1. INTRODUCTION

As basic mathematical objects, graphs have been studied extensively, for they not only demonstrate purely scientific aesthetics but also have many important applications in all branches of science. Very recently, Cabello et al. [1] have discovered that there is a fundamental connection between graphs and the contextual correlations between the outcomes of compatible experiments on quantum systems (i.e., contextuality [2,3]). For any graph one can imagine, there exist a quantum system and a set of projective measurements producing “events”—characterized by the outcomes of a set of compatible projective measurements—which are in one-to-one correspondence with the vertices of the graph, such that whenever there is an edge between the vertices, any pair of these events correspondingly are exclusive (i.e., cannot happen at the same time, or can be distinguished by a projective measurement) [4,5]. More precisely, the central idea of the graph-theoretic approach to quantum correlations is that an arbitrary $N$-vertex graph can always be associated with a noncontextuality inequality (NC inequality):

In fact, the sum of the physically possible probabilities of these events necessarily corresponds to a point of the Grötschel–Lovász–Schrijver theta body of the graph, the latter being a set of purely combinatorial objects [9]. In particular, the maximum value of the sum is exactly the Lovász number (usually denoted by $\vartheta$) of the graph, which was originally introduced as an easy-to-approximate number between two hard-to-approximate numbers in graph theory [10]. Such a correspondence between graphs and quantum-correlation experiments is remarkable, as it (i) places some bound on the quantum set [11], (ii) suggests that correlations in nature are essentially bounded by a simple physical principle (the exclusivity principle [12–14]) valid in all scenarios, instead of the ones only applied within Bell’s scenario, and (iii) has been a key to explore quantum supremacy in quantum computation [15–18]. The correspondence has thus far been proved useful to write out interesting forms of contextual correlation through a series of work [19–25].

In this work, we develop the exact forms of contextual relations from Platonic graphs. By strictly measuring the contextual correlations between compatible observables on quantum four-dimensional states encoded in the spatial modes of single photons generated from a defect in a bulk silicon carbide, we experimentally demonstrate the quantum contextuality of the icosahedron graph, which gives rise to the largest quantum-classical difference.

2. THEORETICAL FRAMEWORK

We start with graphs associated with Platonic solids, or to shorten it, Platonic graphs, i.e., the five graphs that have each of the five Platonic solids as skeletons. The details can be found in Sections 1 and 2 in Supplement 1. To have the Platonic graphs tested in a quantum-correlation experiment with the satisfactory compatibility requirements, we adopt the following alternative NC inequality [26]:

In this work, we choose to experimentally test contextual correlations from the icosahedron graph as shown in Fig. 1. The reasons are threefold. (i) Among the five Platonic graphs, the ratio of the quantum violation ($\vartheta$) to the classical bound ($\alpha$) is maximal for the icosahedron graph. The larger the ratio, the more friendly the experimental observation. (ii) To our knowledge, the ratio $\vartheta /\alpha$ of the icosahedron graph is maximal for all graphs that we have known, at least for all regular polyhedrons in any dimension (including all regular polygons in two dimensions). (iii) For the icosahedron graph, almost all four-dimensional states (except the maximally mixed state) violate the NC inequality, thus one can reveal the contextuality for almost all four-dimensional states by a single inequality. To a certain sense, such an experimental revealing has merit over the state-independent contextuality proof with 18 rays [27,28], because experimentally the former requires fewer projective measurements but with larger quantum violations.

 

Fig. 1. Icosahedron graph. (a) The icosahedron solid with 12 vertices. (b) The 12 blue vertices depict the corresponding icosahedron graph. Four red vertices are added for the need of experimental test for the compatibility conditions. (c) The corresponding optimal quantum initial state $|\psi ⟩$ and measurement settings $|i⟩$. The set $\{|1⟩,|2⟩,\dots ,|12⟩\}$ forms the Lovász optimum orthogonal realization. $|i⟩$ (unnormalized vector for simplicity) denotes $|A_i⟩$. $\tau =\sqrt{\varphi }$, $\varphi =\frac{1}{2}(\sqrt{5}+1)$, ${\tau }^{\prime }=1/\tau$, ${\varphi }^{\prime }=1/\varphi$, $\omega =\sqrt{2+\sqrt{5}}$, and $\sum _{i=1}^{12}|⟨\psi |A_i⟩{|}^2=3(\sqrt{5}-1)$ give the quantum maximum that equals to the Lovász number. (d) The independence number $\alpha$ and the Lovász number $\vartheta$ of the icosahedron.

