Article
Reference
Quantum nonlocality in two three-level systems
ACIN, Antonio, et al.
Abstract
Recently a new Bell inequality has been introduced (CGLMP,KKCZO) that is strongly resistant to noise for maximally entangled states of two d-dimensional quantum systems. We prove that a larger violation, or equivalently a stronger resistance to noise, is found for a non-maximally entangled state. It is shown that the resistance to noise is not a good measure of non-locality and we introduce some other possible measures. The non-maximally entangled state turns out to be more robust also for these alternative measures. From these results it follows that two Von Neumann measurements per party may be not optimal for detecting non-locality. For d=3,4, we point out some connections between this inequality and distillability. Indeed, we demonstrate that any state violating it, with the optimal Von Neumann settings, is distillable.
ACIN, Antonio, et al . Quantum nonlocality in two three-level systems. Physical Review. A , 2002, vol. 65, no. 5
DOI : 10.1103/PhysRevA.65.052325
Available at:
http://archive-ouverte.unige.ch/unige:36791
Disclaimer: layout of this document may differ from the published version.
1 / 1
Quantum nonlocality in two three-level systems
A. Acı´n,1,2T. Durt,3N. Gisin,1and J. I. Latorre2
1GAP-Optique, 20 rue de l’E´ cole-de-Me´decine, CH-1211 Geneva 4, Switzerland
2Departament d’Estructura i Constituents de la Mate`ria, Diagonal 647, E-08028 Barcelona, Spain
3TONA, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium 共Received 29 November 2001; published 8 May 2002兲
Recently a new Bell inequality has been introduced by Collins et al.关Phys. Rev. Lett. 88, 040404共2002兲兴, which is strongly resistant to noise for maximally entangled states of two d-dimensional quantum systems. We prove that a larger violation, or equivalently a stronger resistance to noise, is found for a nonmaximally entangled state. It is shown that the resistance to noise is not a good measure of nonlocality and we introduce some other possible measures. The nonmaximally entangled state turns out to be more robust also for these alternative measures. From these results it follows that two von Neumann measurements per party may be not optimal for detecting nonlocality. For d⫽3,4, we point out some connections between this inequality and distillability. Indeed, we demonstrate that any state violating it, with the optimal von Neumann settings, is distillable.
DOI: 10.1103/PhysRevA.65.052325 PACS number共s兲: 03.67.⫺a, 03.65.Ta
I. INTRODUCTION
Since the seminal work of Bell 关1兴, it is known that no local variable共LV兲theory关2兴can reproduce all the statistical results predicted by quantum mechanics for states of com- posite systems. In fact, it was proven that the correlations observed between two spin-12 particles in the singlet state violate some inequalities, called Bell inequalities, that any LV theory satisfies. This provided a possible definition of quantum nonlocality: a quantum state is said to be nonlocal when it violates a Bell inequality. More recently, it was shown that any pure state that is not separable关3兴, i.e., such that the parties cannot prepare it using only local operations and classical communication 共LOCC兲, violates a Bell in- equality 关4,5兴. Unfortunately, the distinction between local and nonlocal states is not as clear for density matrices. In- deed there exist mixed states that, despite being entangled, do not violate any Bell inequality 关6兴 共see, however, Ref.
关7兴兲. Consequently, there are two different ways of consider- ing quantum nonlocality: entanglement 共quantum nonlocal resources are required for preparing the state兲and violation of Bell inequalities.
The interest in entanglement has dramatically increased during the last two decades due to the fact that entanglement is the key ingredient in many of the recent quantum informa- tion applications. Many efforts have been devoted to quan- tify entanglement 共see, for instance, Refs. 关8,9兴兲 as a re- source, and nowadays bipartite pure-state entanglement is well understood. The maximally entangled state of a bipartite system,兩⌿典苸Cd丢Cd, reads
兩⌿典⫽ 1
冑
ddi兺
⫽⫺01 兩j j典, 共1兲where 兩j典 are the orthonormal bases in each subsystem.
