Relativistic quantum nonlocality for the three-qubit Greenberger-Horne-Zeilinger state Shahpoor Moradi ∗ Department of Physics, Razi University, Kermanshah, Iran (Dated: January 4, 2012) Lorentz transformation of three-qubit Greenberger-Horne-Zeilinger (GHZ) state is studied. Also weobtaintherelativisticspinjointmeasurementforthetransformedstate. Usingtheseresultsitis shown that Bell’s inequality is maximally violated for three-qubit GHZ state in relativistic regime. For ultrarelativistic particles we obtain the critical value for boost speed which Bell’s inequality is notviolated forvelocities smaller thanthisvalue. Wealso showthatinultrarelativistic limit Bell’s inequality is maximally violated for GHZ state. 2 PACSnumbers: 71.10.Ca;41.20.Jb 1 0 2 Relativistic effects on quantum entanglement and Bell’s inequality. Finally we calculate the degree of vi- n quantuminformationisinvestigatedbymanyauthors[1- olation for GHZ state when particles moving with same a 13]. AlsingandMilburn[1]studiedtheLorentztransfor- momentum and particles moving in the center of mass J mationofmaximallyentangledstates. Byexplicitcalcu- frame. Finally we compare our results with two qubit 2 lationoftheWignerrotationtheydescribedtheobserva- case. tionofthe entangledBellstatesfromtwoinertialframes A multipartite state is expressed by ] h moving with the constance velocity with respect to each p other. They concluded that entanglement is Lorentz in- Φ =a (p~ ,σ )a (p~ ,σ )...Φ , (1) t- variant. Terashima, et al. [2] considered to relativistic p~1σ1,p~2σ2,... † 1 1 † 2 2 0 n Einstein-Podolsky-Rosen correlation and Bell’s inequal- where p~ is three momentum vector, σ is spin label, a a i i † ity. TheyshowedthatthedegreeoftheviolationofBell’s is creation operator and Φ is Lorentz invariant vacuum u 0 q inequality decreases with increasing the velocity of the state. Multipartite state(1)hasthe Lorentztransforma- [ observers if the directions of the measurement are fixed. tion property [14] Theyextendedtheseconsiderationsto themasslesscase. 1 Ahn, et al. [3]investigatedtheBellobservableforentan- U(Λ)Φ = v p~1σ1,p~2σ2,... gledstatesinthe restframeseenbythemovingobserver 2 9 and showed that the entangled states satisfy the Bell’s 4 inequalitywhenthe boostspeedapproachesthe speedof D(j1) (W(Λ,p ))D(j2) (W(Λ,p ))...Φ . 0 light. D. Lee, et al. [4] showed that maximal violation σ¯1σ1 1 σ¯2σ2 2 p~1Λσ1,p~2Λσ2,... 1. of the Bell’s inequality can be achieved by properly ad- σ¯1Xσ¯2... (2) 0 justing the directions of the spin measurement even in a Here p~ is the three vector part of Λp , D(j) is the uni- 2 relativistically moving inertial frame. Kim, et al. [5] ob- 1Λ 1 σ¯σ tary spin-j representationof the three dimensional rota- 1 tained an observer-independentBell’s inequality, so that tion group,and W(Λ,p) is Wigner’s little groupelement : v itismaximallyviolatedaslongasitisviolatedmaximally Xi in the rest frame. They showed that the Bell observ- W(Λ,p)=L−1(Λp)ΛL(p), (3) ableandBellstatesfor Bell’sinequalityshouldbe trans- r formed following the principle of relativistic covariance, a where L(p) is the standardboost that takes a particle of which results in a frame independent Bell’s inequality. massmfromresttofour-momentumpµ. Transformation InthispaperwewouldliketostudytheBell’sinequal- of creation operator is ity for three-qubit GHZ state in relativistic regime. For doing this, we need the Lorentz transformation of GHZ U(Λ)a (p~,σ)U 1(Λ)= D(j)(W(Λ,p))a (p~ ,σ ). state and relativistic spin joint measurement. † − σ′σ † Λ ′ σ′ The following paper is organized as follows. First X (4) we review the representation of the Lorentz group and TheWignerrepresentationoftheLorentzgroupforspin- Wignerlittlegroup. ThenwecalculatetheLorentztrans- 1 is 2 formationofthree-qubitGHZstate. Afterthatweobtain tchaelcurelalatteivtihseticdesgprineejooifnvtimolaeatisounrefmorenatsopfecGiaHlZcassteatwehainchd D(W(Λ,p))=cosδp~ +i(~σ ~n)sinδp~ = D00 D01 , 2 · 2 D10 D11 in non relativistic case gives the maximally violation of (cid:18) (cid:19) (5) with δ ξ χ p~ ∗Electronicaddress: [email protected] cot 2 =coth2coth 2 +eˆ·pˆ, (6) 2 where Using a proton (hydrogenatom) in the millikelvin range as an example, condition for distinguishablity is a coshχ=(p0/m), tanhξ =β =v/c, ~n=eˆ pˆ. 100A . ≫ × ◦ (7) Usingrelation(4)LorentztransformationofGHZstate Here eˆ is a normal vector in the boost direction and v becomes is the boost speed. We consider the case in which the 1 boostspeed is perpendicular to momentums ofparticles. GHZ′ = (A000 +B 001 +C 010 +D 011 In this case we have | i √2 | i | i | i | i 1/2 δ (1+ 1 β2)(coshχ+1) cos = − , (8) +E 100 +F 101 +G110 +H 111 )p~1p~2p~3 Λ, (14) 2 " 2(p1 β2+coshχ) # | i | i | i | i | i − with p 1/2 A=D1 D2 D3 +D1 D2 D3 , δ (1 1 β2)(coshχ 1) 00 00 00 01 01 01 sin = − − − , (9) 2 " 2(p1 β2+coshχ) # − B =D1 D2 D3 +D1 D2 D3 , where in ultrarelativisptic limit as β 1 take the forms 00 00 10 01 01 11 → δ 1+sechχ 1/2 C =D1 D2 D3 +D1 D2 D3 , 00 10 00 01 11 01 cos , (10) 2 → 2 (cid:20) (cid:21) D =D1 D2 D3 +D1 D2 D3 , 00 10 10 01 11 11 1/2 δ 1 sechχ sin − . (11) 2 → 2 E =D1 D2 D3 +D1 D2 D3 (cid:20) (cid:21) 10 00 00 11 01 01 Investigations show that exist a family of pure entan- gledN >2qubitstatesthatdonotviolateanyBell’sin- F =D1 D2 D3 +D1 D2 D3 , 10 00 10 11 01 11 equalityforN-particlecorrelationsforthe caseofastan- dard Bell experiment on N qubits [15]. For N = 3, one class is Greenberger-Horne-Zeilinger (GHZ) state given G=D110D120D030+D111D121D031, by GHZ = 1 (000 + 111 ), the other class is repre- | i √2 | i | i sented by the W state W = 1 (110 + 101 + 011 ), H =D1 D2 D3 +D1 D2 D3 , (15) | i √3 | i | i | i 10 10 10 11 11 11 where 0 and 1 represent spins polarized up and down along the z axis. where Di is Wigner representation for particle i. The We can express GHZ state using creation operator in generalizationof the Bell’s type inequality to the case of the rest frame three particlesisthe oneproposedby Merminwhichcan be expressed in terms of correlation functions as follows 1 GHZ = a (p~ ,0)a (p~ ,0)a (p~ ,0) [16] † 1 † 2 † 3 | i √2 (cid:8) ε= E(~a,~b,c~)+E(~a,b~,~c)+E(a~,~b,~c) E(a~,b~,c~) 2, | ′ ′ ′ − ′ ′ ′ |≤ (16) +a (p~ ,1)a (p~ ,1)a (p~ ,1) Φ . (12) † 1 † 2 † 3 0 where For simplicity we assume that momen(cid:9)tum of particles E(~a,~b,~c)= ψ (~a ~σ) (~b ~σ) (~c ~σ)ψ , are sufficiently localized aroundmomentum p~i. Realistic h | · ⊗ · ⊗ · | i situation involve the wave pockets with gaussian form is correlator function, ~a, ~b and ~c are real three- exp( p~2/2∆2) with characteristic spread ∆. Note that − i dimensional vectors of unit length and ~σ = (σ ,σ ,σ ) these particles are indistinguishable. Authors in refer- x y z isthe Paulispinoperator. Foreachmeasurement,oneof ence[12]investigatedthatonecancreatedistinguishable two possible alternative measurement is performed: ~a or qubits