Tight Correlation-Function Bell Inequality for Multipartite $d$-Dimensional System

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by  Jing-Ling Chen

Tight Correlation-Function Bell Inequality for Multipartite d-Dimensional System
Jing-Ling Chen1, and Dong-Ling Deng1
∗
1Theoretical Physics Division, Chern Institute of Mathematics,
Nankai University, Tianjin 300071, People’s Republic of China
(Dated: January 6, 2009)
WegeneralizethecorrelationfunctionsoftheClauser-Horne-Shimony-Holt(CHSH)inequalityto
multipartite d-dimensional systems. All the Bell inequalities based on this generalization take the
same simple form as theCHSH inequality. For small systems, numerical results show that thenew
inequalities are tight and we believe this is also valid for higher dimensional systems. Moreover,
the new inequalities are relevant to the previous ones and for bipartite system, our inequality is
equivalentto the Collins-Gisin-Linden-Masser-Popescu (CGLMP) inequality.
9
0 PACSnumbers: 03.65.Ud,03.67.Mn,03.65.-w
0
2
That local and realistic theories impose certain con- 1 (0 0 + 1 1 . ButforthegeneralizedGHZstates
n √2 | ··· i | ··· i
straints in the form of some inequalities on statistical ψ = cosξ 0 0 +sinξ 1 and N odd, there
a ′ GHZ
| i | ··· i | ···i
J correlations of measurements on multiparticles was first existsoneregionξ (0,1arcsin(1/√2N 1)]inwhichthe
shown by Bell in 1964 [1]. Bell pointed out that any ∈ 2 −
6 MABK inequalities are not violated[13, 14]. Thus the
kind of local hidden variable theory should obey these MABK inequalities may not be the ‘natural’ generaliza-

] inequalities, while they can be violated easily in quan-
h tions of the CHSH inequality to more than two qubits,
tum mechanics. After Bell’s applaudable progress, ex-
p in the sense that the CHSH inequality violates all the
- tensiveworksonBellinequalitieshavebeendone,includ- pure states of two-qubit systems. In 2004, Chen et al.
t
n ing both theoretical analysis and experiment test. For presented a two-setting Bell inequality for three qubits
a instance, the Clauser-Horne-Shimony-Holt (CHSH) [2] which can be seen numerically to be violated by any
u inequality was proposed in 1969, which is more conve-
pure entangled state [15]. In [16], tight Bell inequali-
q
nient for experiment to test the non-locality of two 2-
[ ties for threeparticles withlow dimensionarepresented.
dimensional(qubit)system. However,thereexistsalong-
Nevertheless, up to now, there is no generic tight Bell
2 living open question: “What are the general inequalities
inequality for arbitrary N-qudit systems, even for three
v
for more complicated situations?” i.e., for more particles
3 qudits,nosuchinequalityhasbeenfound. Sincemanyof
and higher-dimensional systems.
1 quantumcommunicationschemes,suchasmultipartykey
7 On the one hand, for higher dimensions of two par- (secret) sharing [17] and quantum communication com-
1 ticles, Collins et al. constructed a CHSH type inequal- plexity problems [18], can be measured with multiparty
9. ity for arbitrary d-dimensional (qudit) systems in 2002, Bell inequalities of some form [19], derivations of multi-
0 now known as the Collins-Gisin-Linden-Masser-Popescu partyBellinequalitiesarethusoneofthemostimportant
8 (CGLMP) inequality [3]. This inequality was shown to and challenging subject in quantum theory.
