Cirel'son inequality states that the absolute value of the combination of
quantum correlations appearing in the Clauser-Horne-Shimony-Holt (CHSH)
inequality is bound by $2 \sqrt 2$. It is shown that the correlations of two
qubits belonging to a three-qubit system can violate the CHSH inequality beyond
$2 \sqrt 2$. Such a violation is not in conflict with Cirel'son's inequality
because it is based on postselected systems. The maximum allowed violation of
the CHSH inequality, 4, can be achieved using a Greenberger-Horne-Zeilinger
state.Comment: REVTeX4, 4 page

A. Aspect

A. Chefles

A. Einstein

A. Peres

Adán Cabello

B. S. Cirel'son

D. Bouwmeester

D. M. Greenberger

G. Weihs

H. P. Stapp

J. F. Clauser

J. S. Bell

J.-W. Pan

L. J. Landau

M. A. Rowe

N. D. Mermin

S. L. Braunstein

S. M. Roy

S. Popescu

English

arXiv

Crossref

Physical Review Letters

Violating Bell's Inequality Beyond Cirel'son's Bound

Cirel’son inequality states that the absolute value of the combination of quantum correlations appearing in the Clauser-Horne-Shimony-Holt (CHSH) inequality is bound by 2 √ 2. It is shown that the correlations of two qubits belonging to a three-qubit system can violate the CHSH inequality beyond 2 √ 2. Such a violation is not in conflict with Cirel’son’s inequality because it is based on postselected systems. The maximum allowed violation of the CHSH inequality, 4, can be achieved using a Greenberger-Horne-Zeilinger state