fromindistinguishable particles by preparingpar- a~ for particle 1, ~b or b~ for particle 2, ~c or c~ for parti- ticles in minimum uncertainty states that are well local- ′ ′ ′ cle 3. For GHZ state, Bell’s inequality is violated if, for ized with a sharp momentum. They show that N-qubit example, measurements are made in the xy plane along product state can be constructed from N single particle some appropriate directions. In this case states as |ψiN =⊗Nn=1e−iaPz|ψi1, (13) E(~a,~b,~c)=cos(φ1+φ2+φ3), (17) where ψ is a single particle state. State (13) describes where we labelled the angles from the x-axis. The corre- 1 a one d|imiensional latices of particles with separation a. lation function E(~a,~b,~c) can take the value either +1 or 3 -1 under both realistic theory and quantum mechanical thenforGHZstateBell’sinequalityismaximallyviolated theory, thus the maximum value of ε is 4. with ε = 4. For set vectors (22) the relativistic Bell | | The normalized relativistic spin observable aˆ is given measurement becomes by [17] ε =4 AH . (24) ′ ∗ ℜ{ } ( 1 β2~a +~a ) ~σ aˆ= − ⊥ k · , (18) We obtain the degree of violation for two cases. p1+β2[(~e ~a)2 1] Cass I. p~ =~p =p~ =pzˆ · − 1 2 3 In this case where the subscriptps and denote the components ⊥ k which are perpendicular and parallel to the boost direc- D(W(Λ,p ))=D(W(Λ,p ))=D(W(Λ,p )) 1 2 3 tion. Operatoraˆ is relatedtothe Pauli-Lubanskipseudo vectorwhichisrelativisticinvariantoperatorcorrespond- ing to spin. Now we are ready to calculate the relativis- = cos2δ −sin2δ , (25) tic Bell’sinequalityforthreeparticlessystem. Spinjoint sinδ cosδ (cid:18) 2 2 (cid:19) measurementfor the transformedstate GHZ for mea- | ′i and Bell observable takes the form surement in xy plane is GHZ′ aˆ ˆb cˆGHZ′ ε′ =cos3δ+3cosδ. (26) h | ⊗ ⊗ | i In ultrarelativistic limit as β 1, (26) reduces to → = [1+β2(a2x−1)][1+β2(b2x−1)][1+β2(c2x−1)] −1/2 ε′ →sech3χ+3sechχ≤4. (27) (cid:8) (cid:9) In this limit amount of violation for very high energy ×ℜ(E∗Daxyb∗xyc∗xy+F∗Caxyb∗xycxy particles goes to zero, but for low energy particles ap- proaches to 4, similar to nonrelativistic limit β 0. Cass II. p~ +p~ +p~ =0 → 1 2 3 +G Ba b c +H Aa b c ), (19) ∗ xy xy ∗xy ∗ xy xy xy The particles are in the center of mass frame with the following momentums where a = a +ia 1 β2 and so on. In ultra rela- xy x y − tivistic limit as β 1 we get p~ = pzˆ, → p 1 − GHZ′ aˆ ˆb cˆGHZ′ h | ⊗ ⊗ | i 1 √3 p~ = zˆ+ yˆ p, 2 2 2 ! a b c x x x AH∗+G∗B+F∗C+E∗D , (20) → a b c ℜ{ } x x x | | 1 √3 which is not correlated. In non-relativistic limit p~ = zˆ yˆ p. (28) 3 2 − 2 ! GHZ aˆ ˆb cˆGHZ ′ ′ h | ⊗ ⊗ | i Wigner representations of the the Lorentz group for particles 1, 2 and 3 respectively are written as a b c a b c a b c a b c x x x y x y y y x x y y → − − − cosδ sinδ D(W(Λ,p ))= 2 2 , (29) 1 sinδ cosδ =cos(φ +φ +φ ). (21) (cid:18)− 2 2 (cid:19) 1 2 3 Here we consider to the vector set inducing the max- D(W(Λ,p ))=D (W(Λ,p )) 2 ∗ 3 imal violation of Bell’s inequality for GHZ state in non relativisticcase. Withthefollowingsuitablychosenmea- surement settings, cosδ +i√3sinδ 1sinδ = 2 2 2 −2 2 . (30) 1sinδ cosδ i√3sinδ ! ~a=~b=~c=yˆ, 2 2 2 − 2 2 In this case Bell observable to be a~′ =b~′ =c~′ =xˆ, (22) ε′ = 1 cos3δ+ 3cos2δ+ 33cosδ+ 3. (31) 16 8 16 2 and using the algebra of pauli matrices we have which for ultrarelativistic limit as β 1 reduces to → (σxσxσx−σyσyσx−σyσxσy−σxσyσy)|GHZi=4|GH(Z23i), ε′ 1 sech3χ+ 