Given an entangled state inCd丢Cd, it is important to know if it can be distilled, i.e., if N copies of it can be transformed by
LOCC into M copies of 兩⌿典. State distillability, or useful quantum correlations, offers an alternative way of analyzing quantum nonlocality. All bipartite entangled pure states can be reversibly transformed using LOCC into 兩⌿典 共in the so- called asymptotic regime兲. The ratio of the conversion is equal to the entropy of entanglement关8兴. For the mixed-state case, however, the picture is again not clear: indeed it is not known when a given entangled density matrix is distillable.
It is also not known how distillability properties of mixed states are connected to Bell inequality violation关10–13兴.
All these questions about nonlocality in mixed states are essentially solved for the simplest case of two two-level sys- tems, also called qubits. There, the Peres-Horodecki criterion of positivity of the partial transposition 关14,15兴 detects if a given state is separable or entangled. Furthermore, all two- qubit entangled states are distillable 关16兴. As far as the Bell inequality is concerned, the CHSH inequality 关17兴 plays a very important role 关18兴, and it is already known when a quantum state violates it关19兴. Its maximal violation is only obtained for the maximally entangled state of two spin-12 particles.
Recently an inequality 关20,21兴 has been found, which generalizes the CHSH inequality to systems of arbitrary di- mension, d, often referred to as qudits. This offers the oppor- tunity of testing some of the concepts seen above for this inequality. This is the scope of the present paper. We see that, surprisingly, the maximal violation of the inequality, under von Neumann measurements, is not obtained for the maxi- mally entangled state of two three-level systems or qutrits 共Sec. III兲. This leads us to analyze if the resistance to noise is a correct measure of nonlocality. By a simple example we see that it is not. Furthermore, we prove that any state vio- lating the inequality with the optimal settings is distillable, and the witness that comes from it 关11兴 is decomposable 共Sec. IV兲.
II. BELL INEQUALITY FOR QUTRITS
In this section we review the Bell inequality for qutrits obtained in Refs. 关20,21兴. The two parties, A and B, are al-
lowed to perform two different three-outcome measure- ments, A1 and A2 for A, and B1 and B2 for B. Denoting by P(Ai⫽Bj⫹k) the probability that the outcomes for parties A and B, measuring Ai and Bj, differ by k modulo d 共in this case d⫽3兲, one can consider the following Bell inequality:
I3⫽P共A1⫽B1兲⫹P共B1⫽A2⫹1兲⫹P共A2⫽B2兲⫹P共B2⫽A1兲
⫺P共A1⫽B1⫺1兲⫺P共B1⫽A2兲⫺P共A2⫽B2⫺1兲
⫺P共B2⫽A1⫺1兲⭐2. 共2兲
The authors of Refs.关20,21兴analyzed the violation of this inequality by the maximally entangled state 共1兲 of two qutrits. They consider the following settings: first the two parties apply a unitary operation on each subsystem with only nonzero terms in the diagonal equal to eia( j)for A and eib( j) for B, with j⫽0,1,2 and a,b⫽1,2. These unitary op- erations are denoted by U(ជa), where ជa
⬅关a(0),a(1),a(2)兴, for party A, and the same for B withinstead of. The values of these phases are
1共j兲⫽0, 2共j兲⫽
3 j , 1共j兲⫽
6 j , 2共j兲⫽⫺ 6 j,
共3兲
with j⫽0,1,2. In this scenario, the freedom in the choice of the measurement the parties apply is given by this first uni- tary transformation. Then, party A carries out a discrete Fou- rier transform, UFT关22兴, and B applies UFT* , and finally they measure in the initial basis兩j典. With these experimental set- tings, quantum mechanics predicts that I3(兩⌿典)⫽4(2)
⫹3)/9⯝2.8729. Numerical simulations 关20,23兴 show that this is the maximum value of I3 achieved starting from the maximally entangled state,兩⌿典. It is then conjectured that the described experimental settings are optimal for兩⌿典with this inequality.
It is possible to define a more absolute measure of nonlo- cality in the following way: the initial entangled state,兩⌿典, is mixed with some amount of noise, the resulting state being equal to
⫽兩⌿典具⌿兩⫹共1⫺兲19, 共4兲
when 0⭐⭐1. The entanglement in decreases with, so one can look for the maximal amount of noise, or minimum
, such that it is still not possible to build a LV model for the predicted probabilities. This measure of nonlocality, known as the resistance to noise, depends on the experimental set- tings, that is, on the number and the type of measurements each party can apply. The inequality 共2兲reproduces for Eq.
