22.3. ENTANGLEDSTATESFOR”QUANTUMRADIO”? 351
where
cnφn(x 2 )=
∫
dx 1 φ∗n(x 1 )Ψ(x 1 ,x 2 ) (22.24)Definedinthisway,thestates{φn(x 2 )}donothavetobeorthogonal,theydonot
evenhavetobedifferentstates. Ifallthe{φn(x 2 )}arethesamestate,thenΨ(x 1 ,x 2 )
isseparable.Ifatleastsomeofthe{φn(x 2 )}aredifferent,thenΨ(x 1 ,x 2 )isentangled.
Now wemakethe measurementof observableA onparticle 1, andoneofthe
eigenvalues,λ=λk say,isobtained. Thismeansthatparticle 1 hasjumpedinto
thestateφk(x 1 ). But,iftheintialstateisentangled,italsomeansthatparticle 2
hasjumpedintostateφk(x 2 ),whichitwasnotinbefore.Inotherwords,theentire
two-particlestateΨ(x 1 ,x 2 )has”jumped”into stateφk(x 1 )φk(x 2 ),evenifparticles
1 and 2 arefarapart,anda measurementisperformed onlyon particle1. This
certainlyseemslikeanon-localinfluence,andBell’sTheorempreventsthe”cheap”
explanationthatparticle 2 wasreallyinstateφk(x 2 )(perhapssupplementedbysome
”hidden”variables)allalong.
Itsnaturaltosupposethatifameasurementatdetector 1 causessomethingto
happentothedistantparticle2,thenthiseffectcouldbeusedtosendmessages,in-
stantaneously,betweentheobserveratdetector 1 andtheobserveratdetector2.But
ifinstantaneous(oringeneral,fasterthanlight)communicationwerepossible,then
accordingtospecialrelativityitshouldalsobepossibletosendmessagesbackwards
intime.Thiswouldhavesomeinterestingpracticalapplications. Atypicalscenario
isasfollows:
Thefouraccompliceshad planneditforyears,after comingintopos-
sessionoftwopairsofsuper-secret-andhighlyillegal -quantumradios.
Nowforthepayoff. John,theringleader,standsatthebettingwindowof
amajor racetrack,whilehis wife, Mary,waitsimpatientlyontheplanet
Mars. Thethirdparticipant,Rajiv,ispilotingarocketshipwhichishead-
ingtowardsEarthataverysubstantialfractionofthespeedoflight. His
partner, Fatima,is on anotherrocketship, movingparallel toRajivand
atequal speedtowardsMars. Theobjectiveof allthese preparations is
forJohntolearn thenameofthewinninghorseattheKentuckyDerby,
beforetheraceisactuallyrun.
Theworldlinesofallfourparticipantsareindicatedonaspacetimediagramshown
inFig.22.4.AtpointAinFig.22.4,theracehasjustended.
”NielsBohr”,alongshotpaying 200 to1,hascomeinfirst. AsRajiv’s
rocket streaksoverhead,John signalsthename of thewinning horse to
him. Rajivthenuses hisquantumradiotosend thisinformation instan-
taneouslytoFatima.
However,asweknowfromrelativitytheory,simultaneityisrelativetoreference
frame. IntheJohn-Maryrestframe,eventsAandA’,aresimultaneous,andBand