0
be tight, i.e., it defines one of the facets of the convex
: ThepurposeofthispaperistopresentgeneralBellin-
v polytope [4] of local-realistic (LR) models [5]. There are
i someotheralternativeformsofthisinequality[6,7],and equalities basedon the correlationfunctions for N-qudit
X
systems. These inequalities, obtained by using the same
themaximalquantumviolationofthisinequalitywasan-
r methodin[6],aretightandrelevanttothepreviousones.
a alyzed in [8], which showed that the maximal violation
What’s more, all the Bell inequalities based on this gen-
ofthis inequality occursatthe non-maximallyentangled
eralization take the same simple form as the CHSH in-
state. More recently, Seung Woo Lee and Dieter Jaksch
equality. For bipartite systems, our inequality is equiva-
introduced another tight Bell inequality which is maxi-
lent to the (CGLMP) inequality. However,to be honest,
mally violated by maximally entangled states [9].
there are two disadvantages for these new inequalities:
On the other hand, there are also various Bell in-
(i)Thequantumviolationsoftheseinequalitiesaresmall
equalities for N (N > 2) particles. In 1990, Mermin,
and they are not as strong as the previous inequalities,
in the first time, produced a series of two setting in-
namely, they are less resistant to noise; (ii) Some pure
equalities for arbitrary many qubits [10]. A comple-
states do not violate these inequalities. It is interesting
mentary series of inequalities was introduced by Arde-
tonotethatthesetwodisadvantagesindicatethatatight
hali [11]. In the next step, Belinskii and Klyshko gave
Bell inequality may not always be the optimal one.
a series of two setting inequalities, which contained the
tight inequalities of Mermin and Ardehali [12]. These The approach to our new tight Bell inequalities for
inequalities, now known as Mermin-Ardehali-Belinskii- N-qudit systems is based on the Gedanken experiment.
Klyshko(MABK)inequalities,aremaximallyviolatedby ConsiderN spatiallyseparatedparties andalloweachof
theGreenberger-Horne-Zeilinger(GHZ)states ψ = them to choose independently between two dichotomic
GHZ
| i
2
observables. Let X[1], X[2] (j = 1,2, ,N) denote dimensionalsystem. fij(m,n)=S M[ε(i j)(m+n),d];
j j ··· − −
the two observables on the jth party, each of them ε(x) = 1 and 1 for x 0 and x < 0, respectively;
− ≥
have d possible outcomes: x[1],x[2] = 0,1, ,d 1 M(x,d)=(xmodd)and0 M(x,d) d 1. Basedon
j j ··· − ≤ ≤ −
(j = 1,2, ,N). The joint probabilities are denoted this correlation functions, a tight Bell inequality for two
···
by P(X[i1], ,X[ij]), which should satisfy the normal- qudits is generalized as:
1 ··· j
ization condition:
[2]
I =Q +Q Q +Q 2. (3)
dX−1 P(X1[i1] =x[1i1],··· ,Xj[ij] =x[jij])=1. (1) d 11 12− 21 22 ≤
x[i1], ,x[ij]=0 Inequality(3)isequivalenttotheCGLMPinequalityand
1 ··· j
its maximal quantum violationis analyzed in Ref. [7, 8].
For two-qudit systems, namely N = 2, Ref. [6] intro-
Inspired by the ideas in Ref. [6], we generalized the
ducedthecorrelationfunctionsQ inthefollowingform:
ij correlation functions for multipartite d-dimensional sys-
tems. For simplicity and convenience, we will focus on
d 1 d 1
Qij = S1 X− X− fij(m,n)P(X1[i] =m,X2[j] =n), (2) tchaeset.hrFeoer-qtuhdriete-cqausediattsfiyrssttemansd, nthamenelaynaNlyz=et3h,ethgeenneerawl
m=0n=0
correlationfunctionsQ canbewritteninthefollowing
ijk
where S =(d 1)/2 is the spin of the particle for the d- form:
−
d 1 d 1d 1
Qijk ≡ S1 X− X− X− fijk(m,n,l)P(X1[i] =m,X2[j] =n,X3[k] =l), (4)
m=0n=0l=0
where S takes the same value as the two qudits case: Now, we should enumerate all the possible cases accord-
S =(d 1)/2,andfijk(m,n,l)=S M[( 1)i j k(m+ ing to the different values of r ,,r , r , and r .