Cabello Quintero, Adán

idUS. Depósito de Investigación Universidad de Sevilla

VOLUME 88, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 11 FEBRUARY 2002060403-1Violating Bell’s Inequality Beyond Cirel’son’s BoundAdán Cabello*Departamento de Física Aplicada II, Universidad de Sevilla, 41012 Sevilla, Spain(Received 21 August 2001; revised manuscript received 12 November 2001; published 29 January 2002)Cirel’son inequality states that the absolute value of the combination of quantum correlations appearingin the Clauser-Horne-Shimony-Holt (CHSH) inequality is bound by 2p2. It is shown that the correlationsof two qubits belonging to a three-qubit system can violate the CHSH inequality beyond 2p2. Such aviolation is not in conflict with Cirel’son’s inequality because it is based on postselected systems. Themaximum allowed violation of the CHSH inequality, 4, can be achieved using a Greenberger-Horne-Zeilinger state.DOI: 10.1103/PhysRevLett.88.060403 PACS numbers: 03.65.Ta, 03.65.UdBell’s theorem [1] has been described as “the most pro-found discovery of science” [2]. It states that, according toquantum mechanics, the value of a certain combination ofcorrelations for experiments on two distant systems can behigher than the highest value allowed by any local-realistictheory of the type proposed by Einstein, Podolsky, andRosen [3], in which local properties of a system determinethe result of any experiment on that system. The mostcommonly discussed Bell inequality, the Clauser-Horne-Shimony-Holt (CHSH) inequality [4], states that in anylocal-realistic theory the absolute value of a combinationof four correlations is bound by 2. Cirel’son’s inequal-ity [5] shows that the combination of quantum correla-tions appearing in the CHSH inequality is bound by 2p2(Cirel’son’s bound). It is widely believed that “[q]uantumtheory does not allow any stronger violation of the CHSHinequality than the one already achieved in Aspect’s ex-periment [6] [2p2]” [7]. However, it has been shown thatexceeding Cirel’son’s bound is not forbidden by relativis-tic causality [8]. Therefore, an intriguing question is whythe CHSH inequality is not violated more. Here it is shownthat, for three-qubit systems (that is, systems composed bythree two-level quantum particles), the correlation func-tions of two suitably postselected qubits violate the CHSHinequality beyond Cirel’son’s bound and that this violationcan even reach 4, the maximum value allowed by the defi-nition of correlation.To introduce the CHSH inequality, let us consider sys-tems with two distant particles i and j. Let A and a (B andb) be physical observables taking values 21 or 1 refer-ring to local experiments on particle i  j. The correlationCA, B of A and B is defined asCA, B  PAB1, 1 2 PAB1, 21 2 PAB21, 11 PAB21, 21 , (1)where PAB1, 21 denotes the joint probability of obtain-ing A  1 and B  21 when A and B are measured. Inany local-realistic theory, that is, in any theory in whichlocal variables of particle i  j determine the results of lo-cal experiments on particle i  j, the absolute value of aparticular combination of correlations is bound by 2:0031-90070288(6)060403(4)$20.00jCA, B 2 mCA, b 2 nCa,B 2 mnCa, bj # 2 ,(2)where m and n can be either 21 or 1. The CHSH in-equality (2) holds for any local-realistic theory, whateverthe values of m and n are, in the allowed set, 21, 1.The bound 2 in inequality (2) can easily be derived asfollows: In a local-realistic theory, for any individualsystem, the observables A, a, B, and b have predefinedvalues yA, ya, yB, and yb , either 21 or 1. Therefore,for an individual system the combination of correlationsappearing in (2) can be calculated asyByA 2 nya 2 mybyA 1 nya , (3)which is either 22 or 2, because one of the expressionsbetween parentheses is necessarily zero and the other iseither 22 or 2. Therefore, the absolute value of the corre-sponding averages is bound by 2, q.e.d.For a two-particle system in a quantum pure state de-scribed by a vector jc, the quantum correlation of A andB is defined asCQA, B  cjÂB̂jc , (4)where Â and B̂ are the self-adjoint operators which repre-sent observables A and B. For certain choices of Â, â, B̂, b̂,and jc, quantum correlations violate the CHSH inequal-ity [4]. Therefore, no local-realistic theory can reproducethe predictions of quantum mechanics [1].Later on, Cirel’son [5] demonstrated that for a two-particle system the absolute value of the combination ofquantum