共1兲the same resistance to noise as it was found numerically in Ref.关23兴, with two von Neumann measurements on each
side. This means that the probabilities resulting from per- forming these measurements on the state 共4兲 admit a LV model when
0⭐⭐ 2
I3共兩⌿典)⯝0.6962. 共5兲
III. MAXIMAL VIOLATION OF THE INEQUALITY The authors of Refs. 关20,21兴 focused their attention onto the violation of this inequality for the maximally entangled state of two qutrits. However, it may happen that a larger value of I3 is found if we consider a different initial state.
For the experimental settings 共3兲, one can derive the Bell operator关24兴associated with this inequality. The joint prob- ability, P(Aa⫽j,Bb⫽k), of detecting result j in A, k in B, when Aa and Bb are measured and the initial state is 兩⌽典 苸C3丢C3, is given by
tr关⌸j丢⌸kV共ជa兲丢V共ជb兲兩⌽典具⌽兩V共ជa兲†丢V共ជb兲†兴
⫽tr关V共ជa兲†丢V共ជb兲†⌸j丢⌸kV共ជa兲丢V共ជb兲兩⌽典具⌽兩兴, 共6兲 where V(ជa)⬅UFTU(ជa) and V(ជb)⬅UFT*U(ជb). From this formula the Bell operator, B, such that
I3共兩⌽典)⫽tr共B兩⌽典具⌽兩兲⫽具B典⌽, 共7兲 is found, and it reads
B⫽
¨
0 0 0 0 2
) 0 0 0 2
0 0 0 0 0 2
) 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 2
) 0 2
) 0 0 0 0 0 0 0 2 )
0 2
) 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 2
) 0 0 0 0 0
2 0 0 0 2
) 0 0 0 0
©
.共8兲
A. ACI´N, T. DURT, N. GISIN, AND J. I. LATORRE PHYSICAL REVIEW A 65 052325
052325-2
The maximal violation of the inequality with these settings corresponds to the maximum eigenvalue of B, which is equal to 1⫹
冑
11/3⯝2.9149. Note that this value is a bit larger than the violation obtained for 兩⌿典. Indeed its corresponding ei- genvector is a nonmaximally entangled state of two qutrits, which reads,兩⌿mv典⫽ 1
冑
n共兩00典⫹␥兩11典⫹兩22典), 共9兲where ␥⫽(
冑
11⫺))/2⯝0.7923, and n⫽2⫹␥2 共the same results are obtained starting from the inequality in Ref.关21兴兲. All the details of the calculation are given in the Appendix.It is natural to ask about the optimality of the chosen set of measurements. We have performed a numerical search for this inequality, varying freely the two von Neumann mea- surements performed by each of the parties and the initial state. The maximal violation is indeed obtained by the con- figuration shown above. Moreover, in the Appendix we prove that these experimental settings give a local maximum for the largest eigenvalue of B.
The Bell inequality共2兲was also extended to arbitrary di- mension in Ref.关20兴. There it was shown that the combina- tion of joint probabilities
Id⫽ k
兺
⫽0 关d/2兴⫺1冉
1⫺d2k⫺1冊
关P共A1⫽B1⫹k兲⫹P共B1⫽A2⫹k⫹1兲⫹P共A2⫽B2⫹k兲⫹P共B2⫽A1⫹k兲⫺P共A1⫽B1⫺k⫺1兲
⫺P共B1⫽A2⫺k兲⫺P共A2⫽B2⫺k⫺1兲
⫺P共B2⫽A1⫺k⫺1兲兴⭐2, 共10兲 for LV models. However, this inequality can be violated if we consider the maximally entangled state 共1兲 and similar experimental settings with 1( j )⫽0, 2( j)⫽j/d, 1( j )
⫽j/(2d), and 2( j)⫽⫺j/(2d), where j⫽0,•••, d⫺1.
Indeed, this inequality reproduces the resistance to noise ob- tained numerically in Ref.关23兴, but now for two Bell multi-
ports关25兴on each side. Starting from Eq.共10兲we can derive the corresponding Bell operator and a larger violation is again found for partially entangled states of two qudits. Table I summarizes these results up to d⫽8. Note that the differ- ence between the violation for兩⌿典and兩⌿mv典 increases with increase in the dimension.
