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

VOLUME 88, NUMBER 6 PHYSICAL REVIEW LETTERS 11F EBRUARY 2002 Violating Bell’s Inequality Beyond Cirel’son’s Bound Adán 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 appearing in the Clauser-Horne-Shimony-Holt (CHSH) inequality is bound by 2p2. It is shown that the correlations of two qubits belonging to a three-qubit system can violate the CHSH inequality beyond 2p2. 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. DOI: 10.1103/PhysRevLett.88.060403 PACS numbers: 03.65.Ta, 03.65.Ud Bell’s theorem [1] has been described as “the most profound discovery of science” [2]. It states that, according to quantum mechanics, the value of a certain combination of correlations for experiments on two distant systems can be higher than the highest value allowed by any local-realistic theory of the type proposed by Einstein, Podolsky, and Rosen [3], in which local properties of a system determine the result of any experiment on that system. The most commonly discussed Bell inequality, the Clauser-Horne- Shimony-Holt (CHSH) inequality [4], states that in any local-realistic theory the absolute value of a combination of four correlations is bound by 2. Cirel’son’s inequality [5] shows that the combination of quantum correlations appearing in the CHSH inequality is bound by 2p2 (Cirel’son’s bound). It is widely believed that “[q]uantum theory does not allow any stronger violation of the CHSH inequality than the one already achieved in Aspect’s experiment [6] [2p2]” [7]. However, it has been shown that exceeding Cirel’son’s bound is not forbidden by relativistic causality [8]. Therefore, an intriguing question is why the CHSH inequality is not violated more. Here it is shown that, for three-qubit systems (that is, systems composed by three two-level quantum particles), the correlation functions of two suitably postselected qubits violate the CHSH inequality beyond Cirel’son’s bound and that this violation can even reach 4, the maximum value allowed by the definition of correlation. To introduce the CHSH inequality, let us consider systems with two distant particles iand j. Let Aand a(Band b) be physical observables taking values 21or 1 referring to local experiments on particle i共j兲. The correlation C共A,B兲of Aand Bis defined as C共A,B兲苷PAB共1, 1兲2PAB共1, 21兲2PAB共21, 1兲 1PAB共21, 21兲,(1) where PAB共1, 21兲denotes the joint probability of obtaining A苷1and B苷21when Aand Bare measured. In any local-realistic theory, that is, in any theory in which local variables of particle i共j兲determine the results of local experiments on particle i共j兲, the absolute value of a particular combination of correlations is bound by 2: jC共A,B兲2mC共A,b兲2nC共a,B兲2mnC共a,b兲j#2, (2) where mand ncan be either 21or 1. The CHSH inequality (2) holds for any local-realistic theory, whatever the values of mand nare, in the allowed set, 兵21, 1其. The bound 2 in inequality (2) can easily be derived as follows: In a local-realistic theory, for any individual system, the observables A,a,B, and bhave predefined values yA,ya,yB, and yb, either 21or 1. Therefore, for an individual system the combination of correlations appearing in (2) can be calculated as yB共yA2nya兲2myb共yA1nya兲,(3) which is either 22or 2, because one of the expressions between parentheses is necessarily zero and the other is either 22or 2. Therefore, the absolute value of the corresponding averages is bound by 2, q.e.d. For a two-particle system in a quantum pure state described by a vector jc典, the quantum correlation of Aand Bis defined as CQ共A,B兲苷具cjˆ Aˆ Bjc典,(4) where ˆ Aand ˆ Bare the self-adjoint operators which represent observables Aand B. For certain choices of ˆ A,ˆa,ˆ B,ˆ b, and jc典, quantum correlations violate the CHSH inequality [4]. Therefore, no local-realistic theory can reproduce the predictions of quantum mechanics [1]. Later on, Cirel’son [5] demonstrated that for a twoparticle system the absolute value of the combination of quantum correlations equivalent to those appearing in the CHSH inequality (2) is bound by 2p2, jCQ共A,B兲2mCQ共A,b兲2 nCQ共a,B兲2mnCQ共a,b兲j#2p2. (5) Cirel’son’s bound can easily be derived as follows [9]: Consider the operator with the same structure as the combination which appears in inequality (5), ˆ C苷ˆ Aˆ B2mˆ Aˆ b2nˆaˆ B2mnˆaˆ b.