354 CHAPTER22. THEEPRPARADOXANDBELL’STHEOREM
=
∑
|bM|^2 <φM|Q|φM>
=∑M[c∗RURM∗ <ψ|r+c∗GUGM∗ <ψ′|r]Q[cRURM|ψ>r+cGUGM|ψ′>r]=
∑M[
|cR|^2 URM∗ URM <ψ|Q|ψ>+|cG|^2 UGM∗ UGM<ψ′|Q|ψ′>+c∗RcGURM∗ UGM<ψ|Q|ψ′>+c.c.]
= |cR|^2 <ψ|Q|ψ>+|cG|^2 <ψ′|Q|ψ′> (22.36)where,inthelaststep,wehaveusedtheunitarityproperty(22.32).
TheconclusionisthatnothingMarycanmeasurewilltellherwhetherJohnhas
turnedonhisapparatus,orwhatobservableheismeasuring. Thisisaverygeneral
result,andcanbeeasilyextendedtomorecomplicatedsituations. Thereisnosuch
thingasa”quantumradio;”entangledstates canneverbeusedtotransfer infor-
mationacrosslongdistancesataspeedfasterthanlight. Onecanneverdetermine,
justbyameasurementononeof theparticles,whetherthe stateof thesystemis
entangled,orwhetherishasbecomedisentangledbyafarawaymeasurement. The
”entangled”natureofentangledstates,andtheirstrange,non-localproperties,can
onlybeappreciatedwhencomparingmeasurementsmadeontheentiresystem,rather
thananyoneofitsparts.
22.4 For Whom the Bell(’s Theorem) Tolls
Entangledstatesarethenorm,nottheexception,inquantummechanics. Generally
speaking, when any two systems comeinto interaction, the resultingstateof the
combinedsystemwillbeentangled.Inthistechnicalsenseofquantuminseparability,
thepoetwasright:nomanisanisland. Noelectronisanislandeither.
Aswe have seen,the existenceof entangledstatesmeans thatameasurement
ofonesystemcancausethesecondsystem,perhapsverydistantfromthefirst,to
jumpintooneoranotherquantumstate. ItistimetoreturntotheEPRcriticism:
”Isn’tthissituationparadoxical? Doesn’titimplythat somethingmustbe wrong
withquantumtheory?”
Nowfirstofall,aspointedoutbyNielsBohrinhisresponsetoEPR,thissituation
isnotreallyaparadox. Thenon-localitypointedoutbyEPRiscertainlyvery,very
strange. Butquantumnon-localityisnotactuallyaparadoxinthesenseofbeinga
logicalcontradiction.
”Wellthen,doesn’tthisnon-localbehaviorviolatetheoryofRelativity?”Accord-
ingtorelativitytheory,afterall,nosignalcan propagatefaster thanthespeedof
light, sohow canthe stateofparticle 2 changeinstantaneously, inthelaboratory
referenceframe, uponmeasurementof particle1? Herewe mustbe careful- the
relativisticprohibitionisagainst informationpropagatingfasterthan thespeedof
light. Ifsuchpropagationwerepossible,itwouldalsobepossibletosendmessages
backwardsintime,whichwouldraisemanyother(authentic!)paradoxes. However,