A Hardy-like proof of quantum contextuality is a compelling way to see the conflict between quantum theory and noncontextual hidden variables (NCHVs), as the latter predict that a particular probability must be zero, while quantum theory predicts a nonzero value. For the existing Hardy-like proofs, the success probability tends to 1/2 when the number of measurement settings goes to infinity. It means the conflict between the existing Hardy-like proof and NCHV theory is weak, which is not conducive to experimental observation. Here we advance the study of a stronger Hardy-like proof of quantum contextuality, whose success probability is always higher than the previous ones generated from a certain -cycle graph. Furthermore, the success probability tends to 1 when goes to infinity. We perform the experimental test of the Hardy-like proof in the simplest case of by using a four-dimensional quantum system encoded in the polarization and orbital angular momentum of single photons. The experimental result agrees with the theoretical prediction within experimental errors. In addition, by starting from our Hardy-like proof, one can establish the stronger noncontextuality inequality, for which the quantum-classical ratio is higher with the same , which provides a new method to construct some optimal noncontextuality inequalities. Our results offer a way for optimizing and enriching exclusivity graphs, helping to explore more abundant quantum properties.
1. INTRODUCTION
Ideal measurements yield the same outcome when repeated, even when other compatible measurements have been performed in between. Some predictions of quantum theory for ideal measurements cannot be explained under the assumption that ideal measurements reveal preexisting outcomes that are independent of the context (i.e., the set of compatible measurements that have been jointly measured), this is known as quantum contextuality or Kochen–Specker (KS) contextuality [1]. Quantum contextuality is an intrinsic signature of nonclassicality as its classical simulation requires memory [2] and thermodynamical cost [3]. Moreover, it has been proven that quantum contextuality is necessary for fault-tolerant quantum computation via magic state distillation and measurement-based quantum computation [4]. Quantum contextuality also plays a fundamental role in some quantum key distribution protocols [5] and is crucial for understanding the underlying physics behind the limitation of quantum correlations [6]. So far, quantum contextuality has been experimentally observed in trapped ions [7], photons [8], ensembles of molecular nuclear spins in the solid state [9], and superconducting systems [10].
KS contextuality is the first theory about contextuality. The KS-type proof [1,11,12] serves as a no-go theorem, indicating that it is impossible for the noncontextual hidden variable (NCHV) models to describe quantum mechanics. For instance, for Peres-33 rays [11], it is impossible to define any consistent assignment of 0s and 1s to the states in the set. In other words, the standard KS sets cannot be colored in a consistent way in classical theory. Beyond all doubt, the KS-type proof is an elegant argument for contextuality, which is similar to the Greenberger–Horne–Zeilinger (GHZ) proof for the Bell nonlocality. Throughout research history, many well-known scientists have done excellent work on KS contextuality. The first proof scheme of KS includes 117 ray quantities in three-dimensional space, and that with less rays is found step by step [11–14]. Those that we are familiar with are Peres-33 rays [11], CEG-18 rays [12], and so on. There are some that also have other famous research, for example, the Peres–Mermin square—a considerable simplification of the original KS argument by Mermin and Peres using nine observables that are organized in a square [15,16]. Later, noncontextuality inequality has been proposed as an experiment-friendly approach to reveal quantum contextuality and has been tested by many experiments [7,12,14,17,18]. We will talk about noncontextuality inequality in detail in the next paragraph. In general, a KS-type proof must be transferred to a noncontextuality inequality to be experimentally validated, and experimental tests of contextuality based on standard KS-type proofs were seldom performed in the literature, which may be due to this reason: noncontextuality inequalities are straightforward for the Klyachko–Can–Binicioğlu–Shumovsky (KCBS) cases [19], but in the KS case, the meaning is less clear because value assignments are logically inconsistent, and it is not clear how to compare them with quantum prediction.
The graph-theoretic approach to quantum correlations bridges a fundamental connection between graph theory and quantum contextuality [20,21]. The central idea is that, for an arbitrary exclusivity graph, [Note: Any graph is made up by vertices and edges, where is a set of vertices and is the set of edges. The exclusivity graph described that the measurement events are represented as the vertices of the graph, and there is an edge between vertex and vertex if events and are mutually exclusive events, and the corresponding measurement vectors and are mutually orthogonal.] one can always associate it with a noncontextuality inequality, for which the classical bound has a definite meaning as the independence number of the graph, and the maximum quantum violation of the inequality is the Lovász number describing the Shannon capacity of the graph. The noncontextuality inequality is a correlation inequality that any theory should satisfy under the assumption of outcome noncontextuality for ideal measurements. The typical examples are the KCBS inequality [12] and its extensions [22] and some state-independent-contextuality (SIC) inequalities [14,23]. The noncontextuality inequalities have these advantages: (i) theoretically, quantum contextuality can be revealed in a direct way by the violations of noncontextuality inequalities; (ii) compared with the KS-type proofs, the violations of noncontextuality inequalities are more feasible to observe quantum contextuality in the experiments.