(6) If ˆ A2苷ˆa2苷ˆ B2苷ˆ b2苷I,where Iis the identity operator, 060403-1 0031-9007兾02兾88(6)兾060403(4)$20.00 © 2002 The American Physical Society 060403-1 VOLUME 88, NUMBER 6 PHYSICAL REVIEW LETTERS 11F EBRUARY 2002 ˆ C2苷4I2mn关ˆ A,ˆa兴关ˆ B,ˆ b兴.(7) Since for all ˆ Fand ˆ Gbounded operators, k关ˆ F,ˆ G兴k#kˆ Fˆ Gk1kˆ Gˆ Fk#2kˆ Fkkˆ Gk,(8) then kˆ C2k#8,orkˆ Ck#2p2, q.e.d. Different derivations of this bound can be found in [10,11]. Violations of the CHSH inequality (2) by 2p2 can be obtained with pure [4] or mixed states [10]. Popescu and Rohrlich [8] raised the question whether relativistic causality could restrict the violation of the CHSH inequality to 2p2instead of 4, which would be the maximum bound allowed if the four correlations in the CHSH inequality (2) were independent. They prove this conjecture false [8] by defining a contrived correlation function which satisfies relativistic causality while still violating the CHSH inequality by the maximum value 4. Here I shall show that violations of the CHSH inequality beyond the 2p2bound can be naturally obtained using only the predictions of quantum mechanics. This does not entail a violation of Cirel’son’s inequality but a violation of the CHSH inequality beyond Cirel’son’s bound. To understand the difference, let us consider three identical brothers. Every morning each of them takes a bus in London. One goes to Aylesbury, one to Brighton, and the third to Cambridge. Two of them wear white coats and the other wears a black one. We are interested in the correlations between the experiments on two of them. Then, the first step is to define which two. One possibility is to choose those brothers arriving in Aylesbury and Brighton. Another possibility is to choose those wearing white coats, regardless of their destination. Both possibilities are legitimate in a theory in which both procedures used for selecting pairs are related to predefined properties. According to Einstein, Podolsky, and Rosen [3], a local system is assumed to have a predefined property if we can predict with certainty the value of that property from the results of experiments on distant systems. Therefore, for a localrealistic theory, both procedures described above for selecting pairs would be legitimate. If we are interested in the correlations between the experiments on the two brothers with white coats, we can see whether the coat of the brother arriving in Cambridge is black. If this is the case, we can legitimately conclude that the other two brothers wear white coats. However, in quantum mechanics it is not meaningful to assume that some physical observables have predefined values before the measurements are made. Therefore, such an inference is not permitted. The key for the understanding of our approach is to realize that, in searching for violations of the CHSH inequality (which is derived assuming local realism, without any mention of quantum mechanics), one is not limited to studying only the correlations of systems prepared in a quantum state, as in Cirel’son’s inequality, but rather that one can use the correlations predicted by quantum mechanics for different subsets of systems previously prepared in a quantum state. Therefore, one can use a procedure like the one described above for selecting pairs. However, one cannot do this if we are interested in violations of Cirelson’s inequality, which is valid for systems prepared in a