Two qubits of a W state violate Bell's inequality beyond Cirel'son's bound

Two qubits of a Wstate violate Bell’s inequality beyond Cirel’son’s bound Ada ´n Cabello* Departamento de Fı ´sica Aplicada II, Universidad de Sevilla, 41012 Sevilla, Spain 共Received 30 May 2002; published 22 October 2002兲 It is shown that the correlations between two qubits selected from a trio prepared in a Wstate violate the Clauser-Horne-Shimony-Holt inequality more than the correlations between two qubits in any quantum state. Such a violation beyond Cirel’son’s bound is smaller than the one achieved by two qubits selected from a trio in a Greenberger-Horne-Zeilinger state 关A. Cabello, Phys. Rev. Lett. 88, 060403 共2002兲兴. However, it has the advantage that all local observers can know from their own measurements whether or not their qubits belong to the selected pair. DOI: 10.1103/PhysRevA.66.042114 PACS number共s兲: 03.65.Ud, 03.65.Ta I. INTRODUCTION The Bell inequality 关1兴proposed by Clauser, Horne, Shimony, and Holt 共CHSH兲关2兴, points out that in any localrealistic theory, that is, in any theory in which the local variables of a particle determine the results of local experiments on this particle, the absolute value of a combination of four correlations is bound by 2, 兩 C共A,B兲⫺mC共A,b兲⫺nC共a,B兲⫺mnC共a,b兲 兩 ⭐2. 共1兲 In inequality 共1兲,Aand aare two observables taking values ⫺1 or 1 on particle i, and Band bare two observables taking values ⫺1 or 1 on a distant particle j;mand ncan be either ⫺1or1. The CHSH inequality 共1兲is violated for certain quantum states and certain choices of the observables A,a,B, and b 关2兴. Therefore, the conclusion is that no local-realistic theory can reproduce the predictions of quantum mechanics 关1兴. Later on, Cirel’son 关3兴showed that, according to quantum mechanics, for any two-qubit system prepared in a quantum state, the absolute value of the combination of correlations appearing in the CHSH inequality 共1兲is bound by 2 冑 2 共Cirel’son’s bound兲. This bound is also the maximum violation predicted by quantum mechanics for the two-qubit singlet state 共or any other two-qubit Bell state兲关2兴. Indeed, this is the violation of Bell’s inequality traditionally tested in real experiments involving systems of two qubits prepared in a quantum state 关4–9兴. However, as shown in Ref. 关10兴, according to quantum mechanics the CHSH inequality can be violated beyond Cirel’son’s bound. The reason is the following. Bell’s inequalities are derived assuming local realism, without any mention to quantum mechanics. Therefore, when searching for violations of a Bell’s inequality, one is not restricted to studying correlations between ensembles of systems prepared in a quantum state; instead, one can study any ensemble of systems, irrespective of whether such an ensemble is meaningful in quantum mechanics or not 共i.e., irrespective of whether it can be described by a quantum state or not兲. For instance, one can consider trios of qubits prepared in a certain quantum state and then assume local realism to select a pair of qubits in each trio, and calculate, using quantum mechanics, the correlations between these two qubits. The whole procedure makes sense and can be translated into real experiments as long as one can obtain the required correlations and probabilities for the two selected qubits from the data obtained in a real experiment with three qubits prepared in a quantum state. In Ref. 