These results are quite surprising. Previous numerical work in Ref.关23兴showed that the resistance to noise for the maximally entangled state of two qutrits with two von Neu- mann measurements per party is indeed the one predicted by this Bell inequality共5兲. Nevertheless, there exists a nonmaxi- mally entangled state, 兩⌿mv典, whose quantum correlations are more resistant to noise, since its violation of Eq. 共2兲 is larger. Let us mention here that the parties, if they start with the maximally entangled state兩⌿典, are always able to prepare by LOCC and with probability one the state兩⌿mv典, and then to check the violation of the Bell inequality. This leads us to analyze more precisely whether the resistance to noise is a good measure of nonlocality共see also Ref.关26兴兲.
Take the following two-qudit state,
兩⌿2典⫽ 1
&共兩00典⫹兩11典), 共11兲 where d⫺2 of the Schmidt coefficients are zero. Now, con- sider the CHSH inequality 关17兴
具A1共B1⫹B2兲⫹A2共B1⫺B2兲典⭐2, 共12兲 where Ai and Bi, i⫽1,2, are measurements of two out- comes, labeled by ⫹1 and ⫺1. The maximum violation of this inequality attained by quantum states is 2& 关27兴. The following choice of measurements achieves this maximum for the state 共11兲:
A1⫹1⫽P0, A2⫹1⫽P/2, B1⫹1⫽P⫺/4, B2⫹1⫽P/4, 共13兲 where P is the projector onto the state 1/&(兩0典⫹ei兩1典), and Xi⫺1⫽1⫺Xi⫹1, with X⫽A,B and i⫽1,2. Since tr(Xi⫹1)
⫽1 and tr(Xi⫺1)⫽d⫺1, the contribution of the maximally mixed noise to Eq.共12兲for these settings is not zero. Indeed it is not difficult to see that in this case the resistance to noise for 兩⌿2典 is
⫽
1⫺
冉
d⫺d2冊
2&⫺
冉
d⫺d2冊
2, 共14兲which tends to zero when d→⬁. One can argue that Eq.共11兲 is not really a two-qudit state, but similar results can be obtained for states of full Schmidt number that are infinitesi- mally close to it. This example shows that the resistance to noise is not a good measure of nonlocality. Let us briefly explore here other alternative candidates.
TABLE I. Violation of the inequality 共10兲 for two qudits, Cd
丢Cd, up to d⫽8. The values obtained for the maximally entangled state共1兲and the maximal violation of the inequality corresponding to the largest eigenvalue of the Bell operator are shown.
Dimension Violation for
兩⌿典
Maximal violation 共for兩⌿mv典兲
Difference 共%兲
3 2.8729 2.9149 1.4591
4 2.8962 2.9727 2.6398
5 2.9105 3.0157 3.6133
6 2.9202 3.0497 4.4345
7 2.9272 3.0776 5.1411
8 2.9324 3.1013 5.7588
The first possibility consists of studying the resistance of entangled states when they are mixed with the state resulting from the tensor product of the reduced density matrices.
Then, given an entangled state 兩⌽典苸Cd丢Cd, we want to determine the minimum value of⬘ such that there is no LV model for the state
⬘⫽⬘兩⌽典具⌽兩⫹共1⫺⬘兲A丢B, 共15兲 where A⬅trB(兩⌽典具⌽兩) and similarly for B. This measure has the advantage of being equal to the resistance to noise for maximally entangled states, and avoids problems as the one previously discussed. A second possibility can be to consider mixtures of the initial entangled states with the closest sepa- rable one, AB. Similarly as for the relative entropy of en- tanglement 关28兴, one can choose the relative entropy as a measure of distance, S(,)⫽tr(ln⫺ln). Therefore
ABis defined as the state minimizing S(兩⌽典具⌽兩,) over the set of separable states 关29兴. Now, we look for the minimum
⬙ such that the state
⬙⫽⬙兩⌽典具⌽兩⫹共1⫺⬙兲AB 共16兲 does not admit a LV description. Remarkably, for the states 共1兲and共9兲, the settings defined above and the inequality共2兲, the three numbers for each state coincide, i.e., min⫽min⬘
⫽min⬙ . Thus, no change is observed by using these alterna- tive measures of nonlocality.
