Abstract

Contextuality, the impossibility of assigning context-independent measurement outcomes, is a critical resource for quantum computation and communication. No-signaling between successive measurements is an essential requirement that should be accomplished in any test of quantum contextuality and that is difficult to achieve in practice. Here, we introduce an optimal quantum state-independent contextuality inequality in which the deviation from the classical bound is maximal. We then experimentally test it using single photons generated from a defect in a bulk silicon carbide, while satisfying the requirement of no-signaling within the experimental error. Our results shed new light on the study of quantum contextuality under no-signaling conditions.

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

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    [Crossref]
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  5. A. Cabello, “Simple Explanation of the Quantum Violation of a Fundamental Inequality,” Phys. Rev. Lett. 110, 060402 (2013).
    [Crossref] [PubMed]
  6. B. Yan, “Quantum Correlations are Tightly Bound by the Exclusivity Principle,” Phys. Rev. Lett. 110, 260406 (2013).
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  7. G. Chiribella and X. Yuan, “Measurement sharpness cuts nonlocality and contextuality in every physical theory,” arXiv: 1404.3348 (2014).
  8. A. Cabello, “Simple Explanation of the Quantum Limits of Genuine n-Body Nonlocality,” Phys. Rev. Lett. 114, 220402 (2015).
    [Crossref] [PubMed]
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    [PubMed]
  10. R. Raussendorf, “Contextuality in measurement-based quantum computation,” Phys. Rev. A 88, 022322 (2013).
    [Crossref]
  11. K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, M. Pawlowski, and M. Bourennane, “Contextuality offers device-independent security,” arXiv: 1006.0468 (2010).
  12. S. Yu and C. H. Oh, “State-Independent Proof of Kochen-Specker Theorem with 13 Rays,” Phys. Rev. Lett. 108, 030402 (2012).
    [Crossref] [PubMed]
  13. M. Kernaghan, “Bell-Kochen-Specker theorem for 20 vectors,” J. Phys. A: Math. Gen. 27, L829–L830 (1994).
    [Crossref]
  14. A. Cabello, J. M. Estebaranz, and G. García-Alcain, “Bell-Kochen-Specker theorem: A proof with 18 vectors,” Phys. Lett. A 212, 183–187 (1996).
    [Crossref]
  15. M. Kernaghan and A. Peres, “Kochen-Specker theorem for eight-dimensional space,” Phys. Lett. A 198, 1–5 (1995).
    [Crossref]
  16. A. Cabello, M. Kleinmann, and C. Budroni, “Necessary and Sufficient Condition for Quantum State-Independent Contextuality,” Phys. Rev. Lett. 114, 250402 (2015).
    [Crossref] [PubMed]
  17. A. Cabello, M. Kleinmann, and J. R. Portillo, “Quantum State-Independent Contextuality Requires 13 Rays,” J. Phys. A: Math. Theor. 49, 38LT01 (2016).
    [Crossref]
  18. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86, 419–478 (2014).
    [Crossref]
  19. C. Zu, Y.-X. Wang, D.-L. Deng, X.-Y. Chang, K. Liu, P.-Y. Hou, H.-X. Yang, and L.-M. Duan, “State-Independent Experimental Test of Quantum Contextuality in An Indivisible System,” Phys. Rev. Lett. 109, 150401 (2012).
    [Crossref] [PubMed]
  20. Y.-F. Huang, M. Li, D.-Y. Cao, C. Zhang, Y.-S. Zhang, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Experimental test of state-independent quantum contextuality of an indivisible quantum system,” Phys. Rev. A 87, 052133 (2013).
    [Crossref]
  21. X. Zhang, M. Um, J. Zhang, S. An, Y. Wang, D.-L. Deng, C. Shen, L.-M. Duan, and K. Kim, “State-Independent Experimental Test of Quantum Contextuality with a Single Trapped Ion,” Phys. Rev. Lett. 110, 070401 (2013).
    [Crossref] [PubMed]
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    [Crossref]
  23. A. Cabello, “Simple method for experimentally testing any form of quantum contextuality,” Phys. Rev. A 93, 032102 (2016).
    [Crossref]
  24. A. Cabello, S. Severini, and A. Winter, “Graph-Theoretic Approach to Quantum Correlations,” Phys. Rev. Lett. 112, 040401 (2014).
    [Crossref] [PubMed]
  25. S. Castelletto, B. C. Johnson, V. Ivády, N. Stavrias, T. Umeda, A. Gali, and T. Ohshima, “A silicon carbide room-temperature single-photon source,” Nature Mat. 13, 151–156 (2014).
    [Crossref]
  26. M. Kleinmann, C. Budroni, J.-Å. Larsson, O. Gühne, and A. Cabello, “Optimal Inequalities for State-Independent Contextuality,” Phys. Rev. Lett. 109, 250402 (2012).
    [Crossref]
  27. R. Ramanathan and P. Horodecki, “Necessary and sufficient condition for state-independent contextual measurement scenarios,” Phys. Rev. Lett. 112, 040404 (2014).
    [Crossref] [PubMed]
  28. C. Monroe and J. Kim, “Scaling the ion trap quantum processor,” Science 339, 1164–1169 (2013).
    [Crossref] [PubMed]
  29. A. Cabello, M. Gu, O. Gühne, J. Larsson, and K. Wiesner, “Thermodynamical cost of some interpretations of quantum theory,” Phys. Rev. A 94, 052127 (2016).
    [Crossref]
  30. A. Tavakoli and A. Cabello, “Cost of classically simulating quantum sequential measurements on entangled systems,” arXiv:1705.07456 (2017).
  31. A. Cabello, M. Gu, O. Gühne, and Z.-P. Xu, “Optimal simulation of state-independent quantum contextuality,” arXiv:1709.07372 (2017).

