiet al.

(1M5>). It may be notedthat in (2) we havenot [6] in thelight of ref. [1] ~‘.

consideredthosecomponentsof the wave function Now to formalisethis discussionwepresentaden-which containdecayproductson the left astheyare sity matrix treatmentof our gedankenexampleinirrelevantfor our subsequentdiscussionwhich is fo- termsof a specificansatzfor “partial collapse”. Incusedon the flux of I M°>on the left. this treatmentwe explicitly takeprobability conser-

It isclearfromeq. (2) thatthe aboveflux involvesa contributiondueto the overlapbetweenthedecay ~ For footnoteseenextpage

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vation into account,circumventingtherebythe ob- basisfor the systemon the left.jectionraisedby Lindblad [8] regardingour earlier Turning now to the case(B), we consider“mea-work. Herewe areinterestedin the totalnumberof surements”on the right pertainingto physicalattri-M°(‘~‘ IML> — I Ms>) on the left in two cases:(A) butesof the non-orthogonalstates I ØL> and I øs>For no measurementperformedon the right; (B) resultingin “partial collapse”to a mixedstatecorn-After “partial collapse”type measurementpertain- posedof I Wi>, I W2>, I W3>, I ~~‘

4>with therespectiveing to IØL> and I øs> on the right. probabilities Pt, P2, P3, P4 (= IC3 12) where

Let us first considerthecase(A). Thedensityop- I Wi> =C1 IMLØs> +C2IMsØL>, I W2> = IMLØs>,eratorP~Rcorrespondingto I ~P(t)> is given by I W3> = IMsØL>, and I W4> = Ix>. Note that in the

limit of no non-orthogonality(i.e. a=0) thereisP~R I C~121MLØS> <MLØs I “total collapse”in which casePt = 0, P2= I C1 12, and

2+ IC2I2IMsØL><MsØLI + IC

3I2Ix><xI p

3=IC2iAfter the“partial collapse”typemeasurement,the

+CTC2 IMsØL> <MLØS I +C’rC3 Ix> <MLØS IdensityoperatorP~Ris given by

+C~C1lMLøs><MsOLI+C~C3Ix><MsøLIPLR=P1IWI><WlI+P2IW2><W2I+P3IW3><W3I

+C~CiIMLØs><xI+C~C2IMsØL><xI. (3)+1C31

21x><xI, (5)The reduceddensityoperatorp~for the undecayed

whencethereduceddensityoperatorp~ correspond-systemon the left is obtainedby taking the traceofP~Rovera completeset of orthonormalstatesof the ing to the undecayedsystemon the left is obtained

to besystemontheright. Thenusing<øsI ØL> = <ØL I øs>*

=a(t) (say), <ØsIML,s>=<ØLIML,s>=O and p~—Cv1IC’1 1

2+P’2)IML><MLI

<ØL,SIØL,S>=l—exp(—yL,St)=FL,s(t),we get±(P1IC’1i

2+P~)IMs><MsIp~=IC’1 2IML>MLI+IC~I2IMs><MsI +PICTC

2aIMS><MLI+pIC~C,a*IML><Ms+C’t’C2aIM~><MLI +C~Cla*IML> <MsI

+IC3I2p

2(x), (6)+IC3I

2PL(x), (4)whereP2.3=P2,3Fs.L(t). If one invokes probability

where conservationin the “partial collapse”measurementthenTr(p~R)=Tr(p~R),whencewe get

PL(X)=~ KR, Ix> <xIR,>R,

P’2 +P~= (1 —p~)(I C’1 12+1C’2 12

with the kets I R> forming a completeorthonormal +C~rC2a/3+C~Cia*fl), (7)basis for the system on the right and

I C’1,2 12= IC1,2 I2Fs.L(t). Notethat<xIx> = 1 implies wherewehaveused<MLIML> = <M

5IM5> = 1, and

~ <Ljp~(,~)JL,>= 1 <MLIMS> = <MSIML> =fl. Note that a and fi areL, relatedand they both vanish in the limit of CF

conservation.where the kets IL,> form a completeorthonormal Now using (7), we obtainfrom (4) and (6)

B A

~ In thisconnection,it is interestingto notethatthewell-known PL —PL = [(Pi — 1)1 C’1 2+p~exampleofproductionof particletracksin a cloudchamberis < ~ ML> <ML I — I M5> <M~I)interpretableas non-orthodoxmeasurementwehere a planewavecollapsesinto gaussianwavepackets(seeref. [7]). An- — 2fl(p — 1) Re(Cl’ C2a) I M5> <M~Iothersimpleexampleof non-orthodoxmeasurementis pro-videdby theStern—Gerlachexperimentfor spin-1/2 atomsin + (p~— 1) (CTC2aIM5> <ML Iwhichthemagneticfield is assumedto bevery weakandthecountersareplacedsoclosetogetherthateachof thetwo sep- + C~C1 a* I ML> KM5 I ) . (8)aratedbeamshasa finiteprobability ofbeingregisteredin boththecounters. It is transparentfrom (8) thatp~~pj’~which isasig-