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Explicitly, we shall test the NC inequality corresponding to the icosahedron graph as follows:

Because the quantum system we utilize in experiment is four dimensional, we further extend the graph with extra projectors, $A_{13}$, $A_{14}$, $A_{15}$, and $A_{16}$, to construct full sets of compatible observables, such that every projector is measured as part of a complete basis [26]. In summary, the sets of observables are $\{A_1,A_9,A_{10},A_{13}\}$, $\{A_2,A_7,A_{12},A_{14}\}$, $\{A_3,A_4,A_6,A_{15}\}$, and $\{A_5,A_8,A_{11},A_{16}\}$. The method to obtain detailed forms of the vectors can be found in Section 2 in Supplement 1. Each of the measurements can then be performed in the same orthogonal basis. The detection probability of each observable such as $A_1$ can be obtained as $P_{|\psi ⟩}(A_1=1)=\frac{N_{|\psi ⟩}(A_1)}{N_{|\psi ⟩}(A_1)+N_{|\psi ⟩}(A_9)+N_{|\psi ⟩}(A_{10})+N_{|\psi ⟩}(A_{13})}$, where $N_{|\psi ⟩}(A_i)$ is the number of counts of obtaining result 1 when the observable $|A_i⟩⟨A_i|$ is measured on the state $|\psi ⟩$. Then, by measuring the contextual correlations between compatible observables on quantum four-dimensional states encoded in the spatial modes of single photons generated from a defect in a bulk silicon carbide, we observe the experimental proof-of-principle of this result.

3. EXPERIMENTAL SETUP AND RESULTS

Figure 2 shows our experimental setup. The information is encoded into the four spatial modes of a single photon. Single photons are generated by exciting an intrinsic defect (carbon antisite-vacancy pair) in a bulk high-purity semi-insulating 4H silicon carbide (SiC) [29]. The carbon antisite-vacancy pair defects in silicon carbides have been shown to be ultrabright, photostable single-photon sources even at room temperature. To match the operation wavelength of the half-wave plates in the contextuality test setup, we used a bandpass filter with a central wavelength of 720 nm and a width of 13 nm to filter the fluorescence of the defects. The total photon count for each set of measurement is about 40,000. The second-order photon correlation function at zero delay without background correction is $g^2(0)=0.268$ (with the fitting to 0.04), which clearly shows the signature of photon antibunching [the inset in Fig. 2(a)].

 

Fig. 2. Experimental setup. (a) The individual system (single photon) is prepared by exciting an intrinsic defect, known as the carbon antisite-vacancy pair in a bulk SiC sample. The insert shows the second-order photon correlation function of the emitting photons. $g^2(0)=0.268$ (with the fitting to 0.04) clearly confirms the character of single-photon emission. The emitted single photon is filtered by a bandpass filter with a central wavelength of 720 nm and a bandwidth of 13 nm, and is sent to the contextuality test setup. (b)–(d) show the contextuality test setup with the same initial state preparation setup and different final measurement settings. The four-dimension states are encoded into the spatial modes of the single photon ($i_1$, $i_2$, $i_3$, and $i_4$), which are prepared by passing the photon through the half-wave plate 1 (HWP1), and two beam displacers (BDs) with two more half-wave plates (HWP2 and HWP3) in each of the spatial modes. To prepare a mixed state, three glasses with different lengths are inserted into paths $|i_1⟩$, $|i_2⟩$, and $|i_3⟩$ (not shown in the figure) to completely destroy the coherence between different spatial modes. Three kinds of projective measurement settings with four half-wave plates (HWP4, HWP5, HWP6, and HWP7) and two BDs are employed, i.e., (b) the projection to the superposition states with spatial modes of 1, 2, and 3 ($1+2+3$), (c) the projection to the superposition states with spatial modes of 1, 2, and 4 ($1+2+4$) and (d) to the superposition states of spatial modes with 1, 3, and 4 ($1+3+4$). The polarization is finally selected by a polarization beam splitter (PBS) and the photon is detected by a single photon avalanche photodiode (SPAD).