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A natural result yielded from the graph-theoretic approach is the so-called “Hardy-like proof” [24], which for quantum contextuality is analogous to Hardy’s proof for Bell nonlocality [25] and is also a particularly compelling way to reveal contextuality. A Hardy-like proof of contextuality may be seen as a particular violation of the noncontextuality inequality, which is more experimentally friendly than noncontextuality inequalities sometimes. The first Hardy-like proof of contextuality was proposed in Ref. [24] by studying the -cycle graphs. In such a proof, the NCHV theory predicts that the success probability of a particular event must be zero, while quantum theory predicts a nonzero value. For the simple Hardy-like proof of the five-cycle graph, has been experimentally observed [26]. Remarkably, for the -cycle graphs, tends to 1/2 when the number of measurement settings goes to infinity [24]. The conflict between the simple Hardy-like proof and NCHV theory has not reached the limit. A natural question is whether there is a Hardy-like proof for a certain -vertex graph, in which tends to 1 when , thus providing the stronger quantum contextuality.
In this work, a Hardy-like proof was presented for the graphs with vertices, in which the success probability tends to 1 when the number of measurement settings goes to infinity. For a fixed , the new Hardy-like proof is stronger than the previous one for the -cycle graph. To illustrate our idea, we experimentally test the Hardy-like proof with in the simplest case of , by using a four-dimensional quantum system encoded in the polarization and orbital angular momentum (OAM) of single photons. For some symmetric graphs, by starting from the Hardy-like proofs, one can establish some stronger noncontextuality inequalities, for which the quantum-classical ratio is higher or optimal, and this provides a novel way to construct some useful noncontextuality inequalities.
2. STRONGER HARDY-LIKE PROOF
Considering ideal measurements represented in quantum theory by the rank-one projectors and with possible outcomes 0 and 1, some of these measurements are jointly measurable, while some of the corresponding events are mutually exclusive. The -vertex exclusivity graph is shown in Fig. 1. Supposing that there is a state (), its probabilities for the triangles in Fig. 1 satisfy the following Hardy-like constraints: i.e., the sum of the probabilities of each triangle is 1. is the event corresponding to outcome 1 obtained for the measurement in the state , no matter which other compatible measurements are performed. Under these assumptions, any NCHV theory predicts that the outcome of the measurement corresponding to the vertex in the center of Fig. 1 must be zero. That is to say, the NCHV theory predicts . This is due to the fact that, in the NCHV theory, Eq. (1) implies that at least one of the events that are mutually exclusive with has happened for any given hidden variable. Thus, the event can never happen whatever the hidden variable is, i.e., . In addition, for the odd number () outer vertices in the polygon, not all of those values can be zero; hence, at least one of the inner values has to be 1, and vertex must be a value of 0. See Appendix A for a detailed proof for the NCHV value of the Hardy-like paradox.
Figure 1.Exclusivity graph of the measurements (with ) used for the Hardy-like proof of contextuality. Each vertex represents a measurement. Vertices connected by an edge are jointly measurable. Each of the outer vertices belongs to two triangles. In total, there are triangles.
In contrast, quantum theory predicts that for a four-dimensional quantum state, when measurements satisfy Eq. (1), the success probability could be Clearly, the probability tends to 1 when and is 1/4 for the simplest case of . Moreover, the probability is very close to 1 even for a relatively small [e.g., for , ]. The state and the measurement vectors for are listed in Table 3 (see Appendix B), and the detailed comparison of with the success probability obtained in the -cycle Hardy-like proof [24] is shown in Table 4 (see Appendix C).
Table 1 lists the state and the corresponding projectors , and we have for . The exclusivity relations between the projection measurements are given by the graph in Fig. 2. In addition, three annex points (denoted by ) are added due to the need of experimental test for the compatibility conditions.
State and the Measurement Vectors (), Where Are Annex Measurement Vectors
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
Figure 2.Exclusivity relations between the projection measurements in the Hardy-like proof for , including the added measurements (8, 9, 10) used in the experiment. The black vertices are twice of the white vertices in weight.
