Abstract

Great effort has been made in the investigation of contextual correlations between compatible observables due to their both fundamental and practical importance. The graph-theoretic approach to correlate events has been proved to be an effective method in the characterization of quantum contextuality, which implies that quantum violations of noncontextual inequalities derived in the noncontextual hidden-variable models should be achievable. Finding experimentally more friendly and theoretically more powerful noncontextual inequalities associated with specific graphs is of particular interest. Here we consider Platonic graphs to vindicate the quantum maximum predicted by graph theory and test the quantum violation against the mixedness of the state. Among these solids we refer particularly to the icosahedron to build the experiment, as it gives rise to the largest quantum-classical difference. The contextual correlations are demonstrated on quantum four-dimensional states encoded in the spatial modes of single photons generated from a defect in a bulk silicon carbide. Our results shed new light on the conflict between quantum and classical physics and may promote deep understanding of the connection between quantum theory, graph theory, and operator theory.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  31. R. Lapkiewicz, P. Li, C. Schaeff, N. K. Langford, S. Ramelow, M. Wieśniak, and A. Zeilinger, “Experimental non-classicality of an indivisible quantum system,” Nature 474, 490–493 (2011).
    [Crossref]

2017 (2)

N. de Silva, “Graph-theoretic strengths of contextuality,” Phys. Rev. A 95, 032108 (2017).
[Crossref]

M. Howard and E. Campbell, “Application of a resource theory for magic states to fault-tolerant quantum computing,” Phys. Rev. Lett. 118, 090501 (2017).
[Crossref]

2016 (2)

G. Cañas, E. Acuña, J. Cariñe, J. F. Barra, E. S. Gómez, G. B. Xavier, G. Lima, and A. Cabello, “Experimental demonstration of the connection between quantum contextuality and graph theory,” Phys. Rev. A 94, 012337 (2016).
[Crossref]

A. Cabello, “Simple method for experimentally testing any form of quantum contextuality,” Phys. Rev. A 93, 032102 (2016).
[Crossref]

2015 (5)

B. Amaral, M. T. Cunha, and A. Cabello, “Quantum theory allows for absolute maximal contextuality,” Phys. Rev. A 92, 062125 (2015).
[Crossref]

Z. P. Xu, H. Y. Su, and J. L. Chen, “Quantum contextuality of a qutrit state,” Phys. Rev. A 92, 012104 (2015).
[Crossref]

R. Kunjwal and R. W. Spekkens, “From the Kochen-Specker theorem to noncontextuality inequalities without assuming determinism,” Phys. Rev. Lett. 115, 110403 (2015).
[Crossref]

A. Acín, T. Fritz, A. Leverrier, and A. B. Sainz, “A combinatorial approach to nonlocality and contextuality,” Commun. Math. Phys. 334, 533–628 (2015).
[Crossref]

N. Delfosse, P. A. Guerin, J. Bian, and R. Raussendorf, “Wigner function negativity and contextuality in quantum computation on rebits,” Phys. Rev. X 5, 021003 (2015).
[Crossref]

2014 (6)

M. Howard, J. Wallman, V. Veitch, and J. Emerson, “Contextuality supplies the ‘magic’ for quantum computation,” Nature 510, 351–355 (2014).
[Crossref]

E. T. Campbell, “Enhanced fault-tolerant quantum computation in d-level systems,” Phys. Rev. Lett. 113, 230501 (2014).
[Crossref]

A. Cabello, S. Severini, and A. Winter, “Graph-theoretic approach to quantum correlations,” Phys. Rev. Lett. 112, 040401 (2014).
[Crossref]

C. Heunen, T. Fritz, and M. L. Reyes, “Quantum theory realizes all joint measurability graphs,” Phys. Rev. A 89, 032121 (2014).
[Crossref]

R. Kunjwal, C. Heunen, and T. Fritz, “Quantum realization of arbitrary joint measurability structures,” Phys. Rev. A 89, 052126 (2014).
[Crossref]

S. Castelletto, B. C. Johnson, V. Ivády, N. Stavrias, T. Umeda, A. Gali, and T. Ohshima, “A silicon carbide room-temperature single-photon source,” Nat. Mater. 13, 151–156 (2014).
[Crossref]

2013 (3)

A. Cabello, “Simple explanation of the quantum violation of a fundamental inequality,” Phys. Rev. Lett. 110, 060402 (2013).
[Crossref]

B. Yan, “Quantum correlations are tightly bound by the exclusivity principle,” Phys. Rev. Lett. 110, 260406 (2013).
[Crossref]

T. Fritz, A. B. Sainz, R. Augusiak, J. B. Brask, R. Chaves, A. Leverrier, and A. Acín, “Local orthogonality as a multipartite principle for quantum correlations,” Nat. Commun. 4, 2263 (2013).
[Crossref]

2012 (3)

E. Nagali, V. D’Ambrosio, F. Sciarrino, and A. Cabello, “Experimental observation of impossible-to-beat quantum advantage on a hybrid photonic system,” Phys. Rev. Lett. 108, 090501 (2012).
[Crossref]

E. Amselem, L. E. Danielsen, A. J. López-Tarrida, J. R. Portillo, M. Bourennane, and A. Cabello, “Experimental fully contextual correlations,” Phys. Rev. Lett. 108, 200405 (2012).
[Crossref]

S. Yu and C. H. Oh, “State-independent proof of Kochen-Specker theorem with 13 rays,” Phys. Rev. Lett. 108, 030402 (2012).
[Crossref]

2011 (2)

R. Lapkiewicz, P. Li, C. Schaeff, N. K. Langford, S. Ramelow, M. Wieśniak, and A. Zeilinger, “Experimental non-classicality of an indivisible quantum system,” Nature 474, 490–493 (2011).
[Crossref]

S. Abramsky and A. Brandenburger, “The sheaf-theoretic structure of non-locality and contextuality,” New J. Phys. 13, 113036 (2011).
[Crossref]

2009 (1)

G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O. Gühne, A. Cabello, R. Blatt, and C. F. Roos, “State-independent experimental test of quantum contextuality,” Nature 460, 494–497 (2009).
[Crossref]

2008 (1)

A. A. Klyachko, M. A. Can, S. Binicioğlu, and A. S. Shumovsky, “Simple test for hidden variables in spin-1 systems,” Phys. Rev. Lett. 101, 020403 (2008).
[Crossref]

1996 (1)

A. Cabello, J. M. Estebaranz, and G. Garcia-Alcaine, “Bell-Kochen-Specker theorem: a proof with 18 vectors,” Phys. Lett. A 212, 183–187 (1996).
[Crossref]

1986 (1)

M. Grötschel, “Relaxations of vertex packing,” J. Comb. Theory B 40, 330–343 (1986).
[Crossref]

1981 (1)

B. Bollobás, “The independence ratio of regular graphs,” Proc. Am. Math. Soc. 83, 433–436 (1981).
[Crossref]

1979 (1)

L. Lovász, “On the Shannon capacity of a graph,” IEEE Trans. Inf. Theory 25, 1–7 (1979).
[Crossref]

1967 (1)

S. Kochen and E. P. Specker, “The problem of hidden variables in quantum mechanics,” J. Math. Mech. 17, 59–87 (1967).
[Crossref]

1966 (1)

J. S. Bell, “On the problem of hidden variables in quantum mechanics,” Rev. Mod. Phys. 38, 447–452 (1966).
[Crossref]

Abramsky, S.

S. Abramsky and A. Brandenburger, “The sheaf-theoretic structure of non-locality and contextuality,” New J. Phys. 13, 113036 (2011).
[Crossref]

Acín, A.

