Two qubi s o a Ws a e iola e Bell’s inequali y beyond Ci el’son’s bound
Ada
´n Cabello*
Depa amen o de Fı
´sica Aplicada II, Uni e sidad de Se illa, 41012 Se illa, Spain
共Recei ed 30 May 2002; published 22 Oc obe 2002兲
I is shown ha he co ela ions be ween wo qubi s selec ed om a io p epa ed in a Ws a e iola e he
Clause -Ho ne-Shimony-Hol inequali y mo e han he co ela ions be ween wo qubi s in any quan um s a e.
Such a iola ion beyond Ci el’son’s bound is smalle han he one achie ed by wo qubi s selec ed om a io
in a G eenbe ge -Ho ne-Zeilinge s a e 关A. Cabello, Phys. Re . Le . 88, 060403 共2002兲兴. Howe e , i has he
ad an age ha all local obse e s can know om hei own measu emen s whe he o no hei qubi s belong
o he selec ed pai .
DOI: 10.1103/PhysRe A.66.042114 PACS numbe 共s兲: 03.65.Ud, 03.65.Ta
I. INTRODUCTION
The Bell inequali y 关1兴p oposed by Clause , Ho ne, Shi-
mony, and Hol 共CHSH兲关2兴, poin s ou ha in any local-
ealis ic heo y, ha is, in any heo y in which he local a i-
ables o a pa icle de e mine he esul s o local expe imen s
on his pa icle, he absolu e alue o a combina ion o ou
co ela ions is bound by 2,
兩
C共A,B兲⫺mC共A,b兲⫺nC共a,B兲⫺mnC共a,b兲
兩
⭐2. 共1兲
In inequali y 共1兲,Aand aa e wo obse ables aking alues
⫺1 o 1 on pa icle i, and Band ba e wo obse ables
aking alues ⫺1 o 1 on a dis an pa icle j;mand ncan be
ei he ⫺1o 1.
The CHSH inequali y 共1兲is iola ed o ce ain quan um
s a es and ce ain choices o he obse ables A,a,B, and b
关2兴. The e o e, he conclusion is ha no local- ealis ic heo y
can ep oduce he p edic ions o quan um mechanics 关1兴.
La e on, Ci el’son 关3兴showed ha , acco ding o quan um
mechanics, o any wo-qubi sys em p epa ed in a quan um
s a e, he absolu e alue o he combina ion o co ela ions
appea ing in he CHSH inequali y 共1兲is bound by 2
冑
2
共Ci el’son’s bound兲. This bound is also he maximum iola-
ion p edic ed by quan um mechanics o he wo-qubi sin-
gle s a e 共o any o he wo-qubi Bell s a e兲关2兴. Indeed, his
is he iola ion o Bell’s inequali y adi ionally es ed in eal
expe imen s in ol ing sys ems o wo qubi s p epa ed in a
quan um s a e 关4–9兴.
Howe e , as shown in Re . 关10兴, acco ding o quan um
mechanics he CHSH inequali y can be iola ed beyond
Ci el’son’s bound. The eason is he ollowing. Bell’s in-
equali ies a e de i ed assuming local ealism, wi hou any
men ion o quan um mechanics. The e o e, when sea ching
o iola ions o a Bell’s inequali y, one is no es ic ed o
s udying co ela ions be ween ensembles o sys ems p e-
pa ed in a quan um s a e; ins ead, one can s udy any en-
semble o sys ems, i espec i e o whe he such an ensemble
is meaning ul in quan um mechanics o no 共i.e., i espec i e
o whe he i can be desc ibed by a quan um s a e o no 兲.
Fo ins ance, one can conside ios o qubi s p epa ed in a
ce ain quan um s a e and hen assume local ealism o selec
a pai o qubi s in each io, and calcula e, using quan um
mechanics, he co ela ions be ween hese wo qubi s. The
whole p ocedu e makes sense and can be ansla ed in o eal
expe imen s as long as one can ob ain he equi ed co ela-
ions and p obabili ies o he wo selec ed qubi s om he
da a ob ained in a eal expe imen wi h h ee qubi s p epa ed
in a quan um s a e.