Every Hardy-like proof of quantum contextuality can be considered as a violation of a noncontextuality inequality [24]. Hence, a Hardy-like proof can be used as the starting point for identifying new noncontextuality inequalities. An interesting issue is identifying situations in which few measurement settings can produce a high degree of contextuality. The issue is of great importance for applications in which the degree of contextuality has one-to-one correspondence with the quantum-versus-classical advantage [27]. To characterize the degree of quantum contextuality, it is defined as the ratio between the maximum quantum violation and the noncontextual bound [27]. When performing an operation by the Hardy-like proof presented above, we select the weights of the vertices in the symmetric graph (Fig. 1) for achieving the optimal degree of quantum contextuality. Thus, we obtain the following noncontextuality inequality as This inequality is violated by the quantum state in the four-dimensional quantum system. Notice that, in Fig. 1, black vertices have a weight of 2 while white vertices have a weight of 1 in Eq. (3). For , the noncontextual bound of is 3, while the maximum quantum violation is . This maximum violation can be achieved by the state , and the measurements correspond to the projectors onto the seven vectors of , , , , , , where , , , , , (see Appendix D for the cases of higher ).
Equation (3) has remarkable properties, as follows. (i) For any , Eq. (3) leads to a higher degree of contextuality than the one obtained by the extended KCBS noncontextuality inequality (see Appendix E). The extended KCBS inequality is the default target inequality in the contextuality experiment [26] and is also the inequality that naturally follows from the Hardy-like proof in Ref. [24]. (ii) For the simplest case (), numerical computation shows that the inequality (3) leads to a higher degree of contextuality than other noncontextuality inequalities in form , where is the weight of the rank-one projector for the th vertex, and is the classical bound. For , the maximum degree of contextuality for the extended KCBS inequality is 1.106 [22], while for inequality , it is 1.124. For some of the discussions, see Appendix F. This property reveals that searching for situations, in which a few measurement settings lead to a high degree of contextuality, may be beneficial for investigating symmetric graphs and attributing different weights to each of the classes of vertices.
4. EXPERIMENTAL RESULTS
The new Hardy-like proof and the corresponding inequality can reveal the contextuality in a new system because they provide better contextuality witnesses than the extended KCBS inequality under the same number of measurement settings. We will test the Hardy-like proof (for the simplest case of ) in a hybrid four-dimensional quantum system defined by the polarization and OAM of single photons. Testing contextuality is especially difficult as sequential measurements are not easy to implement. To overcome this difficulty, a simple method (which needs two measurements only) has been presented, in which the first measurement can be simulated by a demolition measurement followed by a preparation that depends on the outcome [28]. Of course, the necessary no-signaling conditions [29] also need to be satisfied. Based on this method, in order to test the Hardy-like proof in Eqs. (1) and (2) for , we should add mutually orthogonal vertices to form complete sets. Figure 2 shows that , and have formed three complete sets.
The experimental setup (Fig. 3) consists of two parts: the state preparation and the projection measurement. In the state preparation, photon pairs are produced via a type-II spontaneous parametric downconversion in a 10-mm-long periodically poled potassium titanyl phosphate (ppKTP) crystal pumped by a 405 nm cw diode laser, and one photon serves as a trigger. The produced photons can carry the discrete OAM of [30] ( is an arbitrary integer, and is the reduced Planck constant). Therefore, high-dimensional information can be encoded in the OAM of single photons [31]. As the original work on OAM [32], we consider an OAM carried by Laguerre–Gaussian (LG) mode of azimuthal order and radial order in our experiment. Equally weighted superpositions of LG modes with different OAMs ( and ), and the relative phase can be written as
Figure 3.Experimental setup. In the state preparation part, a 405 nm cw laser pumps a type-II ppKTP crystal (not shown) to create photon pairs. One photon serves as a trigger. The other photon is projected into the horizontal polarization state with a polarizing beam splitter (PBS); the spatial light modulator (SLM) combines a Rochi grating (RG) through two 4f systems to generate the ququart subset of OAM. In the projection measurement part, two sets of -plates (QPs, with different topological charges of and ) are sandwiched by two quarter-wave plates (QWPs), followed by a half-wave plate (HWP) and a PBS used to convert OAM mode into a fundamental mode that is coupled into a single mode fiber (SMF); the photons are detected by a single photon avalanche photodiode (SPAD).