关10兴, a violation of the CHSH inequality 共1兲beyond Cirel’son’s bound is presented for certain subensembles of two qubits of an ensemble of trios prepared in a Greenberger-Horne-Zeilinger 共GHZ兲state 关11兴. In this paper, we shall investigate whether a violation beyond Cirel’son’s bound could be found for pairs of qubits selected from trios prepared in a Wstate 关12兴. The structure of the paper is as follows. In Sec. II, we will find that, for a certain choice of observables, the Wstate violates the CHSH inequality 共1兲beyond Cirel’son’s bound. The observables used in Sec. II do not provide the maximum achievable violation using a Wstate. In Sec. III, we explain the reason behind this choice of observables. In addition, the violation in Sec. II is smaller that the one obtained in Ref. 关10兴using a GHZ state. However, in Sec. IV, we will see that there are some reasons that make the violation provided by the Wstate more interesting than that provided by the GHZ state. Finally, in Sec. V, we discuss how to obtain the required probabilities for the two selected qubits from the data obtained in a real experiment with three qubits. II. THE WSTATE VIOLATES THE CHSH INEQUALITY BEYOND CIREL’SON’S BOUND Let us consider three distant qubits 1,2,3, prepared in the Wstate 兩 W 典 ⫽ 1 冑 3共 兩 ⫹⫺⫺ 典 ⫹ 兩 ⫺⫹⫺ 典 ⫹ 兩 ⫺⫺⫹ 典 ), 共2兲 where ␴ z 兩 ⫾ 典 ⫽⫾ 兩 ⫾ 典 . For each three qubits prepared in the Wstate 共2兲, we are going to concentrate our attention on two of them, namely, those two in which, if we had measured ␴ z, we would have obtained the result ⫺1. These two qubits will be called iand jhereafter, while the corresponding third qubit 共the one in which, if we had measured ␴ z, we would *Electronic address: [email protected] PHYSICAL REVIEW A 66, 042114 共2002兲 1050-2947/2002/66共4兲/042114共5兲/$20.00 ©2002 The American Physical Society66 042114-1 have found the result 共1兲will be called k. In quantum mechanics, the result of measuring ␴ zis not predefined and therefore this prescription for choosing pairs is meaningless. However, the prescription makes sense in a local-realistic theory. For reasons that will be explained in Sec. III, we are interested in the correlations when we choose A⫽Zi,a⫽Xi, B⫽Zj, and b⫽Xj, where Zqand Xqare the spin of qubit q along the zand xdirections, respectively. In addition, the particular CHSH inequality 共1兲we are interested in is the one in which m⫽n⫽xk, where xkis one of the possible results, ⫺1or1共although we do not know which one兲, of measuring Xk. With this choice we obtain the following CHSH inequality: 兩 C共Zi,Zj兲⫺xkC共Zi,Xj兲⫺xkC共Xi,Zj兲⫺C共Xi,Xj兲 兩 ⭐2, 共3兲 which holds for any local-realistic theory, regardless of the particular value, either ⫺1or1,ofxk. The next step is to use quantum mechanics to calculate the four correlations appearing in inequality 共3兲for the subensemble of two qubits iand jtaken from three qubits prepared in the Wstate 共2兲. For the subensemble of two qubits iand jdefined above, C共Zi,Zj兲⫽1, 共4兲 because, for the Wstate 共2兲, PZ1Z2Z3共1,⫺1,⫺1兲⫽1 3,共5兲 PZ1Z2Z3共⫺1,1,⫺1兲⫽1 3,共6兲 PZ1Z2Z3共⫺1,⫺1,1兲⫽1 3,共7兲 where PZ1Z2Z3(1,⫺1,⫺1) means the probability of qubit 1 giving the result 1, and qubits 2 and 3 giving the result ⫺1 when measuring ␴ zon all three qubits. By the definition of qubits iand j, C共Zi,Xj兲⫽⫺xk,共8兲 because, for the Wstate 共2兲, PZ1X2X3共⫺1,1,⫺1兲⫹PZ1X2X3共⫺1,⫺1,1兲⫽0, 共9兲 PX1Z2X3共1,⫺1,⫺1兲⫹PX1Z2X3共⫺1,⫺1,1兲⫽0, 共10兲 PX1X2Z3共1,⫺1,⫺1兲⫹PX1X2Z3共⫺1,1,⫺1兲⫽0. 