Two fundamental experimental tests of nonclassicality with qutrits

Two Fundamental Experimental Tests of Nonclassicality with Qutrits Johan Ahrens 1 , Elias Amselem 1 , Ada ´n Cabello 1,2 & Mohamed Bourennane 1 1 Department of Physics, Stockholm University, S-10691, Stockholm, Sweden, 2 Departamento de Fı ´sica Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain. We report two fundamental experiments on three-level quantum systems (qutrits). The first one tests the simplest task for which quantum mechanics provides an advantage with respect to classical physics. The quantum advantage is certified by the violation of Wright’s inequality, the simplest classical inequality violated by quantum mechanics. In the second experiment, we obtain contextual correlations by sequentially measuring pairs of compatible observables on a qutrit, and show the violation of Klyachko et al. ’s inequality, the most fundamental noncontextuality inequality violated by qutrits. Our experiment tests exactly Klyachko et al. ’s inequality, uses the same measurement procedure for each observable in every context, and implements the sequential measurements in any possible order. In classical physics, there is no contradiction in considering that systems like balls and coins have preestablished properties like position and velocity that are independent of whether one actually measures them or not. However, according to quantum mechanics, the results of experiments on systems such as atoms and photons do not correspond to preestablished properties. A natural and fundamental question is: Which is the simplest quantum system in which this difference between classical and quantum physics can be observed? For instance, this difference is shown in the violation of Bell inequalities 1–3 , which requires at least two measurements on, at least, two separate subsystems, thus requiring a physical system of dimension four (i.e., with four perfectly distinguishable states). However, even before Bell inequalities were discovered, Kochen and Specker 4,5 , and Bell 6 pointed out that the classical/quantum conflict occurs even in a simpler three-level quantum system, or qutrit (e.g., a spin-1 particle). This implies that entanglement is not needed for the generation of nonclassical correlations, since, by definition, a single qutrit cannot be in an entangled state. Indeed, by performing a sequence of compatible 7 measurements on the same single qutrit, contextual correlations can be obtained. The classical/quantum conflict can be revealed even without correlations. The simplest test producing classically impossible results was first described by Wright 8 and is based on the probabilities of the results of five single measurements on a qutrit. Later on, Klyachko et al. derived a noncontextuality inequality (i.e., satisfied by all theories in which the measurement results are independent of any compatible measurement) for qutrits using correlations 9 . Klyachko et al.’s inequality is the simplest noncontextuality inequality violated by quantum mechanics, in the sense that there is no conflict for systems of lower dimension or with inequalities with fewer terms. The conflict can be made independent of the initial quantum state of the system by considering more complex inequalities 10–12 . Results In this Letter we report experimental violations of both Wright’s and exactly Klyachko et al.’s inequality with qutrits. We first introduce both inequalities and show that they are actually connected. Then, we describe the experimental setups corresponding to each test. Finally, we present the experimental results and discuss them. Wright’s inequality.Wright’s inequality 8 is the simplest classical inequality violated by quantum mechanics. It can be proven that the simplest set of questions such that the sum of the probabilities of obtaining a yes answer is higher in quantum mechanics than in classical physics is precisely the one in which 5 questions Q i are such that Q i and Q i11 (with the sum modulo 5) are exclusive. Wright’s inequality can be formulated as a game. In each run of the game, one of these questions is picked at random. The goal of the game is to obtain as many yes answers as possible. Below we give an example that illustrates an optimal classical strategy. The system is prepared in a random mixture of 5 classical states 0, 1, 2, 3, and 4, and the player provides the following 5 yes-no questions: Q 0 5‘‘0 or 1?’’ (denoting the question ‘‘Is the system in one of the states 0 or 1?’’), Q 1 5‘‘2 or 3?’’, Q 2 5‘‘0 or 4?’’, Q 3 5‘‘1 or OPEN SUBJECT AREAS: QUANTUM MECHANICS QUANTUM INFORMATION OTHER PHOTONICS INFORMATION THEORY AND COMPUTATION Received 20 February 2013 Accepted 18 June 2013 Published 10 July 2013 Correspondence and requests for materials should be addressed to M.B. ([email protected]) SCIENTIFIC REPORTS | 3 : 2170 | DOI: 10.1038/srep02170 1 2?’’, and Q 4 5‘‘3 or 4?’’. If P(11jQ i ) denotes the probability of obtaining a yes answer when Q i is asked, then, in our example, Pz1Qi j ðÞ~2 5for any i50, …, 4. Therefore, P4 i~0Pz1Qi j ðÞ~2. It can be proven that no other preparation or set of classical questions provides a better solution; for any set of 5 classical questions with the previous exclusiveness constraints, the following inequality holds: W:~X 4 i~0 Pz1Qi j ðÞƒ2:ð1Þ The upper bound follows from the fact that the maximum number of questions Q i that simultaneously can have a yes answer is 2. Inequality (1) is Wright’s inequality. According to quantum mechanics, the system can be in a superposition of states and the questions can refer to such a superposition. This provides an advantage. For example, the quantum system is prepared in the qutrit state y hj ~0,0,1ðÞ ð2Þ and the questions correspond to the 5 operators Qi~2vi ji vi hj {, which in quantum mechanics represent 5 observables with possible outcomes 11 and 21 (corresponding to yes and no, respectively). Q i and Q i11 are compatible and exclusive. Correspondingly, the 5 projectors jv i æÆv i jare such that jv i æÆv i jand jv i11 æÆv i11 jare orthogonal. Specifically, to obtain the maximum quantum value, we chose v0 hj ~N1,0,rðÞ,ð3aÞ v1,4 hj~Nc,+s,r ðÞ ,ð3bÞ v2,3 hj~NC,+S,rðÞ,ð3cÞ with r~ffiffiffiffiffiffiffiffiffiffiffiffiffi cos p 5  q,c~cos 4 p 5  ,s~sin 4 p 5  ,C~cos 2 p 5  ,S~sin 2 p 5  , N~1ffiffiffiffiffiffiffiffiffiffiffi 1zr2 p, and is the identity matrix. Then, quantum mechanics predicts that WQM~X 4 i~0 viy jhijj 2~ffiffiffi 5 p<2:236,ð4Þ which violates Wright’s inequality (1). It can be proven that the simplest set of questions (such that any of them belongs to an exclusive pair) and constraints for which quantum mechanics provides an advantage is precisely the one in which 5 questions Q i are such that Q i and Q i11 (with the sum modulo 5) are exclusive. It can also be proven that the simplest system with quantum advantage has d53. Finally, it can be proven that the maximum quantum violation (for any d) is precisely ffiffiffi 5 p(see Supplemental Material). Klyachko et al.’s noncontextuality inequality.The simplest experiment showing contextual correlations in a qutrit is the violation of Klyachko et al.’s inequality 9 . This inequality defines the only nontrivial facet of the polytope of classical (noncontextual) correlations, and completely separates noncontextual from contextual correlations 9 . The importance of observing contextual correlations between the results of sequential measurements on the same qutrit comes from the fact that these correlations cannot be attributed to entanglement, since, by the definition of entanglement, a single qutrit cannot be in an entangled state. This is not the case in recent experiments showing quantum contextual correlations on two-qubit systems 13–16 . To put Klyachko et al.’s inequality in the frame of the classical game introduced before, we now ask two questions Q i and Q i11 , one immediately after the other. By collecting all answers (yes, yes), (yes, no), (no, yes), and (no, no), we can calculate