There is no anomaly in the nonlocality of two entangled qutrits
aa r X i v : . [ qu a n t - ph ] S e p There is no anomaly in the nonlocality of two entangled qutrits
E. A. Fonseca and Fernando Parisio ∗ Departamento de F´ısica, Universidade Federal de Pernambuco, 50670-901, Recife, Pernambuco, Brazil
There is no doubt about the fact that entanglement and nonlocality are distinct resources. It isacknowledged that a clear illustration of this point is the difference between maximally entangledstates and states that maximally violate a Bell inequality. We give strong evidence that this anomalymay be an artefact of the measures that have been used to quantify nonlocality. By reasoning thatthe numeric value of a Bell function is akin to a witness rather than a quantifier, we define a measureof nonlocality and show that, for pairs of qutrits and of four-level systems, maximal entanglementdoes correspond to maximal nonlocality in the same scenario that gave rise to the discrepancy.
PACS numbers: 03.65.Ud, 03.65.Ta, 03.67.Mn
Entanglement [1, 2] is behind some of the most per-plexing physical effects ever observed. In spite of this,it alone, may be regarded as a purely mathematical con-cept: the failure of a vector in a Hilbert space H to befactorized as a single product of vectors in spaces H i , thattogether form H (= N i H i ) [3]. For mixed states there isa corresponding definition in terms of convex sums of ex-plicitly separable density operators [4]. Nonlocality [5–7],in contrast, also refers to the experimental scheme thatis used to investigate the entangled state and, thus, toangles of Stern-Gerlach apparatuses or to the spatial dis-position of beam splitters, for instance. That is to say,entanglement happens in Hilbert spaces while nonlocal-ity manifests itself in our ordinary (3+1)D space (seehowever [8]). Therefore, from a physical perspective, tocarefully quantify nonlocality is as important as to seekentanglement measures. A faulty estimation of the ex-tent of nonlocality embodied by a physical situation maylead to deceptive conclusions.In an essay in honor of A. Shimony, Gisin provides alist of questions on Bell inequalities [9]. The one closelyrelated to our goal in this work is “Why are almost allknown Bell inequalities for more than 2 outcomes max-imally violated by states that are not maximally entan-gled?” (for exceptions see [10, 11]). This fact originallyreported in [12] and referred to as an anomaly [13] of non-locality, has received a great deal of attention [10, 14–18].To investigate this issue, we begin by addressing the tacitassociation often made between states that maximally vi-olate a Bell inequality and maximally nonlocal states.Given a Bell function I , let us generically denote theassociated inequality by I ≤ ξ , (1) ξ representing the bound imposed by local causality (weuse this terminology instead of “local realism” [19]).Then, if a state satisfies (1) for all possible settings ofthe measurement apparatus, it is local with respect tothe inequality. Otherwise, the state is said to be nonlo-cal. Recall that the Werner matrices [4], which are local ∗ [email protected] with respect to the CHSH inequality, are nonlocal if morecomplex measurements are considered for dimension 5 orhigher [20] . A complete definition of locality must beexhaustive: a state is said to be local, with no furtherqualifiers, if it satisfies all relevant Bell inequalities.So, the notion of Bell nonlocality is quite neat. Thesensitive question is about ordering . What should oneimply by asserting that state ρ is more nonlocal thanstate σ ? A common answer is that ρ is more nonlocalthan σ if I max ( ρ ) is larger than I max ( σ ). The maximabeing determined by scanning all possible settings. Al-though it is known that for any Bell function one can findanother, equivalent function that arbitrarily increases thenumerical value of the maximal violation [6], it is ac-knowledged that carefully normalised Bell inequalitiesmay provide objective figures to quantify nonlocality. Inwhat follows we reason against this view.Insightful alternatives have been put forward in thelast two decades. The tolerance of nonclassical correla-tions against noise has been considered as an operationalmeasure of nonlocality [21, 22], but this approach is notconsensual [12]. In [23], it is shown that optimal Belltests occur for states that are neither maximally entan-gled nor maximally violating. The (statistically) optimalstate found by the authors is the most suitable to un-veil nonlocality, given that the experimentalist can onlyperform a finite number N of realisations, as is alwaysthe case. However, at least in principle, one should beallowed to think of the limit N → ∞ , as we do withmany other concepts in quantum theory. In this limitall nonlocal states can be safely devised. In a differentframework, the communication cost for a local model toreproduce the quantum correlations has also been used asa task-based quantifier of nonlocality [24–26]. However,different tasks usually induce different state orderings [6].One can also ask what is the minimal detector effi-ciency required to evidence nonlocality for a given state.The interesting fact is that the efficiency required for themaximally entangled state of two qubits is larger thanthat of some partially entangled states [27, 28] for theCHSH inequality [29]. This might be considered an inde-pendent instance of the nonlocality anomaly. About thispoint, let us appeal to an analogy with entanglement.There is little doubt that the GHZ state is the maximallyentangled state of 3 qubits, in particular, that it is moreentangled than the W state. However, in measuring the3 particles on | GHZ i , if the 3 detectors have an efficiencyof p <
1, then the probability to witness the entangledcharacter of | GHZ i is only p , since, whenever a sin-gle detector fails one only sees a maximally mixed, non-entangled ensemble for the remaining particles. In con-trast, the probability to perceive entanglement in | W i ,with the same detectors, is p + 3(1 − p ) p × /
3, since,by losing any of the particles we still have a noisy EPRpair (with a fidelity of 2 / | i + | i ) / √ The fact that a state is maximally nonlocal doesnot necessarily mean that its nonlocality is either the eas-iest to detect or the most resistant against imperfections.
