Experimental observation of quantum contextuality beyond Bell nonlocality

PHYSICAL REVIEW A 100, 042118 (2019) Experimental observation of quantum contextuality beyond Bell nonlocality Zheng-Hao Liu,1,2Hui-Xian Meng,3,4Zhen-Peng Xu,5Jie Zhou,3Sheng Ye,1,2Qiang Li,1,2Kai Sun,1,2Hong-Yi Su,6 Adán Cabello,7Jing-Ling Chen,3,*Jin-Shi Xu,1,2,†Chuan-Feng Li ,1,2,‡and Guang-Can Guo1,2 1CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People’s Republic of China 2CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People’s Republic of China 3Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China 4School of Mathematics and Physics, North China Electric Power University, Beijing 102206, People’s Republic of China 5Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, 57068 Siegen, Germany 6Graduate School of China Academy of Engineering Physics, Beijing 100193, People’s Republic of China 7Departamento de Fisica Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain (Received 2 June 2019; published 22 October 2019) Bell nonlocality and quantum contextuality are two important resources which, however, behave very differently in many tasks where the quantum correlation has an edge over its classical counterpart. In this work, we directly compare these two behaviors, which via the Cabello-Severini-Winter graph-theoretical approach, can be associated with a same exclusivity graph. In particular, we consider the exclusivity graph that leads to I3322-type inequalities in the Bell and the contextual scenarios, respectively. We find that maximal violation of Bell inequality is around 6.251, and maximal violation of the noncontextuality inequality is around 6.588 for a five-dimensional system and 6.571 for a four-dimensional system, respectively. The results predict a gap of ≈ 0.32 between quantum contextuality and Bell nonlocality. We then present an experimental observation of the gap by employing both the maximally entangled photon pairs from the spontaneous parametric down-conversion process and the single photons encoded into qudits from an intrinsic defect in gallium nitride. Our results will further deepen the understanding of different types of quantum correlations. DOI: 10.1103/PhysRevA.100.042118 I. INTRODUCTION Bell nonlocality (BN) and quantum contextuality (QC) have been proved to be intrinsic properties of nature. A physical system exhibits Bell nonlocality if the outcomes of ideal measurements (i.e., those that yield the same outcome when repeated and do not disturb compatible measurements) cannot be reproduced by local-hidden-variable (LHV) models, in which measurement outcomes are independent of spacelike separated measurements performed on each distant subsystem [1]. In a similar sense, a physical system exhibits quantum contextuality if the outcomes of ideal measurements cannot be reproduced by noncontextual-hidden variable (NCHV) models in which measurement outcomes are independent of the measurement contexts (i.e., the set of jointly measurable observables that are actually measured) [2]. Both LHV and NCHV models are referred to as classical models in many situations, as they, as well as their breakdowns in quantum mechanics (QM), share several similarities indeed [3]. For instance, (i) both BN and QC reflect the confliction between quantum mechanics and some deterministic theories in which measurement outcomes are well defined even before measurements are performed; and (ii) every Bell inequality becomes *[email protected] †[email protected] ‡[email protected] a noncontextuality inequality if the constraint of spacelike separation between measurements on subsystems is removed [3,4]. As two important correlations, BN and QC have been studied widely, and in most cases separately, in the literature. They have also found significant applications in quantum information processing, such as speeding-up quantum algorithms [5], quantum key distribution [6], communication complexity [7], and randomness certification [8]. Very recently, several fundamental relations between these two quantum correlations were found. First, e.g., a fundamental monogamy relation exists between QC and BN. The theoretical result was presented in [9] and then observed in experiments [10,11]. Second, via the Cabello-Severini-Winter (CSW) graph-theoretical approach [12], the different behaviors between BN and QC can be compared in a same exclusivity graph [13] with which BN is proved to be tightly bounded by QC [14]. Considering an arbitrary Bell inequality written in the probability form, one easily has an exclusivity graph due to the exclusive relations among the probability events. Then, based on the same graph, one can also construct a noncontextuality inequality. Let =IQC −IBN denote the gap between QC and BN, where IQC and IBN denote the maximal violations of the noncontextuality inequality and Bell inequality, respectively. In general, one has ⩾0. For =0, it means Bell nonlocality saturates quantum contextuality, and >0 means quantum contextuality beyond Bell nonlocality. The purpose of this work is to perform an experiment to observe 2469-9926/2019/100(4)/042118(11) 042118-1 ©2019 American Physical Society ZHENG-HAO LIU et al. PHYSICAL REVIEW A 100, 042118 (2019) the gap , thus demonstrating the phenomenon of quantum contextuality beyond Bell nonlocality. The point is to choose an appropriate exclusivity graph for which is sufficiently large such that QC beyond BN can be observed in the experiment. The paper is organized as follows. In Sec. II,we shall study the exclusivity graph associated with the I3322-type inequality, which theoretically gives ≈0.3[4]. In Sec. III, we shall then demonstrate the phenomenon of QC beyond BN by experiment. Within the experimental deviation, the experimental results coincide with the theoretical predictions. Conclusions will be drawn in the last section. II. THEORY: QC BEYOND BN Let us start from the symmetric Bell inequality I3322 ⩽0 [15], a natural generalization of the Clauser-Horne-Shimony- Holt (CHSH) inequality [16] from two measurement settings to three measurement settings. The Bell inequality can be written in the following form [4] (see Appendix A): I=P(0,0|0,1) +P(0,0|0,2) +P(0,0|1,0) +P(0,0|1,2) +P(0,0|2,0) +P(0,0|2,1) +P(0,1|1,1) +P(1,0|1,1) +P(1,1|1,1) +P(0,1|2,2) +P(1,0|2,2) +P(1,1|2,2) +P(1,_|0,_) +P(1,_|1,_) +P(_,1|_,0) +P(_,1|_,1) LHV ⩽6,(1) where Iis a substitute for ICSW 3322 ,P(m,n|i,j)≡P(ai= m,bj=n)=|vAB|AB|2denotes the probability that Alice’s ith measurement outcome is mand Bob’s jth measurement outcome is n,|ABis the quantum state, |vAB= |ai=m⊗|bj=nis the factorizable joint-measurement vector for Alice and Bob, P(m,_|i,_) ≡P(ai=m) denotes the marginal probability of Alice’s ith measurement result being m, and similarly for P(_,n|_,j). For simplicity, we use R1,R2,...,R16 to denote the 16 terms in inequality (1) later. Inequality (1) holds for any two d-dimensional systems (i.e., two qudits). For two qubits, it is a tight inequality with the maximal quantum violation IBN =25/4=6.25, and we give the optimal observables and quantum state in Appendix A. When the dimension dincreases, the maximal quantum violation slowly increases. It has been shown that the maximal violation tends to IBN ≈6.250 875 38 when d→∞[17]. In other words, for the I3322-type inequality, the value of Bell nonlocality is less than 6.251. Based on the exclusive relations among the 16 probability events in the inequality (1), one can immediately plot an exclusivity graph Gas shown in Fig. 1(see also Ref. [4]). The vertices of graph Grepresent probability events, and two connected vertices imply that two events are exclusive with one another, i.e., the two corresponding projectors or jointmeasurement vectors are mutually orthogonal. Accordingly, from graph Gone can have the noncontextuality inequality as follows: I= i∈V Pi NCHV ⩽β QM ⩽ϑ(G),(2) 15 16 1 2 6 8 910 14 11 12 35 4 7 13 FIG. 1. Exclusivity graph associated with I. For Bell nonlocality, the 16 events in inequality (1) correspond to 1: 1,1|2,2, 2: 0,0|2,0, 3: 1,0|2,2, 4: 0,0|2,1, 5: 0,1|2,2, 6: _,1|_,0, 7: _,1|_,1, 8: 0,0|1,0, 9: 1,0|1,1, 10: 0,1|1,1, 11: 1,_|1,_, 12: 0,0|1,2, 13: 1,1|1,1, 14: 0,0|0,1, 15: 1,_|0,_, and 16: 0,0|0,2, and the spacelike measurements between Alice and Bob guarantee the mutual orthogonality. For quantum contextuality, 16 vertices correspond to 16 vectors |Ai’s, and |Ai⊥|Ajwhen vertex iand vertex jare connected. where Vdenotes the vertex of graph G, and Pi≡P|ψ(Ai= 1) denotes the probability of obtaining result 1 when the observable |AiAi|is measured on the state |ψ, namely, P|ψ(Ai=1) =|Ai|ψ|2.Hereβ=6 is the classical bound of the NCHV models, and ϑ(G) is the maximal violation of the noncontextuality inequality and is equal to the Lovász number [18] describing the Shannon capacity of G. The Lovász number is computed by ϑ(G)= max{|ψ,|Ai} i∈V|Ai|ψ|2.InRef.