Fig. 1. Exclusivity graph of the n measurements (with n=7, 11,15, 19, …) 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 (n−1)/2 triangles.

Fig. 2. Exclusivity relations between the projection measurements in the Hardy-like proof for n=7, including the added measurements (8, 9, 10) used in the experiment. The black vertices are twice of the white vertices in weight.

Fig. 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 q-plates (QPs, with different topological charges of q1=1/2 and q2=1) 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).

Fig. 4. Quantum violation of Eq. (3) for n=7. NCHV (=3) represents the classical bound of the NCHV theory. The maximum violation is QT1≈3.372 by using the optimal measurement settings, and the non-maximum violation is QT2=3.250 by using the measurements adopted in the Hardy-like proof.

Fig. 5. Success probabilities PSUC (blue curve) and PCBCB (red curve) versus n.

Fig. 6. Curve of the ratio rn versus n. The red curve represents the relationship of rn with n in our paper, while the blue curve represents the relationship of rn′ with n for the extended KCBS noncontextuality inequality.

Fig. 7. Different regions for αn, where pi are boundary points, p1=(0,0),p2=(0,2/(n−1)), and p3=(2,1).

Fig. 8. Experimental results for δ(_,0|oj,ok) (red circles) and δ(0,_|oj,ok) (black squares) in (a) and for δ(_,1|oj,ok) (red circles) and δ(1,_|oi,oj) (black square) in (b). The numbers of 1 to 24 (even numbers are not marked in figures) represent the settings of (o1,o2), (o1,o3), (o1,o4), (o1,o6), (o2,o1), (o2,o3), (o2,o4), (o2,o5), (o3,o1), (o3,o2), (o3,o5), (o3,o6), (o4,o1), (o4,o2), (o4,o5), (o4,o7), (o5,o2), (o5,o3), (o5,o7), (o6,o1), (o6,o3), (o6,o7), (o7,o4), (o7,o5), and (o7,o6). Error bars are calculated from the counting statistics.

Table 1. State |ϕ⟩ and the Measurement Vectors |νj⟩ (j=1, 2,  ...,  7), Where |ν8⟩,|ν9⟩,|ν10⟩ Are Annex Measurement Vectors

Table 2. Experimental Results of Hardy-Like Proof for n=7^a