All these reasonings suggest that two von Neumann mea- surements are not optimal for detecting nonlocality in two- qutrit systems. A possible way out can be that more general measurements关positive operator valued measures共POVM兲兴 are required for having a larger violation for 兩⌿典. It seems more likely that more observables for each party, and a new Bell inequality, are needed. Another interesting scenario con- sists of the analysis of these quantum correlations under se- quences of measurements as in Ref. 关26兴. Indeed, states 共4兲 are entangled and distillable for 1/(N⫹1)⬍⭐1 关30兴. Fi- nally, it also follows from this result that it is not correct even for pure states to quantify entanglement by means of the violation of a particular Bell inequality关31兴.
IV. DISTILLABILITY AND BELL INEQUALITIES Violation of Bell inequalities is a possible manifestation of nonlocality, but, as it has been discussed in the Introduc- tion, there are other ways of thinking about nonlocality.
From the point of view of quantum information it is interest- ing to know if the correlations in a quantum state are useful, i.e., if the state is distillable. It is usually conjectured that Bell inequality violation implies the distillability of the state 关10兴. In this section we show that, for the experimental set- tings seen above, the corresponding entanglement witness is decomposable. This implies that any bound entangled state with positive partial transposition 关32兴 does not violate this inequality. Moreover, from our construction, it can be proven that any state violating the inequality is distillable.
From Eq. 共2兲it is possible to construct the Bell operator 关24兴, B, and from it the entanglement witness 关11兴, W⫽2
⫺B, such that tr(SW)⭓0 for all separable states S, and there exists an entangled state, , that is detected, i.e., tr(W)⬍0. There exists a class of entanglement witnesses, called decomposable, Wd, that can be written as
Wd⫽P⫹QTA, 共17兲 where P and Q are positive operators, and TAdenotes partial transposition with respect to subsystem A. Note that these entanglement witnesses are not able to detect entangled states with positive partial transposition, since ifTA⭓0 we have tr(W)⫽tr(P)⫹tr(TAQ)⭓0. Thus, they are not very useful for checking the separability of a given state共see, for instance, Refs. 关15,33兴兲, since they do not provide more in- formation than the partial transposition operation. In the fol- lowing lines we show that the witness coming from Eq.共2兲is decomposable.
Our aim is to prove that there exist some positive opera- tors, P and Q satisfying Eq.共17兲. Note that the role of Q is to detect the entangled state, so we will choose an operator Q maximizing tr(兩⌿mv典具⌿mv兩QTA)⫽tr(兩⌿mv典具⌿mv兩TAQ). We will take then Q proportional to the projector onto the space of negative eigenvalues of 兩⌿mv典具⌿mv兩TA, the state that gives the maximal violation of the inequality. If we work in the Schmidt basis of this state, it is easy to see that Q is proportional to the projector onto the antisymmetric space of two qutrits,
Pa⫽1
2
兺
j k兺
⫽j共兩jk典⫺兩k j典)共具jk兩⫺具k j兩兲. 共18兲Our guess then for the decomposition 共17兲of W is
W⫽P⫹k PaTA, 共19兲 where k is a positive number to be determined. Now, we look for a value of this constant such that P⫽W⫺k PaTA⭓0, or equivalently, its minimum eigenvalue is positive. In Fig. 1 we have represented the minimum of these eigenvalues as a function of the constant k. There exists a range of k where P is positive, which means that the witness is decomposable 共for instance, taking k⫽1.2兲.
This decomposition gives us an insight into the nonlocal properties of these states violating the inequality共2兲with the optimal settings described above. First, all the states with positive partial transposition do not violate this inequality.
Moreover, if a given stateis detected by the corresponding witness, tr(W)⬍0, it follows that tr(PaTA)⬍0, and since PaTA⫽1/2(1⫺d兩⌿典具⌿兩), we have
具⌿兩兩⌿典⬎1d, 共20兲
which means that the state is distillable关30兴. Thus, for these settings, Bell inequality violation implies state distillability 关13兴.