[4], the authors computed ϑ(G)=IQC ≈6.588 412 87 without giving |Aiand |ψ that yield the Lovász number. Here we show that the above ϑ(G) can be achieved in a five-dimensional system, and we also list the corresponding optimal measurement settings and optimal state (see Table Iin Appendix A). For the I3322-type inequalities, the gap between QC and BN is given by =IQC −IBN ≈0.337, which is sufficiently large, hence the I3322-type inequality is a very good candidate to experimentally demonstrate QC beyond BN. Remark 1. The difference between QC and BN can be understood in this way: For QC, in order to evaluate the quantity Q=i∈V|Ai|ψ|2, one has to numerate all possible |Ai’s and |ψin the Hilbert space of arbitrary dimensions d, with all orthogonal relations of |Ai’s being satisfied, such that the maximal value Qmax equals the Lovász number ϑ(G). For the graph in Fig. 1, the minimum dimension of obtaining the Lovász number is 5. Nevertheless, for testing the Bell inequality, it is additionally required that |Ai’s have to be factorizable, i.e., |Ai=|vid1⊗|uid2, and so the dimension ought to be factorized to d=d1×d2accordingly. For instance, let us take d1=d2=2. By running over all possible |Ai’s and |ψunder the factorizability requirement, one can obtain the maximal value Qmax =6.25, which exactly is the maximal violation of inequality (1) for two qubits in the Bell scenario. Remark 2. For the vector |Ai’s, the minimal dimension to fulfill the exclusive relations shown in Fig. 1is 4. In a four-dimensional system, the maximal violation of 042118-2 EXPERIMENTAL OBSERVATION OF QUANTUM … PHYSICAL REVIEW A 100, 042118 (2019) SNSPD from single photon source HWP PBS Lens Pinhole SLM SMF QWP EPR Source Φ+ SPAD SPAD Alice Bob Laser 532nm (a) (b) (c) Beam splitter GaN sample -100 -50 0 50 100 0.0 0.5 1.0 1.5 g2(τ) Delay time τ (ns) FIG. 2. Experimental setup. (a) The two-particle setup to check BN. Two maximally polarization-entangled photons were separately sent to Alice and Bob, going through their polarization discrimination system and recorded by single-photon avalanche detectors (SPADs). (b) The single-photon source, prepared by exciting an intrinsic defect in a bulk GaN sample. The second-order photon correlation function at zero delay without background correction was 0.225, clearly exhibiting the signature of photon antibunching and confirming the character of single-photon emission. Single photons were further filtered by a bandpass filter with a central wavelength of 780 nm and a bandwidth of 25 nm, and sent to the contextuality test setup. (c) The single-body contextuality test setup. The single photons from (b), after passing through the beam expanding system with two lenses and a pinhole between them, were sent to a spatial light modulator (SLM) controlled by a computer to prepare the high-dimensional states, which is denoted by a violet box. The measurement setup consists of another SLM and a single-mode fiber (SMF), and the counting apparatus here was a superconducting nanowire single-photon detector (SNSPD). noncontextuality inequality (2)isIQC ≈6.571 52, which is slightly smaller than the Lovász number ϑ(G). The optimal settings (i.e., |Ai,i=1,2,...,16) and the state (|ψ) are given in Table IV of Appendix A. The gap is around ≈0.320. Therefore, in this work we shall choose a fourdimensional system instead of a five-dimensional system to demonstrate QC beyond BN. Remark 3. To experimentally test quantum contextuality, we need to recast inequality (2) to the following noncontextuality inequality (by adding the second term) based on theorem 1ofRef.