Recently, two measures have been experimentally im-plemented in [30]: the first one is based on how far isthe state from the local polytope, and, the second is alsorelated to the amount of communication needed to estab-lish correlations. In [31] a nonlocality quantifier has beendefined, such that in certain scenarios it is inversely re-lated to concurrence. Some other proposals to quantifynonlocality can be found in the literature, focusing onmultipartite systems [32, 33] and presenting nonlocalityas a concept derivable from a notion of “irreality” [34].Common to these previous works is the fact that thedifferent figures of merit associated (or identified) to non-locality attain their maxima for non-maximally entangledstates. This does not violate any logical necessity, but,one should not refrain from a critical assessment of this,arguably, counterintuitive finding. In this work we de-fine a measure of nonlocality which indicates that theanomaly that appears to exist for two entangled three-and four-level systems may well be an artefact of theprevious definitions. Our suggestion seems to have deep,though simply definable, physical and statistical mean-ings.A tenable reasoning about the quantification of nonlo-cality is that some clue might come from nonlocal hid-den variable (NLHV) models capable of reproducing thequantum correlations. For example, one could say thata state is more nonlocal than another if the underly-ing NLHV model violates local causality in different de-grees for these different states. This, however, cannotbe inferred from these models in any obvious way. Thedistance between subsystems does not enter in the Bellfunctions, after all. As a first illustration we refer tothe model developed by Bell in his seminal paper [35].It is simply assumed that “the results of measurementswith one magnet now depend on the setting of the dis-tant magnet [...]”. The mutual influence between thesubsystems being instantaneous, no matter the numeri-cal value assumed by the Bell function. In a more gen-eral picture, consider the NLHV theory par excellence ,Bohmian mechanics [36]. The so-called quantum poten- tial does not react faster or slower, for different states,under a measurement on one of the subsystems. In par-ticular, this holds for two entangled rotors of spin-1/2[37]. The fact that, for a given Bell inequality, some ofthese instantaneous interactions are related to non vio-lating states must be understood in the light of the gen-eralised Gisin’s theorem: all bipartite N × N entangledstates violate some Bell inequality [38, 40]. The action ata distance appears to be equally spooky for all nonlocalstates within these NLHV models.Even for theories relying on finite (superluminal) sig-nalling speed, the relation I max ( ρ ) > I max ( σ ) > ξ doesnot necessarily imply that v ρ > v σ > c , where v is thesignal velocity for each state and c is the speed of light.This reasoning suggests that all violating states for a par-ticular setting are equally nonlocal, that the essential in-formation provided by a Bell inequality is of a seeminglyBoolean nature, a state being either local or nonlocalwith respect to those settings, without gradations.This apparently all-or-nothing picture, however, doesnot lead to a dead-end. On the contrary, it points to aconceptually simple solution.Given a state and a specific Bell inequality, the mostexhaustive experiment one can go through is to inves-tigate local causality for all settings. For simplicity ofpresentation we refer to inequalities associated to non-degenerate von Neumann measurements [39]. Based onour previous discussion, we are lead to state