A. ACI´N, T. DURT, N. GISIN, AND J. I. LATORRE PHYSICAL REVIEW A 65 052325
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It would be nice if the recipe for finding this decomposi- tion was general and if it worked for other choices of the experimental settings, or higher dimension. Unfortunately this is not the case. Indeed we have seen numerically that the same procedure does not work for other settings violating 共2兲, i.e., such that there exists a state that is detected by the corresponding witness or Bell operator. We have also consid- ered higher dimensional systems, with the corresponding Bell inequality 共10兲and optimal settings. A similar decom- position is found for d⫽4 but our method fails for d⫽5,6.
Looking at the variation of the Bell operator spectrum 共for the optimal settings兲, when d increases, we can somehow understand why this procedure does not work anymore. For d⫽3, (d⫽4), the maximum eigenvalue of B is ⯝2.9149 共⯝2.9727兲, while for the maximally entangled state, having the same Schmidt basis, we have ⯝2.8729 共⯝2.8962兲. This may mean that this maximally entangled state is quite close to the region of maximal violation, and then Eq. 共20兲holds for those states violating the inequality. For higher dimension the difference between these two values increases共3.0517 vs 2.9105 for d⫽5兲, what can explain why the decomposition is not possible? Let us mention here that in the case of two- qubits we are able to find P and Q for any witness associated to the CHSH inequality following a similar approach 关34兴 共actually, it is a known result that any two-qubit entangle- ment witness is decomposable 关15兴兲. Whether a better gen- eralization of the CHSH inequality exists also satisfying this construction is an interesting open question.
V. CONCLUDING REMARKS
In this work we analyzed different manifestations of quantum nonlocality in two-qutrit systems. The starting point was the Bell inequality introduced in Refs.关20,21兴.
First, we observed that the largest violation of the inequal- ity over all possible von Neumann measurements and initial states is not obtained for the maximally entangled state of two qutrits. We also proved that the resistance to noise is not a good measure of nonlocality, and we proposed some simple alternatives. However, even for these measures, the maximal resistance seems not to be given by the maximally entangled state共further research in this direction is needed兲. Our results suggest that two von Neumann measurements per site do not detect two-qutrit nonlocality in an optimal way. Notice that the results in Table I may imply that indeed two measure- ments on each side become less efficient for the detection of nonlocality when the dimension increases. It is not excluded that POVMs would give a larger Bell inequality violation for the two-qutrit maximally entangled state. However, it seems more likely that more observables on each side are needed.
In this sense, we still lack a good generalization of the CHSH inequality to qutrits.
We also related the violation of this inequality to other manifestations of nonlocality. For the optimal settings, we demonstrated that the corresponding witness is decompos- able by explicitly deriving its decomposition共17兲in terms of P and Q. Moreover, our construction sheds light on the dis- tillability properties of those states violating the inequality.
In fact, bound entangled states do not violate the inequality for the optimal settings 共see also Ref.关12兴兲.
Our last point is more general and concerns the character- ization of nonlocality. As it has been mentioned, the resis- tance to noise has proven to be an incorrect criterion for the analysis of nonlocality. Is a single number enough for de- scribing all the nonlocal features of quantum states for bipar- tite systems of dimension greater than qubits, even for the FIG. 1. Variation of the mini- mum eigenvalue of P in Eq.共19兲 with k.
pure-state case? Or, as it happens for other questions related to entanglement, are more parameters needed?
ACKNOWLEDGMENTS
We thank Marek Zukowski for many useful comments.
A.A. and N.G. acknowledge financial support from the Swiss FNRS and OFES, within the European project EQUIP, and the Swiss NCCR ‘‘Quantum Photonics.’’ A.A. also thanks the Spanish MEC and ESF-QIT for funding. J.I.L. acknowl- edges financial support from CICYT共AEN99-0766兲, CIRIT 共1999SGR-00097兲, and EQUIP共IST-1999-11053兲.