[19], i.e., I= i∈V P|ψ(Ai=1) − (i,j)∈E P|ψ(Ai=1,Aj=1) NCHV ⩽β QM ⩽ϑ(G),(3) where Edenotes the vertex and edge sets of the graph G, and P|ψ(Ai=1,Aj=1) ≡P|ψ(Ai=1)P|Ai(Aj=1) represents the experimental imprecision of |Ai⊥|Aj when they are measured successively. Introducing the second term in (3) is crucial and necessary, as it is used to evaluate the compatibility requirement in any contextuality test in a practical experiment [19]. Certainly, the second term becomes zero in the ideal case and can thus be neglected. Crucially, the graph Gneeds to be extended with extra projectors that complete the full sets of compatible observables in order that every projector is measured as part of a complete basis [20]. In the work, we make measurements in the following six complete bases: {|A1,|A2,|A3,|A17},{|A1,|A4,|A7,|A18},{|A1,|A5, |A16,|A19},{|A1,|A9,|A12,|A20},{|A1,|A6,|A8, |A21}, and {|A1,|A14,|A15,|A22}. Here, |A17, |A18,...,|A22are uniquely determined by the other three vectors in a complete basis. Thus, the detection probability of each observable, say, |A2A2|, can be obtained as P|ψ(A2=1) =N|ψ(A2)/[N|ψ (A1)+N|ψ(A2)+N|ψ(A3)+N|ψ(A17)], where N|ψ(Ai)is the number of counts of obtaining result 1 when |AiAi|is measured on the state |ψ,and similarly for the measurements of the other observables. III. EXPERIMENTAL SETUP AND RESULTS We are now ready to experimentally test the two distinct quantum values of Iseparately, namely, IBN ⩽6.25 and IQC ⩽6.57, to explicitly demonstrate QC beyond BN. The experimental setup is illustrated in Fig. 2. We start from IBN. Based on the maximally entangled state |+=(|00+ |11)/√2 of two photonic polarization qubits, we checked the maximal violation of inequality (1). A type-II β-barium borate (β-BBO) crystal was pumped by a frequency-doubled 042118-3 ZHENG-HAO LIU et al. PHYSICAL REVIEW A 100, 042118 (2019) ×10-2 0 0.5 1.0 1.5 2.0 14710 13 16 1 4 7 10 13 16 A i Aj 0. 0.2 0.4 0.6 12345678 0. 0.2 0.4 0.6 9 10111213141516 12345678 0. 0.2 0.4 0.6 9 10111213141516 0. 0.2 0.4 0.6 P i R i Theory Expm. Theory Expm. (a) (b) (c) FIG. 3. Experimental results of detection probabilities for computing IBN (a) and IQC (b and c). (a) Coincidence counting probabilities of Alice and Bob. The settings are chosen to produce the 16 terms, namely, R1∼R16 in (1). (b) Detection probabilities of the 16 optimal projectors, representing the 16 vertices of the corresponding exclusivity graph, with the input state |ψ. The probability is equivalent to P1∼P16 in (2). (c) Detection probabilities of nonzero P|Ai(Aj=1), where |Aiand |Ajare connected by 32 edges in the exclusivity graph. femtosecond laser to generate a polarization-entangled biphoton, which was further sent to Alice and Bob, where a polarization beam splitter (PBS), proceeded by a quarter-wave plate (QWP) and a half-wave plate (HWP), was utilized to construct a polarization discrimination system. Two singlephoton avalanche detectors (SPADs) recorded photon counting rate after the polarization discrimination system, and their coincidence counts were proportional to the probability of detecting two photons with a certain setting. For each set of measurements, the total photon count was about 8000, and results are shown in Fig. 3(a). Substituting the measurement results into inequality (1) gives a result of IBN = 6.165 ±0.012. Here, the standard deviation was deduced from the Poisson distribution. The result surpasses prediction of any possible LHV model but, due to imperfections in the experiment, falls sightly below the Tsirelson bound, 6.25. We now turn to investigate IQC. In a four-dimensional space, the single-body quantum violation of inequality (3) was checked. To construct this synthetic space, we exploit the orbital angular momentum (OAM) of light. The manipulation of states was accomplished by modulating the phase of the photon’s wave function. By exciting an intrinsic defect in a bulk GaN sample, we obtained an ultrabright, photostable single-photon source at room temperature [21]. A bandpass filter with a central wavelength of 780 nm and a width of 25 nm was used to filter the fluorescence of the defects and match the operation wavelength of the contextuality test setup. The total photon counting rate past the filter was about 0.94 Mcps. The single photon was directed to the setup by a singlemode fiber (SMF) and spatial filtered into fundamental Gaussian mode. Using a spatial light modulator (SLM), a coordinate-related wave function, originating from displayed holograms on the SLM, was bestowed to photons [22]. The photons would then carry a wave