the average of obtaining the same results (i.e., Q i Q i11 511) or different results (i.e., Q i Q i11 521). For our classical strategy, for each pair of exclusive questions Q i and Q i11 , we obtain QiQiz1 hi~{ 3 5. In other words, when asking Q i and Q i11 sequentially, on average, 4 out of 5 times we obtain different answers, and 1 out of 5 times we obtain equal answers. If we now sum over all possible pairs Q i ,Q i11 , we obtain 23, which can be proven to be the classical lower bound for the sum. Therefore, k:~X 4 i~1 QiQiz1 hi§{3,ð5Þ which is Klyachko et al.’s inequality 9 . The maximum quantum violation of Klyachko et al.’s inequality (5) is attained for the same qutrit state jyæand observables Q i providing the maximum quantum violation of Wright’s inequality (1). The maximum quantum violation (for any d) of inequality (5) is kQM~5{4ffiffiffi 5 p<{3:944:ð6Þ We would like to note that Lapkiewicz et al. 17 have recently performed an experiment aiming to test a noncontextuality inequality with 6 measurements and 6 correlations (4 of them contained in Klyachko et al.’s inequality, plus two more which, added up, act as substitutes for the fifth correlation in Klyachko et al.’s inequality). However, the 6-correlation inequality tested is not a facet of the simplest polytope of noncontextual correlations (i.e., it does not belong to the simplest set of inequalities that separates noncontextual from contextual correlations). Moreover, Lapkiewicz et al.’s experiment cannot be considered a proper test of a noncontextuality inequality, since the same observable is measured with different setups in different contexts. In 17 it is left as an ‘‘open question’’ whether any experimental apparatus can be designed to test exactly Klyachko et al.’s inequality. See the Supplemental Material for further discussion. Experiment 1 (see Fig. 1) tests the quantum violation of inequality (1) and experiment 2 (see Fig. 2) tests the quantum violation of inequality (5). In both experiments, the qutrit is defined by means of the polarization and path degrees of freedom of a single photon. The two spatial modes are labeled aand band, by design, the polarization in mode ais enforced to be horizontal (see Fig. 1). The encoding is 0 ji ~H,b ji ,1 ji ~V,b ji ,2 ji ~H,a ji ,ð7Þ where Hdenotes horizontal polarization and Vdenotes vertical polarization. To ensure that the system stays a qutrit, we make sure that a potential jV,aæcomponent is always associated to loss and thus never expands the Hilbert space. Experimental results.The experimental results for experiment 1 testing inequality (1) are shown in Table I and the obtained value is W~2:29+0:06. We observe a clear violation of inequality (1). It can be noted that the obtained experimental value is slightly higher than the maximum quantum value but in the range of the error bars. This can be explained by the fact that the latter is obtained under the assumption that the measured questions/operators are perfectly exclusive, while perfect exclusiveness is difficult to guarantee experimentally. In our implementation, a perfect question Q i would need an interference visibility of 100% to guarantee a perfect exclusiveness with Q i11 . The main sources of systematic errors were the imperfection of the polarization components and the limited precision of their settings. The errors DP det and DP sys were respectively deduced from propagated Poissonian counting statistics of the raw detection events and the limited precision of the setting h i of the HWP plate. See the Supplemental Material for more details. The experimental results for experiment 2 testing inequality (5) are presented in Table II. The experimental results show a clear violation of inequality (5), in good agreement with the quantum mechanics prediction. The measurements are performed in all possible orders. Due to the experimental imperfections, our results show a violation that slightly depends on the order. www.nature.com/scientificreports SCIENTIFIC REPORTS | 3 : 