quantum state (without any further postselection). The important point for physics is not whether a quantum state violates the CHSH inequality but rather whether the predictions of quantum mechanics violate the CHSH inequality and the extent of this violation. We will show that the correlations of a postselected subsystem of a threequbit system prepared in a Greenberger-Horne-Zeilinger (GHZ) state [12–14] as described by quantum mechanics allow the maximum violation. Let us consider systems of three distant qubits prepared in the GHZ state: jC典苷 1 p2共j111典1j222典兲 ,(9) where 1and 2denote, respectively, spin-up and spindown in the ydirection. For each three-qubit system prepared in the state (9), let us denote as qubits iand jthose giving the result 21when measuring the spin in the zdirection on all three qubits; the third qubit will be denoted as k. If all three qubits give the result 1, qubits iand j could be any pair of them. Since no other combination of results is allowed for state (9), qubits iand jare well defined for every three-qubit system. Generally, qubits iand jwill be in a different location for each three-qubit system. For instance, if we denote the three possible locations as 1, 2, and 3, in the first three-qubit system, qubits iand jcould be in locations 1 and 2; in the second three-qubit system, they could be in locations 1 and 3, etc. However, we can force qubits iand jto be those in 1and 2, just by measuring the spin in the zdirection on the qubit in 3 and then selecting only those events in which the result of this measurement is 1. We are interested in the correlations between two observables Aand aof qubit iand two observables Band bof qubit j. In particular, let us choose A苷Zi,a苷Xi, B苷Zj, and b苷Xj, where Zqand Xqare the spin of qubit qalong the zand xdirections, respectively. The particular CHSH inequality (2) we are interested in is the one in which m苷n苷xk, where xkis one of the possible results, 21or 1 (although we do not know which one), of measuring Xk. With this choice we obtain the CHSH inequality: jC共Zi,Zj兲2xkC共Zi,Xj兲2 xkC共Xi,Zj兲2C共Xi,Xj兲j#2, (10) which holds for any local-realistic theory, regardless of the particular value, either 21or 1, of xk. Now let us use quantum mechanics to calculate the four correlations appearing in (10). By the definition of qubits iand j, and taking into account that state (9) is an eigenstate of the self-adjoint operator ˆ Ziˆ Zjˆ Zkwith eigenvalue 1, we obtain C共Zi,Zj兲苷1, (11) 060403-2 060403-2 VOLUME 88, NUMBER 6 PHYSICAL REVIEW LETTERS 11F EBRUARY 2002 since the only possible results are Zi苷Zj苷1and Zi苷 Zj苷21. By taking into account that state (9) is an eigenstate of ˆ Ziˆ Xjˆ Xkwith eigenvalue 21, we obtain C共Zi,Xj兲苷2xk,(12) since the only possible results are Zi苷1,Xj苷2xkand Zi苷21,Xj苷xk. By taking into account that state (9) is an eigenstate of ˆ Xiˆ Zjˆ Xkwith eigenvalue 21, we obtain C共Xi,Zj兲苷2xk,(13) since the only possible results are Xi苷xk,Zj苷21and Xi苷2xk,Zj苷1. Finally, by the definition of qubit k as the one in which zk苷1, and taking into account that state (9) is an eigenstate of ˆ Xiˆ Xjˆ Zkwith eigenvalue 21, we obtain C共Xi,Xj兲苷21, (14) since the only possible results are Xi苷2Xj苷1and Xi苷2Xj苷21. Therefore, the left-hand side of inequality (10) is 4, which is the maximum value allowed by the definition of correlation. Other choices of three-qubit entangled quantum states and observables lead to violations of the CHSH inequality in the 2p2to 4 range. This result opens the possibility of using sources of quantum entangled states of three or more particles [15] to experimentally test [16] local realism using not only proofs of Bell’s theorem without inequalities [12–14] or Bell inequalities involving correlations between three or more particles [17,18], but also the CHSH inequality (10). However, it must be stressed that, since