Do all pure entangled states violate Bell's inequalities for correlation functions

a r X i v :q u a nt-ph /0110095v 2 16 S e p 2002Do all pure entangled states violate Bell’s inequalities for correlation functions?Marek ˙Zukowski 1,ˇCaslav Brukner 2,Wies l aw Laskowski 3and Marcin Wie´s niak 31Instytut Fizyki Teoretycznej i Astrofizyki,Uniwersytet Gda´n ski,PL-80-952Gda´n sk,Poland2Institut f¨u r Experimentalphysik,Universit¨a t Wien,Boltzmanngasse 5,A–1090Wien,Austria3Wydzia l Matematyki i Fizyki,Uniwersytet Gda´n ski,PL-80-952Gda´n sk,PolandAny pure entangled state of two particles violates a Bell inequality for two-particle correlationfunctions (Gisin’s theorem).We show that there exist pure entangled N >2qubit states that do not violate any Bell inequality for N particle correlation functions for experiments involving two dichotomic observables per local measuring station.We also find that Mermin-Ardehali-Belinskii-Klyshko inequalities may not always be optimal for refutation of local realistic description.PACS numbers:3.65.Ud,03.67.-a,42.50.-pQuantum mechanics violates Bell type inequalities [1,2,3,4],which hold for any local realistic theory.In a realistic model the measurement results are deter-mined by ”hidden”properties the particles carry prior to and independent of observation.In a local model the results obtained at one location are independent of any measurements or actions performed at space-like separa-tion.The theorem of Gisin [5]states that any pure non-product state of two particles violates a Clauser-Horne-Shimony-Holt (CHSH)[2]inequality,which involves only two-particle correlation functions,for two alternative di-chotomic measurements for each of the local observers.Can Gisin’s theorem be generalized to all N -particle pure entangled states?We show here that this is not the case for Bell inequalities involving only correlation func-tions in experiments in which local observers can choose between two dichotomic observables.We find a family of pure entangled states of N qubits which do not vio-late any such Bell inequality.This family is a subset of a larger one of the generalized GHZ states given by|ψ =cos α|0,...,0 +sin α|1,...,1 .(1)with 0≤α≤π/4.The GHZ states [3]are for α=π/4Scarani and Gisin [9]noticed a surprising feature of such states.They show that for sin 2α≤1/√2N −1the generalized GHZ states satisfy all possible Bell in-equalities for N -particle correlation functions,which in-volve two alternative dichotomic observables at each local measurement station.Note also,that since the reduced density matrices of all proper subsystems of the N -qubit system described by (1)are separable,no Bell inequality for K <N particle correlation functions can be violated.Thus,Gisin’s theorem cannot be straightforwardly gen-eralized in this case.We also find that for all N even the generalized GHZ state always violates a Bell inequality for N -particle correlation functions.Interestingly,this inequality,as conjectured in [9],is not a MABK one.This implies that MABK inequalities may not always be optimal for refutation of local realism.It is important to notice that the measurements on qubits in the states (1)can violate Bell inequalities if (i)one makes an additional postselection of measurements,and/or (ii)not all N observers are separated.Popescu and Rohrlich [6]showed that no local realist descrip-tion is possible for any pure multipartite entangled state,provided additional manipulations are allowed.Specifi-cally,for the state (1)a pair of observers can perform an analysis of their correlations conditioned on a spe-cific set of results of other N −2observers.This can lead to a violation of a CHSH inequality for the pair of observers.For example,each of N −2observers mea-sures his qubits in the basis |0′=1/√2(|0 −|1 ).Then if all of them obtain result0′,the first two observers are left with the state |ψ =cos α|0,0 +sin α|1,1 ,which always violates a CHSH in-equality [5].If (ii)is allowed,then the N observers can be separated into two spatially separate groups of M and N −M observers,which make joint measurements.For both groups one can always find dichotomic observables,such that a CHSH inequality for correlations between the measurements of the two groups is violated.Simply,the state (1)can be considered as a ”two-qubit”state |ψ =cos α|0M 0N −M +sin α|1M 1N −M where |0M ,|1M ,and |0N −M ,|1N −M are states of the qubits of the group with M and with N −M observers,respectively.Thus Gisin’s theorem applies.Note also that,Bennett et al.