QMGreensite_merged

23.1. MIXTURESANDPURESTATES 359

andreflectsthefactthatthepurestateψ,unlikethemixture(23.2),isnotreally
inoneor theotherstateαz orβz. Ofcourse, theinterferencetermwill vanishif
αz andβzareeigenstatesofQ,e.g. ifQ=Sz. Otherwise,theinterferencetermis
notzero. Thus,allMaryhastodoisrunthebeamofelectronsemergingfromthe
firstdetectorintoaseconddetector,whichmeasureselectronspininthex-direction;
fromtheresultingshecanlearnwhetherornotthefirstdetectorhasbeen
switchedoff.

Problem - For Q = Sx, calculate < Q >pure forthe purestate (23.1)with
A=−B= 1 /

√

2,andforthecorrespondingmixture.

Thefactthatmixturescanalwaysbedistinguishedfrompurestates, atleastin
principle,alsoholdsgoodforentangledpurestates. Consider,forexample,thepure
state
Ψ=

∑

i

ciφi(x 1 )φi(x 2 ) (23.6)

andthemixture

M={particleinoneofthestates φi(x 1 )φi(x 2 ) withprobability |ci|^2 } (23.7)

Suppose,forsimplicity,thatallthesestatesareorthonormal,i.e.

<φi|φj>=<φi|φj>=δij (23.8)

andthatAisanobservableofparticle1,andBanobservableofparticle2.Itiseasy
toseethatifwemeasureonlyA,oronlyB,thatthemixtureM andthepurestate
Ψgiveexactlythesameanswer,namely

<A> =

∑

i

|ci|^2 <φi|A|φi>

<B> =

∑

i

|ci|^2 <φi|B|φi> (23.9)

However,ifwemeasuretheexpectationvalueoftheproductAB,thenwefindthat
thepurestatepredicts

<AB>pure=

∑

ij

c∗icj<ψi|A|ψj><φi|B|φj> (23.10)

whileforthemixture

<AB>mix=

∑

i

|ci|^2 <ψi|A|ψi><φi|B|φi> (23.11)