Bounding quantum theory with the exclusivity principle in a two-city experiment
Mohamed Nawareg, Fabrizio Bisesto, Vincenzo D'Ambrosio, Elias Amselem, Fabio Sciarrino, Mohamed Bourennane, Adan Cabello
BBounding quantum theory with the exclusivity principle in a two-city experiment
Mohamed Nawareg,
1, 2
Fabrizio Bisesto, Vincenzo D’Ambrosio, EliasAmselem, Fabio Sciarrino,
3, 4
Mohamed Bourennane, and Ad´an Cabello Department of Physics, Stockholm University, S-10691 Stockholm, Sweden Instytut Fizyki Teoretycznej i Astrofizyki Uniwersytet Gda´nski, PL-80-952 Gda´nsk, Poland Dipartimento di Fisica, “Sapienza” Universit`a di Roma, I-00185 Roma, Italy Istituto Nazionale di Ottica (INO-CNR), Largo E. Fermi 6, I-50125 Firenze, Italy Departamento de F´ısica Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain (Dated: October 3, 2018)
Why do correlations between the results ofmeasurements performed on physical systems vi-olate Bell [1–7] and other non-contextuality in-equalities [8–16] up to some specific limits butnot beyond them ? The answer may follow fromthe observation that in quantum theory, unlike inother theories, whenever there is an experimentto measure A simultaneously with B , another tomeasure B with C , and another to measure A with C , there is always an experiment to measure all ofthem simultaneously [17, 18]. This property im-plies that quantum theory satisfies a seeminglyirrelevant restriction called the exclusivity (E)principle: the sum of the probabilities of any setof pairwise exclusive events cannot exceed 1 [19–23], which, surprisingly, explains the set of quan-tum correlations in some fundamental scenarios[19, 23]. A problem opened in [19] is whetherthe E principle explains the maximum quantumviolation of the Bell-CHSH inequality [1, 2] andquantum correlations in other scenarios. Here weshow experimentally that the E principle imposesan upper bound to the violation of the Bell-CHSHinequality that matches the maximum predictedby quantum theory. For that, we use the resultof an independent experiment testing a specificnon-contextuality inequality [21–23]. We performboth experiments: the Bell-CHSH inequality ex-periment on polarization-entangled states of pairsof photons in a laboratory in Stockholm and, todemonstrate independence, the non-contextualityinequality experiment on single photons’ orbitalangular momentum states in a laboratory inRome. The observed results provide the firstexperimental evidence that the E principle de-termines the limits of quantum correlations forboth scenarios and prove that hypothetical super-quantum violations for either experiment wouldviolate the E principle. This supports the conclu-sion that the E principle captures a fundamen-tal limitation of nature. If this is true, much ofquantum theory trivially follow from merely tak-ing the E principle to be a fundamental truth,and various information-theoretic postulates arealso simplified and/or strengthened. Quantum theory (QT) is the most successful scientific theory of all times. However, the reason for its success isnot clearly understood, as it is not known how to deriveQT from fundamental physical principles. Recently, thisproblem has been addressed using different approaches,including reconstructing QT from information-theoreticaxioms [24–29] and looking for principles for quantumnon-local correlations [30–33]. One of these approaches[19–23, 34] seeks principles for explaining the specific wayin which QT is contextual, i.e., the manner it violates Belland other non-contextuality (NC) inequalities.For years, violations of NC inequalities have been usedto emphasize the conflict between QT and local hiddenvariable theories [1] and between QT and non-contextualhidden variable theories [8, 9] (i.e., theories in which mea-surement outcomes are determined before measurementsare performed and are independent of which combinationof jointly measurable observables is considered). The ob-servation that the specific way in which QT violates NCinequalities may contain valuable information about theprinciples of QT is relatively recent [34].The explanation of the limits of the quantum viola-tions of the different NC inequalities may follow from anobservation made long ago [17]: In QT, whenever there isone experiment that jointly measures observables A and B , one experiment that jointly measures B and C , andone experiment that jointly measures A and C , there isalways one experiment that jointly measures A , B and C . This observation implies (see the proof in the Sup-plementary Material) a condition that can be taken asa principle, the exclusivity (E) principle: the sum of theprobabilities of any set of pairwise exclusive events can-not exceed 1 . By ‘event’ we mean the outcome of someexperimental test, such that two events are mutually ex-clusive when they are represented by different outcomesof the same test.So far, the E principle has shown an extraordinary pre-dictive power: For certain experimental scenarios, theE principle singles out the entire set of quantum corre-lations [19, 23]. Furthermore, for all other scenarios itsingles out the entire set of quantum correlations underthe assumption that the correlations for a specific inde-pendent experiment are given by QT [21, 23]. Here wecapitalize on the recent theoretical developments estab-lished in [23] to experimentally explain the maximum vi-olation of the Bell-CHSH inequality. We will show thatthe quantum maximum of the Bell-CHSH inequality is a r X i v : . [ qu a n t - ph ] N ov FIG. 1: (a) Graph representing the mutual exclusivity rela-tions between 8 pairwise exclusive events w i . (b) Graph repre-senting the mutual exclusivity relations between the 8 eventsin S . (c) Graph representing the mutual exclusivity relationsbetween the 8 events in R . Important for the argumentationis that (b) and (c) are vertex-transitive and mutually comple-mentary graphs. determined by the E principle given the result of a spe-cific independent experiment. Moreover, the experimen-tal observation of the maximum value predicted by QTfor either of the two experiments directly eliminates thepossibility of super-quantum correlation in the other, byvirtue of the E principle.The idea can be explained as follows: Consider, forexample, 8 pairwise mutually exclusive events w i , with i = 0 , . . . ,
7. The E principle states that the sum of theirprobabilities satisfy (cid:80) i =0 p ( w i ) E ≤
1. Each event can berepresented as a node in the graph of Fig. 1(a) in whichedges represent mutual exclusivity.Now imagine two independent experiments S and R and suppose that event w i is defined as the one in whichevent u i occurs in experiment S and event v i occurs inexperiment R . Independence implies that p ( w i ) = p ( u i ) p ( v i ) . (1)Therefore, the E principle establishes that W ≡ (cid:88) i =0 p ( u i ) p ( v i ) E ≤ . (2)Now suppose that experiments S and R are devisedin such a way that the graph of Fig. 1(b) represents therelations of mutual exclusivity between the events u i andthe graph of Fig. 1(c) represents the relations of mutual Event p | ψ (cid:105) ( a, b | i, j ) Experimental value Expected u p | ψ (cid:105) (1 , | ,
0) 0 . ± . . u p | ψ (cid:105) (1 , | ,
0) 0 . ± . . u p | ψ (cid:105) (1 , − | ,
1) 0 . ± . . u p | ψ (cid:105) ( − , − | ,
1) 0 . ± . . u p | ψ (cid:105) ( − , − | ,
0) 0 . ± . . u p | ψ (cid:105) ( − , − | ,
0) 0 . ± . . u p | ψ (cid:105) ( − , | ,
1) 0 . ± . . u p | ψ (cid:105) (1 , | ,
1) 0 . ± . . S . ± .