correlations equivalent to those appearing in theCHSH inequality (2) is bound by 2p2,jCQA, B 2 mCQA, b 2nCQa, B 2 mnCQa, bj # 2p2 . (5)Cirel’son’s bound can easily be derived as follows [9]:Consider the operator with the same structure as the com-bination which appears in inequality (5),Ĉ  ÂB̂ 2 mÂb̂ 2 nâB̂ 2 mnâb̂ . (6)If Â2  â2  B̂2  b̂2  I, where I is the identityoperator,© 2002 The American Physical Society 060403-1VOLUME 88, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 11 FEBRUARY 2002Ĉ2  4I 2 mnÂ, â	 B̂, b̂	 . (7)Since for all F̂ and Ĝ bounded operators,kF̂, Ĝ	k # kF̂Ĝk 1 kĜF̂k # 2kF̂k kĜk , (8)then kĈ2k # 8, or kĈk # 2p2, q.e.d.Different derivations of this bound can be found in[10,11]. Violations of the CHSH inequality (2) by 2p2can be obtained with pure [4] or mixed states [10].Popescu and Rohrlich [8] raised the question whetherrelativistic causality could restrict the violation of theCHSH inequality to 2p2 instead of 4, which would bethe maximum bound allowed if the four correlations in theCHSH inequality (2) were independent. They prove thisconjecture false [8] by defining a contrived correlationfunction which satisfies relativistic causality while stillviolating the CHSH inequality by the maximum value 4.Here I shall show that violations of the CHSH inequal-ity beyond the 2p2 bound can be naturally obtained us-ing only the predictions of quantum mechanics. This doesnot entail a violation of Cirel’son’s inequality but a vio-lation of the CHSH inequality beyond Cirel’son’s bound.To understand the difference, let us consider three identi-cal brothers. Every morning each of them takes a bus inLondon. One goes to Aylesbury, one to Brighton, and thethird to Cambridge. Two of them wear white coats andthe other wears a black one. We are interested in the cor-relations between the experiments on two of them. Then,the first step is to define which two. One possibility is tochoose those brothers arriving in Aylesbury and Brighton.Another possibility is to choose those wearing white coats,regardless of their destination. Both possibilities are legiti-mate in a theory in which both procedures used for select-ing pairs are related to predefined properties. Accordingto Einstein, Podolsky, and Rosen [3], a local system isassumed to have a predefined property if we can predictwith certainty the value of that property from the resultsof experiments on distant systems. Therefore, for a local-realistic theory, both procedures described above for se-lecting pairs would be legitimate. If we are interested inthe correlations between the experiments on the two broth-ers with white coats, we can see whether the coat of thebrother arriving in Cambridge is black. If this is the case,we can legitimately conclude that the other two brotherswear white coats. However, in quantum mechanics it isnot meaningful to assume that some physical observableshave predefined values before the measurements are made.Therefore, such an inference is not permitted.The key for the understanding of our approach is torealize that, in searching for violations of the CHSH in-equality (which is derived assuming local realism, withoutany mention of quantum mechanics), one is not limitedto studying only the correlations of systems prepared in aquantum state, as in Cirel’son’s inequality, but rather thatone can use the correlations predicted by quantum mechan-ics for different subsets of systems previously prepared in a060403-2quantum state. Therefore, one can use a procedure like theone described above for selecting pairs. However, one can-not do this if we are interested in violations of Cirelson’sinequality, which is valid for systems prepared in a quan-tum state (without any further postselection).The important point for physics is not whether a quan-tum state violates the CHSH inequality but rather whetherthe predictions of quantum mechanics violate the CHSHinequality and the extent of this violation. We will showthat the correlations of a postselected subsystem of a three-qubit system prepared in a Greenberger-Horne-Zeilinger(GHZ) state [12–14] as described by quantum mechanicsallow the maximum violation.Let us consider systems of three distant qubits preparedin the GHZ state:jC 1p2j 1 1 1 1 j 2 2 2 , (9)where 1 and 2 denote, respectively, spin-up and spin-down in the y direction. For each three-qubit system pre-pared in the state (9), let us denote as qubits i and j thosegiving the result 21 when measuring the