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348 CHAPTER22. THEEPRPARADOXANDBELL’STHEOREM

Nolocalhiddenvariabletheorycanagreewithallofthepredictionsofquantum
mechanics.

Therestrictiontoalocal theorymeansthattheorydoesnotcontain anynon-
local”action-at-a-distance”interactionsor,equivalently,anyeffectswhichpropagate
fromoneplaceto anotherat greaterthanthe speedof light. Thismeansthat a
measurementofLcannotaffectthehiddenvariablesofR;atleast,notbeforethe
timerequiredforlighttotravelfromtheleftdetectortotheparticleontheright.
Onlyoneexampleisneededofaquantum-mechanicalpredictionthatcannotbe
reproducedbyahiddenvariabletheory;thisisenoughtoprovethetheorem. The
exampleIwillgivehereisduetoMermin.
ConsidertheexperimentalsetupshowninFig. 22.2. Thisisverymuchlikethe
previoussetup,inthatwhenabuttonispushed,twoparticlesemerge,onetraveling
tothedetectorontheright,andonetothedetectorontheleft. Oneachdetector
thereis aswitch, andtwolights, oneredand onegreen. Whenthe switchisin
position 1 thenthe detectormeasuressomeobservable,O 1 say,and,dependingon
thevaluefound,thelightflasheseitherredorgreen.Likewise,whentheswitchisin
position2,thedetectormeasuressomeotherobservable,O 2 say. Again,depending
onthevaluefound,thelightflashesredorgreen.Inourpreviousexample,O 1 =Sz,
andO 2 =Sx,andwecouldhavearrangedthatthelightflashesredforspinup,and
greenforspindown.
Aftermanyruns,thefollowingresultsarefound:

A.Inrunsinwhichthedetectorshavedifferentswitchsettings,thelightsoneach

detectorneverbothflashgreen;i.e. 12 GGand 21 GGneveroccur.^1

B.Inrunsinwhichbothswitchesaresetto2,onefindsthat9%ofthetimeboth

lightsflashgreen(22GG).

C.Inrunsinwithbothswitchesaresetto1,bothlightsneverflashred,i.e. 11 RR

neveroccurs.

Now we’llimagine that theseresults can be explained by assumingthat each
particlecomesoutoftheapparatusinadefinitestate,withsomedefinitevaluesof
thehiddenvariables. Fromresult(A),wecanconcludethatifoneoftheparticles
isinastatewhichwouldcause aswitch 2 detectorto flashgreen, thentheother
particleisinastatewhichwouldcauseaswitch 1 detectortoflashred. Ifthiswere
notthecase,thenoccasionallyonewouldfind 12 GGand 21 GG,which,itisasserted,
neveroccurs.Itfollowsthatinthose 22 GGrunswherebothlightsflashgreen,each

(^1) Where,e.g., 12 RGmeans: theswitchontheleft-handdetectorissetto1,theswitchonthe
right-handdetectorissetto2,thelightontheleft-handdetectorflashesred,thelightonthe
right-handdetectorflashesgreen.