APPENDIX: MAXIMIZATION OF THE BELL INEQUALITY VIOLATION BY ENTANGLED QUTRITS
Consider the situation that was described in Sec. II. Two three-dimensional subsystems A and B are prepared in an arbitrary entangled state兩⌽典苸C3丢C3. Then, the two parties apply a unitary operation on each subsystem with only non- zero terms in the diagonal equal to eia( j) for A and eib( j) for B, with j⫽0,1,2 and a,b⫽1,2. The setting of the six phasesa( j ) andb( j˜) ( j⫽0,1,2) defines an experimental configuration. Later, A carries out a discrete Fourier trans- form, UFT, and B applies UFT* . Finally, at the output of such devices, the states are detected in the local bases 兩kA典 and
兩lB典. When the state is prepared initially in the pure state 兩⌽典⫽兺j, j2˜⫽0␣j, j˜兩j j˜典 the probability of a joint detection in the kth detector in A and in the lth detector in B is equal to
P共k,l兲⫽1
9
冏
j, j兺
˜2⫽0 exp冋
i冉
a共j兲⫹b共˜j兲⫹共jk⫺˜lj 兲22冊册
␣j, j˜冏
2.共A1兲 It was shown in Ref. 关23兴 by numerical methods that when the state is maximally entangled 关Eq. 共1兲兴, the maximal re- sistance of nonlocality against noise, when all the possible phases a( j ),b( j ) are considered 共with a,b⫽1,2 and j
⫽0,1,2兲corresponds to the optimal phase settings共3兲. Let us now consider the associated Bell inequality 共2兲. When Eq. 共3兲is satisfied and we let vary the state兩⌽典苸C3
丢C3, the violation of this inequality is maximal when the state 兩⌽典 is, up to normalization, 兩00典⫹␥兩11典⫹兩22典, with
␥⫽(
冑
11⫺))/2⯝0.7923. The violation is equal to 1⫹
冑
11/3⬇2.914 85.Indeed, take the general state兩⌽典⫽兺2j, j˜⫽0␣j, j˜兩j j˜典共where 兺j, j2˜⫽0兩␣j, j˜兩2⫽1 by normalization兲. Then, Eq.共2兲can be re- written as follows:
I3共兩⌽典)⫽1
9
兺
j, j˜,m,m˜⫽0 2
␣m,m*˜␣j, j˜k
兺
⫽0 2exp
再
i23关k共j⫺m兲⫺k共˜j⫺m˜兲兴冎再
exp兵i关1共j兲⫺1共m兲⫹1共˜j兲⫺1共m˜兲兴其⫻
冋
1⫺exp冉
⫺i23共j⫺m兲冊册
⫹exp兵i关2共j兲⫺2共m兲⫹1共˜j兲⫺1共m˜兲兴其冋
exp冉
⫺i23共˜j⫺m˜兲⫺1册
⫹exp兵i关2共j兲⫺2共m兲⫹2共˜j兲⫺2共m˜兲兴其
冋
1⫺exp冉
⫺i23共j⫺m兲冊册
⫹exp兵关1共j兲⫺1共m兲⫹2共˜j兲⫺2共m˜兲兴其
冋
1⫺exp冉
i23共˜j⫺m˜兲册冎
. 共A2兲I3(兩⌽典) can be expressed as具⌽兩B兩⌽典 where B is the共self- adjoint兲 Bell operator. The maximal value of I3 is thus reached when兩⌽典is the eigenstate associated with the maxi- mal eigenvalue of B,兩⌿mv典. We must now determine what this eigenvalue is. The problem is considerably simplified if we note that 兺k⫽0
2 exp (i2/3k( p⫺q)⫽3␦pq
(3), where ␦pq (3)
⫽1 when p⫽q modulo 3 and 0 otherwise. This means that B can be decomposed into the sum of three operators that are decoupled and act individually inside the subspaces spanned by the vectors 兵兩00典兩11典,兩22典其, 兵兩01典,兩12典,兩20典其, and 兵兩02典,兩10典,兩21典其, respectively. Inside the subspace spanned by the vectors兵兩00典,兩11典,兩22典其, j⫺m⫽˜j⫺m˜ and the reduced Bell operator obeys the following equation:
共Bred1兲m j⫽2
3
再
exp冉
i23 共j⫺4m兲冊
⫺exp冉
23 共⫺2兲共4j⫺m兲冊
⫹exp
冉
i23 共⫺1兲共4j⫺m兲冊
⫺exp
冉
i23 3共j⫺4m兲冊
, 共A3兲and in matricial notation,
A. ACI´N, T. DURT, N. GISIN, AND J. I. LATORRE PHYSICAL REVIEW A 65 052325
052325-6
Bred1⫽2
3
冉
)03 ))0 )30冊
. 共A4兲Inside the two other subspaces, we obtain in a similar fashion the following expression for the reduced Bell operator:
