Abstract
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:
Figure 1.Exclusivity graph of the
In contrast, quantum theory predicts that for a four-dimensional quantum state, when measurements satisfy Eq. (1), the success probability could be
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 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
3. STRONGER NONCONTEXTUALITY INEQUALITY
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
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
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,
Figure 4.Quantum violation of Eq. (
5. CONCLUSION
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,
The conditions in Eq. (
APPENDIX B: MEASUREMENTS OF THE STATE FOR THE HARDY-LIKE PROOF OF QUANTUM CONTEXTUALITY WITH n MEASUREMENTS
For the -vertex exclusivity graph (Fig.
Remark 1. The measurement vectors can be obtained based on the symmetry of our exclusivity graph (Fig.
APPENDIX C: COMPARISON BETWEEN THE SUCCESS PROBABILITIES PSUC AND PCBCB
The success probability is described by Eq. (
Table
Figure 5.Success probabilities
From Table
APPENDIX D: STATE AND MEASUREMENTS FOR THE MAXIMUM QUANTUM VALUE OF In
For the noncontextuality inequality with , Quantum State Here Numerical Value of Quantum Bound For the optimal values of the parameters 1 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0 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 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
Remark 2. The maximum quantum violation in Eq. (
APPENDIX E: COMPARISON OF DEGREE OF QUANTUM CONTEXTUALITY BETWEEN OUR INEQUALITY AND THE EXTENDED KCBS INEQUALITY
The degree of quantum contextuality is given by [
The extended KCBS inequality reads [
Then one can obtain the degree of quantum contextuality for the extended KCBS inequality as
In Table Numerical Values of the Ratios – 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
For any , the inequality in Eq. (
APPENDIX F: OPTIMAL NONCONTEXTUALITY INEQUALITY WITH WEIGHTS OF c=2 AND d=1
The exclusivity graph in Fig.
Now we would like to discuss on the optimization of Eq. (
For the proof, the state , and the measurement vectors shown in Table
Figure 7.Different regions for
Remark 3. Here we would like to provide a detail proof for the case of . In this case, the quantum bound becomes
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The quantum-classical ratio is
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. (
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. (
APPENDIX G: EXPERIMENTAL DETAILS
Equally weighted superpositions of LG modes with different OAMs can be written as
First, we test the efficiencies of two cascaded QPs with topological charges and in our experiment, as shown in Table Efficiencies of the Two Cascaded QPs Sandwiched between Two QWPsInput State Output State Efficiency 66.9% 66.8% 66.5% 66.7%
Second, it is necessary to satisfy the no-signaling conditions [
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. (
Figure 8.Experimental results for
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