共11兲 Analogously, using Eqs. 共9兲–共11兲, C共Xi,Zj兲⫽⫺xk.共12兲 Finally, for the Wstate 共2兲, PX1X2X3共1,1,1兲⫽3 8,共13兲 PX1X2X3共⫺1,⫺1,⫺1兲⫽3 8,共14兲 PX1X2X3共1,1,⫺1兲⫽1 24,共15兲 PX1X2X3共⫺1,⫺1,1兲⫽1 24,共16兲 PX1X2X3共1,⫺1,1兲⫽1 24,共17兲 PX1X2X3共⫺1,1,⫺1兲⫽1 24,共18兲 PX1X2X3共⫺1,1,1兲⫽1 24,共19兲 PX1X2X3共1,⫺1,⫺1兲⫽1 24.共20兲 From Eqs. 共13兲and 共14兲, the contribution of cases x1⫽x2 ⫽x3⫽1 is cancelled by the contribution of cases x1⫽x2 ⫽x3⫽⫺1; from Eqs. 共15兲and 共16兲, the contribution of cases x1⫽x2⫽⫺x3⫽1 is cancelled by the contribution of cases x1⫽x2⫽⫺x3⫽⫺1, etc. Therefore, irrespective of whether iand jare qubits 1 and 2, or 1 and 3, or 2 and 3, we conclude that C共Xi,Xj兲⫽0. 共21兲 Correlations 共4兲,共8兲,共12兲, and 共21兲violate the CHSH inequality 共3兲. The violation 共3vs2兲goes beyond Cirel’son’s bound (2 冑 2). III. WHY XAND Z? A particular type of local-realistic theories are those in which the only local experiments whose results are assumed to be predetermined are those which satisfy the criterion for ‘‘elements of reality’’ proposed by Einstein, Podolsky, and Rosen 共EPR兲:‘‘If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity’’ 关13兴. As can be easily checked, the violation reported in Sec. II is not the maximal violation of the CHSH inequality 共3兲for two qubits in the Wstate 共2兲. For instance, considering local spin observables on plane x-zand assuming A⫽Band a ⫽b, we find a maximum violation of 3.046 关by choosing A⫽cos(0.628) ␴ x⫺sin(0.628) ␴ zand a⫽cos(1.154) ␴ x ⫹sin(1.154) ␴ z]. Why then have we chosen A⫽Zi,a⫽Xi, B⫽Zj, and b⫽Xj? The reason is that these observables are not only local observables but, for the Wstate 共2兲, they also satisfy EPR’s criterion of elements of reality. ADA ´N CABELLO PHYSICAL REVIEW A 66, 042114 共2002兲 042114-2 From Eqs. 共5兲–共7兲, it can be immediately seen that z1,z2, and z3are elements of reality, since any of them can be predicted with certainty from spacelike separated measurements of ␴ zon the other two qubits. In addition, from Eqs. 共9兲–共11兲, it can easily be seen that, if zi⫽⫺1 then, with certainty, xj⫽xk. Therefore, if zi⫽⫺1, then by measuring xj(xk) one can predict xk(xj) with certainty. Therefore, if zi⫽⫺1, then xjand xkare elements of reality. If zi⫽1 then, using Eqs. 共5兲–共7兲, it can immediately be seen that zj⫽ ⫺1. Therefore, following the previous reasoning, xiand xk are elements of reality 共although xicould have ceased to be an element of reality after measuring ␴ zon particle i). In conclusion, for trios of qubits in the Wstate 共2兲,z1,z2,z3, x1,x2, and x3are EPR elements of reality and thus, according to EPR, they should have predefined values ⫺1or1 before any measurement. The violation of the CHSH inequality 共3兲presented in Sec. II is thus not only a proof of the impossibility of local hidden variables, but also proves a more powerful result: the apparently mild condition proposed by EPR is inconsistent with quantum mechanics. IV. WHY W? As was shown in Ref. 