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The single photon is then directed to the contextuality test setup constructed by several half-wave plates (HWPs) and beam displacers (BDs). A beam displacer is a birefringent crystal, in which light beams with horizontal and vertical polarizations are separated by a certain displacement. The polarization of the photon can be rotated using half-wave plates, and the relative amplitudes of different spatial modes can be conveniently adjusted.

The polarization of the single photon initially in the spatial modes $i_1$ is first rotated by HWP1, which is horizontally separated into $i_1$ and $i_3$ after the first beam displacer. The two half-wave plates (HWP2 and HWP3) further rotate the polarizations of the photon in the corresponding spatial modes, which are vertically separated by the second beam displacer. The final state we prepared can be the superposition of four spatial modes $i_1$, $i_2$, $i_3$, and $i_4$. The amplitudes of the four spatial modes can be changed by adjusting the angles of the HWP1, HWP2, and HWP3. To prepare a diagonal mixed state, three glasses with different lengths are inserted into paths $|i_1⟩$, $|i_2⟩$, and $|i_3⟩$ (not shown in Fig. 2) to completely destroy the coherence between spatial modes. We build different kinds of measurement settings. Three of them are shown in Fig. 2, i.e., the projection to the superposition states of spatial modes of $i_1$, $i_2$, and $i_3$ [$1+2+3$, Fig. 2(b)], the projection to $i_1$, $i_2$, and $i_4$ [$1+2+4$, Fig. 2(c)] and to $i_1$, $i_3$, and $i_4$ [$1+3+4$, Fig. 2(d)]. There, the quantum information encoded in spatial modes is converted back into polarization-encoded quantum information. The angle of HWP4 is set to be 45° such that the paths $i_1$ and $i_3$ ($i_2$ and $i_4$) are combined to one path after the third beam displacer. The unwanted path in the corresponding measurement is denoted by the dashed line. The relative amplitudes of the projected state can be adjusted by the last three half-wave plates, i.e., HWP5, HWP6, and HWP7. The polarization of the photon is finally rotated to be horizontal which is determined by a polarization beam splitter (PBS), and the photon is detected by a single photon avalanche photodiode (SPAD).

To test Inequality (3), we need to check the contextuality between two successive measurements. As proposed in Ref. [26], the statistics of the second measurements affected by the first measurements can be calculated as

In our experiment, the input state is prepared as

To verify Inequality (3), we need to check the measurement statistical effect for the state $\rho$. Because $\rho$ is classically mixed by four orthogonal bases, the corresponding probabilities of the four bases can be experimentally measured by counting photons in each path. We test the statistical effect when the four bases of $|i_1⟩$, $|i_2⟩$, $|i_3⟩$, and $|i_4⟩$ are used as the input states. The experimental results of the measurement statistical effect are shown in Section 4 in Supplement 1. For the four input states, one can have $ϵ(\_,0|\_,A_j)\approx ϵ(0,\_|A_i,\_)$ and $ϵ(\_,1|\_,A_j)\approx ϵ(1,\_|A_i,\_)$ within the experimental precision. Error bars are calculated from the counting statistics.

