Preliminary analysis of a recent experiment by F. A. Bovino

An analysis is made of the results of a recent polarization correlation experiment by Bovino (unpublished) where about 60,000 data have been obtained. I assume that the state of the photon pairs produced in the source (a non-linear crystal) are in a (sligh

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:0viXraPreliminaryanalysisofarecentexperimentbyF.A.BovinoEmilioSantosDepartamentodeF´ sica.UniversidaddeCantabria.Santander.SpainFebruary2,2008AbstractAnanalysisismadeoftheresultsofarecentpolarizationcorrela-tionexperimentbyBovino(unpublished)whereabout60,000datahavebeenobtained.Iassumethatthestateofthephotonpairsproducedinthesource(anon-linearcrystal)areina(sightly)non-maximallyentangledstateandthemostrelevantnon-idealitiesoftheset-uparetakenintoaccount.Acomparisonismadebetweenthepredictionsofquantummechanicsandasimplefamilyoflocalhiddenvariablesmodelswiththeresultthattheformerisviolatedbymorethan4standarddeviationswhilstthedataarecompatiblewiththefamilyoflocalmodels.Thepossibilityofmakingatruediscriminationbetweenlocalrealismandquantummechanics,vialoophole-freetestsofBell´sinequality,seemsextremelydi cultorimpossibleasshownbythefa¨ lureofthee ortsmade

duringmorethan40yearsafterBell´swork.Thusthestandardpracticehasbeentotestinequalitiesderivedfromlocalrealismplustheassumptionoffairsampling.TheseinequalitieshavebeenviolatedinmanyHoweverfairsamplingisanassumptiontheoreticallyabsurd(becauseitex-cludesallsensiblehiddenvariablesandithasbeenclaimedtobeempiricallyThus,inordertomakerealprogressweshouldtestinequalitesvalidforrestricted,butsensible,familiesoflocalhiddenvariables(LHV)theoriessuchastheonesderivedinandsumarizedinInthislightIwillanalyzetheresultsofarecentexperimentbyBovino[6].

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An analysis is made of the results of a recent polarization correlation experiment by Bovino (unpublished) where about 60,000 data have been obtained. I assume that the state of the photon pairs produced in the source (a non-linear crystal) are in a (sligh

IntheBovinoexperimentthedetectione ciencyofphotoncountershasbeen55%andtheoveralle ciencyabout17%[6].Thus,assuminglossesoforder15%in lters[7],Imayestimatethatthereisacollectione ciency(“e ective bercouplingcoe cients”)about36%.Bovinousedtwo-channelpolarizersandhehasrecordedahugeamountofdatawhichcorrespondto4coincidenceratesand4singleratesfor45anglesofAlice´spolarizerscombinedeachwith45anglesofBob´spolarizers.

Intheidealcasequantummechanicspredictsforthecoincidencerates,R12,asafunctionoftheangle,φ,betweenAlice´sandBob´spolarizers,acosinuscurveoftheform

QR12(φj)= R12 [1+Vcos(2φ)],(1)

exp(φj),withthispredictionThedisagreementoftheexperimentaldata,R12maybemeasuredbythequantity

=

An analysis is made of the results of a recent polarization correlation experiment by Bovino (unpublished) where about 60,000 data have been obtained. I assume that the state of the photon pairs produced in the source (a non-linear crystal) are in a (sligh

thecollectione ciency(“e ective bercouplingcoe cients”)andthe ltersbutnottodetectors,seemstobeviolated.Itpredictsthat shouldbelargerthan0.04,aconstraintnotful lledinabouthalfthecases(thedocumentSantos3.xlssentbyBovinoreportsthat -labelledD(eta)there-isabout0.12butmycalculationgivesavalueabout1/3thatofBovino).Inanycasetheviolationmightbeattributedtothenon-idealitiesoftheexperimentalset-up.

Insummarytheprevious(ratherpoor)analysisoftheexperimentseemstoimplythatitiscompatiblewiththefamilyLHV1(andthereforeLHV0)butdisagreeswithboththefamilyLHV2andquantummechanics.Inordertogetmoreinterestinginformationabetteranalysisofthedataisrequired,whichismadeinthefollowing.

Thenon-idealitiesoftheexperimentalset-uparequiteimportant,asisshownbythefactthatthe4singleratesareverydi erent,rangingfromabout70000to110000countsinthe(unspeci ed)timewindow.Furthermore,fora xedpositionofBob´sanalyzerAlice´ssingleratesdependontheangleofAlice´sanalyzers,withavariationupto7%betweenthemaximumandtheminimumvalue.Similarvariationsexistforthesinglerateswhenweconsiderdi erentvaluesofBob´sanalyzerswithAlice´sanalyzers xed.(Nosigni cantvariationexistsinthesinglecountsofAlice(Bob)whenBob(Alice)polarizerisrotated,asisexpectedbythe“no-signallingprinciple”whichforbidssendinginformationatadistance.)Ishallmakethequantum-mechanicalanalysisoftheexperimentbystudyingthestateproducedinthenon-linearcrystalandhowthisstateevolvesinthetravelofphotonsuntilthedetectors,asfollows.

1.Ishouldassumethatinthenonlinearcrystalthephotonsareproducedalwaysinpairs(nosingle-photonproductions)andthatthetwophotonsinapairareinsomepure,entangled,quantum-mechanicalstate.Theentangle-mentmaynotbemaximal,althoughclosetomaximal.ThusIwillassumethatthestateisoftheform

|ψ>=1

1+(1+γ)2[|H>1 |V>2 (1+γ)|V>1 |H>2],(3)

whereγisarealnumbersuchthat|γ|<<1.

2.Onlyafractionµa(µb)ofthephotonsgoingtoAlice(Bob)arecol-lected,sothatthestateofthephotonsintheoptical bersbecomesasta-tisticalmixtureof

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An analysis is made of the results of a recent polarization correlation experiment by Bovino (unpublished) where about 60,000 data have been obtained. I assume that the state of the photon pairs produced in the source (a non-linear crystal) are in a (sligh

a)theinitialtwo-photonstate(3)withweightµaµb,

b)asingle-photonstatehorizontallypolarizedforAlicewithweightµa(1 µb)/(2+2γ+γ2),

c)asingle-photonstateverticallypolarizedforAlicewithweightµa(1 µb)(1+γ)2/(2+2γ+γ2)),

d)asingle-photonstatehorizontallypolarizedforBobwithweightµb(1 µa)(1+γ)2/(2+2γ+γ2)),

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