function resembling superpositions of Laguerre-Gaussian modes LGl pwith OAM number of l[23] and be prepared to desired initial fourdimensional states |ψafter being diffracted by the first SLM. The four eigenvectors are set to be |i1=|1,|i2= |3,|i3=|−1, and |i4=|−3.A4fsystem mapped the first and the second SLM at its input and output plane, with an aperture inserted at the focal plane to filter off the unwanted zero- and higher-order diffraction terms. The second SLM converted specific OAM states |αback to the Gaussian mode |0, which was subsequently postselected by a SMF, as photons carrying nonzero OAM cannot be focused to a spot and collected by a SMF. In addition, two telescope lenses were inserted between the second SLM and SMF to adjust beam size and optimize fiber coupling efficiency. Photons were then sent to a superconducting nanowire single-photon detector (SNSPD), and the counting rate was proportional to the probability of successful postselection, i.e., |α|ψ|2. The input state was prepared to be |ψ=(|i1+ |i2)/√2, and the 16 corresponding projectors |AjAj|(j= 1,2,...,16) were then measured on six sets of complete bases. To test the inequality (3), we also need to check the no-signaling condition, which requires the local marginal probabilities of successive measurements to be irrelevant to the other settings being chosen, either forward or backward in time, as suggested in [19]. Our experimental results confirm the no-signaling condition in which the detailed information is shown in Appendix B. The first term of IQC inequality was evaluated by directly referring to detect probabilities of the 16 corresponding projectors on six full sets of compatible observables. In order to obtain all probabilities needed for the exclusivity graph, there were in total 19 independent projectors to be measured. The measurement settings of these projectors were shown in Appendix B. The counting rate of each projector was then recorded and the detection probability was calculated. The integration time for each projection measurement was 10 s. Shown in Fig. 3(b), the results were fairly close to prediction. The experimental result of the first term of IQC was 6.548 ±0.020, an error estimated by assuming Poisson distribution. The second term of IQC inequality represents experimental imperfections of orthogonality between states denoted by connected vertices in the exclusivity graph. Similar to the way the 042118-4 EXPERIMENTAL OBSERVATION OF QUANTUM … PHYSICAL REVIEW A 100, 042118 (2019) first term was calculated, the second term was also evaluated by adding up mutual projection probabilities of the states mentioned above, and their fluctuation was also estimated by Poisson distribution. The measurement result is displayed in Fig. 3(c), yielding a value of −0.060 ±0.005 without correcting background noise from SNSPD and stray light. By adding the two terms together, the final result of quantum contextuality is IQC =6.488 ±0.025, exceeding the classical bound by about 19 standard deviations and the Tsirelson’s bound derived from Bell nonlocality by over 9 standard deviations. We thus successfully observed quantum contextuality beyond Bell nonlocality. IV. CONCLUSION We have studied the different behaviors between Bell nonlocality and quantum contextuality associated with the same exclusivity graph by comparing quantum violations of the I3322-type Bell inequality and the noncontextuality inequality. Quantum contextuality has been investigated in the five- and four-dimensional systems, and we have found that the gap between QC and BN is ≈0.3, which is experimentally observable. We have performed an experimental observation of the gap in a four-dimensional system to demonstrate the phenomenon of QC beyond BN. Within experimental deviation, the results are in good agreement with the theoretical predictions. Our results indicate a deep-rooted connection between graph theory and quantum correlations, and will further deepen the understanding of different types of quantum correlations. ACKNOWLEDGMENTS This work was supported by the National Key Research and Development Program of China (Grant No. 2016YFA0302700), the National Natural Science Foundation of China (Grants No. 61725504, No. 61490711, No. 11774335, and No. 11821404), the Key Research Program of Frontier Sciences, Chinese Academy of Sciences (CAS) (Grant No. QYZDY-SSW-SLH003), Science Foundation of the CAS (Grant No. ZDRW-XH-2019-1), Anhui Initiative in Quantum Information Technologies (Grants No. AHY060300 and No. AHY020100), and the Fundamental Research Funds for the Central Universities (Grants No. WK2030380017 and No. WK2470000026). H.