In Re . 关10兴, a iola ion o he CHSH inequali y 共1兲be-
yond Ci el’son’s bound is p esen ed o ce ain subensembles
o wo qubi s o an ensemble o ios p epa ed in a
G eenbe ge -Ho ne-Zeilinge 共GHZ兲s a e 关11兴. In his pape ,
we shall in es iga e whe he a iola ion beyond Ci el’son’s
bound could be ound o pai s o qubi s selec ed om ios
p epa ed in a Ws a e 关12兴.
The s uc u e o he pape is as ollows. In Sec. II, we will
ind ha , o a ce ain choice o obse ables, he Ws a e
iola es he CHSH inequali y 共1兲beyond Ci el’son’s bound.
The obse ables used in Sec. II do no p o ide he maximum
achie able iola ion using a Ws a e. In Sec. III, we explain
he eason behind his choice o obse ables. In addi ion, he
iola ion in Sec. II is smalle ha he one ob ained in Re .
关10兴using a GHZ s a e. Howe e , in Sec. IV, we will see ha
he e a e some easons ha make he iola ion p o ided by
he Ws a e mo e in e es ing han ha p o ided by he GHZ
s a e. Finally, in Sec. V, we discuss how o ob ain he e-
qui ed p obabili ies o he wo selec ed qubi s om he da a
ob ained in a eal expe imen wi h h ee qubi s.
II. THE WSTATE VIOLATES THE CHSH INEQUALITY
BEYOND CIREL’SON’S BOUND
Le us conside h ee dis an qubi s 1,2,3, p epa ed in he
Ws a e
兩
W
典
⫽
1
冑
3共
兩
⫹⫺⫺
典
⫹
兩
⫺⫹⫺
典
⫹
兩
⫺⫺⫹
典
), 共2兲
whe e
␴
z
兩
⫾
典
⫽⫾
兩
⫾
典
. Fo each h ee qubi s p epa ed in he
Ws a e 共2兲, we a e going o concen a e ou a en ion on wo
o hem, namely, hose wo in which, i we had measu ed
␴
z,
we would ha e ob ained he esul ⫺1. These wo qubi s will
be called iand jhe ea e , while he co esponding hi d
qubi 共 he one in which, i we had measu ed
␴
z, we would
*Elec onic add ess: [email p o ec ed]
PHYSICAL REVIEW A 66, 042114 共2002兲
1050-2947/2002/66共4兲/042114共5兲/$20.00 ©2002 The Ame ican Physical Socie y66 042114-1
ha e ound he esul 共1兲will be called k. In quan um me-
chanics, he esul o measu ing
␴
zis no p ede ined and
he e o e his p esc ip ion o choosing pai s is meaningless.
Howe e , he p esc ip ion makes sense in a local- ealis ic
heo y.
Fo easons ha will be explained in Sec. III, we a e in-
e es ed in he co ela ions when we choose A⫽Zi,a⫽Xi,
B⫽Zj, and b⫽Xj, whe e Zqand Xqa e he spin o qubi q
along he zand xdi ec ions, espec i ely. In addi ion, he
pa icula CHSH inequali y 共1兲we a e in e es ed in is he one
in which m⫽n⫽xk, whe e xkis one o he possible esul s,
⫺1o 1共al hough we do no know which one兲, o measu -
ing Xk. Wi h his choice we ob ain he ollowing CHSH
inequali y:
兩
C共Zi,Zj兲⫺xkC共Zi,Xj兲⫺xkC共Xi,Zj兲⫺C共Xi,Xj兲
兩
⭐2,
共3兲
which holds o any local- ealis ic heo y, ega dless o he
pa icula alue, ei he ⫺1o 1,o xk.
The nex s ep is o use quan um mechanics o calcula e
he ou co ela ions appea ing in inequali y 共3兲 o he sub-
ensemble o wo qubi s iand j aken om h ee qubi s p e-
pa ed in he Ws a e 共2兲.