× × 111 222 121 212
− − −
n + l),d]. Then the Bell inequality for three particles Case 1: Both r and r are less than d. From (6),
111 222
d-dimensional systems reads: there are two cases for the rest:(i) none of r and r
212 121
Id[3] =Q111−Q222+Q121+Q212 ≤2. (5) irsla)rger1t]h/aSn=d. th1e/nSw(enhotaevethIda[3t] =d [=r1121S++r2122);−((iir)21o2n+e
121
Obviously, I[3] is upper bounded by 4 since the extreme of r −and r −is equal to or larger than d. Then after
d 212 121
valuesofQijk are 1anditcanneverreachthisvaluebe- some simple calculating, we get I[3] =(d 1)/S =2.
causethatthefour±functionsinEq.(5)arestronglycorre- Case 2: r < d and d r d< 2d or−d r < 2d
111 222 111
lated. Infact,forlocalhiddenvariabletheories,itiseasy and r < d. From (6),≤there are four c≤ases for the
222
toprovethatthe maximumvalue ofI[3] is2. We usethe rest: (i) both r andr areless thand. then wehave
d 212 121
samemethodasfortwod-dimensionalsystemsinRef.[6]. I[3] =[r +r d (r +r ) 1]/S = 2(S+1)/S;
The essential idea of this proof is to enumerate all the d 111 222− − 212 121 − −
(ii)oneofr andr isequaltoorlargerthand. Then
212 121
possible relations between pairs of operators. Defining after some simple calculating, we get I[3] = 1/S;(iii)
r X[1] + X[1] + X[1], r X[2] + X[2] + X[2], d −
111 ≡ 1 2 3 222 ≡ 1 2 3 Both r212 and r121 are larger than d and less than 2d,
r121 ≡X1[1]+X2[2]+X3[1], and r212 ≡X1[2]+X2[1]+X3[2]. thenId[3] =2;(iv) oneofr212 andr121 is lessthan d,and
Then the constraint follows immediately: the other is largerthan2d,then we canalsogetI[3] =2.
d
r111+r222 =r121+r212. (6) Case 3: d r111 < 2d and d r222 < 2d. From
≤ ≤
(6), there are four cases for the rest: (i) one of r and
For convenience, we define two functions: g (x) = 212
1
S−MS(x,d),g2(x) = M(x,dS)−S−1. Then, for a given choice rth12a1nis2dle.ssthtehnanwdeahnadvethIe[3]ot=her2is(Sla+rge1r)/tSha;n(idi)abnodthleossf
of r111, r222, r121, and r212, the correlation functions in d −
themarelargerthandandlessthan2d. Thenobviously,
Eq.(5) can be rewritten as: Q = g (r ), Q =
g1(r222), Q121 = g1(r121), and Q121112 =g12(r211121). A2d2i2rect Id[3] = −1/S; (iii) one of them is larger than 2d and the
calculation shows that: other is less than d, then I[3] = 1/s; (iv) one of them
d −
1 is larger than 2d and the other is larger than d and less
[3]
Id = S [ M(r111,d)+M(r222,d) than 2d, then I[3] =2.
d
M(r ,d) M(r ,d) 1]. (7) Case 4: Both r and r are equal to or larger than
121 212 111 222
− − −
3
2d . From (6), there are two cases for the rest:(i) one of inequality (5). We will restrict the considerations to
r andr islargerthan2dandtheotherislargerthan multi-port beamsplitters since the software takes too
212 121
d and less than 2d. then we have I = 2(S+1)/S; (ii) long to run on our computer if the most general mea-
d
−
both of them are larger than 2d, then obviously, I[3] = surements are employed. Actually, for low dimensional
d
1/S. systems (d 3), we have used the most general mea-
−Thus, we have proved that I[3] 2 for local realistic surements b≤ut no larger violations are founded. In a
d ≤ Gedanken experiment [20], the matrix elements of an
theories (Note that for d = 2, I[3] has only two possible
2 unbiased symmetric multi-port beamsplitter are given
values 2sincenotallthepossibilitiesenumeratedabove
can oc±cur). Moreover, we have found computationally by Ukl(ϕ~) = √1dαklexp(iϕl), here α =exp(2diπ) and
ϕl (l = 0,1, ,d 1) are the settings of the appro-
that the inequality (5) is tight for d 10, and suspect
··· −
that this will generalize. If we set X[≤1] = 0 and X[2] = priate phase shifters, for convenience we denote them as
3 3 a d dimensional vector ϕ~ = (ϕ0,ϕ1,ϕ2, ,ϕd 1). For
0, then the inequality (5) reduces to a two qudits Bell ··· −
state ψ3 ofthree-quditsystems,thequantumprediction
inequality which is an alternative form of inequality (3) | di
for the probabilities of obtaining the outcome (m,n,l)is
and equivalent to the CGLMP inequality.
