Physics Letters A 307 (2003) 262–264

www.elsevier.com/locate/pla

A new Bell inequality for three spin-half particle system

Xiao-Hua Wua,b,∗, Hong-Shi Zongb

a Department of Physics, Sichuan University, Chengdu 610064, Chinab Department of Physics and Center for Theoretical Physics, Nanjing University, Nanjing 210093, China

Received 28 September 2002; received in revised form 18 November 2002; accepted 20 November 2002

Communicated by P.R. Holland

Abstract

In this Letter, we derive a inequality for the Greenberger–Horne–Zeilinger state. If the state is perfect correlated, the largestviolation of the inequality is 2. While the state is imperfect correlated, the violation of the inequality can still be shown exceptfor the product state. 2002 Elsevier Science B.V. All rights reserved.

PACS: 03.65.Bz

In 1965, Bell demonstrated that an interpretationof quantum theory in terms of local hidden-variables(LHV) is impossible [1]. Using quantum systems con-sisting at least three particles, Greenberger, Horne andZeilinger (GHZ) showed that the incompatibility ofLHV theories with quantum mechanics is strongerthan the one previously revealed for two-particle sys-tems by Bell’s inequalities [2]. Besides playing a cru-cial role in fundamental tests of quantum mechanicsversus local realism, the GHZ states also has importantapplications in many quantum information and com-putation schemes [3,4], this has inspired interest in thefundamental results of GHZ states. Recently, the three-photon GHZ entanglement has been observed [5] andit is shown that the three-particle GHZ theorem can betransformulated in terms of inequalities, allowing im-perfect correlations due to detector inefficiencies [6].

* Corresponding author.E-mail address: wxhscu@yahoo.com.cn (X.-H. Wu).

However, in the application of GHZ states to show thecontradiction of LHV models, only the noncontextualLHV theories are concerned.

In 1991, Gisin has proved that Bell inequality holdsfor all nonproduct states for two spin-half particlessystem [7]. A natural related question is: if the statefor three spin-half particle system is

(1)|ψ〉 = cosk|+++〉 + sink expiδ|−−−〉,whether the GHZ’s nonlocality proof [2] and Mer-min’s inequalities [8] forN spin-half particle systemstill hold? When|cosk| �= |sink|, Gisin concludedthat no such results hold. Since the fact that the Bell’sinequalities for two-particle systems have been studiedthroughly either in a theoretical or experimental way,one may wonder if the Gisin’s argument can also beapplied to the three-particle case in the sense that us-ing only two particles, say, particle 1 and particle 2,to reject local realism. However, this cannot be done.The reason is like this: letE(n̂1, n̂2) to be the expec-

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.doi:10.1016/S0375-9601(02)01672-9

X.-H. Wu, H.-S. Zong / Physics Letters A 307 (2003) 262–264 263

tation value of the operator̂σ · n̂1 ⊗ σ̂ · n̂2 ⊗ I (n3),from Eq. (1), we can easily showE(n̂1, n̂2) equals tothe one when particle 1 and particle 2 are describedby a mixture of direct product states|++〉and |−−〉with probability cos2 k and sin2 k, respectively, andthere should be no contradiction for such a mixture inGisin’s theorem. Recently, it was found that Mermin’sinequalities may not always be optical for refutationof local realistic description [9], in present Letter, wefirstly derive a Bell inequality for the three spin-halfparticle system, then we shall show that the inequalityis violated for the state (1). Our inequality is a gener-alization of Mermin’s.

Considering the local-hidden variables (LHV) mod-els discussed in Bell’s theorem for two-particle sys-tems, we can put it into a form for three spin-half-particle case: suppose along the directionn̂i , a mea-surementσ̂ (n̂i) will be performed on particlei (i =1,2,3), and get the result̂σ (n̂i)= ±1. Assuming thatthe state of this three-particle system can be describedby a set of parametersλ, LHV models give

(2)

E(n̂1, n̂2, n̂3)=∫Λ

dλρ(λ)σ̂ (n̂1, λ)σ̂ (n̂2, λ)σ̂ (n̂3, λ),

where theρ(λ) is the density function forλ over asetΛ, while the natural conditions|σ̂ (n̂i , λ)| � 1 and∫Λρ(λ) dλ= 1 are required.In order to deduce inequality, introducing the Bell

operator

B̂ = σ̂(n1

1

) ⊗ σ̂(n1

2

) − σ̂(n1

1

) ⊗ σ̂(n2

2

)(3)+ σ̂

(n2

1

) ⊗ σ̂(n1

2

) + σ̂(n2

1

) ⊗ σ̂(n2

2

),

and B̂ ′ is defined asB̂(nij → n′ij ), we have the

following inequality∣∣B̂(λ)∣∣(σ̂ (n3, λ)+ σ̂ (n′3, λ)

)(4)+ ∣∣B̂ ′(λ)

∣∣(σ̂ (n3, λ)− σ̂ (n′3, λ)

)� 4,

since|σ̂ (ni , λ)| � 1, which is the same condition usedin Bell’s theorem. Multiplying this result byρ(λ) andintegrating overλ, then

S =E(|B̂|(σ̂ (n3)+ σ̂ (n′

3)) + |B̂ ′|(σ̂ (n3)− σ̂ (n′

3)))(5)� 4,

shall hold for the LHV model. We can easily showthis inequality should be violated by using a GHZ state

in Eq. (1). Defining the eigenstate|n̂i ,+〉 and|n̂i ,−〉for the operator̂σ · n̂i , and the eigenvalue is+1 and−1, respectively. Letθi andφi are polar and azimuthalangles ofn̂i , there is

E(σ̂ (n1)⊗ σ̂ (n2)⊗ σ̂ (n3)

)= cos 2k cosθ1 cosθ2 cosθ3

+ sin2k sinθ1 sinθ2 sinθ3

(6)× cos(φ1 + φ2 + φ3 − δ).