A. Acín, T. Fritz, A. Leverrier, and A. B. Sainz, “A combinatorial approach to nonlocality and contextuality,” Commun. Math. Phys. 334, 533–628 (2015).
[Crossref]

T. Fritz, A. B. Sainz, R. Augusiak, J. B. Brask, R. Chaves, A. Leverrier, and A. Acín, “Local orthogonality as a multipartite principle for quantum correlations,” Nat. Commun. 4, 2263 (2013).
[Crossref]

Acuña, E.

G. Cañas, E. Acuña, J. Cariñe, J. F. Barra, E. S. Gómez, G. B. Xavier, G. Lima, and A. Cabello, “Experimental demonstration of the connection between quantum contextuality and graph theory,” Phys. Rev. A 94, 012337 (2016).
[Crossref]

Amaral, B.

B. Amaral, M. T. Cunha, and A. Cabello, “Quantum theory allows for absolute maximal contextuality,” Phys. Rev. A 92, 062125 (2015).
[Crossref]

Amselem, E.

E. Amselem, L. E. Danielsen, A. J. López-Tarrida, J. R. Portillo, M. Bourennane, and A. Cabello, “Experimental fully contextual correlations,” Phys. Rev. Lett. 108, 200405 (2012).
[Crossref]

Augusiak, R.

T. Fritz, A. B. Sainz, R. Augusiak, J. B. Brask, R. Chaves, A. Leverrier, and A. Acín, “Local orthogonality as a multipartite principle for quantum correlations,” Nat. Commun. 4, 2263 (2013).
[Crossref]

Barra, J. F.

G. Cañas, E. Acuña, J. Cariñe, J. F. Barra, E. S. Gómez, G. B. Xavier, G. Lima, and A. Cabello, “Experimental demonstration of the connection between quantum contextuality and graph theory,” Phys. Rev. A 94, 012337 (2016).
[Crossref]

Bell, J. S.

J. S. Bell, “On the problem of hidden variables in quantum mechanics,” Rev. Mod. Phys. 38, 447–452 (1966).
[Crossref]

Bian, J.

N. Delfosse, P. A. Guerin, J. Bian, and R. Raussendorf, “Wigner function negativity and contextuality in quantum computation on rebits,” Phys. Rev. X 5, 021003 (2015).
[Crossref]

Binicioglu, S.

A. A. Klyachko, M. A. Can, S. Binicioğlu, and A. S. Shumovsky, “Simple test for hidden variables in spin-1 systems,” Phys. Rev. Lett. 101, 020403 (2008).
[Crossref]

Blatt, R.

G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O. Gühne, A. Cabello, R. Blatt, and C. F. Roos, “State-independent experimental test of quantum contextuality,” Nature 460, 494–497 (2009).
[Crossref]

Bollobás, B.

B. Bollobás, “The independence ratio of regular graphs,” Proc. Am. Math. Soc. 83, 433–436 (1981).
[Crossref]

Bourennane, M.

E. Amselem, L. E. Danielsen, A. J. López-Tarrida, J. R. Portillo, M. Bourennane, and A. Cabello, “Experimental fully contextual correlations,” Phys. Rev. Lett. 108, 200405 (2012).
[Crossref]

Brandenburger, A.

S. Abramsky and A. Brandenburger, “The sheaf-theoretic structure of non-locality and contextuality,” New J. Phys. 13, 113036 (2011).
[Crossref]

Brask, J. B.

T. Fritz, A. B. Sainz, R. Augusiak, J. B. Brask, R. Chaves, A. Leverrier, and A. Acín, “Local orthogonality as a multipartite principle for quantum correlations,” Nat. Commun. 4, 2263 (2013).
[Crossref]

Cabello, A.