013 3 . p | ψ (cid:105) ( a, b | i, j ) denotes the joint probability of theevent “outcome a is obtained when test i is performed onthe first particle and outcome b is obtained when test j isperformed on the second particle” when the initial state is | ψ (cid:105) . exclusivity between the events v i . Note that every w i and w j are mutually exclusive because either u i and u j are mutually exclusive, or else v i and v j are mutuallyexclusive.Let’s define S ≡ (cid:88) i =0 p ( u i ) , (3a) R ≡ (cid:88) i =0 p ( v i ) . (3b)Since S and R are independent, if we rotate Fig. 1(b) by kπ/ k = 0 , . . . ,
7, and make the spec-ular reflection with respect to the axis v - v of Fig. 1(c) m times, with m = 0 ,
1, and then we merge the tworesulting figures, we end up with the graph in Fig. 1(a)but now representing the mutual exclusivity relations be-tween the events ( u i , v j ). This gives us 16 conditions like(2) (which corresponds to k = 0, m = 0). For instance,for k = 1, m = 0, we have W ≡ (cid:88) i =0 p ( u i ) p ( v i +1 ) E ≤ , (4)where the sum in i + 1 is taken modulo 8. Summing allthese 16 inequalities and dividing by 2, we obtain S × R E ≤ . (5)Now notice that the Bell-CHSH inequality can be writ-ten as S ≡ (cid:88) i =0 p ( u i ) NCHV ≤ , (6)where events u i are defined in table I and have the rela-tions of mutual exclusivity of Fig. 1(b). NCHV ≤ S for non-contextual hiddenvariable theories is 3. In QT, the maximum value of S is2 + √ ≈ . R and take its result, R exp , as a lower boundfor R , the E principle leads to the following upper boundfor S : S E ≤ /R exp . (7)To experimentally measure R we need 8 events with therelations of mutual exclusivity represented in Fig. 1(c).A set of events v i with this property is given in table II.These events satisfy the following NC inequality: R = (cid:88) i =0 p ( v i ) NCHV ≤ , (8)where the upper bound follows from the fact that 2 is themaximum number of events that can have probability 1while satisfying the relations of exclusivity in Fig. 1(c).In QT, the maximum value of R is 8 − √ ≈ . S and take its result, S exp , as a lower bound for S , the E principle leads to the following upper bound for R : R E ≤ /S exp . (9)The aim of this work is to exploit relation (7) to ex-perimentally obtain the tightest possible upper bound for S and exploit relation (9) to experimentally obtain thetightest possible upper bound for R . Event p | φ (cid:105) (0 , , | i − , i − , i ) Experimental value Expected v p | φ (cid:105) (0 , , | , ,
0) 0 . ± . . v p | φ (cid:105) (0 , , | , ,
1) 0 . ± . . v p | φ (cid:105) (0 , , | , ,
2) 0 . ± . . v p | φ (cid:105) (0 , , | , ,
3) 0 . ± . . v p | φ (cid:105) (0 , , | , ,
4) 0 . ± . . v p | φ (cid:105) (0 , , | , ,
5) 0 . ± . . v p | φ (cid:105) (0 , , | , ,
6) 0 . ± . . v p | φ (cid:105) (0 , , | , ,
7) 0 . ± . . R . ± .
011 2 . p | φ (cid:105) (0 , , | i − , i − , i ) denotes the probability of the event“outcomes 0 , , i − , i − , i are measured” when the initial stateis | φ (cid:105) . In the experiment we have first checked that events1 | i −
2, 1 | i − | i are pairwise mutually exclusive andthen used this to conclude that p | φ (cid:105) (0 , , | i − , i − , i ) isequal to p | φ (cid:105) (1 | i ), since observables i only have two outcomes. For that, we perform a Bell-CHSH inequality experi-ment. Its aim is to reach the maximum possible value for S b(see Methods). The results obtained in this ex-periment are shown in table I. We also perform a seriesof additional tests to check that the 8 events in S havethe 12 relations of mutual exclusivity shown in Fig. 1(b)(see Supplementary Material).On the other hand, we perform an independent NCinequality experiment in a different physical system ina distant laboratory. Its aim is to reach the maximumpossible value for R (see Methods). The results obtainedin this experiment are shown in table II. We also checkthat the 8 events in R satisfy the 16 relations of mutualexclusivity in Fig. 1(c) and check that the 16 inequalities W i E ≤ S can be as high as 2+ √ ≈ . S , thenthe E principle would restrict R to be upper-bounded byexactly its maximal quantum prediction. This is exactlythe meaning of (5) and (9). When we actually performedthe experiment we found S exp = 3 . ± . R E ≤ . ± . , (10)per the E principle, this by direct observation instead of assuming achievability of the quantum maximum predic-tion for S . This result is a significant improvement overan earlier limit of R ≤ √ ≈ . R exp = 2 . ± .