then given by:
Let us now focus on the quantum violation of the
P(X[i] = m,X[j] =n,X[k] =l)= mnl U(ϕ~ ) U(ϕ~ ) U(ϕ~ )ψ3 2
1 2 3 |h | X1[i] ⊗ X2[j] ⊗ X3[k] | di|
= Tr[(U (ϕ~ ) U (ϕ~ ) U (ϕ~ ))mnl mnl (U(ϕ~ ) U(ϕ~ ) U(ϕ~ ))ψ3 ψ3 ]. (8)
† X1[i] ⊗ † X2[j] ⊗ † X3[k] | ih | X1[i] ⊗ X2[j] ⊗ X3[k] | dih d|
Substituting Eq. (8) in to the inequality (5), one get the exist states which violate inequality (5) but do not vio-
expression of I[3] in quantum mechanics. For the gener- late the MABK inequality. For instance, one may check
d
alized GHZ state of three qubits: that the state: Ψ = 0.169414000 +0.0461131100 +
| i | i | i
0.161369101 +0.193624110 +0.951652111 donotvio-
|ψ23i=cosθ|000i+sinθ|111i, (9) latetheM|ABiKinequality| buititdoviolat|eineiquality(5),
and the violation is 2.00382.
numerical results show that when we set θ = π/4,
ϕ~X1[1] = (0,−π/12), ϕ~X1[2] = (0,π/4), ϕ~X2[1] = (0,−π/6), For the generalized GHZ state of three qutrits:
ϕ~ = (0,π/3), ϕ~ = (0,0), ϕ~ = (0,π/6), we
X[2] X[1] X[2]
2 3 3
get the maximal violation 2√2, which is the same of the
ψ3 =sinθ sinθ 000 +sinθ cosθ 111 +cosθ 222 ,
maximal violation of CHSH inequality for two qubits. | 3i 1 2| i 1 2| i 1| i
For θ (0,π/8], the state (9) doest not violate the
∈
inequality. To measure the strength of violation of lo-
numerical results shows that when we set θ = 0.9066,
cal realistic theories, we may consider the mixed state 1
ρ(F)=(1−F)|ψ23ihψ23|+F8I⊗I⊗I,whereF (0≤F ≤1) θ2 = 0.6663, ϕ~X1[1] = (0,−π/5,π/24), ϕ~X1[2] =
is the amountofthe noise presentin the system[21] and (0,π/24, 5π/12), ϕ~ = (0,0,π/12), ϕ~ =
I isa2 2identitymatrix. Accordingtotheproposalin- (0,π/3, −π/4), ϕ~ X2[1=] (0,π/30,π/20), ϕ~X2[2] =
troduce×dinRef.[21],thereexistssomethresholdvalueof − X3[1] X3[2]
(0,π/8,π/6), we get the maximal violation 2.915, which
F, denoted by F , such that for every F F , local
thr ≤ thr is the same of the maximal violation of the CGLMP in-
and realistic description does not exist. For inequality
equalityfortwoqutrits. Ontheotherhand,forthemax-
(5), the threshold fidelity is 0.29289, which is smaller
imal entangled state for three qutrits, namely θ = π/4,
2
than 1/2, the threshold fidelity for MABK inequality
θ =arccos(1/√3),thequantumviolationis2.873,which
1
for three qubits. This indicate that our inequality is
is smaller than 2.915. This indicts that the maximal
not as strong as the MABK inequality. Another set of
violation of inequality (5) occurs at the nonmaximally
states considered are the generalizedW states: ψ3 =
| 2iW entangled state. For higher dimensions, our numerical
sinβsinξ 001 +sinβcosξ 010 +cosβ 100 . The maxi-
| i | i | i results show that the maximal violation is similar to the
malviolationofthis setofstatesisalso2√2. Thisresult
CGLMPinequalityandtheinequality(5)isalsorelevant
is surprising since for the previous inequalities, the vi-
to the inequalities presented in Ref.[16].