We generate the ququart states based on the interferometric superposition of horizontally (H) and vertically (V) polarized photons carrying OAMs, which is similar to that in Ref. [33]. The collimated photons at 810 nm are split into two paths by a beam splitter (BS). Two computer-generated holographic gratings (HG1 and HG2) displayed on a spatial light modulator (SLM) () diffract the H-polarized light into different diffraction orders. Then a Rochi grating (RG) is used to recombine the st diffraction order of HG1 and the st diffraction order of HG2 into the single one, as shown in Fig. 3. The weight and phase of two different OAMs can be controlled by adjusting the HGs. Thus, we use the two-dimensional orthogonal polarizations and four-dimensional OAMs ( and ) to build four-dimensional subspace . Finally, the prepared state can be expressed as follows:
In the projection measurement, we perform the projection measurements by two -plates (QPs) with topological charges of and , respectively. The function of the QP can be described as , [34], where and denote left- and right-handed circular polarizations, respectively. Two cascaded QPs sandwiched between two quarter-wave plates (QWPs) are used to convert into a mode, which is easily coupled to a single photon avalanche photodiode (SPAD) through the single mode fiber (SMF).
In order to measure the probability for in Eq. (2), we need to introduce two additional vertices (, ) to form a complete set , where and . Here we describe the observables by . For the sake of simplicity, is replaced by . We first perform the verification of no-signaling between two measurements. We use the equations given in Ref. [22] to characterize the influence, which represent the statistical effects of the first measurement on the second measurement, indicating that the result of projecting the state onto is 0 and 1, respectively. In the same way, the statistical effects of the second measurement on the first should be To obtain , we need to prepare the orthogonal state based on the Lüder rule [35]. Our experiment meets the no-signaling condition within the experimental accuracy, and (see Appendix G). The projection probability and the joint measurement probability should be measured in their corresponding complete sets.
Figure 4.Quantum violation of Eq. (3) for . NCHV () represents the classical bound of the NCHV theory. The maximum violation is by using the optimal measurement settings, and the non-maximum violation is by using the measurements adopted in the Hardy-like proof.
Stronger quantum contextuality is particularly significant as it reveals a sharper contradiction between the NCHV theory and the quantum theory. In this work, we have advanced the study of a stronger Hardy-like proof of quantum contextuality. We have theoretically presented a new Hardy-like proof, which is stronger than the previous one given in Ref. [24]. For the simplest case (), the success probability equals 1/4, and it tends to 1 when . We have also performed the experimental test for the new Hardy-like proof by the four-dimensional quantum system encoded in the polarization and OAM of single photons. The experimental result gives , which agrees with the theoretical prediction within the experimental errors. Importantly, the stronger Hardy-like proof can yield stronger noncontextuality inequality and has an advantage over the extended KCBS noncontextuality inequality for the -cycle graphs. Our results not only advance the study of the Hardy-like proof for quantum correlations but also open a new approach to observe quantum contextuality in new systems. That also paves a way for further research on fundamental quantum resources.
APPENDIX A: PROOF FOR THE NCHV VALUE FOR THE HARDY-LIKE PARADOX
For simplicity, we use to represent .
For the NCHV case, once the hidden variable is given, each ray, which can be viewed as a dichotomic projective measurement, can be either associated with the value 0 or 1. To put it more explicitly, where is the set of hidden variables, , and if , are connected in the graph,
The conditions in Eq. (1) can be rewritten as In the NCHV case, it implies that, for any hidden variable , we have By taking the sum of the above equations, we have According to the relation in Eq. (A2), we can know that which implies that That is, for any hidden variable , there are always at least one such that . Again, because of the relation in Eq. (A2), we have . Consequently,
APPENDIX B: MEASUREMENTS OF THE STATE FOR THE HARDY-LIKE PROOF OF QUANTUM CONTEXTUALITY WITH n MEASUREMENTS
For the -vertex exclusivity graph (Fig. 1), based on whose symmetry and the mutually exclusive relationship of vertices, we can set the measurements form as shown in Table 3. Via calculation, we can get the parameters. Then, there are four-dimensional quantum state and measurements , which are listed in Table 3. Based on that, one can have the success probability for the Hardy-like proof as From Eq. (B1), we can find that the success probability tends to 1 when goes to infinity.
Remark 1. The measurement vectors can be obtained based on the symmetry of our exclusivity graph (Fig. 1). The state and the measurements listed in Table 3 for are equivalent to the ones given in Table 1 under the unitary transformation.
