QMGreensite_merged

22.3. ENTANGLEDSTATESFOR”QUANTUMRADIO”? 353

Iftheright-handparticleisinstate|ψ>,theexpectationvalueforobservableQis
<ψ|Q|ψ>;whileiftheparticleisinstate|ψ′>,thevalueisis<ψ′|Q|ψ′>. After
manyruns,theaveragevalueobtainedmustbe

<Q> = Prob.tofind1R×<ψ|Q|ψ>+Prob.tofind1G×<ψ′|Q|ψ′>

= |cR|^2 <ψ|Q|ψ>+|cG|^2 <ψ′|Q|ψ′> (22.29)

whichisthesameasresult(22.26).ThereforeMarycan’ttellifJohnhadturnedon
hisdetector,iftheswitchsettingisat1. WhatifJohnsetstheswitchtosetting2?
Ingeneral,the”switch1”eigenstates{| 1 R>, | 1 G>}andthe”switch2”eigen-
states{| 2 R>, | 2 G>}aresimply differentbasisvectorsforthe left-handparticle
HilbertSpace,relatedbyaunitarytransformation

| 1 M>=

∑

M=R,G

UMN| 2 N> (22.30)

wherewerecallthattheunitarityproperty

δMN=

∑

J

UNJUM∗J (22.31)

followsfromorthonormality

δMN = < 1 M| 1 N>

=

∑

K

UM∗K< 2 K|

∑

J

UNJ| 2 J>

=

∑

J

UM∗JUNJ (22.32)

Wecanthereforewritetheentangledstate|Ψ>as

|Ψ> =

∑

M=R,G

[cRURM|ψ>r+cGUGM|ψ′>r]| 2 M>l

=

∑

M=R,G

bM| 2 M>l|φM>r (22.33)

where
bM|φM>≡cRURM|ψ>r+cGUGM|ψ′>r (22.34)

Then,afterJohnmakeshismeasurement,theparticleswillbe

eitherinstate | 2 R>|φR>, withprobability |bR|^2

orelseinstate | 2 G>|φG>, withprobability |bG|^2

(22.35)

Aftermanyruns,theexpectationvalueisfoundbyMarytobe

<Q> = Prob.tofind2R×<φR|Q|φR>+Prob.tofind2G×<φG|Q|φG>