Bred2⫽Bred3⫽2
3
冉
)00 )00 000冊
. 共A5兲The problem consists now of determining the maximal ei- genvalues of these 3⫻3 matrices. One can check directly that in this matrical notation兩⌿典is not an eigenstate of Bred1 so that it does not certainly maximize the violation of the Bell inequality. The eigenvalues of Bred1 are equal to⫺2, 1
⫺
冑
11/3, and 1⫹冑
11/3, while for Bred2,3we have⫺2/), 0, and 2/), so that the maximal violation is equal to 1⫹
冑
11/3⬇2.914 85. It is easy to check that the corresponding eigenvector is, up to normalization,兩00典⫹␥兩11典⫹兩22典, with␥⫽(
冑
11⫺))/2⯝0.7923.Note that at first sight it could seem strange that the maxi- mally entangled state does not maximize the violation of the inequality共2兲, because it seems that the states兩00典,兩11典, and 兩22典are indistinguishable in our approach. The discrete Fou- rier transforms are well known for their cyclic properties, and in this inequality all the detectors are treated on an equal footing 共the Bell operator contains cyclic summations of probabilities of coincident firings兲. Nevertheless, if we con- sider Eq. 共A2兲, we can notice that the matrix coefficient 具mm˜兩B兩j j˜典 contains expressions of the type exp兵i关a(j)
⫺a(m)⫹b(j˜)⫺b(m˜ )兴其wherea( j ) and b( j˜) are locally adjustable phases. Due to the presence of factors of this type, the cyclic invariance is broken 共in the sense that when j
⫺m⫽j⬘⫺m⬘ modulo 3 and j˜⫺m˜⫽˜j⬘⫺m˜⬘ modulo 3, it is not necessarily true that具mm˜兩B兩j j˜典⫽具m⬘m˜⬘兩B兩j⬘˜j⬘典). Note that when the phase settings are optimal, they depend lin- early on the indices j, m, j˜, m˜ according to Eq. 共3兲 and 具mm˜兩B兩j兩˜j典 depends on j, m, j˜, m˜ only through the combi-
nations j⫺m and j˜⫺m˜ . Thus, the matrix coefficients 具mm˜兩B兩j j˜典and具(m⫹␦)(m˜⫹˜ )␦ 兩B兩( j⫹␦)( j˜⫹˜ )␦ 典are equal.
This explains why Bred2⫽Bred3 and also why the states兩00典 and兩22典appear symmetrically in the matrix Bred1. Neverthe- less, the cyclic invariance is still broken共in the sense made precise above兲, which explains why Bred1 is singularized relatively to Bred2 and Bred3, as 兩11典 relatively to 兩00典 and 兩22典.
Finally, we prove that we must not expect a larger viola- tion of the inequality 共2兲 if we modify the phase settings defined in Eq. 共3兲. Indeed, let us varyជ1, keeping the other phases fixed,
1共0兲⫽0, 1共1兲⫽␣, 1共2兲⫽,
2共j兲⫽
3 j, 1共j兲⫽
6 j, 2共j兲⫽⫺
6 j . 共A6兲 Then,
Bvariedred ⫽1
3
冉
)03 ))0 )30冊
⫹1
3
冉
)3ee0⫺⫺ii␣ ))ei0共e␣⫺兲i␣ )e3ei共⫺␣兲0i冊
.共A7兲 It is easy to check that the two matrices that appear in Eq.
共A7兲have the same eigenvalue equation and thus the same spectrum, which is ⫺1, (1⫺
冑
11/3)/2, and (1⫹冑
11/3)/2.The maximal eigenvalue of these matrices is also equal to their norm in the present case, so that the norm of Bvariedred1 , and thus its maximal eigenvalue, is certainly not larger than 1
⫹
冑
11/3. Entirely similar results can be obtained whenជ2,ជ1, or ជ2 are varied. This proves that the phase settings defined in Eq.共3兲maximize locally共in the space of all pos- sible phase settings兲the violation of the Bell inequality共2兲.
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