Fo he subensemble o wo qubi s iand jde ined abo e,
C共Zi,Zj兲⫽1, 共4兲
because, o he Ws a e 共2兲,
PZ1Z2Z3共1,⫺1,⫺1兲⫽1
3,共5兲
PZ1Z2Z3共⫺1,1,⫺1兲⫽1
3,共6兲
PZ1Z2Z3共⫺1,⫺1,1兲⫽1
3,共7兲
whe e PZ1Z2Z3(1,⫺1,⫺1) means he p obabili y o qubi 1
gi ing he esul 1, and qubi s 2 and 3 gi ing he esul
⫺1 when measu ing
␴
zon all h ee qubi s.
By he de ini ion o qubi s iand j,
C共Zi,Xj兲⫽⫺xk,共8兲
because, o he Ws a e 共2兲,
PZ1X2X3共⫺1,1,⫺1兲⫹PZ1X2X3共⫺1,⫺1,1兲⫽0, 共9兲
PX1Z2X3共1,⫺1,⫺1兲⫹PX1Z2X3共⫺1,⫺1,1兲⫽0, 共10兲
PX1X2Z3共1,⫺1,⫺1兲⫹PX1X2Z3共⫺1,1,⫺1兲⫽0. 共11兲
Analogously, using Eqs. 共9兲–共11兲,
C共Xi,Zj兲⫽⫺xk.共12兲
Finally, o he Ws a e 共2兲,
PX1X2X3共1,1,1兲⫽3
8,共13兲
PX1X2X3共⫺1,⫺1,⫺1兲⫽3
8,共14兲
PX1X2X3共1,1,⫺1兲⫽1
24,共15兲
PX1X2X3共⫺1,⫺1,1兲⫽1
24,共16兲
PX1X2X3共1,⫺1,1兲⫽1
24,共17兲
PX1X2X3共⫺1,1,⫺1兲⫽1
24,共18兲
PX1X2X3共⫺1,1,1兲⫽1
24,共19兲
PX1X2X3共1,⫺1,⫺1兲⫽1
24.共20兲
F om Eqs. 共13兲and 共14兲, he con ibu ion o cases x1⫽x2
⫽x3⫽1 is cancelled by he con ibu ion o cases x1⫽x2
⫽x3⫽⫺1; om Eqs. 共15兲and 共16兲, he con ibu ion o
cases x1⫽x2⫽⫺x3⫽1 is cancelled by he con ibu ion o
cases x1⫽x2⫽⫺x3⫽⫺1, e c. The e o e, i espec i e o
whe he iand ja e qubi s 1 and 2, o 1 and 3, o 2 and 3,
we conclude ha
C共Xi,Xj兲⫽0. 共21兲
Co ela ions 共4兲,共8兲,共12兲, and 共21兲 iola e he CHSH
inequali y 共3兲. The iola ion 共3 s2兲goes beyond Ci el’son’s
bound (2
冑
2).
III. WHY XAND Z?
A pa icula ype o local- ealis ic heo ies a e hose in
which he only local expe imen s whose esul s a e assumed
o be p ede e mined a e hose which sa is y he c i e ion o
‘‘elemen s o eali y’’ p oposed by Eins ein, Podolsky, and
Rosen 共EPR兲:‘‘I , wi hou in any way dis u bing a sys em,
we can p edic wi h ce ain y (i.e., wi h p obabili y equal o
uni y) he alue o a physical quan i y, hen he e exis s an
elemen o physical eali y co esponding o his physical
quan i y’’ 关13兴.
As can be easily checked, he iola ion epo ed in Sec. II
is no he maximal iola ion o he CHSH inequali y 共3兲 o
wo qubi s in he Ws a e 共2兲. Fo ins ance, conside ing local
spin obse ables on plane x-zand assuming A⫽Band a
⫽b, we ind a maximum iola ion o 3.046 关by choos-
ing A⫽cos(0.628)
␴
x⫺sin(0.628)
␴
zand a⫽cos(1.154)
␴
x
⫹sin(1.154)
␴
z]. Why hen ha e we chosen A⫽Zi,a⫽Xi,
B⫽Zj, and b⫽Xj? The eason is ha hese obse ables a e
no only local obse ables bu , o he Ws a e 共2兲, hey also
sa is y EPR’s c i e ion o elemen s o eali y.
