22.2. BELL’STHEOREM 349
particlemusthavebeeninastatewhichwouldcauseatype 1 detectortoflashred.
So,ifwesetbothswitchesto1,wewouldexpecttosee 11 RRexactlyasmanytimes
asonesees 22 GG,namely,9%ofthetime. Butaccordingtoresult(C), 11 RRnever
occurs! Therefore,nohiddenvariabletheorycanbeconsistentwithallthreeresults
(A),(B),and(C).Canquantumtheoryeverpredictsuchresults?
LetsdenotetheeigenstatesofobservableO 1 bythekets
| 1 G> and | 1 R> (22.17)Themeaningofthenotationisthatwhenaparticleisinaneigenstate| 1 G>,itcauses
thelightonaswitch 1 detectortoflashgreen,andsoon.NowsupposeoperatorO 2
ischosensuchthatitseigenstatesarerelatedtotheeigenstatesofO 1 inthefollowing
way:
| 2 G> =
√
3
5| 1 G>+
√
2
5| 1 R>
| 2 R> = −
√
2
5| 1 G>+
√
3
5| 1 R> (22.18)
Supposefurtherthat theapparatusisdesigned suchthat theparticlesleavingthe
apparatusareinthestate
|Ψ>=
√
3
8| 1 R>l| 1 G>r+√
3
8| 1 G>l| 1 R>r−1
2
| 1 G>l| 1 G>r (22.19)wherethesubscriptrmeansthatthestatereferstotheparticleontheright,andl
referstotheparticleontheleft.
WithobservableO 2 andstateΨchoseninthisway,thenbyfollowingthestandard
rulesofquantumtheorywecancomputetheprobabilitiesforfindingoutcomesGG,
RR,RG,andGR,forswitchsettings12,21, 22,and 11. Theseprobabilitiesare
showninFig.22.3Theyareincompleteagreementwiththeassertedresults(A),(B)
and(C)above.
Inthisway, Bell’sTheorem is proven. There are physical states inquantum
mechanicswhichleadtopredictionsthatcanneverbereproducedbyalocalhidden
variablestheory. Thesephysicalstatesarealwaysofthe”entangled”form,inwhich
itisimpossibletodeducethestateofeitherparticleseparately.
Ofcourse, theexampleconstructedhere, althoughsufficienttoprove theorem,
israther special. Bell’s theorem, inmore generality, states that if themeasured
valuesofvariousobservablesviolatecertaininequalities,thenthoseresultscannotbe
explainedbyanylocalhiddenvariablestheory. Thisbringsusto anexperimental
question: Maybe quantum mechanics is wrong! Can we actuallyobserve, in the
laboratory, resultswhichviolatetheBellinequalities (i.e. cannot beexplainedby
localhiddenvariabletheories)?