011 as a lower bound of what nature can reach for R ,the E principle leads to the conclusion that S E ≤ . ± . . (11)This result similarly improves upon an earlier limit of S ≤ √ ≈ . R puts a tighter upper bound on S , and the result of theexperiment measuring S puts a tighter upper bound on R .More importantly, the limits on S and R determinedin our experiments are not just necessary but also suf-ficient, that is to say, these bounds derived using the Eprinciple are completely tight. Not only do the bounds(10) and (11) together saturate the inequality (5), thusexploiting the full restrictive power of the E principle, butthey coincide with the maxima in QT. These are preciselimits of nature itself: They can be reached, as we haveobserved, but they cannot be exceeded, as proven by theE principle.Result (11) is the first empirical evidence that the Eprinciple explains the maximum quantum violation of theBell-CHSH inequality. We relied here on the E princi-ple, together with an observation of nature, to single outthe limits of the quantum correlations of the Bell-CHSHequality. We find this to be promising support for thestill-unproven conjecture [19–23] that the E principle, byitself , can recover this quantum bound. In addition, thefact that even slightly higher values of S would violate theE principle indicates the impossibility of super-quantumcorrelations satisfying the E principle for the Bell-CHSHinequality experiment.Similarly, result (10) is the first empirical evidence thatthe E principle explains the maximum quantum violationof the NC inequality (8) and supports the conjecture thatthe E principle, by itself, singles out the limits of quan-tum correlations for (8).The two specific experimental scenarios we have con-sidered are particularly important, since graphs inFig. 1(b) and (c) are the only simple vertex-transitivegraphs for which no previous proof or strong evidenceexisted that their maximum quantum bound were deter-mined by the E principle [20, 36].There are logically consistent universes in which theE principle does not hold [17, 18] and theoretical ma-chines that violate the E principle, such as nonlocal boxes[19, 33, 35]. However, the E principle holds in QT for pro-jective measurements and any generalized measurementin QT is a projective measurement in an enlarged Hilbertspace. Previous works have proved that the E principle,by itself, singles out the set of quantum correlations insome scenarios [23] and, at least, the maximum quantumcorrelations in others [19, 20, 36]. Now our experimentshows that, by the mechanism proven in [23], the E prin-ciple determines the observed limits for correlations, intwo relevant scenarios for which theoretical proofs wereheretofore inconclusive. This suggests that the E princi-ple is a key to a deeper understanding of QT and nature.The possibility that the E principle is fundamental hasfundamental implications for both QT and informationtheory.In the E principle - unlike in QT - the basic concept isprobability rather than a complex probability amplitude.The E principle looks as an extra axiom to Kolmogorov’saxioms of probability theory. This extra axiom wouldbe significant if we accept as the zero principle of QTthat “unperformed experiments have no results” (sinceresults do not correspond to intrinsic properties) [37].From this follows that it may not be possible, in general,to measure all observables jointly. Then, the fact thatpairwise jointly measurable observables are jointly mea-surable and, consequently, the E principle is highly non-trivial. With the E principle, QT appears more like anevolution from Kolmogorov’s probability theory rather FIG. 2: Setup for the Bell-CHSH inequality experiment. ASPDC source is used to generate a maximally entangled state.To prepare state (12), a HWP in mode a is set to − . ◦ .The measurement is performed by HWPA, HWPB plates,PBS and single photon detectors. than a development from Newtonian and Maxwellianphysics. The same way Kolmogorov’s probability the-ory is not about coins and dice, QT would not really beabout electrons and photons. This would explain whyQT successfully applies to most branches of physics andthe flexibility of QT for dealing with new phenomena.The E principle also provided valuable a-priori restric-tions on information processing and communication. Forexample, non-local boxes, whose information processingcapabilities have been extensively investigated [38], turnout to be impossible under the E principle, in the sameway as perpetual motion machines are impossible with-out violating the principles of thermodynamics. Anotherexample is secure communication: if the E principle isa fundamental one, then communication with securityguaranteed by fundamental principles would be easier toaccomplish. This is so because current security is basedon the assumption that the adversary’s capabilities arelimited by only the impossibility of signalling betweencausally unconnected parties [39]. The possible eaves-dropping strategies that a cryptographer needs to con-sider are vastly reduced by relying on the E principle,which is notoriously more restrictive.All this follows from a simple observation: (in QT)pairwise joint measurability implies joint measurability[17]. Unfortunately, the apparent irrelevance of this ob-servation suppressed awareness of its significant implica-tions, until only recently. Methods
Setup for the Bell-CHSH inequality experi- ment . S is maximized by the following two-qubit state: | ψ (cid:105) = 12 (cid:112) − √ (cid:104) | (cid:105) − | (cid:105) − (1 − √