that ρ ismore nonlocal than σ if the former violates the inequal-ity, by any extent , for a larger amount of setting param-eters than the latter. This statement can be cast in verysimple statistical terms: ρ is more nonlocal than σ if, foran unbiased random choice of settings, the probability toobtain a violation is larger for ρ .To formalise this idea, we define the space X = { x , . . . , x n } of all possible parameters determining thesettings for a given (preferably tight) Bell inequality. Fora particular state ρ , let Γ ρ ⊂ X be the set of points thatlead to violation and V ( ρ ) be proportional to the vol-ume of Γ ρ . We say that if V ( ρ ) > V ( σ ), then ρ is morenonlocal than σ , with V ( ρ ) ≡ N Z Γ ρ d n x , (2)where N is a normalisation constant. The measure ofintegration is such that every setting (set of parame-ters) has equal weight. For instance, one setting cor-responding to a direction in space demands two param-eters one polar ( ϕ ) and one azimutal ( θ ) angle, leadingto d x = dΩ = sin θ d θ d ϕ . If, on the other hand, the set-tings are defined by the plane angles of n polarisers, e.g.,then we simply have d n x = d ϕ . . . d ϕ n . We call V the volume of violation . Hereafter we focus on the importantcase where the settings are such that X is a boundedset. We remark that the numeric calculations needed todetermine the volume of violation are the paradigmaticproblem for which Monte Carlo methods are intended[41]. The above definition has no relation to the volumeof the set of separable states defined in [43], the volumeof violation is an integration over the experimental pa-rameters that can be varied within the context of a givenBell inequality.A more fundamental definition should not invoke a par-ticular Bell inequality, but rather, the set of conditionalprobabilities P ( ab | xy ) (also called behaviors), where a and b are outputs and x and y are inputs, see [42]. Thisamounts to an integration similar to (2), but over theexterior, no signalling part of the local polytope, which,however is an exponentially hard computational problem.In addition, in this first account we intend to address thevery same situation that gave rise to the anomaly. In [42],some criteria are given that reasonable measures shouldfulfil in terms of operations in the space of behaviours.In this more general picture one interesting question iswhether V satisfies those criteria.As an initial test, we consider the CHSH inequal-ity [29] for two entangled qubits in pure and mixedstates. In this case the Bell function depends on fourunit vectors: I CHSH (ˆ a, ˆ b, ˆ c, ˆ d ) = | E (ˆ a, ˆ b ) − E (ˆ a, ˆ d ) | + E (ˆ c, ˆ d ) + E (ˆ c, ˆ b ), with E being a correlation function de-fined for a pair of directions. We can write it more ex-plicitly in terms of eight angular parameters, I CHSH = I ( θ a , ϕ a , θ b , ϕ b , θ c , ϕ c , θ d , ϕ d , ), X corresponding to thecartesian product of four unit spheres, yielding d n x =dΩ a dΩ b dΩ c dΩ d , with dΩ i = sin θ i d θ i d ϕ i . We found thatthe maximally entangled state maximises, both, I and V .In Fig. 1 (a) we show these quantities along with the en-tropy of entanglement for the family of pure states | ψ α i = α | i + p − α | i , (3)as functions of α . The volume V is rather sensitive tovariations of α , presenting the steepest descent from itsmaximum at α = 1 / √
2. In Fig. 1 (b) we plot the con-currence C ( α ) [44, 45] and V ( α ) ≡ V ( ρ α ) of the noisystate ρ α = (1 − F ) | ψ α ih ψ α | + F I /
4, where I is the 4 × F is the noise fraction. The volumeof violation is more fragile against noise than entangle-ment. Around a noise fraction of F ≈ .