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natureof non-locality at the statisticallevel, i.e. the surementsin thecontextoftheexamplediscussedinstatisticalpropertiesof the undecayedsystemon the this paperneedsfurther examination.For instance,left would changedueto the “partial collapse”type it maybeprobedwhetherit is possibleto exploit themeasurementon the right. In the limit of no non-or- differencein the life-times of the states I ML> andthogonality (no CF violation), a= fl= 0 andPt = 0, I M~>to selectoutpartially the decayproductscor-

P2= IC1 12, whencep~=p~. respondingto, say,the IØL> state,tinkeringthereby,

Now, to be morespecific,we computei~M°,the the interferencebetween I ØL> and I cbs>.Thiswoulddifferencein the numberof M°observedon the left correspondto a non-orthodox“measurement”in-for (A) and (B). Using therelevantformulaegiven volving selectionof decayproductswith their timein ref. [1] we obtain from eq. (8) of origin restrictedwithin a specific interval.

RecentlyHall [9] andGhirardiet al. [6] havear-I (pE—pt) I M°> guedon the basisof the operation-effectformalism

— (1 ~ (1 ,~f~’*,-’ (using the first representationtheorem [10]) that—k ~P ~ —Pu e~1~2a, even for non-orthodoxmeasurements,the reduced= ~(p~ —1 )$e~”(cosL~~mt_e~

t). (9) densityoperatorfor oneparticleremainsunaffectedif the measurementis restrictedto its partner.They

In the secondline we havesubstitutedthe actual thenconcludethat the type of collapseenvisagedinexpressionsfor C

1, C2 anda from ref. [1]. Notice our examplenecessarilycorrespondsto a “measure-that for CFinvariance,fl= 0 andhencei~M°= 0. The ment” which affects both the particlessimultane-non-localeffect, therefore,crucially hinges on the ously. However, applicability of this abstractnon-orthogonalitybetweenthe states I ML> and argumentbasedon the first representationtheoremI M5>. Thisis thecruxof theessentialresultalsoob- for all typesof non-orthodoxmeasurementsneedstotamedin ref. [11. A commenton the probabilities be carefully examinedbeforedrawinganyfirm con-p, (i= 1, 2, 3, 4) is in order.Theseprobabilitiescan clusion.Namiki has pointedout to one of us (pri-be calculatedunambiguouslyin thelimit of CF con- vate discussion) that the many-Hilbert-spacesservation.In the presenceof CF violation their pre- formulation of the quantum measurementtheorycisevaluesare,however,not calculabledue to non- [11] appearsto providea suitableframeworkto dealorthogonalityof the basisstatesandwe treatthem with “partial collapse” type measurements.Thisas phenomenologicalparameters.It is, however,in- schememay be studiedto analysethe exampledis-terestingto note that the non-locality at the statis- cussedhere.tical level (eq. (9)) doesnotdependon thesedetails Thecuriousresultdiscussedin thispaperandref.andis non-vanishingunlessPt = 1, a valuewhich is [1] gives rise to the following questions:ruled out for obviousreasons(seeeq. (7)). (a) Is the peculiarityof this exampleessentially

The genesisof this intriguing non-locality at the due to theincompletenessof theconventionalquan-statisticallevel lies in the possibility of partially dis- tum mechanicalformalism (with its inherent ap-tinguishing the non-orthogonalstates I cbL>, I cbs> proximations)usedtodescribethedecayingsystemsthroughtheir physical attributes.Of course,if one in the presenceof CPnoninvariance?chooses to confineone’sattentiononly to orthodox (b) Doesthis exampleindicatethat the notionofquantummeasurementsinvolving, for example,the non-orthodoxmeasurement,by itself, canleadto ainvariant massesof the individual decay product new featureof the EPR paradox?componentsof I cbL> and I cbs> which are mutuallyorthogonal,then the non-locality at the statistical Oneof us (DH) wishesto thankJ. CorbettandS.level will notbe manifestedas shownin refs. [2,31; Senguptafor helpful remarks.DH alsogratefullyac-whatweenvisagehereis a generalisedmeasurement knowledgesvaluablediscussionswith the partici-in the sensediscussedin ref. [4]. It shouldinvolve pants at the International Conference tonon-standardmeasurementsin contrast to ideal commemorateSchrodinger’sbirth centenaryheld atmeasurementsentailing orthogonalprojections.A Delphi, Greece. AD thanks the Alexander-von-concreteoperationalschemefor realisingsuchmea- Humboldt-Stiftung,Bonn,for financialsupport.The

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researchof AR is supportedby the Indian National orthodoxmeasurementspertainingto non-orthogo-Science Academy and the University Grants nal stateshavebeencritically examinedby SrinivasCommision. andHome [19].