关10兴, two qubits belonging to a three-qubit system in a GHZ state can provide a higher violation 共4 vs 2, instead of 3 vs 2兲of the CHSH inequality 共3兲, even using observables that satisfy EPR’s criterion of elements of reality. Why then use a Wstate? One reason is because a test of the violation of Bell’s inequalities beyond Cirel’son’s bound could be achieved in practice in the near future. Sources of Wstates based on parametric down-converted photons are now available for real experiments 关14兴and some new proposals to prepare W states via cavity quantum electrodynamics have recently been presented 关15兴. Another reason is because this violation beyond Cirel’son’s bound is, in one sense, surprising. The Wstate is the genuine three-qubit entangled state whose entanglement has the highest robustness against the loss of one qubit 关12兴. In particular, from a single copy of the reduced density matrix for any two qubits belonging to a three-qubit Wstate, one can always obtain by means of a filtering measurement a state that is arbitrarily close to a Bell state. Therefore, one might think that any two qubits belonging to a Wstate will not lead to a higher violation of the CHSH inequality 共3兲 than that for two qubits in a Bell state, and thus it is of interest to realize that this is not the case. There is, however, another subtler reason for preferring the Wstate instead of the GHZ state for a test of violation of Bell’s inequalities beyond Cirel’son’s bound. Any test of this kind requires a prescription for selecting a pair of qubits from each trio prepared in a quantum state. Such a prescription assumes local realism. In the violation of the CHSH inequality 共3兲presented in Sec. II, this prescription is simple: qubits iand jare those two in which, if we had measured ␴ z, we would have obtained the result ⫺1. However, in the violation of the CHSH inequality 共3兲using a GHZ state described in Ref. 关10兴, the prescription is not so simple: there, qubits iand jare either those two in which, if we had measured ␴ z, we would have obtained the result ⫺1, or any two, if we had obtained the result 1 for all three qubits if we had measured ␴ z. This means that, for the Wstate, any local observer could know whether or not his qubit belonged to the selected pair just by measuring ␴ z, while for the GHZ state, the fact that whether or not a qubit belongs to the selected pair cannot be decided with certainty from a measurement on that qubit, but requires knowledge of the results of measurements on the other two qubits. From the perspective of local realism, for the Wstate, one of the elements of reality carried by each qubit determines whether or not it belongs to the selected pair; while for the GHZ state, this information is not local since it is distributed among distant elements of reality. V. EXPERIMENTAL CH INEQUALITY The result in Sec. II opens the possibility of using sources of three-qubit Wstates 关14,15兴to experimentally test the CHSH inequality. The main advantage of an experiment like this 共or that proposed in Ref. 关10兴兲 is that it will admit a direct comparison with the dozens of previous experiments with two qubits 关4–9兴and thus goes beyond any previous experiments to test local realism using sources of three qubits 关16,17兴inspired by proofs of Bell’s theorem without inequalities 关11兴or by Bell’s inequalities for three qubits 关18,19兴. However, in any real experiment using three qubits, the experimental data consist of the number of simultaneous detections by three detectors NABC(a,b,c) for various observables A,B, and C. This number is assumed to be proportional to the corresponding joint probability, PABC(a,b,c). Therefore, in order to make