-X.M. was supported by a project funded by the National Natural Science Foundation of China (Grant No. 11901317). H.-Y.S. was supported by the National Natural Science Foundation of China (Grant No. 11905209) and the China Postdoctoral Science Foundation (Grant No. 2018M630063). J.-L.C. was supported by the National Natural Science Foundations of China (Grant No. 11875167) and the Fundamental Research Funds for the Central Universities, Nankai University (Grant No. 63191507). J.Z. was supported by the Nankai Zhide Foundation. The processing of GaN samples was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication. Z.-H.L., H.-X.M., and Z.-P.X. contributed equally to this work. APPENDIX A: THE THEORETICAL PART 1. The Bell inequality and noncontextuality inequality associated with ICSW 3322 The symmetric version of the inequality I3322 ⩽0 proposed in Ref. [15]is I3322 LHV ⩽0,(A1) with I3322 =P(0,0|0,1) +P(0,0|0,2) +P(0,0|1,0) +P(0,0|1,2) +P(0,0|2,0) +P(0,0|2,1) −P(0,0|1,1) −P(0,0|2,2) −P(0,_|0,_) −P(0,_|1,_) −P(_,0|_,0) −P(_,0|_,1).(A2) Using the equation −P(Ax=a,Bx=b)=−1+ (a,b)=(a,b) P(Ax=a,By=b) (A3) to replace probabilities with minus signs by the corresponding positive probabilities, we obtain I3322 =I−6,(A4) where I≡ICSW 3322 =P(0,0|0,1) +P(0,0|0,2) +P(0,0|1,0) +P(0,0|1,2) +P(0,0|2,0) +P(0,0|2,1) +P(0,1|1,1) +P(1,0|1,1) +P(1,1|1,1) +P(0,1|2,2) +P(1,0|2,2) +P(1,1|2,2) +P(1,_|0,_) +P(1,_|1,_) +P(_,1|_,0) +P(_,1|_,1).(A5) TABLE I. Numerically optimal settings and optimal state in a five-dimensional system for IQC ≈6.588 412 87. |Aii1i2i3i4i5 |A100 0 01 |A20.538648 −0.458664 0.0322917 −0.706005 0 |A30.672155 −0.188775 −0.359722 0.619009 0 |A40 0.95651 0 0.2917 0 |A50.188775 −0.672155 −0.359722 −0.619009 0 |A60.192831 −0.671551 −0.44115 0.563224 0 |A70.912455 0 −0.409176 0 0 |A8−0.946087 0 0 0.323911 0 |A901 0 00 |A1010 0 00 |A110 0.912455 0.409176 0 0 |A120.95651 0 0 0.2917 0 |A1300 1 00 |A140−0.946087 0 0.323911 0 |A15−0.671551 0.192831 0.44115 0.563224 0 |A16−0.458664 0.538648 −0.0322917 −0.706005 0 |ψ−0.674033 0.674033 0.302255 0 0 042118-5 ZHENG-HAO LIU et al. PHYSICAL REVIEW A 100, 042118 (2019) Therefore, the inequality I3322 ⩽0 can be written as I LHV ⩽6.(A6) The maximal Bell nonlocality (BN) of inequality (A6) for two qubits is exactly IBN =6.25, which can be obtained in the following case: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ A0=B0=⎛ ⎝ √2−√3 2 √2+√3 2 √2+√3 2 √2−√6 4 ⎞ ⎠,A1=B2=⎛ ⎝ √2+√3 2 √2−√3 2 √2−√3 2−√2+√3 2 ⎞ ⎠,A2=B1=−1 √2 1 √2 1 √2 1 √2, ρ=1 2⎛ ⎜ ⎝ 1001 0000 0000 1001 ⎞ ⎟ ⎠=|++|,|+= 1 √2(|00+|11),I2=10 01 , P(a,b|i,j)=TrI2+(−1)aAi 2⊗I2+(−1)bBj 2ρ. (A7) For the exclusivity graph G(see Fig. 1) associated with I, a noncontextuality (NC) inequality is constructed with the form I= i∈V P|ψ(Ai=1) − (i,j)∈E P|ψ(Ai=1,Aj=1) NCHV ⩽6,(A8) where Vand Erespectively denote the vertex and edge sets of graph G,P|ψ(Ai=1) denotes the probability of obtaining result 1 when the observable |AiAi|is measured on the state |ψ, i.e., P|ψ(Ai=1) =|Ai|ψ|2,P|ψ(Ai=1,Aj=1), i.e., P|ψ(Ai=1)P|Ai(Aj=1), represents the experimental imprecision of |Ai⊥|Ajwhen vertex iand vertex jare connected in the graph. The optimal quantum contextuality (QC) of inequality (3) is just the Lovász number of G, i.e., IQC =ϑ(G)≈6.588 412 87, which is obtained in the five-dimensional quantum system. In Table I, we list the optimal settings and the optimal state numerically. Remark 4. If we take the following analytical settings and state in Table II , then the maximal QC is I= 16  i=1|i|ψ|2−0=[355 +209 cos(2τ)+112 sin(2τ)]/90,(A9) whose maximal point is τ=arctan[(−209 +5√2249)/112] with the maximum (71 +√2249)/18 ≈6.579 09. And if we choose τ=2π/25, then 16 i=1|i|ψ|2≈6.578 94. It is easy to see that this numerical contextuality is close to the optimal one with an error of 0.0014. 