G. Cañas, E. Acuña, J. Cariñe, J. F. Barra, E. S. Gómez, G. B. Xavier, G. Lima, and A. Cabello, “Experimental demonstration of the connection between quantum contextuality and graph theory,” Phys. Rev. A 94, 012337 (2016).
[Crossref]

A. Cabello, “Simple method for experimentally testing any form of quantum contextuality,” Phys. Rev. A 93, 032102 (2016).
[Crossref]

B. Amaral, M. T. Cunha, and A. Cabello, “Quantum theory allows for absolute maximal contextuality,” Phys. Rev. A 92, 062125 (2015).
[Crossref]

A. Cabello, S. Severini, and A. Winter, “Graph-theoretic approach to quantum correlations,” Phys. Rev. Lett. 112, 040401 (2014).
[Crossref]

A. Cabello, “Simple explanation of the quantum violation of a fundamental inequality,” Phys. Rev. Lett. 110, 060402 (2013).
[Crossref]

E. Amselem, L. E. Danielsen, A. J. López-Tarrida, J. R. Portillo, M. Bourennane, and A. Cabello, “Experimental fully contextual correlations,” Phys. Rev. Lett. 108, 200405 (2012).
[Crossref]

E. Nagali, V. D’Ambrosio, F. Sciarrino, and A. Cabello, “Experimental observation of impossible-to-beat quantum advantage on a hybrid photonic system,” Phys. Rev. Lett. 108, 090501 (2012).
[Crossref]

G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O. Gühne, A. Cabello, R. Blatt, and C. F. Roos, “State-independent experimental test of quantum contextuality,” Nature 460, 494–497 (2009).
[Crossref]

A. Cabello, J. M. Estebaranz, and G. Garcia-Alcaine, “Bell-Kochen-Specker theorem: a proof with 18 vectors,” Phys. Lett. A 212, 183–187 (1996).
[Crossref]

Campbell, E.

M. Howard and E. Campbell, “Application of a resource theory for magic states to fault-tolerant quantum computing,” Phys. Rev. Lett. 118, 090501 (2017).
[Crossref]

Campbell, E. T.

E. T. Campbell, “Enhanced fault-tolerant quantum computation in d-level systems,” Phys. Rev. Lett. 113, 230501 (2014).
[Crossref]

Can, M. A.

A. A. Klyachko, M. A. Can, S. Binicioğlu, and A. S. Shumovsky, “Simple test for hidden variables in spin-1 systems,” Phys. Rev. Lett. 101, 020403 (2008).
[Crossref]

Cañas, G.

G. Cañas, E. Acuña, J. Cariñe, J. F. Barra, E. S. Gómez, G. B. Xavier, G. Lima, and A. Cabello, “Experimental demonstration of the connection between quantum contextuality and graph theory,” Phys. Rev. A 94, 012337 (2016).
[Crossref]

Cariñe, J.

G. Cañas, E. Acuña, J. Cariñe, J. F. Barra, E. S. Gómez, G. B. Xavier, G. Lima, and A. Cabello, “Experimental demonstration of the connection between quantum contextuality and graph theory,” Phys. Rev. A 94, 012337 (2016).
[Crossref]

Castelletto, S.

S. Castelletto, B. C. Johnson, V. Ivády, N. Stavrias, T. Umeda, A. Gali, and T. Ohshima, “A silicon carbide room-temperature single-photon source,” Nat. Mater. 13, 151–156 (2014).
[Crossref]

Chaves, R.

T. Fritz, A. B. Sainz, R. Augusiak, J. B. Brask, R. Chaves, A. Leverrier, and A. Acín, “Local orthogonality as a multipartite principle for quantum correlations,” Nat. Commun. 4, 2263 (2013).
[Crossref]

Chen, J. L.

Z. P. Xu, H. Y. Su, and J. L. Chen, “Quantum contextuality of a qutrit state,” Phys. Rev. A 92, 012104 (2015).
[Crossref]

Cunha, M. T.

B. Amaral, M. T. Cunha, and A. Cabello, “Quantum theory allows for absolute maximal contextuality,” Phys. Rev. A 92, 062125 (2015).
[Crossref]

D’Ambrosio, V.