The above expectation value can be simplified if wechoose the anglesθ3 = θ ′

3 = 2k, φ3 = δ, φ′3 = δ−π/2,

now,

E(σ̂ (n1)⊗ σ̂ (n2)⊗

(σ̂ (n3)+ σ̂ (n′

3)))

= 2 cos2 2k cosθ1 cosθ2

+ sin2 2k sinθ1 sinθ2

(7)× (cos(φ1 + φ2)+ sin(φ1 + φ2)

),

E(σ̂ (n′

1)⊗ σ̂ (n′2)⊗

(σ̂ (n3)− σ̂ (n′

3)))

= sin2 2k sinθ ′1 sinθ ′

2

(8)× (cos(φ′

1 + φ′2)− sin(φ′

1 + φ′2)

).

Let θ ′1 = θ ′

2 = π/2, φ′11 = φ′1

2 = 0, φ′21 = −π/2,

φ′22 = π/2, there is

(9)E(B̂ ′ ⊗ (

σ̂ (n3)− σ̂ (n′3)

)) = 4 sin2 2k.

For the perfect correlated GHZ state, wheresin2 2k = 1, choosingθ1 = θ2 = π/2, andθ1 = θ2 =π/2,φ1

1 = φ′12 = 0,φ2

1 = −π/2,φ22 = π/2, we get

(10)E(B̂ ⊗ (

σ̂ (n3)+ σ̂ (n′3)

)) = 4 sin2 2k

and

S =E(B̂ ′ ⊗ (

σ̂ (n3)− σ̂ (n′3)

))(11)+E

(B̂ ⊗ (

σ̂ (n3)+ σ̂ (n′3)

)) = 8.

The violation of the inequality is 2, which is the sameas the result given by Mermin’s inequality for threespin-half particles case.

If the state is imperfect correlated, wheresin2 2k �= 1, the violation of the inequality still holdsexcept for the direct product state in which sin2 2k = 0.The proof is arrived at by following steps: letting

264 X.-H. Wu, H.-S. Zong / Physics Letters A 307 (2003) 262–264

φi1 = 0,φj2 = π/4 (i, j = 1,2), there is

E(B̂ ⊗ (

σ̂ (n3)+ σ̂ (n′3)

))= 2 cos2 2k

(cosθ1

1

(cosθ1

2 − cosθ22

)+ cosθ2

1

(cosθ1

2 + cosθ22

))+ √

2 sin2 2k(sinθ1

1

(sinθ1

2 − sinθ22

)(12)+ sinθ2

1

(sinθ1

2 + sinθ22

)),

choosingθ11 = 0, θ2

1 = π/2, and

cosθ12 = −cosθ2

2 = 2 cos2 2k√4 cos4 2k+ 2 sin4 2k

,

(13)sinθ12 = sinθ2

2 =√

2 sin2 2k√4 cos4 2k+ 2 sin4 2k

.

So, we have

S =E(B̂ ⊗ (

σ̂ (n3)+ σ̂ (n′3)

))+E

(B̂ ′ ⊗ (

σ̂ (n3)− σ̂ (n′3)

))(14)= 2

√4 cos4 2k + 2 sin4 2k+ 4 sin2 2k � 4,

the inequality is violated except for the product statewhere sin2 2k = 1.

In conclusion, for the general GHZ-state (1) ofthree spin-half particle system, we have derived ainequality as Bell’s inequality for the two spin-halfparticle case. When the state is perfect correlated,the largest violation is 2; if the state is imperfectcorrelated, the violation can also be shown except forthe product state.

References

[1] J.S. Bell, Physics 1 (1965) 195.[2] D.M. Greenberger, M. Horne, A. Zeilinger, in: M. Kafatos

(Ed.), Bell’s Theorem, Quantum Theory, and Conceptions ofthe Universe, Kluwer, Dordrecht, 1989.

[3] S. Bose, V. Verdal, P.L. Knight, Phys. Rev. A 57 (1998) 822.[4] M. Hillery, V. Bu z̆ek, A. Berthiaume, Phys. Rev. A 59 (1999)

1829.[5] D. Bouwmeester, J.-W. Pan, M. Danial, H. Weingurter,

A. Zeilinger, Phys. Rev. Lett. 82 (1999) 1354.[6] J.A. Barros, P. Suppes, Phys. Rev. Lett. 84 (2000) 793.[7] N. Gisin, Phys. Lett. A 154 (1991) 201.[8] N.D. Mermin, Phys. Rev. Lett. 65 (1990) 3373.[9] M. Z̆ukowski, C. Brukner, W. Laskowski, M. Wiensiak, Phys.

Rev. Lett. 88 (2002) 210402.