spin in the z di-rection on all three qubits; the third qubit will be denotedas k. If all three qubits give the result 1, qubits i and jcould be any pair of them. Since no other combination ofresults is allowed for state (9), qubits i and j are well de-fined for every three-qubit system. Generally, qubits i andj will be in a different location for each three-qubit sys-tem. For instance, if we denote the three possible locationsas 1, 2, and 3, in the first three-qubit system, qubits i andj could be in locations 1 and 2; in the second three-qubitsystem, they could be in locations 1 and 3, etc. However,we can force qubits i and j to be those in 1 and 2, just bymeasuring the spin in the z direction on the qubit in 3 andthen selecting only those events in which the result of thismeasurement is 1.We are interested in the correlations between two ob-servables A and a of qubit i and two observables B andb of qubit j. In particular, let us choose A  Zi , a  Xi ,B  Zj, and b  Xj, where Zq and Xq are the spin ofqubit q along the z and x directions, respectively. Theparticular CHSH inequality (2) we are interested in is theone in which m  n  xk, where xk is one of the pos-sible results, 21 or 1 (although we do not know whichone), of measuring Xk . With this choice we obtain theCHSH inequality:jCZi, Zj 2 xkCZi, Xj 2xkCXi , Zj 2 CXi , Xjj # 2 , (10)which holds for any local-realistic theory, regardless ofthe particular value, either 21 or 1, of xk. Now let ususe quantum mechanics to calculate the four correlationsappearing in (10). By the definition of qubits i and j, andtaking into account that state (9) is an eigenstate of theself-adjoint operator ẐiẐjẐk with eigenvalue 1, we obtainCZi , Zj  1 , (11)060403-2VOLUME 88, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 11 FEBRUARY 2002since the only possible results are Zi  Zj  1 and Zi Zj  21. By taking into account that state (9) is an eigen-state of ẐiX̂j X̂k with eigenvalue 21, we obtainCZi , Xj  2xk , (12)since the only possible results are Zi  1, Xj  2xk andZi  21, Xj  xk. By taking into account that state (9)is an eigenstate of X̂i ẐjX̂k with eigenvalue 21, we obtainCXi, Zj  2xk , (13)since the only possible results are Xi  xk, Zj  21 andXi  2xk, Zj  1. Finally, by the definition of qubit kas the one in which zk  1, and taking into account thatstate (9) is an eigenstate of X̂iX̂jẐk with eigenvalue 21,we obtainCXi , Xj  21 , (14)since the only possible results are Xi  2Xj  1 andXi  2Xj  21. Therefore, the left-hand side of in-equality (10) is 4, which is the maximum value allowed bythe definition of correlation. Other choices of three-qubitentangled quantum states and observables lead to viola-tions of the CHSH inequality in the 2p2 to 4 range.This result opens the possibility of using sources ofquantum entangled states of three or more particles [15]to experimentally test [16] local realism using not onlyproofs of Bell’s theorem without inequalities [12–14] orBell inequalities involving correlations between three ormore particles [17,18], but also the CHSH inequality (10).However, it must be stressed that, since the correlations(12) and (13) between qubits i and j depend on qubit k, theexperimental test cannot be simply a test on, for instance,those pairs arriving in locations 1 and 2 when a particularmeasurement on the qubit arriving in 3 gives a particularresult, but, as we shall see below, it requires treating allthree qubits in a completely symmetrical way.On the other hand, in real experiments using threequbits, the experimental data consist of the number ofcoincidences (that is, of simultaneous detections by threedetectors) NABCa, b, c for various observables A, B, andC. This number is proportional to the corresponding jointprobability, PABCa, b, c.Therefore, in order to make inequality (10) useful forreal experiments, we must first translate it into the languageof joint probabilities and then we must show how the jointprobabilities of qubits i and j are related to the probabili-ties of coincidences of qubits arriving in 1, 2, and 3.For the first step, it is useful to note that, by assumingphysical locality (that is, that the expected value of any lo-cal observable cannot be affected by anything done to a dis-tant particle), the CHSH inequality (10) can be transformedinto a more convenient experimental inequality [19,20]:21 # PZiZj 21, 21 2 PZiXj 21, 2xk2 PXiZj 2xk, 21 2 PXiXj xk , xk # 0 . (15)060403-3The bounds l of inequalities (2) and (10) are transformedinto the bounds l 2 24 of inequality (15). Therefore,the local-realistic bound in (15) is 0, Cirel’son’s bound isp2 2 12, and the maximum value is 12. For qubits iand j of a system in the state (9),PZiZj 21, 21  34 , (16)since, in the state (9), the four possible results satisfyingzizjzk  1 (where zi denotes the result of measuring Zi ,etc.) have probability 14 and in three of them 21 appearstwice;PZiXj 21, 2xk  0 , (17)since, in the state (9), zixjxk  21;PXiZj 2xk , 21  0 , (18)since, in the state (9), xizjxk  21;PXiXj xk, xk  14 , (19)since, in the state (9), both results xi  xj  xk  1 andxi  xj  xk  21 have probability 18. Therefore, asexpected, the maximum allowed violation of inequality(15) occurs for the same choices in which the maximumviolation of the CHSH inequality (10) does.The second step consists of showing how the four jointprobabilities (16)–(19) are related to the probabilities ofcoincidences in an experiment with three spatial locations,1, 2, and 3. As can easily be seen,PZiZj 21, 21  PZ1Z2Z3 1, 21, 21 1 PZ1Z2Z321, 1, 211 PZ1Z2Z3 21, 21, 11 PZ1Z2Z3 21, 21, 21 , (20)where, in the state (9), the first three probabilities inthe right-hand side of (20) are expected to be 14 andthe fourth is expected to be zero. On the other hand,PZiXj 21, 2xk  and PXiZj 2xk , 21 are both less than orequal toPZ1X2X3 21, 1, 21 1 PZ1X2X3 21, 21, 11 PX1Z2X3 1, 21, 21 1 PX1Z2X3 21, 21, 11 PX1X2Z3 1, 21, 21 1 PX1X2Z3 21, 1, 21 , (21)where, in the state (9), the six probabilities in (21) areexpected to be zero. Finally,PXiXj xk , xk  PX1X2X31, 1, 1 1 PX1X2X3 21, 21, 21 ,(22)where, in the state (9), the two probabilities in the right-hand side of (22) are expected to be 18.The experimental data of previous tests using three-photon systems prepared in a GHZ state [16] or possiblenew experiments over large distances with spacelikeseparated randomly switched measurements [21] or withthree-ion systems and almost-perfect detectors [22] couldexperimentally confirm this violation of local realismpredicted by quantum mechanics.060403-3VOLUME 88, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 11 FEBRUARY 2002I thank J. L. Cereceda and K. Svozil for commentsand the Junta de Andalucía Grant No. FQM-239 andthe Spanish Ministerio de Ciencia y Tecnología GrantsNo. BFM2000-0529 and No. BFM2001-3943 for support.*Electronic address: adan@us.es[1] J. S. Bell, Physics (Long Island City, N.Y.) 1, 195 (1964).[2] H. P. Stapp, Nuovo Cimento Soc. Ital. Fis. 29B, 270 (1975).[3] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777(1935).[4] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt,Phys. Rev. Lett. 23, 880 (1969).[5] B. S. Cirel’son, Lett. Math. Phys. 4, 93 (1980).[6] A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49,1804 (1982).[7] A. Peres, Quantum Theory: Concepts and Methods(Kluwer, Dordrecht, 1993), p. 174.[8] S. Popescu and D. Rohrlich, Found. Phys. 24, 379 (1994).[9] L. J. Landau, Phys. Lett. A 120, 54 (1987).[10] S. L. Braunstein, A. Mann, and M. Revzen, Phys. Rev. Lett.68, 3259 (1992).060403-4[11] A. Chefles and S. M. Barnett, J. Phys. A 29, L237 (1996).[12] D. M. Greenberger, M. A. Horne, and A. Zeilinger, inBell’s Theorem, Quantum Theory, and Conceptions of theUniverse, edited by M. Kafatos (Kluwer, Dordrecht,1989), p. 69.[13] N. D. Mermin, Phys. Today 43, No. 6, 9 (1990); Am. J.Phys. 58, 731 (1990).[14] D. M. Greenberger, M. A. Horne, A. Shimony, and A.Zeilinger, Am. J. Phys. 58, 1131 (1990).[15] D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, andA. Zeilinger, Phys. Rev. Lett. 82, 1345 (1999).[16] J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, andA. Zeilinger, Nature (London) 403, 515 (2000).[17] N. D. Mermin, Phys. Rev. Lett. 65, 1838 (1990).[18] S. M. Roy and V. Singh, Phys. Rev. Lett. 67, 2761 (1991).[19] J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526(1974).[20] N. D. Mermin, Ann. N.Y. Acad. Sci. 755, 616 (1995).[21] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A.Zeilinger, Phys. Rev. Lett. 81, 5039 (1998).[22] M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M.Itano, C. Monroe, and D. J. Wineland, Nature (London)409, 791 (2001).060403-4

Violating bell's inequality beyond Cirel'son's bound

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Violating Bell's inequality beyond Cirel'son's bound

Violating Bell's inequality beyond Cirel'son's bound

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