2) ( | (cid:105) + | (cid:105) ) (cid:105) , (12)with measurements performed on both qubits in the ba-sis spanned by the Pauli observables, 0 = σ z and 1 = σ x .The 8 measurement outcomes needed for S are describedin table I. We encode our states in the optical polar-izations of pairs of photons as follows: | (cid:105) = | HH (cid:105) , | (cid:105) = | HV (cid:105) , | (cid:105) = | V H (cid:105) and | (cid:105) = | V V (cid:105) , where | (cid:105) = | HV (cid:105) is the two-photon state for which the firstphoton is vertically polarized and the second photon ishorizontally polarized.The setup is shown in Fig. 2. Ultraviolet light centeredat wavelength of 390 nm is focused inside a beta bar-ium borate (BBO) nonlinear crystal to produce photonpairs emitted into two spatial modes (a) and (b) throughthe degenerate emission of spontaneous parametric down-conversion (SPDC) [40]. The source is engineered to pre-pare the singlet state | Ψ (cid:105) = ( | HV (cid:105) − | V H (cid:105) ) / √
2. Toobtain the state | ψ (cid:105) given in (2), we prepare a singletstate and then rotate a half-wave plate (HWP) placed inmode (a) to − . ◦ (angle between the waveplate’s opti-cal axes direction and horizontal polarization direction).The polarization measurement is performed using HWPsand polarizing beam splitters (PBSs). At the input of thePBS, we place narrow-bandwidth interference filters (F)( δλ = 1 nm) to guarantee well defined spectral modes. Atthe output of the PBS, the photons are coupled into 2 msingle mode optical fibers followed by actively quenchedSi-avalanche photodiodes D ij . The measurement timefor each pair of local settings is 200 seconds. Setup for the NC inequality experiment . R ismaximized by a single five-dimensional five-dimensionalquantum system in the state | φ (cid:105) = (cid:115) − √ | (cid:105) + (cid:115) − √ | (cid:105) + (cid:115) − √ | (cid:105) + (cid:115) √ − | (cid:105) , (13)and perform the 8 tests i = | v i (cid:105)(cid:104) v i | , with | v i (cid:105) defined as FIG. 3: Setup for the NC inequality experiment. By means ofa single mode fiber, the beam from a single photon source issent to a first SLM (Generation), which provides the states tobe measured. Then, projective measurements are performedby means of a second SLM (Analysis), in combination witha single mode fiber and a single photon detector. To avoidGouy phase shift effect, we have realized an imaging system(not reported in figure) between the screens of the two SLMs. follows: | v (cid:105) = | (cid:105) , | v (cid:105) = | (cid:105) , | v (cid:105) = | (cid:105) , | v (cid:105) = (2 − √ | (cid:105) + (cid:113) √ − | (cid:105) − (cid:113) √ − | (cid:105) , | v (cid:105) = (3 − √ | (cid:105) + (2 − √ | (cid:105) + (cid:114) (cid:16) √ − (cid:17) | (cid:105) + (cid:113) √ − | (cid:105) , | v (cid:105) = (2 − √ | (cid:105) + (3 − √ | (cid:105) + (2 − √ | (cid:105)− (cid:113) √ − | (cid:105) , | v (cid:105) = ( √ − | (cid:105) + (2 √ − | (cid:105) − (cid:114) (cid:16) √ − (cid:17) | (cid:105) + (cid:113) √ − | (cid:105) , | v (cid:105) = ( √ − | (cid:105) − (cid:113) √ − | (cid:105) − (cid:113) √ − | (cid:105) . (14)We encode the states in photon orbital angular mo-mentum (OAM) space of dimension 5. Each state is alinear combination of the OAM basis states {| m (cid:105)} m = − .The experimental setup is illustrated in Fig. 3. Singlephotons in fundamental TEM00 Gaussian state ( m = 0)are prepared in the desired OAM superposition state bymeans of a spatial light modulator (SLM) (Generation).This device modulates the phase wave front according tocomputer generated holograms. After the state gener-ation, a second SLM (Analysis) is used in combinationwith a single mode fiber and single photon detector toperform a projective measurement on the photon state.This setup allows us to generate and project over allthe states needed for the experiment. The generationand measurement processes are completely automatizedand computer controlled. We adopt a hologram genera-tion technique that maximizes the fidelity of the states byintroducing losses in the beam [41]. This results in differ-ent hologram diffraction efficiencies (see SupplementaryMaterial). Acknowledgments
The authors thank B. Amaral, M. Ara´ujo, M. Klein-mann, J. R. Portillo, R. W. Spekkens and M. TerraCunha for useful conversations, J.-˚A. Larsson, A. J.L´opez-Tarrida and E. Wolfe for suggestions to improvethe manuscript, and E. Wolfe for the figure in theSupplementary Material. This work was supportedby Project No. FIS2011-29400 (MINECO, Spain) withFEDER funds, the FQXi large grant project “The Natureof Information in Sequential Quantum Measurements”,the Brazilian program Science without Borders, FIRBFuturo in Ricerca-HYTEQ, the Swedish Research Coun-cil (VR), the Linnaeus Center of Excellence ADOPT, theERC Advanced Grant QOLAPS and Starting Grant 3D- QUEST (3D-Quantum Integrated Optical Simulation;Grant Agreement No. 307783): . M.N. is supported by the international PhD projectgrant MPD/2009-3/4 from the Foundation for Polish Sci-ence.
Authors’ contributions
M.N. conducted the Bell-CHSH inequality experimentand processed the data assisted by M.B. and A.C., M.N.and E.A. built the two-photon source. F.B. and V.D.conducted the NC inequality experiment and processedthe data assisted by F.S. and A.C., V.D. and F.S. builtthe OAM setup. A.C. conceived the experiment. Allauthors wrote the paper and agree on its content.
Additional information
Correspondence and requests for materials should beaddressed to F.S., M.B. or A.C. The authors declare nocompeting financial interests. [1] Bell, J. S. On the Einstein Podolsky Rosen paradox.
