344 CHAPTER22. THEEPRPARADOXANDBELL’STHEOREM
systemisinstate
ψ(x 1 ,x 2 )=ψC(x 1 )ψD(x 2 ) (22.2)
Butwhatdowemakeofthe(normalized)superposition
ψ(x 1 ,x 2 )=aψA(x 1 )ψB(x 2 )+bψC(x 1 )ψD(x 2 ) (22.3)Accordingto theprinciplesof quantum mechanics, physicalstates arenormalized
vectorsinHilbertspace,andthesumoftwovectorsisanothervector.Ifthatvector
isnormalized,itisaphysicalstatetoo.Sothewavefunction(22.3)is,fromthispoint
ofview,entirelykosher:itisaphysicallyallowedstateofthetwo-particlesystem.But
thenifweask:whatisthestateofparticle1,whenthetwo-particlesystemisinstate
(22.3),wediscoverthatwecan’tanswerthequestion! Particle 1 isnotinstateψA,
norisitinstateψC;neitherisitinalinearcombinationsuchascψA(x 1 )+dψC(x 1 ).
Thefactis,thestateofparticle 1 hasbecomeentangledwiththatofparticle2;itis
impossibletospecifythestateofeitherparticleseparately,evenifthetwoparticles
arenon-interacting,andveryfarapart.
Thefactthatthe quantumstates ofcompositesystems canbeinseparable (or
entangled)inthiswaywasfirstnoticedbyEinstein,Podolsky,andRosen(”EPR”)
in1933.Theconsequencesofthisentanglementaretrulymind-bending,evenbythe
generousmind-bendingstandardsofquantummechanics.
22.1 The EPR Paradox
Einstein,PodolskyandRosenwereoftheopinionthatquantummechanicsisinsome
wayincomplete,becauseitleadsto(whattheyconsideredtobe)aparadoxincertain
circumstances. Toillustratetheirpoint,weconsiderthefollowingexperiment(Fig.
22.1).Everytimeabuttonispushed,twospin 1 / 2 particlesareejected,inopposite
directions,eachparticleeventuallypassingthrough adetector whichmeasuresthe
spinoftheparticlealongagivenaxis.Thedetectorscanberotated,sothatthespin
alonganyaxis,inparticularalongthex-andz-axes,canbemeasured. Aftermany
trials,thefollowingresultisreported:
A.Bothdetectorsaresettomeasurespinalongthez-axis.Wheneverthespinofthe
particleontheleftisfoundtobespinup,andthisoccurs50%ofthetime,the
spinoftheparticleontherightisfoundtobespindown. Likewise,whenever
theparticleontheleftisfoundtobespindown,whichistheother50%ofthe
time,theparticleontherightisfoundtobespinup.Thismeansthatbymeasuringthez-componentofthespinoftheparticleonthe
left,wecandeterminethez-componentofthespinoftheparticleontheright,without
actuallyinteractingwiththeparticleontheright.Suppose,inaparticularrun,that
theparticlewhichmovestotheleft(whichwe’llnowcall”L”)isfoundtobespin-
up. Sincethemeasurementontheleft(goestheEPRargument)couldn’tpossibly