E. Nagali, V. D’Ambrosio, F. Sciarrino, and A. Cabello, “Experimental observation of impossible-to-beat quantum advantage on a hybrid photonic system,” Phys. Rev. Lett. 108, 090501 (2012).
[Crossref]

Danielsen, L. E.

E. Amselem, L. E. Danielsen, A. J. López-Tarrida, J. R. Portillo, M. Bourennane, and A. Cabello, “Experimental fully contextual correlations,” Phys. Rev. Lett. 108, 200405 (2012).
[Crossref]

de Silva, N.

N. de Silva, “Graph-theoretic strengths of contextuality,” Phys. Rev. A 95, 032108 (2017).
[Crossref]

Delfosse, N.

N. Delfosse, P. A. Guerin, J. Bian, and R. Raussendorf, “Wigner function negativity and contextuality in quantum computation on rebits,” Phys. Rev. X 5, 021003 (2015).
[Crossref]

Emerson, J.

M. Howard, J. Wallman, V. Veitch, and J. Emerson, “Contextuality supplies the ‘magic’ for quantum computation,” Nature 510, 351–355 (2014).
[Crossref]

Estebaranz, J. M.

A. Cabello, J. M. Estebaranz, and G. Garcia-Alcaine, “Bell-Kochen-Specker theorem: a proof with 18 vectors,” Phys. Lett. A 212, 183–187 (1996).
[Crossref]

Fritz, T.

A. Acín, T. Fritz, A. Leverrier, and A. B. Sainz, “A combinatorial approach to nonlocality and contextuality,” Commun. Math. Phys. 334, 533–628 (2015).
[Crossref]

R. Kunjwal, C. Heunen, and T. Fritz, “Quantum realization of arbitrary joint measurability structures,” Phys. Rev. A 89, 052126 (2014).
[Crossref]

C. Heunen, T. Fritz, and M. L. Reyes, “Quantum theory realizes all joint measurability graphs,” Phys. Rev. A 89, 032121 (2014).
[Crossref]

T. Fritz, A. B. Sainz, R. Augusiak, J. B. Brask, R. Chaves, A. Leverrier, and A. Acín, “Local orthogonality as a multipartite principle for quantum correlations,” Nat. Commun. 4, 2263 (2013).
[Crossref]

Gali, A.

S. Castelletto, B. C. Johnson, V. Ivády, N. Stavrias, T. Umeda, A. Gali, and T. Ohshima, “A silicon carbide room-temperature single-photon source,” Nat. Mater. 13, 151–156 (2014).
[Crossref]

Garcia-Alcaine, G.

A. Cabello, J. M. Estebaranz, and G. Garcia-Alcaine, “Bell-Kochen-Specker theorem: a proof with 18 vectors,” Phys. Lett. A 212, 183–187 (1996).
[Crossref]

Gerritsma, R.

G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O. Gühne, A. Cabello, R. Blatt, and C. F. Roos, “State-independent experimental test of quantum contextuality,” Nature 460, 494–497 (2009).
[Crossref]

Gómez, E. S.

G. Cañas, E. Acuña, J. Cariñe, J. F. Barra, E. S. Gómez, G. B. Xavier, G. Lima, and A. Cabello, “Experimental demonstration of the connection between quantum contextuality and graph theory,” Phys. Rev. A 94, 012337 (2016).
[Crossref]

Grötschel, M.

M. Grötschel, “Relaxations of vertex packing,” J. Comb. Theory B 40, 330–343 (1986).
[Crossref]

Guerin, P. A.

N. Delfosse, P. A. Guerin, J. Bian, and R. Raussendorf, “Wigner function negativity and contextuality in quantum computation on rebits,” Phys. Rev. X 5, 021003 (2015).
[Crossref]

Gühne, O.

G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O. Gühne, A. Cabello, R. Blatt, and C. F. Roos, “State-independent experimental test of quantum contextuality,” Nature 460, 494–497 (2009).
[Crossref]

Heunen, C.