Figure 3 further show the detailed probabilities for the input state of $|i_1⟩$. Red dots represent the detection probabilities of the corresponding measurement settings of $P_{|i_1⟩}(A_i=1)$ ($i\in \{\mathrm{1,12}\}$) with the theoretical prediction of 0.309. To show the experimental precision in preparing two orthogonal states, the sum orthogonal probability of $A_1$ is defined as $P(A_1\perp )=P(A_1=1,A_2=1)+P(A_1=1,A_7=1)+P(A_1=1,A_8=1)+P(A_1=1,A_9=1)+P(A_1=1,A_{10}=1)$. The other eleven probabilities can be defined in the same way. In the experiment, the implementation of the projective measurement and the joint probability measurement correspond to detect the probabilities of the photon in several bases with different prepared states. The detailed method to calculate the probabilities can be found in Section 4 in Supplement 1. Black squares in Fig. 3 show the corresponding experimental results with the solid line representing the theoretical prediction of 0. As one has expected, for the pure input state, the summation of the 12 probabilities $P(A_i)$s approximately recover the Lovász number, thus coinciding with the theoretical prediction. The fluctuation of the polarization of background light in the imperfect single photon emission would affect the measurement result, which is shown to be small in our experiment.

 

Fig. 3. Probabilities for the input state $|i_1⟩$. Red dots represent the detection probabilities of the corresponding measurement settings of $P_{|i_1⟩}(A_i=1)$ ($i\in \{\mathrm{1,12}\}$) with the theoretical prediction of 0.309. Black squares represent the probabilities of $P(A_1\perp )$ to $P(A_{12}\perp )$, with the solid line representing the theoretical prediction of 0, which indicates the compatibility requirements are satisfied. The error bars deduced from the Poisson photon distribution are smaller than the symbols and cannot be seen clearly in the figure.

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Furthermore, we experimentally demonstrate that all four-dimensional states except the maximally mixed state violate single Inequality (3). To reach the purpose, we prepare four kinds of initial states, which are ${\rho }_1=\mathrm{Diag}(x,y,\mathrm{0,0})$, ${\rho }_2=\mathrm{Diag}(x,x,y,0)$, ${\rho }_3=\mathrm{Diag}(x,x,x,y)$, and ${\rho }_4=\mathrm{Diag}(x,y,y,y)$, here $x$ and $y$ represent the corresponding amplitudes and $x\ge y$. Figure 4 indicates that the experimental results coincide with our theoretical predictions. The $x$ axis represents the linear entropy of the states, which is calculated as $\ell =4(1-\mathrm{Tr}({\rho }^2))/3$. In particular, for the pure input state of $|i_1⟩$ ($x=1$ and $y=0$), the value of $S$ we obtained is $S=3.671\pm 0.026$, which violated NCHV prediction 3 by about 25 standard deviations.

 

Fig. 4. Experimental results for the contextuality test. Red dots, black squares, blue upward triangles, and purple downward triangles represent the experimental results with the initial input states of ${\rho }_1=\mathrm{Diag}(x,y,\mathrm{0,0})$, ${\rho }_2=\mathrm{Diag}(x,x,y,0)$, ${\rho }_3=\mathrm{Diag}(x,x,x,y)$, and ${\rho }_4=\mathrm{Diag}(x,y,y,y)$ ($x\ge y$), respectively. The dashed line, dotted line, dashed–dotted line, and solid line represent the corresponding theoretical predictions (only for the maximally mixed state $\mathrm{Diag}(\mathrm{1,1},\mathrm{1,1})/4$ is Inequality (3) not violated). Error bars are deduced from the Poisson photon distribution.

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4. DISCUSSION

It has been shown [23] that only about one-third of the quantum states of a three-dimensional system can be revealed by the simplest Klyachko–Can–Binicioğlu–Shumovsky inequality [30], which contains merely five measurements. Such kind of contextual correlation related to the simplest graph, pentagon, has been experimentally demonstrated by encoding a qutrit state (three-state system) into the path information of a trigger single photon [31]. However, Inequality (3) can detect almost all quantum states of a four-dimensional system. This fact reflects that different graphs have different powers to demonstrate contextual correlations. In our work, we have selected the icosahedron graph to test contextuality. Our quantum-correlation experiment sheds new light on the conflict between quantum and classical physics. Our results open a way for exploring quantum properties with graphs as a starting point and indicate that graph theory presents more physics than precedented thoughts.