Physics , 195–200 (1964).[2] Clauser, J. F., Horne, M. A., Shimony, A. & Holt, R. A.Proposed experiment to test local hidden-variable theo-ries. Phys. Rev. Lett. , 880–884 (1969).[3] Freedman, S. J. & Clauser, J. F. Experimental test oflocal hidden-variable theories. Phys. Rev. Lett. , 938–941 (1972).[4] Aspect, A., Dalibard, J. & Roger, G. Experimental testof Bell’s inequalities using time-varying analyzers. Phys.Rev. Lett. , 1804–1807 (1982).[5] Weihs, G., Jennewein, T., Simon, C., Weinfurter, H. &Zeilinger, A. Violation of Bell’s inequality under strictEinstein locality conditions. Phys. Rev. Lett. , 5039–5043 (1998).[6] Rowe, M. A., Kielpinski, D., Meyer, V., Sackett, C. A.,Itano, W. M., Monroe, C. & Wineland, D. J. Experimen-tal violation of a Bell’s inequality with efficient detection. Nature , 791–794 (2001).[7] Giustina, M., Mech, A., Ramelow, S., Wittmann, B.,Kofler, J., Beyer, J., Lita, A., Calkins, B., Gerrits, T.,Nam, S. W., Ursin, R. & Zeilinger, A. Bell violation usingentangled photons without the fair-sampling assumption.
Nature , 227-230 (2013).[8] Klyachko, A. A., Can, M. A., Binicio˘glu, S. & Shu-movsky, A. S. Simple test for hidden variables in spin-1systems.
Phys. Rev. Lett. , 020403 (2008).[9] Cabello, A. Experimentally testable state-independentquantum contextuality.
Phys. Rev. Lett. , 210401(2008).[10] Kirchmair, G., Z¨ahringer, F., Gerritsma, R., Kleinmann,M., G¨uhne, O., Cabello, A., Blatt, R. & Roos, C. F. State-independent experimental test of quantum contex-tuality.
Nature , 494–497 (2009).[11] Amselem, E., R˚admark, M., Bourennane, M. & Cabello,A. State-independent quantum contextuality with singlephotons.
Phys. Rev. Lett. , 160405 (2009).[12] (cid:32)Lapkiewicz, R., Li, P., Schaeff, C., Langford, N.,Ramelow, S., Wie´sniak, M. & Zeilinger, A. Experimentalnon-classicality of an indivisible quantum system.
Nature , 490–493 (2011).[13] Amselem, E., Danielsen, L. E., L´opez-Tarrida, A. J., Por-tillo, J. R., Bourennane, M. & Cabello, A. Experimen-tal fully contextual correlations.
Phys. Rev. Lett. ,200405 (2012).[14] Zhang, X., Um, M., Zhang, J., An, S., Wang, Y.,Deng, D.-L., Shen, C., Duan, L.-M. & Kim, K. State-independent experimental test of quantum contextualitywith a single trapped ion.
Phys. Rev. Lett. , 070401(2013).[15] D’Ambrosio, V., Herbauts, I., Amselem, E., Nagali, E.,Bourennane, M., Sciarrino, F. & Cabello, A. Experimen-tal implementation of a Kochen-Specker set of quantumtests.
Phys. Rev. X , 011012 (2013).[16] Ahrens, J., Amselem, E., Cabello, A. & Bourennane, M.Two fundamental experimental tests of nonclassicalitywith qutrits. Sci. Rep. , 2170 (2013).[17] Specker, E. P. Die Logik Nicht Gleichzeitig Entscheid-barer Aussagen. Dialectica , 239–246 (1960). Englishtranslation: E-print arXiv:1103.4537.[18] Liang, Y.-C., Spekkens, R. W. & Wiseman, H. M. Phys.Rep. , 1–39 (2011).[19] Cabello, A. Simple explanation of the quantum violationof a fundamental inequality.
Phys. Rev. Lett. , 060402 (2013).[20] Cabello, A., Danielsen, L. E., L´opez-Tarrida, A. J. &Portillo, J. R. Basic exclusivity graphs in quantum cor-relations.
Phys. Rev. A , 032104 (2013).[21] Yan, B. Quantum correlations are tightly bound bythe exclusivity principle. Phys. Rev. Lett. , 260406(2013).[22] Cabello, A. Proposed experiment to exclude higher-than-quantum violations of the Bell inequality.arXiv:1303.6523.[23] Amaral, B., Terra Cunha, M. & Cabello, A. The exclu-sivity principle forbids sets of correlations larger than thequantum set. E-print arXiv:1306.6289.[24] Hardy, L. Quantum theory from five reasonable axioms.E-print quant-ph/0101012.[25] Hardy, L. Reformulating and reconstructing quantumtheory. E-print arXiv:1104.2066.[26] Daki´c, B. & Brukner, ˇC. In
Deep Beauty. Understandingthe Quantum World through Mathematical Innovation ,edited by H. Halvorson (Cambridge University Press,New York, 2011), pp. 365–392.[27] Masanes, L. & M¨uller, M. P. A derivation of quantumtheory from physical requirements.
New J. Phys. ,063001 (2011).[28] Chiribella, G., D’Ariano, G. M. & Perinotti, P. Informa-tional derivation of quantum theory. Phys. Rev. A ,012311 (2011).[29] Masanes, L., M¨uller, M. P., Augusiak, R. & P´erez-Garc´ıa,D. Existence of an information unit as a postulate ofquantum theory. PNAS , 16373-16377 (2013).[30] Van Dam, W.
Nonlocality and Communication Complex-ity , Ph.D. thesis, Department of Physics, University ofOxford, 2000; Implausible consequences of superstrongnonlocality.
Nat. Comput. , 9–12 (2013).[31] Paw(cid:32)lowski, M., Paterek, T., Kaszlikowski, D., Scarani,V., Winter, A. & ˙Zukowski, M. Information causality asa physical principle. Nature , 1101–1104 (2009).[32] Navascu´es, M. & Wunderlich, H. A glance beyond thequantum model.