3, nonlocality,as rendered by V , completely disappears. We also ap- α α FIG. 1: (color online) In (a) we show the entropy ofentanglement (circles), I max (triangles), and V (squares) versus α for the pure state | ψ α i . In (b) noiseis considered and the plots correspond to V ( α )compared to the concurrence C ( α ) for different valuesof the noise fraction F . plied our measure to the first Bell [35] inequality (CHSHwith ˆ c = ˆ d ) and, importantly, to the inequality 3322 [46](inequivalent to CHSH), with similar results. So far, thevolume of violation gives no sensible new information incomparison to the maximum of the Bell functions, yet, itis consistent with our expectations on what should be anonlocality measure in the safe terrain of two entangledqubits. This agreement between V and I max ceases tohappen when two higher dimensional systems are con-sidered, even in the pure case.Now we consider two entangled qutrits, i. e., acomposite system with Hilbert space H = H ⊗ H ,dim H =dim H =3. Let {| i i , | i i , | i i } be an orthonor-mal basis in H i ( i = 1 ,
2) and consider the three-outcomeobservables A a , for system 1, and B b for system 2 ( a, b =1 , I = P ( A = B ) + P ( B = A + 1) + P ( A = B )+ P ( B = A ) − P ( A = B − − P ( B = A ) − P ( B = A − ≤ , (4)where the arguments of the probabilities above are takenmodulo 3, e.g., P ( B = A + 1) = P ( B = 0 , A =1) + P ( B = 1 , A = 2) + P ( B = 2 , A = 0). As in [12]let us focus on the family of pure states | Ψ γ i = 1 p γ ( | i + γ | i + | i ) . (5)The maximal entropy of entanglement is, naturally, givenby γ = 1, while it was shown that (4) is maximally vio-lated by the state with γ ≡ ˜ γ ≈ .
792 [12]. Entanglementand Bell nonlocality are, indeed, physically distinct, but,the fact is that the former constitutes the sole source ofthe latter (we exclusively refer to Bell nonlocality [48]).Thus, it is not unreasonable to expect that the maximashould coincide.Let us apply measure (2) to this problem. Note care-fully that general Stern-Gerlach-type measurements ona pair of spin-1 particles only demand eight parameters.However, these measurements do not reveal the wholerichness of the Hilbert space [21]. For this reason, inorder to calculate V ( γ ) ≡ V ( | Ψ γ ih Ψ γ | ) one must per-form an integration in a twelve-dimensional space, as wewill see. General unitary operations are achievable in thelaboratory via multiport beam splitters [49–51]. In thisoptical context the whole space of parameters can be vis-ited by varying the reflectivity of beam splitters and theangle of phase shifters, for instance. From the family ofstates (5), with these linear optical elements, we can get: | Ψ ′ i = 13 X j,k,l =0 α j e i [ φ a ( j )+ ϕ b ( j )] e i π j ( k + l ) | k l i , (6)with a, b = 1 ,
2, and α = α = 1, α = γ . The optimalparameters for violations of (4) by the maximally entan-gled state have been determined in [47, 52] and reads γ γ FIG. 2: (color online) Entropy of entanglement (circles), I max (triangles), and V (squares) as functions of γ forstate (5). All quantities are normalised such that theirmaximal value is 1. The inset shows a zoom in of theregion marked by the rectangle in dashed lines. φ ( j ) = 0, φ ( j ) = πj/ ϕ ( j ) = πj/ − ϕ ( j ). Thepuzzling situation arises when one sees that the maximalviolation has a peak at γ = ˜ γ .In Fig. 2 we compare our numeric calculations of V ( γ )to the normalised entropy of entanglement E and to themaximum of I . The maxima of E and V coincide exactlyat γ = 1, as can be seen in the inset, while I max attainsits maximum at γ = ˜ γ . This shows that the anomaly inthe nonlocality of two entangled qutrits does not exist,if one adopts the volume of violation as the measure ofnonlocality.It is easy to understand what is going on. Although | Ψ ˜ γ i presents a more pronounced maximum of I max incomparison to | Ψ i , the nonlocality of the former is lessrobust, for, as we get farther away from the optimal set-ting in X , I (Ψ ˜ γ ) falls off faster than I (Ψ ). This effecton the volume of violation is clearly illustrated in Fig.3, where two-dimensional sections [ φ (0) − ϕ (2)] of Γare shown for | Ψ i (a) and for | Ψ ˜ γ i (b). The other pa-rameters are set as φ (0) = φ (1) = πj/ ϕ ( j ) = 0,the remaining angles keeping the optimal values. In thisparticular example the violation area for γ = 1 is about14 % larger than that for γ = ˜ γ . The scales are identicalin both figures. Finally, to be sure that this conciliationbetween the maxima of entanglement and nonlocality isnot an unlikely coincidence, we addressed the problem oftwo four-dimensional Hilbert spaces. We considered thefollowing family of entangled states | Ψ λ ,λ i = 1Λ ( | i + λ | i + λ | i + | i ) , (7)with Λ = p λ + λ . The CGLMP inequality is max-imally violated by a state that is not maximally entan-gled, given by λ = λ ≈ . I ≈ . φ (0) − ϕ (2) of the12-dimensional space X . Some parameters were setaway from the optimal values. The area of violation for γ = 1 (a) is 14% larger than that for γ = 0 .