[7]showed that the states (1)are reversibly generated from the GHZ states and D¨u r [8]proved that all pure2 entangled states of N qubits are distillable.However,in the present paper,as in the standard Bell-GHZ typesituation,we are interested in the correlations betweenmeasurements on N separated qubits without postselec-tion.We base our analysis on the results of Ref.[10].Hereis a summary.A single generalized Bell inequality wasderived,which is equivalent to the full set of22N Bellinequalities for the correlation functions between mea-surements on N particles,which involve two alternativedichotomic observables at each local measurement sta-tion[10,11,12].The correlation functions for such mea-surements cannot be described by a local realistic model,if and only if,the generalized Bell inequality is violated.In parallel,in Ref.[10]a necessary and sufficient condi-tion was derived for the correlations of N qubits in anarbitrary state to violate the generalized Bell inequality.(Recently it was shown[13],that all states violating thegeneralized Bell inequality are distillable.)The condition is as follows.Consider N observers andallow each of them to choose a local coordinate system.An arbitrary quantum stateρof N qubits can be repre-sented by tensor products of local Pauli operators in thefollowing way1ρ=2N−1and N odd,thecorrelations between measurements on qubits in the gen-eralized GHZ state(1)satisfy all Bell inequalities for cor-relation functions,which involve two dichotomic observ-ables per local measurement station.The main idea of the proof is to show that condition(4)is satisfied for the range ofαgiven above.We givethe proof for N=3.We show that,for this range ofα,the value of i,j,k=x,y T2ijk after an arbitrary set of localrotations is performed is never larger than one.The prooffor general odd N is a straightforward generalization.3The correlation tensor of state (1),for N=3,readsˆT=cos 2α z 1⊗ z 2⊗ z 3+sin 2αx 1⊗ x 2⊗ x 3(6)− x 1⊗ y 2⊗ y 3− y 1⊗ x 2⊗ y 3− y 1⊗ y 2⊗ x 3.We first rotate the local coordinate systems of each ofthe observers around the local z axes.Such a set of rota-tions of course leaves i,j,k =x,y T 2ijk invariant,thereforeit stays put at its initial value 4sin 22α≤1.The corre-lation tensor in the new set of local coordinate systems is given byˆT ′=cos 2α z 1⊗ z 2⊗ z 3+sin 2αi,j,k =x,yT ′ijk i ′1⊗ j ′2⊗ k ′3(7)The values for T ′ijk can be obtained by replacing x i →cos φi x ′i +sin φi y ′i and y i →cos φi y ′i −sin φi x ′i in Eq.(6)for ˆT.The components satisfy the following relations T ′2xxx=T ′2yyx =T ′2yxy =T ′2xyy =T 2xxx cos 23 i =1φi ,(8)T ′2yxx =T ′2xyx =T ′2xxy =T ′2yyy =T 2xxx sin 23 i =1φi .(9)Next,we rotate the local coordinate systems of each ofthe observers around the local x ′by the angle θi .Nowthe specific values for T ′′ijk can be obtained by replacingz ′i→cos θi z ′′i +sin θi y ′′i and y ′i →cos θi y ′′i −sin θi z ′′i in Eq.(7).The new components in the xy sector of the correlation tensor satisfy the following relationsT ′′2xxx =T ′2xxx ,T ′′2xxy =c 23T ′2xxy ,T ′′2xyx =c 22T ′2xyx (10)T ′′2yxx =c 21T ′2yxx ,T ′′2xyy =c 22c 23T ′2xyy ,T ′′2yxy =c 21c 23T ′2yxyT ′′2yyx =c 21c 22T ′2yyx ,T ′′2yyy =(T zzz s 1s 2s 3+T ′yyy c 1c 2c 3)2,where s i ≡sin θi and c i ≡cos θi .By applying the Cauchy inequality (Ac 1+Bs 1)2≤A 2+B 2to the last relation in(10),we obtain that T ′′2yyy ≤T 2zzz s 22s 23+T ′2yyy c 22c 23.Therefore we obtain for the sum i,j,k =x,yT ′′2ijk ≤c 21c 22T ′2yyx +c 22c 23T ′2xyy +c 21c 23T ′2yxy +c 21T ′2yxx+c 22T ′2xyx +c 23T ′2xxy +T ′2xxx +T 2zzz s 22s 23+T ′2yyy c 22c 23.This expression is a linear function of its arguments c 21,c 22,and c 23.Thus,its maximal value is reached at the border of the region for which the function is defined,i.e.at c 2i =0or 1.For c 1=c 2=c 3=1one hasi,j,k =x,yT ′′2ijk ≤4sin 22α(11)and for all other casesi,j,k =x,y T ′′2ijk ≤1provided sin 2α≤1/2.Thus,for the considered range of α,af-ter arbitrary subsequent local rotations along three localz axes and arbitrary local rotations along three x ′axes,one has i,j,k =x,y T ′′2ijk ≤1.The final stage of our proof rest upon the observation,that the final Euler rotation of the local coordinate sys-tems of each of the three observers around axes z ′′leaves the sum of squares of the components in the xy sector of the tensor invariant .The proof for an arbitrary odd N follows the same pat-tern.For αsatisfying sin 2α≤1/√√√√√1+sin 22α x ′′1⊗ x ′′2⊗ x ′′3⊗ x ′′4(15)4 Therefore wefind local coordinate systems in which ourmodified tensor has a component of a value higher than1for all non-vanishing values of sin2α.The criterion(3)is violated.No local realistic description is possible.To generalize the proof to an arbitrary even N it isenough to notice,that the sub-tensor built out of the zxcomponents of the full tensor has the characteristic formˆT[zx]= z1⊗ z2...