Proc. Royal Soc. A , 881–890 (2009).[33] Fritz, T., Sainz, A. B., Augusiak, R., Bohr Brask, J.,Chaves, R., Leverrier, A. & Ac´ın, A. Local orthogonality:A multipartite principle for correlations.
Nature Commu-nications , 2263 (2013).[34] Cabello, A. Correlations without parts. Nature , 456-458 (2011).[35] Popescu, S. & Rohrlich, D. Quantum nonlocality as anaxiom.
Found. Phys. , 379–385 (1994).[36] Cabello, A. et al. In preparation.[37] Peres, A. Unperformed experiments have no results.
Am.J. Phys. , 745–747 (1978).[38] Barrett, J. & Pironio, S. Popescu-Rohrlich correlations asa unit of nonlocality. Phys. Rev. Lett. , 140401 (2005).[39] Scarani, V., Gisin, N., Brunner, N., Masanes, L., Pino,S. & Ac´ın, A. Secrecy extraction from no-signaling cor-relations. Phys. Rev. A , 042339 (2006).[40] Kwiat, P. G., Mattle, K., Weinfurter, H., Zeilinger, A.,Sergienko, A. V. & Shih, Y. New high-intensity source ofpolarization-entangled photon pairs. Phys. Rev. Lett. ,4337-4341 (1995).[41] D’Ambrosio, V., Cardano, F., Karimi, E., Nagali, E.,Santamato, E., Marrucci, L. & Sciarrino, F. Test of mu-tually unbiased bases for six-dimensional photonic quan-tum systems. Sci. Rep. , 2726 (2013). Appendix A: Specker’s observation and theexclusivity principle within a general framework ofoperational theories
We formulate Specker’s observation about quantumtheory and the E principle within a general frameworkof operational theories and prove that the E principleis inherently satisfied by any theory in which Specker’sobservation holds.
1. Preparations and tests
Preparations and tests are taken as primitive notionswith the following meaning:A preparation is a sequence of unambiguous and repro-ducible experimental procedures.A test is a preparation followed by a step in whichoutcome information is supplied to an observer. Thisinformation is not trivial since tests that follow identicalpreparations may not have identical outcomes.An operational theory is one that specifies the probabil-ities of each possible outcome X of each possible test M given each preparation P . We denote these probabilitiesby p ( X | M ; P ).The presented framework is independent of the inter-pretation of probability used; the reader is free to use,e.g., frequentist, propensity, or Bayesian interpretations.
2. States and observables
Two preparations are operationally equivalent if theyyield identical outcome probability distributions for ei-ther test. Each equivalence class of preparations is calleda state . For instance, the state associated with a partic-ular preparation P is ρ ≡ { P | ∀ M : p ( X | M ; P ) = p ( X | M ; P ) } . (A1)Two tests are operationally equivalent if they yieldidentical outcome probability distributions for eitherpreparation. Each equivalence class of tests is called an observable . For instance, the observable associated witha particular test M is µ ≡ { M | ∀ P : p ( X | M ; P ) = p ( X | M ; P ) } . (A2)
3. Joint measurability of observables
Two observables µ and µ are jointly measurable ifthere exists an observable µ such that: (i) the outcomeset of µ , σ ( µ ), is the Cartesian product of the outcomesets of µ and µ , i.e., σ ( µ ) ≡ { ( X i , X j ) | X i ∈ σ ( µ ) , X j ∈ σ ( µ ) } , (A3)and (ii) for all states ρ , the outcome probability distri-butions for every measurement of µ or µ are recovered as marginals of the outcome probability distribution of µ , i.e., ∀ ρ, ∀ X i ∈ σ ( µ ) : p ( X i | µ ; ρ ) = (cid:88) X j ∈ σ ( µ ) p (( X i , X j ) | µ ; ρ ) , (A4a) ∀ ρ, ∀ X j ∈ σ ( µ ) : p ( X j | µ ; ρ ) = (cid:88) X i ∈ σ ( µ ) p (( X i , X j ) | µ ; ρ ) . (A4b) N observables µ , . . . , µ N are jointly measurable ifthere exists an observable µ such that: (i’) the outcomeset of µ is the Cartesian product of the outcome sets of µ , . . . , µ N and (ii’) for all states ρ , the outcome probabil-ity distributions for every joint measurement of any sub-set S ≡ { µ i | i ∈ I } ⊂ { µ , . . . , µ N } , with I = { , . . . , N } ,are recovered as marginals of the outcome probability dis-tribution of µ . Denoting by µ S an observable associatedwith a joint measurement of the subset S , its outcome setby σ ( µ S ) and one of its outcomes by X S , the conditioncan be expressed as ∀ S, ∀ ρ, ∀ X S ∈ σ ( µ S ) : p ( X S | µ S ; ρ ) = (cid:88) X t : t/ ∈ I p (( X , . . . , X N ) | µ ; ρ ) . (A5)Joint measurability of a set of observables implies pair-wise joint measurability of them (i.e., joint measurabilityof any pair of them). The converse is not necessarily true.A joint probability distribution for N observables µ , . . . , µ N exists if, for all subsets S ≡ { µ i | i ∈ I } ⊂{ µ , . . . , µ N } , with I = { , . . . , N } , for all states ρ andfor all X S ∈ σ ( µ S ), where µ S ia an observable associatedwith a joint measurement of S , there exists a probabilitydistribution p ( X , . . . , X N | ρ ) such that p ( X S | µ S ; ρ ) = (cid:88) X t : t/ ∈ I p ( X , . . . , X N | ρ ) . (A6)If some observables are jointly measurable then thereexists a joint probability distribution for them. The ex-istence of a joint probability distribution for some ob-servables does not imply that they are jointly measur-able. The nonexistence of a joint probability distributionfor some observables indicates the impossibility of jointlymeasuring them.