3sech2χ+ 33sechχ+ 3. (32) → 16 8 16 2 4 For very high energy particles amount of violation is this limit. This result is not same as three-qubit case. ε = 1.5, but for low energy particles ε = 4 which is For very high energy particles (36) reduces to ′ ′ maximally violation of Bell’s inequality. Fromthetwoprecedingcasesitisobviousthat,thede- 2 ε (1+ 1 β2 2β2), (37) greeofviolationdecreasesunderLorentztransformation. ′ ≈ 2 β2 − − ThisisbecauseBellobservableisevaluatedwiththesame − p spinmeasurementdirectionsasinthenon-relativisticlab p the critical value for violation of Bell’s inequality in this frame. By finding a new set of spin measurement di- case is β =0.86, which is smaller than three-qubit case rections, for example by rotating the spin measurement c when particles move in the center of mass frame. directions with Wigner rotation, Bell’s inequality is still maximally violated in a Lorentz-boosted frame [2, 4, 5]. In conclusion using Bell’s inequality, we studied the It’sinterestingtoexpressBellobservableinorderfunc- nonlocalquantum properties of GHZ state in relativistic tion of β for ultrarelativistic particles. In this situation formalism. First we obtained the relativistic spin joint relations (8) and (9) reduce to measurements for Lorentz transformed three-qubit GHZ state. We show that in ultrarelativistic limit joint mea- 1/2 surement is uncorrelated. We also investigated the de- δ (1+ 1 β2) cos − , (33) gree of violation for particles moving with same momen- 2 ≈" p2 # tum and particles moving in the center of mass frame. Bell’sinequalityismaximallyviolatedinrestframeorin moving frame with rest particles, but as seen by moving 1/2 δ (1 1 β2) observeris not alwaysviolated,because the degreeof vi- sin − − , (34) olationofBell’sinequalitydependsonthevelocityofthe 2 ≈" p2 # particles and observer. In non relativistic case the spin degreesoffreedomandmomentumdegreesoffreedomare then the amount of violation (26) takes the form independent. ButinrelativisticregimeLorentztransfor- mationofspinofparticledependsonitsmomentum. For ε 1 β2(4 β2). (35) ′ ≈ − − GHZ state we show that in ultrarelativistic limit Bell’s p inequality is maximally violated which is not same as It’s obvious that critical value β for satisfying Bell’s in- c two-qubitcase. Finally,forveryhighenergyparticleswe equality is 0.8. Critical value for case II is 0.97. obtained a critical value for satisfying Bell’s inequality. Now we compare our results with two-qubit case ob- tainedbyAhn, et al [3]. TheycalculatedrelativisticBell The critical value for three-qubit state is greater than two-qubit case. observable for two qubit entangled Bell state, when par- ticles move in the center of mass frame, and found 2 ε′ = ( 1 β2+cos2δ). (36) ACKNOWLEDGMENTS 2 β2 − − p In ultrarelativistpic limit β 1: ε 4sech2χ 2 2 It is a pleasure to thank Professor E. Solano for his ′ → → | − | ≤ which indicates the Bell’s inequality is not violated in valuable suggestions. [1] P. M. Alsing and G. J. Milburn Quant. Inf. Comput. 2, 052114 (2002) 487 (2002) [9] P.Caban,J.Rembielinski,K.A.SmolinskiandZ.Wal- [2] H. Terashima and M. Ueda Quant. Inf. Comput. 3, 224 czak Phys. Rev.A 67, 012109 (2003) (2003); H.Terashima andM.UedaInt.J.Quant.Inf.1, [10] J. Pachos and E. Solano Quant. Inf. Comput. 3, 115 93 (2003) (2003) [3] D. Ahn, H-J Lee, Y. H. Moon and S. W. Hwang Phys. [11] D. R.Terno Phys. 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