2170 | DOI: 10.1038/srep02170 2 Figure 1 | Experimental setup for experiment 1. (a) Scheme for single projective measurement Q i . The red and blue lamps correspond to unsuccessful (no) and successful (yes) projection, respectively. (b) Setup for creating a qutrit and performing Q i . A pulse generator, P in the figure, is trigging the attenuated diode laser in the source, S in the figure. The setup consists of a source of horizontally polarized single photons followed by a half wave plate (HWP) and a polarizing beam splitter (PBS), allowing us to prepare any probability distribution of a photon in modes aand b. The orientation of the HWP in mode bsets the polarization state of that mode. The output of the source is connected to the input of operator Q i , which has detectors in its output. (c) Symbol definition of the optical elements used in the setup: polarizing beam splitter (PBS), mirror (M), half wave plate (HWP), quarter wave plate (QWP), and single photon detector (D). Figure 2 | Experimental setup for experiment 2. (a) Scheme for the sequential measurement on pairwise compatible observables Q i and Q i11 . The red and blue lamps correspond to the eigenvalues 21 (no) and 11 (yes), respectively. (b) Setup for performing the sequential measurements Q i and Q i11 . The device for Q i is exactly the same as described in Fig. 1; the device for the second measurement Q i11 is also the same, except for a longer time delay of 2Dt. A click at the detectors at the time slots t 0 ,t 1 ,t 2 , and t 3 corresponds to, respectively, the answer (no, no), (yes, no), (no, yes), and (yes, yes). The preparation of the qutrit state and the symbols are the same as in Fig. 1. www.nature.com/scientificreports SCIENTIFIC REPORTS | 3 : 2170 | DOI: 10.1038/srep02170 3 The main source of systematic error was due to the optical interferometers involved in the measurements, the imperfect overlapping and coupling of the light modes, and the polarization components. Errors were inferred from propagated Poissonian counting statistics of the raw detection events and from 10 measurement samples to capture the error due to the drift over the measurement time. The number of detected photons was approximately 3 310 4 per second and the total measurement time for each of the 5 pairs of observables was 1 s for each run. Our measurement procedure was to first calibrate the interferometers and then start the measurement where all terms in (5) are measured one after the other. In Table II, the second part has higher values than the first, since it was impossible to replicate exactly the calibration of the interferometers between the two runs. Discussion We have reported two experiments on qutrits showing nonclassical properties. Unlike previous experiments on pairs of qubits 13–16 , here the nonclassical properties cannot be attributed to entanglement. Experiment 1 tests Wright’s inequality 8 , which is the simplest inequality based on outcome probabilities of 5 exclusive yes-no questions, that provides a better-than-classical solution. Experiment 2 tests Klyachko et al.’s inequality 9 , which is the most fundamental correlation inequality satisfied by noncontextual theories and violated by single qutrits. Unlike Lapkiewicz et al.’s experiment 17 , our experiment 2 tests exactly Klyachko et al.’s inequality with 5 observables and 5 correlations, and tests the correlations in any possible order. Unlike previous experiments with sequential measurements on photons, in our experiment 2, the results of the sequential measurements are encoded in different time slots, avoiding much more complicated alternatives 15 , and allowing us to show that the violation of the inequality does not depend on the order of the measurements. Both experiments are of fundamental importance to understand the difference between quantum and classical physics in situations where entanglement is not present, and open the door to new applications in quantum information processing based on simple quantum systems. Methods