⊗ z N+sin2α x1⊗ x2....⊗ x N,and proceed as in the case N=4.The correlations between the measurements on an evennumber of qubits in the GHZ generalized state do notallow any local realistic model.Since the generalized Bellinequality is violated,i.e.at least one out of the full setof22N[10,12]inequalities implied by the generalized onemust be violated.Surprisingly,the violated inequality isnot a MABK one but a generalized CHSH inequality.Consider a Bell experiment in which observers1and2choose between two dichotomic observables and the otherN−2ones keep their settings at z unchanged.The states|0 and|1 in Eq.(1)are eigenstates of z· σ.Since N−2is also even,the product of the local results of the N−2observers for the case of the generalized GHZ state(1)isalways1.Thus their results effectively do not contributeto the value of the total correlation function.One hasE( n k1, n k2, z,..., z)=E( n k1, n k2).Therefore within thelocal realism these correlation functions have to satisfy the CHSH inequality.Of course the generalized GHZ state(1)violates it for the whole range ofα=0.What are the reasons for the completely different be-havior for N even and N odd?The expression on the left-hand side of(4)can be understood as a”total measure of the strength of correlations”in mutually complementary sets of local measurements(as defined by the summation over x and y)[14].Then the unity on the right-hand side of(4)is the classical limit for the amount of corre-lations.Specifically,pure product states cannot exceed the limit of1,as they can show perfect correlations in one set of local measurements directions only.In con-trast,entangled states can show perfect correlations for more than one such set[14].Now,only if N is even, the states(1)already show perfect correlation between measurements along z-directions(as the product is then always+1)reaching therefore the classical limit.Yet, they also show additional correlations in other,comple-mentary,directions.In the case of N odd,however,there is no perfect correlation between measurements along z-directions and the amount of correlations in complemen-tary directions are not enough to violate(4).In summary,we have shown that Gisin’s theorem can-not be straightforwardly generalized to all multi-particle systems,for the case of Bell inequalities involving corre-lation functions,in which local observers can choose be-tween two dichotomic observables(see Statement1and 2).However,the question posed in the title of this letter mayfind a different answer,if,e.g.,more than two di-chotomic observables per local measurement station are allowed.Our results may shell a new light on the connection be-tween the violation of Bell inequalities and the quantum information tasks[13,15].In the problem of classifica-tion of entangled states our results reveal a new class of pure entangled states which do not violate any Bell inequalities for correlation functions.ˇC.B.is supported by the Austrian FWF,project F1506,and by the QIPC Program of the European Union.The work is a part of the Austrian-Polish pro-gram24/00.M.˙Z.acknowledges KBN grant No.5P03B 08820.W.L.and M.W.are supported by the University of Gda´n sk grant BW5400-5-0236-2.Anton Zeilinger and Jian-Wei Pan are warmly thanked for discussions.[1]J.S.Bell,Physics1,195(1964).[2]J.Clauser,M.Horne,A.Shimony,and R.Holt,Phys.Rev.Lett.23,880(1969).[3]D.M.Greenberger,M.Horne, A.Shimony,and A.Zeilinger,Am.J.Phys.58,1131(1990).[4]N.D.Mermin,Phys.Rev.Lett.65,1838(1990);M.Ardehali,Phys.Rev.A46,5375(1992);A.V.Belinskii and D.N.Klyshko,p.36,653(1993).[5]N.Gisin,Phys.Lett.A154,201(1991);N.Gisin and A.Peres,Phys.Lett.A162,15-17(1992).[6]S.Popescu and D.Rohrlich,Phys.Lett.A166,293(1992).[7]C.H.Bennett,S.Popescu,D.Rohrlich,J.A.Smolin andA.V.Thapliyal,Phys.Rev A63,012307(2001).[8]W.D¨u r,Phys.Rev.Lett.87,230402(2001).[9]V.Scarani and N.Gisin,J.Phys.A346043(2001).[10]M.˙Zukowski andˇC.Brukner,Phys.Rev.Lett.88,210401(2002).[11]H.Weinfurter and M.˙Zukowski,Phys.Rev.A64,010102(R)(2001).[12]R.F.Werner and M.M.Wolf,Phys.Rev.A64,032112(2001).[13]A.Acin and V.Scarani,e-print quant-ph/0112102.[14]ˇC.Brukner,M.˙Zukowski,and A.Zeilinger,e-printquant-ph/0106119.[15]V.Scarani and N.Gisin,Phys.Rev.Lett.87,117901(2001).。