4. Events
We are interested in a specific type of preparations:those resulting from a test M with outcome X on a pre-vious preparation P . We denote these preparations by P (cid:48) ≡ X | M ; P .Two of these preparations are operationally equivalentif they yield identical outcome probability distributionsfor either subsequent test M (cid:48) . Each equivalence classof these preparations is called an event . For instance,the event associated with a particular preparation P (cid:48) ≡ X | M ; P is (cid:15) ≡ { P (cid:48) | ∀ M (cid:48) : p ( X (cid:48) | M (cid:48) ; P (cid:48) ) = p ( X (cid:48) | M (cid:48) ; P (cid:48) ) } . (A7)Notice that the term “event”, which is usually restrictedto designate the outcome X of test M on preparation P ,here designates the state after test M with outcome X onpreparation P . The probability of an event is thereforethe probability of transforming one state (e.g., the oneassociated with P ) into another (e.g., the one associatedwith P (cid:48) ). In a given non-contextuality (NC) inequality,all probabilities (of events) are probabilities of differenttransformations of the same state.
5. Mutual exclusivity of events
Two events (cid:15) and (cid:15) are mutually exclusive if thereexist two jointly measurable observables µ , univocallydefined by (cid:15) , and µ , univocally defined by (cid:15) , that dis-tinguish between them, i.e., if there exists an observ-able µ associated with a joint measurement of µ and µ such that there are X ⊂ σ ( µ ) and X ⊂ σ ( µ ) with X ∩ X = ∅ such that p ( X | µ ; (cid:15) ) = 1 , (A8a) p ( X | µ ; (cid:15) ) = 1 . (A8b)The N events of a set E = { (cid:15) , . . . , (cid:15) N } are jointlyexclusive if there exists a set of N jointly measurableobservables M = { µ , . . . , µ N } that distinguish betweenthe events in any subset of E .Joint exclusivity of a set of events implies mutual exclu-sivity of any pair of them. The converse is not necessarilytrue.
6. Specker’s observation and the E principle
Specker’s observation.
Specker pointed out that, inquantum theory, pairwise joint measurability of a set M of observables implies joint measurability of M , while inother theories this implication does not need to hold [17].Later, Specker conjectured that this is “the fundamentaltheorem” of quantum theory (see http://vimeo.com/52923835 ). The E principle states that any set of pairwise mutu-ally exclusive events is jointly exclusive. Therefore, fromKolmogorov’s axioms of probability, the sum of theirprobabilities cannot be higher than 1.
Lemma:
In any theory in which pairwise joint measur-ability of observables implies joint measurability of ob-servables, pairwise mutual exclusivity of events impliesjoint exclusivity of events.
Proof:
If the events in a set E = { (cid:15) , . . . , (cid:15) N } are pair-wise exclusive, there exists a set M = { µ , . . . , µ N } of pairwise jointly measurable observables that permits todistinguish between any two events in E . If pairwise jointmeasurability of M implies joint measurability of M ,then M permits to distinguish between the events in anysubset of E .The converse implication, namely, that in any theoryin which pairwise mutual exclusivity implies joint exclu-sivity also pairwise joint measurability implies joint mea-surability, is not necessarily true. Appendix B: Experimental test of the relations ofmutual exclusivity
As a complement to the tests of the Bell-CHSH andNC inequalities, we make several tests to check that the8 events whose probabilities are tested in the Bell-CHSHinequality experiment satisfy the 12 relations of mutualexclusivity represented in Fig. 1(b) and the 8 eventswhose probabilities are tested in the NC inequality sat-isfy the 16 relations of mutual exclusivity represented inFig. 1(c).
1. Mutual exclusivity between the events in theBell-CHSH inequality
Two events are mutually exclusive if they correspondto different outcomes of an observable µ . We identify atest defining µ for each pair of events. There are twocases:4 of the relations of mutual exclusivity occur betweenevents in which Alice implements µ A and Bob imple-ments µ B . For them, µ is simply an observable associatedwith a joint measurement of µ A and µ B .The other 8 relations of mutual exclusivity occur be-tween events in which one of the observers, e.g. Alice,implements µ A while the other observer implements µ B or µ B (cid:48) depending on the outcome of µ A . This gives atest defining µ , since µ A and µ B are jointly measurableand µ A and µ B (cid:48) are jointly measurable.As an additional test, we experimentally show thatconditions (A8a) and (A8b) are satisfied for any pair ofmutually exclusive events. This is shown in table III.