Experiment 1.The setup in Fig. 1(b) allows for the preparation of all required states and the projections on the eigenstates of all required operators. The questions Q i are implemented through a time multiplexing scheme. A yes answer to Q i corresponds to a successful projection and is indicated by the arrival time t 1 of the photon (or, equivalently, by a blue lamp flashing; see Fig. 1), the no answer corresponds to an unsuccessful projection and is indicated by the arrival time t 0 of the photon (or a red lamp flashing; see Fig. 1). The goal of the measurement is to distinguish the eigenstates corresponding to different eigenvalues of Q i . Any qutrit pure state can be expressed as ajH,bæ1bjV,bæ 1cjH,aæ, where a,b, and care complex numbers. To implement Q i we prepare the eigenstate jv i æ, which corresponds to the positive eigenvalue. The polarization in mode bwill first be rotated to obtain the state b9jV,bæ1cjH,aæ(this can always be done, since the rotation of the polarization is performed only in mode b). Then, the part of the state in mode ais transferred into mode b; this process also flips the polarization. This leads to the state b9jH,bæ1cjV,bæ. The state can now be rotated to jH,bæ, which is coupled to a delay line adding a delay of Dt550 ns so that it can be distinguished from the orthogonal states by the time slots t 1 and t 0 . All of the previous steps (except for the time delay) are then performed in reverse order to reprepare the eigenstate of the observable for further processing. A useful property of our implementation of the 5 operators Q i is that they are exactly the same, up to a half wave plate rotation; see Fig. 1(b). To ask the question Q i , one rotates both the first and last half wave plate by an angle h i 545u, 117u,9u,81u, and 153ufor i50, 1, 2, 3, and 4, respectively. Experiment 2.When we perform two sequential measurements corresponding to pairwise compatible observables Q i and Q i61 , the first measurement is exactly the same one as described above. The eigenstates of observable Q i with eigenvalue 11or 21 are again mapped to different time slots using the same time multiplexing detection method. The setup for the second measurement is the same as that for the first, except for the delay line, which is twice as long, i.e., 2Dtsee Fig. 2(b). After passing the devices for Q i and Q i11 , the photon can be registered by a detector at four equally distributed time slots. These time slots are t 0 ,t 1 5t 0 1Dt,t 2 5t 0 12Dt,andt 3 5t 0 13Dt, and correspond, respectively, to the answers (no, no), (yes, no), (no, yes), and (yes, yes). Another distinguishing feature of this approach is that it allows both the measurement of Q i followed by Q i11 , and also the measurement of Q i11 followed by Q i . This allows us to test each of the correlations in inequality (5) in every possible order. In both experiments, the single-photon source was a pulsed diode laser emitting at 780 nm with a pulse width of 3 ns and a repetition rate of 100 kHz; see Fig. 1(b) and Fig. 2(b). The laser was attenuated so that the two-photon coincidences were negligible. The visibility ranges achieved were between 80% and 90% for each Q i .A single-photon detector was placed in each output mode aand b. All detector signals and timing trigger signals (which define the measurement time slots) were registered using a multichannel coincidence logic with a time window of 1.7 ns. 1. Aspect, A., Dalibard, J. & Roger, G. Experimental test of Bell’s inequalities using time-varying analyzers. Phys. Rev. Lett. 49, 1804 (1982). 2. Weihs, G., Jennewein, T., Simon, C., Weinfurter, H. & Zeilinger, A. Violation of Bell’s inequality under strict Einstein locality conditions. Phys. Rev. Lett. 81,5039 (1998). 3. Matsukevich, D. N., Maunz, P., Moehring, D. L., Olmschenk, S. & Monroe, C. Bell inequality violation with two remote atomic qubits. Phys. Rev. Lett. 100, 150404 (2008). 4. Specker, E. P. Die Logik nicht gleichzeitig entscheidbarer Aussagen. Dialectica 14, 239 (1960). 