2. QC of Iin a four-dimensional system In the four-dimensional quantum system, we take |ψ=(cos [θ],sin [θ] cos [φ],sin [θ]sin[φ] cos [τ],sin [θ]sin[φ]sin[τ]) |Ai=(cos [θi],sin [θi] cos [φi],sin [θi]sin[φi] cos [τi],sin [θi]sin[φi]sin[τi]),(A10) with i=1,2,...,16, then we have P|ψ(Ai=1) =|Ai|ψ|2 =(cos [θi] cos [θ]+sin [θi]sin[θ](cos [φi] cos [φ]+sin [φi]sin[φ](cos [τi] cos [τ]+sin [τi]sin[τ])))2(A11) and max{I:θ,φ,τ,θi,φ i,τ i}≈6.571 52.(A12) To obtain Eq. (A12), it is sufficient to assume the settings and state have the form in Table III, and the angles are as follows: φ2≈1.015 602 970 716 640 5,φ 4≈1.180 253 821 474 319 2, φ3=arctan 2cosφ4 sin2φ4,φ 1=−arctan cos φ2cos φ3−sin φ2sin φ3sin φ4 cos φ4sin φ3,φ 5=−arctan cos φ1cos φ2+φ4 sin φ1.(A13) 042118-6 EXPERIMENTAL OBSERVATION OF QUANTUM … PHYSICAL REVIEW A 100, 042118 (2019) TABLE II. Analytical settings and state in a five-dimensional system for I. |Aii1i2i3i4i5 |A1000 01 |A21 2 1 20−1 √20 |A33 2√5 1 2√5 1 √10 2 50 |A402√2 30−1 30 |A51 2√5 3 2√5 1 √10 −2 50 |A61 2√5 3 2√5 1 √10 2 50 |A72√2 301 300 |A82√2 300−1 30 |A9010 00 |A10100 00 |A1102√2 3 1 300 |A122√2 300 1 30 |A13001 00 |A1402√2 301 30 |A153 2√5 1 2√5 1 √10 −2 50 |A161 2 1 201 √20 |ψcos τ √2 cos τ √2sin τ00 In this case, Table III becomes the following Table IV. In experiment, one can choose φ2=1.015,φ 4=1.180, and then the orthogonal relationship can be satisfied with errors 10−32, i.e.,  (i,j)∈E P|ψ(Ai=1,Aj=1) ≈9.629 65 ×10−33 <10−32, (A14) and max I≈6.571 52. TABLE III. Settings and state in a four-dimensional system for I. |Aii1i2i3i4 |A10010 |A2cos φ1cos φ2sin φ10−cos φ1sin φ2 |A3cos φ3cos φ4sin φ30sinφ3sin φ4 |A40sinφ40−cos φ4 |A5cos φ4sin φ3cos φ30−sin φ3sin φ4 |A6cos φ4cos φ5sin φ50cosφ5sin φ4 |A71000 |A8sin φ400−cos φ4 |A90100 |A101000 |A110100 |A12sin φ400cosφ4 |A130010 |A140sinφ40cosφ4 |A15sin φ5cos φ4cos φ50−cos φ5sin φ4 |A16sin φ1cos φ1cos φ20cosφ1sin φ2 |ψ1 √2 1 √200 TABLE IV. Numerically optimal measurement settings and state in a four-dimensional system for IQC ≈6.571 52. |Aii1i2i3i4 |A10010 |A20.469723 0.453743 0 0.757283 |A30.746839 0.253161 0 0.614931 |A40 0.924703 0 −0.38069 |A50.253161 0.746839 0 −0.614931 |A60.249899 0.754381 0 0.607009 |A71000 |A80.924703 0 0 −0.38069 |A90100 |A101000 |A110100 |A120.924703 0 0 0.38069 |A130010 |A140 0.924703 0 0.38069 |A150.754381 0.249899 0 −0.607009 |A160.453743 0.469723 0 0.757283 |ψ1 √2 1 √200 Remark 5. In fact, for φ2=π/3,φ 4=7π/18, we can obtain that the orthogonal relationship is strictly satisfied [(i,j)∈EP|ψ(Ai=1,Aj=1) =0] and max I≈6.562 33. APPENDIX B: THE EXPERIMENTAL PART 1. Testing experimental reliability of orbital angular momentum qudit The two experimental tests rely on a photonic polarization qubit and an OAM qudit, and for the latter one it is vital to confirm that the measurements performed in experiment ideally represent the terms in (3). Due to the complexity of a four-dimensional system, we opted out of a complete state tomography and, instead, chose to test the orthonormality of the six sets of complete bases used in our experiment. The Born rule declares that for an observable corresponding to a self-adjoint operator A, the probability of obtaining result λion state |ψis ψ|i|ψ, where iis the projection operator onto the eigenspace of Acorresponding to the eigenvalue of λi. It follows immediately from the Born rule that when we prepare an eigenstate of i, the probability of getting λiin the measurement Ais 1, while the probabilities of getting other λj=i’s are 0. For an ideal measurement, the complete set of bases should then be orthonormal, regardless of the exact bases settings we choose. Figure 4shows the result of testing orthonormality of the six sets of complete bases. When we prepared the state corresponding to one specific outcome and then measured the corresponding projector, we (almost) always obtained that certain outcome. Since the projectors to orthogonal bases commute with each other, the null counting rates between orthogonal states and projectors can also serve as proof of nondisturbance of compatible measurements. We thus infer that the measurements performed in the experiment are overall approximately ideal, and the data is suitable to be analyzed in the CSW framework using (3). 042118-7 ZHENG-HAO LIU et al. PHYSICAL REVIEW A 100, 042118 (2019) 0.2 0.4 0.6 0.8 1. 