2. Mutual exclusivity between the events in theNC inequality
Two events are mutually exclusive if they correspondto different outcomes of an observable µ . We identifya test defining µ for each pair of events. For that, weprepare 16 additional states | w i (cid:105) that allows us to definean 5-outcome observable for each of the 8 triangles inFig. 1(b). These states are specified in table IV. Each ofthese 5-outcome observables distinguishes between eachpair of events in the corresponding triangle.0 Probability Experimental value p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 1 . × − ± . × − p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 2 . × − ± . × − p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 3 . × − ± . × − p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 1 . × − ± . × − p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 6 . × − ± . × − p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 1 . × − ± . × − p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 3 . × − ± . × − p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 2 . × − ± . × − p (1 | µ ; u ) 0 . ± . p (1 | µ ; u ) 0 . ± . p (1 | µ j ; u i ) denotes the probabilityof obtaining the result 1 when the observable µ j (which cor-responds in quantum theory to | u j (cid:105)(cid:104) u j | ) is measured on thestate | u i (cid:105) . As an additional test, we experimentally show thatconditions (A8a) and (A8b) are satisfied for any pair ofmutually exclusive events. For this, we measure the fi-delity of each state | v i (cid:105) , defined by p (1 | µ (cid:48) i ; v i ) and theprobabilities p (1 | µ (cid:48) i ; v j ) where µ (cid:48) i is the observable withoutcome set { , } represented in quantum theory by | v i (cid:105)(cid:104) v i | . The experimental results are shown in tables IVand V. Appendix C: Exclusivity inequalities
We use the data in tables I and II to check that the 16exclusivity inequalities W i E ≤
1, with i = 1 , . . . ,
16, aresatisfied in our experiment. The results are in table VI.The fact that we observe an experimental value com-patible with 1 for each of the 16 inequalities indicatesthat the experimental results for S and R are both in thelimit allowed by the E principle. Appendix D: Exclusivity graph of the completetwo-city experiment
Fig. 1(b) shows the exclusivity graph of the 8 eventsin the Bell-CHSH inequality experiment performed inStockholm, Fig. 1(c) shows the exclusivity graph of the8 events in the NC inequality experiment performed inRome and Fig. 1(a) shows a subgraph of the exclusivitygraph of the set of 8 × Basis State Components Fidelity EfficiencyI | v (cid:105) (1,0,0,0,0) (98 . ± . ± | v (cid:105) (0 , − . , − . , − . , . . ± . . ± . | v (cid:105) (0 , , − . , − . , − . . ± . . ± . | w (cid:105) (0 , − . , . , . , − . . ± . . ± . | w (cid:105) (0 , . , . , − . , − . . ± . . ± . | v (cid:105) (1,0,0,0,0) (99 . ± . ± | v (cid:105) (0,1,0,0,0) (98 . ± . ± | v (cid:105) (0,0,1,0,0) (98 . ± . ± | w (cid:105) (0 , , , . , . . ± . . ± . | w (cid:105) (0 , , , . , − . . ± . . ± . | v (cid:105) (0 . , , , . , − . . ± . . ± . | v (cid:105) (0 . , . , , . , . . ± . . ± . | v (cid:105) (0 . , . , . , − . ,
0) (98 . ± . . ± . | w (cid:105) (0 . , − . , . , . , . . ± . . ± . | w (cid:105) (0 . , − . , − . , − . , . . ± . . ± . | v (cid:105) (1 , , , ,
0) (99 . ± . ± | v (cid:105) (0 , , , ,
0) (99 . ± . ± | v (cid:105) (0 , , − . , − . , − . . ± . . ± . | w (cid:105) (0 . , − . , . , . , . . ± . . ± . | w (cid:105) (0 . , − . , − . , − . , . . ± . . ± . | v (cid:105) (0 , , , ,
0) (99 . ± . ± | v (cid:105) (0 , , , ,
0) (98 . ± . ± | v (cid:105) (0 . , , , . , − . . ± . . ± . | w (cid:105) (0 . , − . , . , . , . . ± . . ± . | w (cid:105) (0 . , − . , − . , − . , . . ± . . ± . | v (cid:105) (0 , , , ,
0) (98 . ± . ± | v (cid:105) (0 . , , , . , − . . ± . . ± . | v (cid:105) (0 . , . , , . , . . ± . . ± . | w (cid:105) (0 . , − . , . , . , . . ± . . ± . | w (cid:105) (0 . , − . , − . , − . , . . ± . . ± . | v (cid:105) (0 . , . , , . , . . ± . . ± . | v (cid:105) (0 . , . , . , − . ,
0) (98 . ± . . ± . | v (cid:105) (0 , − . , − . , − . , . . ± . . ± . | w (cid:105) (0 . , − . , . , . , . . ± . . ± . | w (cid:105) (0 . , − . , − . , − . , . . ± . . ± . | v (cid:105) (0 . , . , . , − . ,
0) (98 . ± . . ± . | v (cid:105) (0 , − . , − . , − . , . . ± . . ± . | v (cid:105) (0 , , − . , − . , − . . ± . . ± . | w (cid:105) (0 . , − . , . , . , . . ± . . ± . | w (cid:105) (0 . , − . , − . , − . , . . ± . . ± . TABLE IV: Complete measurement bases for the NC inequality experiment. The table reports, for each state, both the classicalfidelity and the relative diffraction efficiency with respect to the state | v (cid:105) , corresponding to TEM00 Gaussian state. Meanbase fidelities are (98 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . Probability Experimental value p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) ; v ) 0 . ± . p (1 | µ (cid:48) j ; v i ) denotes the probability of obtainingthe result 1 when the observable µ (cid:48) j (which corresponds inquantum theory to | v j (cid:105)(cid:104) v j | ) is measured on the state | v i (cid:105) . u u u u u u u u Experimental value W v v v v v v v v . ± . W v v v v v v v v . ± . W v v v v v v v v . ± . W v v v v v v v v . ± . W v v v v v v v v . ± . W v v v v v v v v . ± . W v v v v v v v v . ± . W v v v v v v v v . ± . W v v v v v v v v . ± . W v v v v v v v v . ± . W v v v v v v v v . ± . W v v v v v v v v . ± . W v v v v v v v v . ± . W v v v v v v v v . ± . W v v v v v v v v . ± . W v v v v v v v v . ± . u i , v j ) in each W k are given by the table by combining event u i of the first rowwith event v j in the intersection between W k ’s row and u i ’scolumn. For example, W is also defined in (2). (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:45) (cid:45) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) (cid:200) FIG. 4: Exclusivity graph of the two-city experiment. The 64 dots represent the 64 global events ( u i , v j ). Two dots areconnected by an edge if the corresponding events are mutually exclusive. Each ( u i , v j ) is indicated by providing the explicitexpression of u i (for instance, u = 1 , | ,
0; see table I) followed by the one of v j (see table II). The 8-vertex subgraph withred edges corresponds to the correlations in W9