5. Kochen, S. & Specker, E. P. The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59 (1967). 6. Bell, J. S. On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447 (1966). Table I | Experimental results for the violation of inequality Wƒ2, the theoretical quantum bound for an ideal experiment is WQM~ffiffiffi 5 p<2:236:P(11 | Q i ) th and P(11 | Q i ) exp indicate the theoretical value for an ideal experiment and the value obtained in the experiment, respectively. DP det ,DP sys , and DP tot indicate the errors deduced from propagated Poissonian counting statistics of the raw detection events, the limited precision of the setting h i of the HWP plate, and the total errors respectively Q i P(11 | Q i ) th P(11 | Q i ) exp DP sys DP det DP tol Q 0 0.447 0.460 0.011 0.001 0.011 Q 1 0.447 0.454 0.011 0.001 0.011 Q 2 0.447 0.460 0.011 0.002 0.011 Q 3 0.447 0.461 0.011 0.001 0.011 Q 4 0.447 0.457 0.011 0.001 0.011 W2.236 2.29 0.055 0.003 0.06 Table II | Experimental results for the violation of inequality k#3, the theoretical quantum bound for an ideal experiment is kQM~ 5{4ffiffiffi 5 p<{3:944.ÆQ i Q i11 æ th and ÆQ i Q i11 æ exp indicate the theoretical value for an ideal experiment and the value obtained in the experiment, respectively. DÆQ i Q i11 æ det ,DÆQ i Q i11 æ sys , and DÆQ i Q i11 æ tot indicate the errors deduced from propagated Poissonian counting statistics of the raw detection events, the limited precision of the setting of the polarization components, and the total errors respectively Q i Q i11 ÆQ i Q i11 æ th ÆQ i Q i11 æ exp DÆQ i Q i11 æ sys DÆQ i Q i11 æ det DÆQ i Q i11 æ tot ÆQ 0 Q 1 æ20.798 20.712 0.023 0.002 0.023 ÆQ 1 Q 2 æ20.798 20.706 0.023 0.002 0.023 ÆQ 2 Q 3 æ20.798 20.704 0.023 0.002 0.022 ÆQ 3 Q 4 æ20.798 20.708 0.023 0.002 0.022 ÆQ 4 Q 1 æ20.798 20.706 0.023 0.002 0.024 k23.94 23.53 0.115 0.005 0.11 Q i11 Q i ÆQ i11 Q i æ th ÆQ i11 Q i æ exp DÆQ i11 Q i æ sys DÆQ i11 Q i æ det DÆQ i Q i11 æ tot ÆQ 1 Q 0 æ20.798 20.785 0.022 0.003 0.022 ÆQ 2 Q 1 æ20.798 20.781 0.023 0.003 0.023 ÆQ 3 Q 2 æ20.798 20.774 0.023 0.003 0.023 ÆQ 4 Q 3 æ20.798 20.774 0.022 0.003 0.022 ÆQ 1 Q 4 æ20.798 20.782 0.021 0.003 0.021 k23.94 23.90 0.115 0.006 0.11 www.nature.com/scientificreports SCIENTIFIC REPORTS | 3 : 2170 | DOI: 10.1038/srep02170 4 7. 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Experimental test of quantum contextuality in neutron interferometry. Phys. Rev. Lett. 103, 040403 (2009). 15. Amselem, E., Ra ˚dmark, M., Bourennane, M. & Cabello, A. State-independent quantum contextuality with single photons. Phys. Rev. Lett. 103, 160405 (2009). 16. Moussa, O., Ryan, C. A., Cory, D. G. & Laflamme, R. Testing contextuality on quantum ensembles with one clean qubit. Phys. Rev. Lett. 104, 160501 (2010). 17. Lapkiewicz, R. et al. Experimental non-classicality of an indivisible quantum system. Nature 474, 490 (2011). Acknowledgments The authors thank I. Bengtsson, A. Klyachko, J.-A ˚. Larsson, R.W. Spekkens, and K. Svozil for discussions. This work was supported by the Swedish Research Council (VR), the Linnaeus Center of Excellence ADOPT, the Spanish Projects Nos. FIS2008-05596 and FIS2011-29400, and the Wenner-Gren Foundation. Author contributions J.A., E.A., A.C. and M.B. designed the experiment and wrote the paper. J.A. and E.A. conducted the experiment and analyzed the data. Additional information Supplementary information accompanies this paper at http://www.nature.com/ scientificreports Competing financial interests: The authors declare no competing financial interests. How to cite this article: Ahrens, J., Amselem, E., Cabello, A. & Bourennane, M. Two Fundamental Experimental Tests of Nonclassicality with Qutrits. Sci. Rep. 3, 2170; DOI:10.1038/srep02170 (2013). This work is licensed under a Creative Commons Attribution- NonCommercial-NoDerivs 3.0 Unported license. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/3.0 www.nature.com/scientificreports SCIENTIFIC REPORTS | 3 : 2170 | DOI: 10.1038/srep02170 5