the correlations (12) and (13) between qubits iand jdepend on qubit k, the experimental test cannot be simply a test on, for instance, those pairs arriving in locations 1 and 2 when a particular measurement on the qubit arriving in 3 gives a particular result, but, as we shall see below, it requires treating all three qubits in a completely symmetrical way. On the other hand, in real experiments using three qubits, the experimental data consist of the number of coincidences (that is, of simultaneous detections by three detectors) NABC共a,b,c兲for various observables A,B, and C. This number is proportional to the corresponding joint probability, PABC共a,b,c兲. Therefore, in order to make inequality (10) useful for real experiments, we must first translate it into the language of joint probabilities and then we must show how the joint probabilities of qubits iand jare related to the probabilities of coincidences of qubits arriving in 1, 2, and 3. For the first step, it is useful to note that, by assuming physical locality (that is, that the expected value of any local observable cannot be affected by anything done to a distant particle), the CHSH inequality (10) can be transformed into a more convenient experimental inequality [19,20]: 21#PZiZj共21, 21兲2PZiXj共21, 2xk兲 2PXiZj共2xk,21兲2PXiXj共xk,xk兲#0. (15) The bounds lof inequalities (2) and (10) are transformed into the bounds 共l22兲兾4of inequality (15). Therefore, the local-realistic bound in (15) is 0, Cirel’son’s bound is 共p221兲兾2, and the maximum value is 1兾2. For qubits i and jof a system in the state (9), PZiZj共21, 21兲苷3兾4, (16) since, in the state (9), the four possible results satisfying zizjzk苷1(where zidenotes the result of measuring Zi, etc.) have probability 1兾4and in three of them 21appears twice; PZiXj共21, 2xk兲苷0, (17) since, in the state (9), zixjxk苷21; PXiZj共2xk,21兲苷0, (18) since, in the state (9), xizjxk苷21; PXiXj共xk,xk兲苷1兾4, (19) since, in the state (9), both results xi苷xj苷xk苷1and xi苷xj苷xk苷21have probability 1兾8. Therefore, as expected, the maximum allowed violation of inequality (15) occurs for the same choices in which the maximum violation of the CHSH inequality (10) does. The second step consists of showing how the four joint probabilities (16)–(19) are related to the probabilities of coincidences in an experiment with three spatial locations, 1, 2, and 3. As can easily be seen, PZiZj共21, 21兲苷PZ1Z2Z3共1, 21, 21兲1PZ1Z2Z3共21, 1, 21兲 1PZ1Z2Z3共21, 21, 1兲 1PZ1Z2Z3共21, 21, 21兲,(20) where, in the state (9), the first three probabilities in the right-hand side of (20) are expected to be 1兾4and the fourth is expected to be zero. On the other hand, PZiXj共21, 2xk兲and PXiZj共2xk,21兲are both less than or equal to PZ1X2X3共21, 1, 21兲1PZ1X2X3共21, 21, 1兲 1PX1Z2X3共1, 21, 21兲1PX1Z2X3共21, 21, 1兲 1PX1X2Z3共1, 21, 21兲1PX1X2Z3共21, 1, 21兲,(21) where, in the state (9), the six probabilities in (21) are expected to be zero. Finally, PXiXj共xk,xk兲苷PX1X2X3共1, 1, 1兲1PX1X2X3共21, 21, 21兲, (22) where, in the state (9), the two probabilities in the righthand side of (22) are expected to be 1兾8. The experimental data of previous tests using threephoton systems prepared in a GHZ state [16] or possible new experiments over large distances with spacelike separated randomly switched measurements [21] or with three-ion systems and almost-perfect detectors [22] could experimentally confirm this violation of local realism predicted by quantum mechanics. 060403-3 060403-3 VOLUME 88, NUMBER 6 PHYSICAL REVIEW LETTERS 11F EBRUARY 2002 I thank J. L. Cereceda and K. Svozil for comments and the Junta de Andalucía Grant No. FQM-239 and the Spanish Ministerio de Ciencia y Tecnología Grants No. BFM2000-0529 and No. BFM2001-3943 for support. *Electronic address: [email protected] [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. 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