0. {1, 2, 3, 17} {1, 4, 7, 18} {1, 5, 16, 19} {14, 15, 23, 24} {1, 9, 12, 20} {6, 8, 21, 22} Eigenstate Projector #1 #2 #3 #4 #1 #2 #3 #4 #1 #2 #3 #4 #1 #2 #3 #4 #1 #2 #3 #4 {#1, #2, #3, #4} Representing: FIG. 4. Testing the orthonormality of the six sets of complete measurements used in the experiment by enumerating the detection probability between bases within each set. For example, the (#2,#3) slot in the darkest blue histogram shows the measured detection probability for |A7|A4|2. 2. Details about the single-photon source The sample used in this experiment was a 2-μm-thick magnesium (Mg)-doped gallium nitride (GaN) film on a 2-mm undoped GaN layer grown on sapphire substrate. This kind of sample was reported in [21], and we followed their practice to pretreat the sample with acetone and isopropanol (1:1) solution and clean in ultrasound bath. The defects distributed over the sample surface, and a confocal microscopy system was exploited to excite the sample and collect emitted single photons. In our experiment, a 532-nm continuous-wave laser was used to excite the GaN sample. The laser was reflected by a dichroic mirror focused onto the sample by an objective lens with high numeric aperture of 1.25. The fluorescence was collected by the same objective and filtered by the dichroic mirror and the bandpass filter. To examine attributes of the photoluminescence (PL) signal, it was coupled to a single-mode fiber and either guided to a SPAD for photon counting and spectrum measurement (the bandpass filter was removed for this measurement) or a Hanbury Brown-Twiss (HBT) interferometer setup for singlephoton verification. (1) The second-order autocorrelation was measured to confirm the single-photon attribute. The experimental value of g2(τ)atτ=0 was 0.225. As the dip fell below the threshold of 0.5, we confirmed that the PL emitter was a genuine singlephoton source. 760 770 780 790 800 0 1500 3000 4500 6000 PL Intensity Wavelength (nm) (a) (b) 0 50 100 150 200 0.0 0.4 0.8 1.2 1.6 Intensity (106counts/s) Time (s) 1mW 0.5 mW FIG. 5. The properties of the single-photon source used in the contextuality experiment. (a) The spectrum. The central wavelength of the emitter was 779.61 nm and the FWHM was 4.97 nm. (b) Photostability stats with respect to two different excitation powers, 0.5mW and 1 mW. 042118-8 EXPERIMENTAL OBSERVATION OF QUANTUM … PHYSICAL REVIEW A 100, 042118 (2019) 1.0 2.0 3.0 4.0 1.0 2.0 3.0 4.0 1.0 2.0 3.0 4.0 1.0 2.0 3.0 4.0 FIG. 6. Eliminating the possibility of signaling between sequential measurements. By exhibiting null results for the four signaling factors ε(0,_|vi,_),ε(1,_|vi,_),ε(_,0|_,vj), and ε(_,1|_,vj), the nonsignaling condition required in [19] is preserved. In each plot, the horizontal coordinate corresponds to experimental settings, i.e., different choices of states and projectors represented by edges of the exclusivity graph. The settings with primed notation swapped eigenstates and projectors. The four signaling factors are shown in four different colors. (2) Spectrum. Figure 5(a) shows the spectrum of the emitter we used in the contextuality experiment with the background subtracted. It had a central wavelength of 779.61 nm and a FWHM of about 4.97 nm. (3) Photostability. The PL intensity versus time under continuous-wave laser excitation was recorded and plotted in Fig. 5(b). The standard deviation of photon counting during a span of 200 s was below 1.5%. As a result, the photon source was narrow linewidth, bright, and stable. The photon source was then connected to the contextuality experiment setup by SMF. 3. Testing nonsignaling condition of sequential measurements The nonsignaling condition, indicating that local marginal probabilities of Alice are independent of Bob’s measurement setting, and vice versa [24], was also checked for testing contextuality in the CSW approach, as suggested in [19,20]. From [19], the statistics of the second measurements vjaffected by the first measurements vican be calculated as ε(_,0|_,vj)=|P|ψ(vj=0) −P|ψ(vi=0,vj=0) −P|ψ(vi=1,vj=0)|,(B1a) ε(_,1|_,vj)=|P|ψ(vj=1) −P|ψ(vi=0,vj=1) −P|ψ(vi=1,vj=1)|,(B1b) ε(0,_|vi,_) =|P|ψ(vi=0) −P|ψ(vi=0,vj=0) −P|ψ(vi=0,vj=1)|,(B1c) ε(1,_|vi,_) =|P|ψ(vi=1) −P|ψ(vi=1,vj=0) −P|ψ(vi=1,vj=1)|.(B1d) 042118-9