QMGreensite_merged

352 CHAPTER22. THEEPRPARADOXANDBELL’STHEOREM

B’aresimultaneous,butAislaterthanBandB’.NotsointheRajiv-Fatimarest
frame:forthem,eventsAandB’aresimultaneous.

WhenRajivsignalsthenameofthewinninghorse toFatima,shere-
ceivesthisinformationatspacetimepointB’,justasherrocketispassing
Mars. FatimarelaystheimportantinformationtoMary,and thenMary
contactsJohn,againbyquantumradio,andtransfersthemessageinstan-
taneously(intheJohn-Maryrestframe)toEarth. Themessagereaches
Johnat spacetimepoint B, several minutes before therace isto begin.
Johnplacesaverylargebeton”NielsBohr”towin,and...

Letsleavethestorythere. Couldentangledstatesbeusedtosendinstantaneous
messagesinthisway,eveninprinciple?Toanswerthisquestion,weagainconsiderthe
apparatusshowninFig.22.2.Johnisatdetector1.Maryisatdetector2,buthasan
arsenaloflaboratoryequipmentavailabletoher,andcanmeasureanyobservableshe
likes. Istheresomeobservationwhichshecouldperformontheright-handparticles,
whichwouldtellher

I. whetherJohnhasturnedonhisdetector;or

II.ifJohn’sdetectorison,whethertheswitchisonsetting 1 orsetting2?

Theentangledstateof theparticles comingoutof thedetector can alwaysbe
writtenintheform

|Ψ>=cR| 1 R>l|ψ>r+cG| 1 G>l|ψ′> (22.25)

where,sincethisisanentangledstate,ψ+=ψ′. Maryhasdecidedtomeasurethe
observableQ.SupposefirstthatJohnhasturnedoffhisdetector.InthatcaseMary
finds

= <Ψ|Q|Ψ>

= [c∗R< 1 R|l<ψ|r+c∗G< 1 G|l<ψ′|r]Q[cR| 1 R>l|ψr>+cG| 1 G>l|ψ′>r]

= |cR|^2 <ψ|Q|ψ>+|cG|^2 <ψ′|Q|ψ′> (22.26)

wherewehaveusedtheorthogonality

< 1 M| 1 N>=δMN M=RorG, N=RorG (22.27)

NowJohnturnsonhisdetector,withtheswitchsettingat1. Whenhemakesa
measurement,thetwoparticleswillbe

eitherinstate | 1 R>|ψ>, withprobability |cR|^2

orelseinstate | 1 G>|ψ′>, withprobability |cG|^2

(22.28)