APPENDIX C: COMPARISON BETWEEN THE SUCCESS PROBABILITIES PSUC AND PCBCB
The success probability is described by Eq. (B1) or Eq. (2). In Ref. [24], Cabello, Badzia, Cunha, and Bourennane (CBCB) have proposed a simple Hardy-like proof of contextuality, which is generated from the -cycle graph. The corresponding success probability is given by where and .
Table 4 lists the numerical values of the success probabilities and for different , where . From Table 4, we have also plotted the curves of these two success probabilities versus in Fig. 5.
Figure 5.Success probabilities (blue curve) and (red curve) versus .
From Table 4 and Fig. 5, we can find that the success probability is higher than . In Ref. [24], the probability tends to 1/2 when . However, in our work, the probability has exceeded 1/2 for , i.e., . In addition, the success probability tends to be 1 when goes to infinity. Interestingly, for a finite number of settings, is already very close to 1. For example, for .
APPENDIX D: STATE AND MEASUREMENTS FOR THE MAXIMUM QUANTUM VALUE OF In
For the noncontextuality inequality with , The noncontextual bound is , and the maximum quantum violation can be achieved when we choose the four-dimensional quantum system state (here ) and measurements , as shown in Table 5. The vectors in Tables 3 and 5 share the similar structure. By optimizing two parameters, and , one can obtain the numerical value of quantum bound as shown in Table 6.
Quantum State and Measurements , for the Maximum Quantum Value of a
1
0
1
0
1
0
1
0
1
0
0
0
0
1
0
0
Here are the projectors onto the vectors (non-normalized) and , , , and .
Numerical Value of Quantum Bounda
7
11
15
19
23
27
31
0.269
0.198
0.158
0.131
0.112
0.098
1.106
1.219
1.285
1.329
1.362
1.386
5.730
7.849
9.903
11.932
13.950
15.962
35
39
43
63
83
103
0.087
0.079
0.071
−0.049
−0.037
0.030
1.406
1.421
1.434
1.666
1.644
1.512
17.970
19.975
21.980
31.990
41.994
51.996
For the optimal values of the parameters and , one can obtain the maximum quantum violation in Eq. (D1).
Remark 2. The maximum quantum violation in Eq. (D1) is obtained in the four-dimensional quantum system in Table 5. The state and the measurement vectors are obtained based on the symmetry of our compatibility graph. Except for the inequality for , the other cases are hard to find analytical results. We list the maximum quantum violation in Table 6.
APPENDIX E: COMPARISON OF DEGREE OF QUANTUM CONTEXTUALITY BETWEEN OUR INEQUALITY AND THE EXTENDED KCBS INEQUALITY
The degree of quantum contextuality is given by [27] which is the ratio between the maximum quantum violation and the noncontextual bound.
The extended KCBS inequality reads [24] where and is odd. Here the classical bound is and the maximum quantum value can be realized in the three-dimensional quantum system as
Then one can obtain the degree of quantum contextuality for the extended KCBS inequality as
In Table 7, we list and with different for the inequalities in Eqs. (D1) and (E2), respectively. From Table 7, we plot the curves of versus for the two inequalities in Eqs. (D1) and (E2), as shown in Fig. 6.
Numerical Values of the Ratios and with for the Inequalities in Eqs. (D1) and (E2)
5
7
11
15
19
23
27
31
35
39
43
63
83
103
–
1.124
1.146
1.121
1.100
1.085
1.073
1.064
1.057
1.051
1.047
1.032
1.024
1.020
1.118
1.106
1.077
1.060
1.048
1.041
1.035
1.031
1.027
1.025
1.022
1.016
1.011
1.010
Figure 6.Curve of the ratio versus . The red curve represents the relationship of with in our paper, while the blue curve represents the relationship of with for the extended KCBS noncontextuality inequality.
For any , the inequality in Eq. (D1) leads to a degree of quantum contextuality higher than that obtained by the extended KCBS noncontextuality inequality in Eq. (E2). Interestingly, the maximum degree of quantum contextuality of the KCBS inequality with is also lower than of our inequality with .
APPENDIX F: OPTIMAL NONCONTEXTUALITY INEQUALITY WITH WEIGHTS OF c=2 AND d=1
The exclusivity graph in Fig. 1 of the main text is a family of symmetric -vertex graphs. Since the symmetry, vertices in the outer circle should possess the same weight, which is denoted by . Similarly, the vertices in the inner circle should possess the same weight, which is denoted by . We then have a more general noncontextuality inequality as where and the weights . Notice that in Eq. (F1), for simplicity, three weights have been reduced, and the weight of is set to 1.