olations of the generalized W states are always smaller
than that of the GHZ states. Moreover, inequality (5) TheBellinequalitiescanbeeasilygeneralizedforarbi-
is relevant to three-qubit MABK inequality, i.e., there trary N-qudit systems. The correlationfunctions in this
4
case are in the following form: tight general Bell inequalities for arbitrary N-qudit sys-
tems and they are relevant to the previous known Bell
d 1 d 1
Qi1,···,iN = S1 X− ··· X− fi1···ij(x[1i1],··· ,x[NiN]) ipnreoqoufaoliftitehse. tFirgahntnkleyssspoefatkhiensge,nweewdionenqoutahliatviees.aIgnedneeerda,l
x[i1]=0 x[iN]=0 we have only checked that for small systems (namely,
1 N
P(X[i1] =x[i1], ,X[iN] =x[iN]), three qudits for d 10, four qudits for d 7, and five
× 1 1 ··· 1 1 quditsford 5). U≤nfortunately,wehaveto≤leavethisas
≤
wMh[e(re1)Sχ(=N(dx−[ij1])),/d2],, fwih1·i·c·ihj(ixs[1i1s]i,m·i·l·ar,xt[NoiN]t)he=deSfin−i- aqnueonptleyn. qSuinecsetiothnehvearreioaunsduwseeosfhBalelllininveesqtuigaalitteyiitnsquubasne--
− Pj=1 j tum information, our results may be very useful for the
N
tionofthree-quditcorrelationfunctionsandχ= i .
Qj=1 j study of other Bell inequalities, quantum entanglement
Based on these correlation functions, the tight Bell in-
measurement, distillation protocols, etc.
equality can be written as:
This work was supported in part by NSF of China
Id[2N] =Q1···1+Q1212···12+Q2121···21−Q2···2 ≤2, (10) (GrantNo. 10605013),Programfor New CenturyExcel-
I[2N+1] =Q +Q +Q Q 2. lent Talents in University, and the Project-sponsoredby
d 1···1 1212···21 2121···12− 2···2 ≤ SRF for ROCS, SEM.
Using the same method as for the case of three qudits,
one may check that the the above inequalities (10) are
valid for local hidden variable theory and they are tight.
For instance, we give two tight Bell inequalities. The
∗
Electronic address: [email protected]
first example is the tight Bell inequality for four qudits:
[1] J.S.Bell,Physics(LongIslandCity,N.Y.)1,195(1964).
[4] [2] J. Clauser, M. Horne, A. Shimony, R. Holt, Phys. Rev.
I =Q +Q +Q Q 2, (11)
d 1111 1212 2121− 2222 ≤ Lett. 23, 880 (1969).
[3] D. Collins, N. Gisin, N. Linden, S. Massar, and S.
Numerical results show that when d = 2, the inequal-
Popescu, Phys.Rev. Lett. 88, 040404 (2002).
ity (11) is maximally violated by the maximally en- [4] A. Peres, Found.Phys.29, 589 (1999).
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[9] S. W. Lee and D. Jaksch, arxiv:quant-ph/0803.3097v1.
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[11] M. Ardehali, Phys.Rev. A 46, 5375 (1992).
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| 2i √2 | i | i
ϕ~ = (0, π/12), ϕ~ = (0,π/3), ϕ~ = (0, π/6), [15] J. L. Chen, C. F. Wu,L. C. Kwek, and C. H. Oh,Phys.
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ϕ~ = (0,π/3), ϕ~ = (0,0), ϕ~ = (0,π/12),
X[2] X[1] X[2] [16] J. L. Chen, C. F. Wu, L. C. Kwek, and C. H. Oh,
2 3 3
ϕ~ = (0,0), ϕ~ = (0,0), ϕ~ = (0,0), and arxiv:quant-ph/0506230v1.
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(2002).
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