ADA
´N CABELLO PHYSICAL REVIEW A 66, 042114 共2002兲
042114-2
F om Eqs. 共5兲–共7兲, i can be immedia ely seen ha z1,z2,
and z3a e elemen s o eali y, since any o hem can be
p edic ed wi h ce ain y om spacelike sepa a ed measu e-
men s o
␴
zon he o he wo qubi s. In addi ion, om Eqs.
共9兲–共11兲, i can easily be seen ha , i zi⫽⫺1 hen, wi h
ce ain y, xj⫽xk. The e o e, i zi⫽⫺1, hen by measu ing
xj(xk) one can p edic xk(xj) wi h ce ain y. The e o e, i
zi⫽⫺1, hen xjand xka e elemen s o eali y. I zi⫽1 hen,
using Eqs. 共5兲–共7兲, i can immedia ely be seen ha zj⫽
⫺1. The e o e, ollowing he p e ious easoning, xiand xk
a e elemen s o eali y 共al hough xicould ha e ceased o be
an elemen o eali y a e measu ing
␴
zon pa icle i). In
conclusion, o ios o qubi s in he Ws a e 共2兲,z1,z2,z3,
x1,x2, and x3a e EPR elemen s o eali y and hus, acco d-
ing o EPR, hey should ha e p ede ined alues ⫺1o 1
be o e any measu emen .
The iola ion o he CHSH inequali y 共3兲p esen ed in
Sec. II is hus no only a p oo o he impossibili y o local
hidden a iables, bu also p o es a mo e powe ul esul : he
appa en ly mild condi ion p oposed by EPR is inconsis en
wi h quan um mechanics.
IV. WHY W?
As was shown in Re . 关10兴, wo qubi s belonging o a
h ee-qubi sys em in a GHZ s a e can p o ide a highe io-
la ion 共4 s 2, ins ead o 3 s 2兲o he CHSH inequali y 共3兲,
e en using obse ables ha sa is y EPR’s c i e ion o ele-
men s o eali y. Why hen use a Ws a e?
One eason is because a es o he iola ion o Bell’s
inequali ies beyond Ci el’son’s bound could be achie ed in
p ac ice in he nea u u e. Sou ces o Ws a es based on
pa ame ic down-con e ed pho ons a e now a ailable o
eal expe imen s 关14兴and some new p oposals o p epa e W
s a es ia ca i y quan um elec odynamics ha e ecen ly
been p esen ed 关15兴.
Ano he eason is because his iola ion beyond
Ci el’son’s bound is, in one sense, su p ising. The Ws a e is
he genuine h ee-qubi en angled s a e whose en anglemen
has he highes obus ness agains he loss o one qubi 关12兴.
In pa icula , om a single copy o he educed densi y ma-
ix o any wo qubi s belonging o a h ee-qubi Ws a e,
one can always ob ain by means o a il e ing measu emen a
s a e ha is a bi a ily close o a Bell s a e. The e o e, one
migh hink ha any wo qubi s belonging o a Ws a e will
no lead o a highe iola ion o he CHSH inequali y 共3兲
han ha o wo qubi s in a Bell s a e, and hus i is o
in e es o ealize ha his is no he case.