R. Kunjwal, C. Heunen, and T. Fritz, “Quantum realization of arbitrary joint measurability structures,” Phys. Rev. A 89, 052126 (2014).
[Crossref]

C. Heunen, T. Fritz, and M. L. Reyes, “Quantum theory realizes all joint measurability graphs,” Phys. Rev. A 89, 032121 (2014).
[Crossref]

Howard, M.

M. Howard and E. Campbell, “Application of a resource theory for magic states to fault-tolerant quantum computing,” Phys. Rev. Lett. 118, 090501 (2017).
[Crossref]

M. Howard, J. Wallman, V. Veitch, and J. Emerson, “Contextuality supplies the ‘magic’ for quantum computation,” Nature 510, 351–355 (2014).
[Crossref]

Ivády, V.

S. Castelletto, B. C. Johnson, V. Ivády, N. Stavrias, T. Umeda, A. Gali, and T. Ohshima, “A silicon carbide room-temperature single-photon source,” Nat. Mater. 13, 151–156 (2014).
[Crossref]

Johnson, B. C.

S. Castelletto, B. C. Johnson, V. Ivády, N. Stavrias, T. Umeda, A. Gali, and T. Ohshima, “A silicon carbide room-temperature single-photon source,” Nat. Mater. 13, 151–156 (2014).
[Crossref]

Kirchmair, G.

G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O. Gühne, A. Cabello, R. Blatt, and C. F. Roos, “State-independent experimental test of quantum contextuality,” Nature 460, 494–497 (2009).
[Crossref]

Kleinmann, M.

G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O. Gühne, A. Cabello, R. Blatt, and C. F. Roos, “State-independent experimental test of quantum contextuality,” Nature 460, 494–497 (2009).
[Crossref]

Klyachko, A. A.

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Supplementary Material (1)

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» Supplement 1       more experimental and theoretical details

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Figures (4)

Fig. 1.
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 |ψ and measurement settings |i. The set {|1,|2,,|12} forms the Lovász optimum orthogonal realization. |i (unnormalized vector for simplicity) denotes |Ai. τ=ϕ, ϕ=12(5+1), τ=1/τ, ϕ=1/ϕ, ω=2+5, and i=112|ψ|Ai|2=3(51) give the quantum maximum that equals to the Lovász number. (d) The independence number α and the Lovász number ϑ of the icosahedron.
Fig. 2.
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. g2(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 (i1, i2, i3, and i4), 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 |i1, |i2, and |i3 (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).
Fig. 3.
Fig. 3. Probabilities for the input state |i1. Red dots represent the detection probabilities of the corresponding measurement settings of P|i1(Ai=1) (i{1,12}) with the theoretical prediction of 0.309. Black squares represent the probabilities of P(A1) to P(A12), 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.
Fig. 4.
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 ρ1=Diag(x,y,0,0), ρ2=Diag(x,x,y,0), ρ3=Diag(x,x,x,y), and ρ4=Diag(x,y,y,y) (xy), respectively. The dashed line, dotted line, dashed–dotted line, and solid line represent the corresponding theoretical predictions (only for the maximally mixed state Diag(1,1,1,1)/4 is Inequality (3) not violated). Error bars are deduced from the Poisson photon distribution.

Equations (5)

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S=i=1NP^iα,
S=iVP|ψ(Ai=1)(Ai,Aj)EP|ψ(Ai=1,Aj=1)NCHVα(G)QMϑ(G),
S=i=112P|ψ(Ai=1)(i,j)EP|ψ(Ai=1,Aj=1)NCHV3.
ϵ(_,0|_,Aj)=|P|ψ(Aj=0)k=01P|ψ(Ai=k,Aj=0)|,ϵ(_,1|_,Aj)=|P|ψ(Aj=1)k=01P|ψ(Ai=k,Aj=1)|,
ρ=a|i1i1|+b|i2i2|+c|i3i3|+d|i4i4|,

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