Now we would like to discuss on the optimization of Eq. (F1) within the framework related to the symmetric graphs. For the noncontextuality inequality [Eq. (F1)], a question is raised: what are the optimal values of and when the quantum-classical ratio is the highest? The answer will be and as the following proof.
For the proof, the state , and the measurement vectors shown in Table 3 in the main text, from the left-hand side of Eq. (F1), we can directly obtain the quantum value as By analyzing the deterministic probability assignments for the NCHV case, we can have So the quantum-classical ratio is Notice that is a linear function of and for any given as shown in Eq. (F2) and is also a linear function of and in different regions as illuminated in Fig. 7. Therefore, is a monotone function of and in each region. This implies that the maximum value of for given should appear at a certain boundary point within a region. A direct calculation shows that for all the cases except for . When and , Eq. (F1) degenerates into Eq. (3) in the main text, and then the quantum violation of the Hardy-like proof implies that which is easily obtained directly by adding Eqs. (1) and (2) in the main text. Since in this case, . Thus, we prove that the maximum value of can be obtained when only.
Figure 7.Different regions for , where are boundary points, , and .
Remark 3. Here we would like to provide a detail proof for the case of . In this case, the quantum bound becomes and the noncontextual bound is Thus, the quantum-classical ratio reads
From the above formula, we know that the larger , the smaller . Then can obtain the maximum value when . The quantum-classical ratio is
This is the same as Eq. (F15), and the maximum value of can be obtained when and .
Now we have finished the proof for the case of .
Remark 4. In addition, for the case of , we can also calculate the values of the inequality of the other seven vertices, and it shows that Eq. (3) in the main text leads to a higher degree of quantum contextuality than any other noncontextuality inequality in the form , where is the weight of the rank-one projector for the th vertex and is the classical bound. More intuitively, the maximum degree of quantum contextuality for the extended KCBS inequality with is 1.106 [19], while the maximum degree of quantum contextuality for our inequality with is 1.124, which is a higher degree of quantum contextuality under the same number of measurement settings.
APPENDIX G: EXPERIMENTAL DETAILS
Equally weighted superpositions of LG modes with different OAMs can be written as where denotes the relative phase between the two modes. In our experiment, the st and st diffraction orders of the blazed gratings carry the OAMs we needed. A non-optimized initial state we obtained experimentally can be expressed as where the two weights ( and , here be not normalized) and the phase can be controlled by the holographic grating, and the phase is controlled by the RG. After optimizing the initial state, we obtain the state we needed, as described in Eq. (5).
First, we test the efficiencies of two cascaded QPs with topological charges and in our experiment, as shown in Table 8. We can see that the two cascaded QPs have almost the same efficiency for the OAMs needed in our experiment. This is the guarantee of realizing projection measurements. In the experiment, the detection efficiency of SPAD is about 60% at 810 nm. Therefore, the overall detection efficiency of the OAM system is about 40%. The detection efficiency can be improved by high efficiency detector and coating -plates.
Efficiencies of the Two Cascaded QPs Sandwiched between Two QWPs
Input State
Output State
Efficiency
66.9%
66.8%
66.5%
66.7%
Second, it is necessary to satisfy the no-signaling conditions [29] in any quantum contextuality. As mentioned in Ref. [22], the influences can be written as where Equation (G3) represents the statistical effect of the first measurement on the second measurement, indicating that the result of projecting the state onto is 0 and 1, respectively. Meanwhile, Eq. (G4) indicates the statistical effect of the second measurement on the first measurement, implying that the result of projecting the state onto is 0 and 1, respectively. In order to obtain , we need to prepare the orthogonal state based on the Lüder rule [30], where is the identity matrix. In our experiment, the orthogonal states are shown as
As stated in the main text, all the measurements should be implemented in a complete set. In our experiment, the statistics of each count is assumed to follow the Poisson distribution. The measured value for any quantity is obtained by averaging the 100 randomly grouped counting sets. The errors are evaluated by the standard deviation. Based on the experimental data, we easily calculate the probabilities. Substituting Eq. (G5) into Eq. (G4), we get . The errors described in Eq. (G4) determine the experimental precision. The influence results shown in Fig. 8 mean no-signaling between the first and second measurements.
Figure 8.Experimental results for (red circles) and (black squares) in (a) and for (red circles) and (black square) in (b). The numbers of 1 to 24 (even numbers are not marked in figures) represent the settings of , , , , , , , , , , , , , , , , , , , , , , , , and . Error bars are calculated from the counting statistics.
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