The e is, howe e , ano he sub le eason o p e e ing
he Ws a e ins ead o he GHZ s a e o a es o iola ion o
Bell’s inequali ies beyond Ci el’son’s bound. Any es o his
kind equi es a p esc ip ion o selec ing a pai o qubi s
om each io p epa ed in a quan um s a e. Such a p esc ip-
ion assumes local ealism. In he iola ion o he CHSH
inequali y 共3兲p esen ed in Sec. II, his p esc ip ion is simple:
qubi s iand ja e hose wo in which, i we had measu ed
␴
z,
we would ha e ob ained he esul ⫺1. Howe e , in he
iola ion o he CHSH inequali y 共3兲using a GHZ s a e de-
sc ibed in Re . 关10兴, he p esc ip ion is no so simple: he e,
qubi s iand ja e ei he hose wo in which, i we had mea-
su ed
␴
z, we would ha e ob ained he esul ⫺1, o any
wo, i we had ob ained he esul 1 o all h ee qubi s i we
had measu ed
␴
z. This means ha , o he Ws a e, any local
obse e could know whe he o no his qubi belonged o
he selec ed pai jus by measu ing
␴
z, while o he GHZ
s a e, he ac ha whe he o no a qubi belongs o he
selec ed pai canno be decided wi h ce ain y om a mea-
su emen on ha qubi , bu equi es knowledge o he esul s
o measu emen s on he o he wo qubi s. F om he pe spec-
i e o local ealism, o he Ws a e, one o he elemen s o
eali y ca ied by each qubi de e mines whe he o no i
belongs o he selec ed pai ; while o he GHZ s a e, his
in o ma ion is no local since i is dis ibu ed among dis an
elemen s o eali y.
V. EXPERIMENTAL CH INEQUALITY
The esul in Sec. II opens he possibili y o using sou ces
o h ee-qubi Ws a es 关14,15兴 o expe imen ally es he
CHSH inequali y. The main ad an age o an expe imen like
his 共o ha p oposed in Re . 关10兴兲 is ha i will admi a
di ec compa ison wi h he dozens o p e ious expe imen s
wi h wo qubi s 关4–9兴and hus goes beyond any p e ious
expe imen s o es local ealism using sou ces o h ee qu-
bi s 关16,17兴inspi ed by p oo s o Bell’s heo em wi hou
inequali ies 关11兴o by Bell’s inequali ies o h ee qubi s
关18,19兴.
Howe e , in any eal expe imen using h ee qubi s, he
expe imen al da a consis o he numbe o simul aneous de-
ec ions by h ee de ec o s NABC(a,b,c) o a ious obse -
ables A,B, and C. This numbe is assumed o be p opo ional
o he co esponding join p obabili y, PABC(a,b,c). The e-
o e, in o de o make inequali y 共3兲use ul o eal expe i-
men s, i would be con enien o ansla e i in o he lan-
guage o join p obabili ies.
Taking in o accoun ha
PZiZj共⫺1,⫺1兲⫽1
4关1⫺C共Zi兲⫺C共Zj兲⫹C共Zi,Zj兲兴,
共22兲
PZiXj共⫺1,⫺xk兲⫽1
4关1⫺C共Zi兲⫺xkC共Xj兲⫹xkC共Zi,Xj兲兴,
共23兲
PXiZj共⫺xk,⫺1兲⫽1
4关1⫺xkC共Xi兲⫺C共Zj兲⫹xkC共Xi,Zj兲兴,
共24兲
PXiXj共xk,xk兲⫽1
4关1⫹xkC共Xi兲⫹xkC共Xj兲⫹xk
2C共Xi,Xj兲兴,
共25兲
whe e C(Zi) is he mean o he esul s o measu ing
␴
zon
qubi i, and assuming physical locali y 关i.e., assuming ha
C(Zi) is independen o whe he
␴
zo
␴
xis measu ed on
qubi j, ha is, assuming ha he alue o C(Zi) is he same
in Eqs. 共22兲and 共23兲, e c.兴, he CHSH inequali y 共3兲be ween
co ela ions can be ans o med in o a Clause -Ho ne 共CH兲
inequali y 关20兴be ween join p obabili ies,
TWO QUBITS OF A WSTATE VIOLATE BELL’s... PHYSICAL REVIEW A 66, 042114 共2002兲
042114-3
⫺1⭐PZiZj共⫺1,⫺1兲⫺PZiXj共⫺1,⫺xk兲⫺PXiZj共⫺xk,⫺1兲
⫺PXiXj共xk,xk兲⭐0. 共26兲
As can be easily checked, he bounds lo he CHSH inequal-
i y 共3兲a e ans o med in o he bounds (l⫺2)/4 o he co -
esponding CH inequali y 共26兲. The e o e, he local- ealis ic
bound in he CH inequali y 共26兲is 0 and Ci el’son’s bound is
(
冑
2⫺1)/2⬇0.207.