inequality 共3兲useful for real experiments, it would be convenient to translate it into the language of joint probabilities. Taking into account that PZiZj共⫺1,⫺1兲⫽1 4关1⫺C共Zi兲⫺C共Zj兲⫹C共Zi,Zj兲兴, 共22兲 PZiXj共⫺1,⫺xk兲⫽1 4关1⫺C共Zi兲⫺xkC共Xj兲⫹xkC共Zi,Xj兲兴, 共23兲 PXiZj共⫺xk,⫺1兲⫽1 4关1⫺xkC共Xi兲⫺C共Zj兲⫹xkC共Xi,Zj兲兴, 共24兲 PXiXj共xk,xk兲⫽1 4关1⫹xkC共Xi兲⫹xkC共Xj兲⫹xk 2C共Xi,Xj兲兴, 共25兲 where C(Zi) is the mean of the results of measuring ␴ zon qubit i, and assuming physical locality 关i.e., assuming that C(Zi) is independent of whether ␴ zor ␴ xis measured on qubit j, that is, assuming that the value of C(Zi) is the same in Eqs. 共22兲and 共23兲, etc.兴, the CHSH inequality 共3兲between correlations can be transformed into a Clauser-Horne 共CH兲 inequality 关20兴between joint probabilities, TWO QUBITS OF A WSTATE VIOLATE BELL’s... PHYSICAL REVIEW A 66, 042114 共2002兲 042114-3 ⫺1⭐PZiZj共⫺1,⫺1兲⫺PZiXj共⫺1,⫺xk兲⫺PXiZj共⫺xk,⫺1兲 ⫺PXiXj共xk,xk兲⭐0. 共26兲 As can be easily checked, the bounds lof the CHSH inequality 共3兲are transformed into the bounds (l⫺2)/4 of the corresponding CH inequality 共26兲. Therefore, the local-realistic bound in the CH inequality 共26兲is 0 and Cirel’son’s bound is ( 冑 2⫺1)/2⬇0.207. For qubits iand jof a system in the Wstate 共2兲, PZiZj共⫺1,⫺1兲⫽1, 共27兲 PZiXj共⫺1,⫺xk兲⫽0, 共28兲 PXiZj共⫺xk,⫺1兲⫽0, 共29兲 PXiXj共xk,xk兲⫽3 4.共30兲 Therefore, probabilities 共27兲–共30兲violate the CH inequality 共26兲. Such a violation 共0.25 vs 0兲is beyond the corresponding Cirel’son’s bound 共0.207兲. On the other hand, since we do not know which ones are qubits iand j, we cannot obtain the four joint probabilities 共27兲–共30兲just by performing measurements on two qubits. Therefore, we must show how the joint probabilities of qubits iand jare related to the probabilities of the three qubits. As can easily be seen from the definition of qubits iand j, PZiZj共⫺1,⫺1兲⫽PZ1Z2Z3共1,⫺1,⫺1兲⫹PZ1Z2Z3共⫺1,1,⫺1兲 ⫹PZ1Z2Z3共⫺1,⫺1,1兲 ⫹PZ1Z2Z3共⫺1,⫺1,⫺1兲.共31兲 Therefore, in order to experimentally obtain PZiZj(⫺1, ⫺1), we must measure the four probabilities in the righthand side of Eq. 共31兲.IntheWstate 共2兲, the first three probabilities in the right-hand side of Eq. 共31兲are expected to be 1/3 and the fourth is expected to be zero. On the other hand, PZiXj(⫺1,⫺xk) and PXiZj(⫺xk,⫺1) are both less than or equal to PZ1X2X3共⫺1,1,⫺1兲⫹PZ1X2X3共⫺1,⫺1,1 ⫹PX1Z2X3共1,⫺1,⫺1兲⫹PX1Z2X3共⫺1,⫺1,1兲 ⫹PX1X2Z3共1,⫺1,⫺1兲⫹PX1X2Z3共⫺1,1,⫺1兲. 共32兲 Therefore, in order to experimentally obtain PZiXj(⫺1, ⫺xk) and PXiZj(⫺xk,⫺1), we must measure 共using three different setups兲all six probabilities in sum 共32兲.IntheW state 共2兲, each of these six probabilities is expected to be zero. Finally, PXiXj共xk,xk兲⫽PX1X2X3共1,1,1兲⫹PX1X2X3共⫺1,⫺1,⫺1兲. 共33兲 Therefore, in order to experimentally obtain PXiXj(xk,xk), we must measure the two probabilities in the right-hand side of Eq. 共33兲.IntheWstate 共2兲, each of them is expected to be 3/8. VI. CONCLUSIONS Two qubits selected from a trio prepared in a Wstate violate the CHSH inequality, or the corresponding CH inequality, more than two qubits prepared in any quantum state. Such violations beyond Cirel’son’s bound are smaller than those achieved by two qubits selected from a trio in a GHZ state 关10兴. However, for the Wstate the argument is simpler, since all local observers can know from their own measurements whether or not their qubits belong to the selected pair. 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