792 (b).We surveyed the volume of violation associated to I inthe region ( λ , λ ) ∈ [0 . , . × [0 . , . V is maximal for λ = λ = 1 among all investigatedstates. In particular, the ratio of the volumes V of themaximally entangled and maximally violating states isaround 1 .
24. Exhaustive numerical investigations in inthe whole Hilbert spaces in the spirit of [53] would bevery welcome. However, these demand a large amount ofcomputational resources.We argue that, given a state, a Bell inequality, and a particular setting, there should be no gradations of nonlo-cality, the inequality functioning as a witness. However,by “tracing over the settings”, attributing equal weightto all those that violate the inequality and weight zero tothose that do not lead to violations, we showed that it ispossible to quantify Bell nonlocality in a consistent way.In particular, within the context of our proposal, there isno discrepancy between maximally entangled and maxi-mally nonlocal states, at least for entangled qutrits andalso for systems composed of two four-level subsystems.In this work the normalisation constant N in Eq. (2)played no important role. We simply set it such that0 ≤ V ≤
1, with V = 1 for the maximally nonlocal purestate. We remark that there is a more absolute definition,which, however, would make the presented results a littlecumbersome to analyse. This definition is N = (vol. of X ), leading to V ( ρ ) = (vol. of Γ ρ ) / (vol. of X ). In thisway the volumes of violation associated to the same statebut different inequalities can be numerically compared.An interesting point, that is presently under consid-eration, is the possibility to compare the nonlocality ofthe maximally entangled state as the dimension of theHilbert space varies. In [21] it is stated that violations inthe principle of locality are stronger for two qunits thanfor two qubits. This conclusion was based on resistanceto noise and, thus, an interesting question is wether ornot we can reach a similar conclusion by employing thevolume of violation.Another question that has gained relevance is whyquantum mechanics presents weaker nonlocality than“probability boxes”, like PR-boxes [54]? Under the lightof our results, the statement implied in the questionshould be reassessed, since it is based solely on the nu-merical value associated to the violation for a particularsetting. It is an exciting perspective to check whether ornot the volume of violation of the “super singlet” pre-sented in [54], in fact supports a stronger nonlocality.To obtain the results described after Eq. (7) an inte-gration (split in a cluster with 20 cores) in a 16-D spacetook a couple of days. We, thus, finish with the hopethat analytical properties of V , Eq. (2), can be derived,helping to reduce the computational effort to calculatethe volume of violation involving states in Hilbert spaces of higher dimensions. ACKNOWLEDGMENTS
We thank R. M. Angelo for his suggestions on severalaspects of this work. Financial support from ConselhoNacional de Desenvolvimento Cient´ıfico e Tecnol´ogico(CNPq) via the Instituto Nacional de Ciˆencia e Tecnolo-gia - Informa¸c˜ao Quˆantica (INCT-IQ), Coordena¸c˜ao deAperfei¸coamento de Pessoal de N´ıvel Superior (CAPES),and Funda¸c˜ao de Amparo `a Ciˆencia e Tecnologia do Es-tado de Pernambuco (FACEPE) is acknowledged. [1] M. B. Plenio and S. Virmani, Quant. Inf. Comput. , 1(2007). A version is available at arXiv:quant-ph/0504163.[2] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki,Rev. Mod. Phys. , 865 (2009).[3] By adopting this view one should consistently agree thattask-based entanglement measures, although insightful,quantify some other resource, closely related to entangle-ment, but not identical to it.