In the light of all thesestudiesit is hopedthat theNoteadded.The specific model for “partial col- model for “partial collapse”used in this papercan

lapse” (partialloss of coherenceof theoriginal pure befurther refinedandmademoreconcretewith ref-stateinvolving non-orthogonalcomponents)usedin erenceto specific measurementalprocedures.Thisthis paperenvisagestransition from the purestate would help to clarify the issue whether the peculi-given by eq. (2) into a mixed state composedof arity of the examplediscussedin ourpaperis an in-

Iw~>, 1W2>, 1W3>, 1W4> (eq. (5)), presumingthe dicativeof a genuinenon-localeffect predictedbypossibility that at leastfor someof the events,the quantummechanicsat the statisticallevel.statesI cbL> and I cbs> canbedistinguishedsothatforthe selectedsub-ensemblesdesignatedby the states

I W3> and I W2> the projectionoperatorscorrespond- Referencesing to the states I cbL> and I cbs> havedefinitevalues(= + 1). For simplicity, we havetaken the coeffi- [1]A. Datta,D. HomeandA. Raychaudhuri,Phys.Lett.A 123

cientsC~,C2 in Iw~>to be the sameas thosein eq. (1987)4.(2) but this is not essentialfor the non-localeffect [2] E. SquiresandD. Siegwart,Phys.Lett. A 126 (1987)73.

[3] J.FinkelsteinandH.P.Stapp,Phys.Lett. A 126 (1987)159.obtainedin our treatment.[4] I. Block andD.A. Burba,Phys.Rev.D 10 (1974) 3206;

After submissionof the manuscript,our attention K. Kraus in: Foundationsof quantummechanicsandor-

hasbeendrawntothepaperby Clifton andRedhead deredlinear spaces,eds.A. HartkamperandH. Neumann[12] relatedto ref. [1], which like refs. [2] and [3] (Springer,Berlin, 1974);

emphasizestrhatwithin the frameworkof the stan- E.B. Davies,Quantumtheoryof opensystems(AcademicPress,NewYork, 1976);dardtheoryofmeasurementsin quantummechanics H.P. Yuen,Phys.Lett.A 91(1982)101;in: Proc. 2nd mt.thereis no scopefor non-localeffectof thetype dis- Symp. on Foundationsof quantummechanics(Phys. Soc.

cussedin our paper.What we contendis that since Japan,Tokyo,1987).

the quantummechanicaltreatmentof CF non-in- [5] ID. Ivanovi~,Phys.Lett. A 123 (1987) 257.varianceprovidesan exampleof physicallyrelevant [6] G.C. Ghirardi, R. Grassi,A. Rimini and T. Weber, pre-print,Universityof Trieste,Italy, 1987.non-orthogonalstatesfor which one may consider [7] W. Heisenberg,The physicalprinciplesofthequantumthe-

applying the notion of “partial distinction” (exploit- ory (Universityof ChicagoPress,Chicago1930);

ing the differencesin their physical attributes), it LI. Schiff,Quantummechanics(McGraw-Hill, NewYork,

raisesthe issueof “non-orthodox”measurementsin 1968)p. 335.

the contextof the EPR example— anarenahitherto [8]G. Lindblad,Phys.Lett. A 126 (1987)71.[9] M.J.W.Hall, Phys.Lett. A 125 (1987)89.left unexplored and which lies beyond the ambit of [10] K. Kraus, States,effectsandoperations(Springer,Berlin,

the standardtheory of quantum measurements. 1983).

“Partial distinction”betweennon-orthogonalstates, Eli] S. MachidaandM. Namiki, in: Proc. 1st mt. Symp. onin a sense,involves “unsharp” or “imprecise” si- Foundationsof quantummechanics,eds.S. Kamefuchiet

multaneousmeasurementof noncommutingobserv- al. (Phys. Soc.ofJapan,Tokyo, 1984);M. Namiki, Ann. N.Y. Acad. Sci. 480 (1986)78.ables, a conceptwhosetenability hasbeenanalysed [12] R.K. Clifton andM.L.G. Redhead,Phys.Lett. A 126 (1988)by variousauthorslike thosementionedin refs. [4,5] 295.

andalso by WoottersandZurek [13], Busch [14], [13] W.K. WootttersandW.H. Zurek, Phys.Rev.D 19 (1979)

Mittelstaedtet al. [15] andGreenbergerandYasin[16]. Onemay also noteherethe recentpapersby [14] P. Busch,Phys.Rev.D33 (1986) 2253.[15] P. Mittelstaedtetal., Found.Phys.17 (1987)391.Dieks [17] andCorbett [18] which studyin depth [16] D.M. GreenbergerandA. Yasin, Phys.Lett. A. 128 (1988)

the particularquestionof “partial discrimination” 391.

betweennon-orthogonalstates.Possiblelimitations [17] D. Dieks,Phys.Lett.A 126 (1988) 303.of the operation-effectformalism (discussedin refs. [18]J. Corbett,Phys.Lett.A 130 (1988),lobepublished.

[6,9,10]) in orderto coverall possiblemodelsofnon- [191M.D. SrinivasandD. Home,UniversityofMadraspreprint.

191