Fo qubi s iand jo a sys em in he Ws a e 共2兲,
PZiZj共⫺1,⫺1兲⫽1, 共27兲
PZiXj共⫺1,⫺xk兲⫽0, 共28兲
PXiZj共⫺xk,⫺1兲⫽0, 共29兲
PXiXj共xk,xk兲⫽3
4.共30兲
The e o e, p obabili ies 共27兲–共30兲 iola e he CH inequali y
共26兲. Such a iola ion 共0.25 s 0兲is beyond he co espond-
ing Ci el’son’s bound 共0.207兲.
On he o he hand, since we do no know which ones a e
qubi s iand j, we canno ob ain he ou join p obabili ies
共27兲–共30兲jus by pe o ming measu emen s on wo qubi s.
The e o e, we mus show how he join p obabili ies o qu-
bi s iand ja e ela ed o he p obabili ies o he h ee qubi s.
As can easily be seen om he de ini ion o qubi s iand j,
PZiZj共⫺1,⫺1兲⫽PZ1Z2Z3共1,⫺1,⫺1兲⫹PZ1Z2Z3共⫺1,1,⫺1兲
⫹PZ1Z2Z3共⫺1,⫺1,1兲
⫹PZ1Z2Z3共⫺1,⫺1,⫺1兲.共31兲
The e o e, in o de o expe imen ally ob ain PZiZj(⫺1,
⫺1), we mus measu e he ou p obabili ies in he igh -
hand side o Eq. 共31兲.In heWs a e 共2兲, he i s h ee p ob-
abili ies in he igh -hand side o Eq. 共31兲a e expec ed o be
1/3 and he ou h is expec ed o be ze o.
On he o he hand, PZiXj(⫺1,⫺xk) and PXiZj(⫺xk,⫺1)
a e bo h less han o equal o
PZ1X2X3共⫺1,1,⫺1兲⫹PZ1X2X3共⫺1,⫺1,1
⫹PX1Z2X3共1,⫺1,⫺1兲⫹PX1Z2X3共⫺1,⫺1,1兲
⫹PX1X2Z3共1,⫺1,⫺1兲⫹PX1X2Z3共⫺1,1,⫺1兲.
共32兲
The e o e, in o de o expe imen ally ob ain PZiXj(⫺1,
⫺xk) and PXiZj(⫺xk,⫺1), we mus measu e 共using h ee
di e en se ups兲all six p obabili ies in sum 共32兲.In heW
s a e 共2兲, each o hese six p obabili ies is expec ed o be
ze o.
Finally,
PXiXj共xk,xk兲⫽PX1X2X3共1,1,1兲⫹PX1X2X3共⫺1,⫺1,⫺1兲.
共33兲
The e o e, in o de o expe imen ally ob ain PXiXj(xk,xk),
we mus measu e he wo p obabili ies in he igh -hand side
o Eq. 共33兲.In heWs a e 共2兲, each o hem is expec ed o be
3/8.
VI. CONCLUSIONS
Two qubi s selec ed om a io p epa ed in a Ws a e
iola e he CHSH inequali y, o he co esponding CH in-
equali y, mo e han wo qubi s p epa ed in any quan um
s a e. Such iola ions beyond Ci el’son’s bound a e smalle
han hose achie ed by wo qubi s selec ed om a io in a
GHZ s a e 关10兴. Howe e , o he Ws a e he a gumen is
simple , since all local obse e s can know om hei own
measu emen s whe he o no hei qubi s belong o he se-
lec ed pai .
The impo ance o hese a gumen s elies on he ac ha
hey sugges how o use sou ces o h ee-qubi quan um en-
angled s a es o expe imen ally e eal iola ions o he a-
milia wo-qubi Bell inequali ies beyond hose ob ained us-
ing sou ces o wo-qubi quan um s a es.
ACKNOWLEDGMENTS
I acknowledge H. Wein u e o aluable discussion, C.
Se a o commen s, and he Spanish Minis e io de Ciencia y
Tecnologı
´a, G an No. BFM2001-3943, and he Jun a de An-
dalucı
´a, G an No. FQM-239, o suppo .
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