[4] R. F. Werner, Phys. Rev. A , 4277 (1989).[5] N. D. Mermin, Rev. Mod. Phys. , 803 (1993).[6] N. Brunner et al, Rev. Mod. Phys. , 419 (2014).[7] S. Popescu, Nature Phys. , 264 (2014).[8] F. J. Tipler, Proc. Nat. Accad. Sci. , 11281 (2014).[9] In essays in honour of Abner Shimony, Eds Wayne C.Myrvold and Joy Christian, The Western Ontario Se-ries in Philosophy of Science, pp 125-140, Springer 2009(arXiv:quant-ph/0702021v2).[10] S.-W. Lee and D. Jaksch, Phys. Rev. A , 010103(R)(2009).[11] J. Lim et al, N. J. Phys. , 103012 (2010).[12] A. Ac´ın, T. Durt, N. Gisin, and J. I. Latorre, Phys. Rev.A , 052325 (2002).[13] A. A. M´ethot and V. Scarani, Quant. Inf. Comp. , 157(2007).[14] M. Gunge and C. Palazuelos, Comm. Math. Phys. ,695 (2001).[15] B. C. Hiesmayr, Eur. Phys. J. C , 73 (2007).[16] T. Vidick and S. Wehner, Phys. Rev. A , 052310(2011).[17] S. W. Ji, J. Lee, J. Lim, J. K. Nagata, and H. W. Lee,Phys. Rev. A , 052103 (2008).[18] N. Brunner, Physica E , 354 (2010).[19] N. Gisin, Found. Phys. , 80 (2012).[20] S. Popescu, Phys. Rev. Lett. , 2619 (1995).[21] D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Mik-laszewski, and A. Zeilinger, Phys. Rev. Lett. , 4418(2000).[22] W. Laskowski, J. Ryu, and M. Zukowsky, J. Phys. A ,424019 (2014).[23] A. Ac´ın, R. Gill, and N. Gisin, Phys. Rev. Lett. ,210402 (2005).[24] G. Brassard, R. Cleve, and A. Tapp, Phys. Rev. Lett. , 1874 (1999).[25] M. Steiner, Phys. Lett. A , 239 (2000).[26] D. Bacon and B. F. Toner, Phys. Rev. Lett. , 157904 (2003).[27] P. H. Eberhard, Phys. Rev. A, , 747(R) (1993).[28] A. G. White, D. F. V. James, P. H. Eberhard, and P. G.Kwiat, Phys. Rev. Lett. , 3103 (1999).[29] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt,Phys. Rev. Lett. , 880 (1969).[30] C. Bernhard et al, J. Phys. A: Math. Theor. , 424013(2014).[31] G. Vallone et al, Phys. Rev. A , 012102 (2014).[32] C. Branciard and N. Gisin, Phys. Rev. Lett. , 020401(2011).[33] J. D. Bancal, C. Branciard, N. Gisin, and S. Pironio,Phys. Rev. Lett. , 090503 (2009).[34] A. L. O. Bilobran and R. M. Angelo, arXiv:1411.7811(2014).[35] J. S. Bell, Physics , 195 (1966).[36] D. Bohm, Phys. Rev. , 166 (1952).[37] A. Ramsak, J. Phys. A: Math. Theor. , 115310 (2012).[38] N. Gisin, Phys. Lett. A , 201 (1991).[39] A possible extension to include POVM’s would requirecare. Apparently nonlocal correlations in the system ofinterest may turn out to be perfectly local when the an-cilla is considered.[40] J.-L. Chen, D.-L. Deng, and M.-G. Hu, Phys. Rev. A ,060306(R) (2008).[41] R. H. Landau, J. Paez , and C. C. Bordeianu, A Surveyof Computational Physics (2008), Princeton UniversityPress.[42] J. I. de Vicente, J. Phys. A: Math. Teor. , 424017(2014).[43] K. Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewen-stein, Phys. Rev. A , 883 (1998).[44] S. Hill and W. K. Wootters, Phys. Rev. Lett. , 5022(1997).[45] W. K. Wootters, Phys. Rev. Lett. , 2245 (1998).[46] D. Collins and N. Gisin, J. Phys. A: Math. Gen. , 1775(2004).[47] D. Collins, N. Gisin, N. Linden, S. Massar, and S.Popescu, Phys. Rev. Lett. , 040404 (2002).[48] C. H. Bennett et al, Phys. Rev. A , 1070 (1999).[49] D. N. Klyshko, Phys. Lett. A , 299 (1988).[50] M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani,Phys. Rev. Lett. , 58 (1994).[51] M. Zukowski, A. Zeilinger, and M. A. Horne, Phys. Rev.A , 2564 (1997). [52] D. Kaszlikowski, L. C. Kwek, J. L. Chen, M. Zukowski,and C. H. Oh, Phys. Rev. A , 032118 (2002).[53] J. Gruca, W. Laskowski, and M. Zukowski, Phys. Rev. A , 022118 (2012).[54] S. Popescu and D. Rohrlich, Found. Phys.24