2016 (3)

A. Cabello, M. Kleinmann, and J. R. Portillo, “Quantum State-Independent Contextuality Requires 13 Rays,” J. Phys. A: Math. Theor. 49, 38LT01 (2016).
[Crossref]

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

A. Cabello, M. Gu, O. Gühne, J. Larsson, and K. Wiesner, “Thermodynamical cost of some interpretations of quantum theory,” Phys. Rev. A 94, 052127 (2016).
[Crossref]

2015 (2)

A. Cabello, M. Kleinmann, and C. Budroni, “Necessary and Sufficient Condition for Quantum State-Independent Contextuality,” Phys. Rev. Lett. 114, 250402 (2015).
[Crossref] [PubMed]

A. Cabello, “Simple Explanation of the Quantum Limits of Genuine n-Body Nonlocality,” Phys. Rev. Lett. 114, 220402 (2015).
[Crossref] [PubMed]

2014 (5)

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

N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86, 419–478 (2014).
[Crossref]

R. Ramanathan and P. Horodecki, “Necessary and sufficient condition for state-independent contextual measurement scenarios,” Phys. Rev. Lett. 112, 040404 (2014).
[Crossref] [PubMed]

A. Cabello, S. Severini, and A. Winter, “Graph-Theoretic Approach to Quantum Correlations,” Phys. Rev. Lett. 112, 040401 (2014).
[Crossref] [PubMed]

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

2013 (6)

C. Monroe and J. Kim, “Scaling the ion trap quantum processor,” Science 339, 1164–1169 (2013).
[Crossref] [PubMed]

Y.-F. Huang, M. Li, D.-Y. Cao, C. Zhang, Y.-S. Zhang, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Experimental test of state-independent quantum contextuality of an indivisible quantum system,” Phys. Rev. A 87, 052133 (2013).
[Crossref]

X. Zhang, M. Um, J. Zhang, S. An, Y. Wang, D.-L. Deng, C. Shen, L.-M. Duan, and K. Kim, “State-Independent Experimental Test of Quantum Contextuality with a Single Trapped Ion,” Phys. Rev. Lett. 110, 070401 (2013).
[Crossref] [PubMed]

R. Raussendorf, “Contextuality in measurement-based quantum computation,” Phys. Rev. A 88, 022322 (2013).
[Crossref]

A. Cabello, “Simple Explanation of the Quantum Violation of a Fundamental Inequality,” Phys. Rev. Lett. 110, 060402 (2013).
[Crossref] [PubMed]

B. Yan, “Quantum Correlations are Tightly Bound by the Exclusivity Principle,” Phys. Rev. Lett. 110, 260406 (2013).
[Crossref] [PubMed]

2012 (3)

S. Yu and C. H. Oh, “State-Independent Proof of Kochen-Specker Theorem with 13 Rays,” Phys. Rev. Lett. 108, 030402 (2012).
[Crossref] [PubMed]

C. Zu, Y.-X. Wang, D.-L. Deng, X.-Y. Chang, K. Liu, P.-Y. Hou, H.-X. Yang, and L.-M. Duan, “State-Independent Experimental Test of Quantum Contextuality in An Indivisible System,” Phys. Rev. Lett. 109, 150401 (2012).
[Crossref] [PubMed]

M. Kleinmann, C. Budroni, J.-Å. Larsson, O. Gühne, and A. Cabello, “Optimal Inequalities for State-Independent Contextuality,” Phys. Rev. Lett. 109, 250402 (2012).
[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]

1996 (1)

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

1995 (1)

M. Kernaghan and A. Peres, “Kochen-Specker theorem for eight-dimensional space,” Phys. Lett. A 198, 1–5 (1995).
[Crossref]

1994 (1)

M. Kernaghan, “Bell-Kochen-Specker theorem for 20 vectors,” J. Phys. A: Math. Gen. 27, L829–L830 (1994).
[Crossref]

1967 (1)

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

1966 (1)

J. B. Bell, “On the Problem of Hidden Variables in Quantum Mechanics,” Rev. Mod. Phys. 38, 447 (1966).
[Crossref]

1960 (1)

E. P. Specker, “Die Logik nicht gleichzeitig entscheidbarer Aussagen,” Dialectica 14, 239–246 (1960).
[Crossref]

An, S.

X. Zhang, M. Um, J. Zhang, S. An, Y. Wang, D.-L. Deng, C. Shen, L.-M. Duan, and K. Kim, “State-Independent Experimental Test of Quantum Contextuality with a Single Trapped Ion,” Phys. Rev. Lett. 110, 070401 (2013).
[Crossref] [PubMed]

Bell, J. B.

J. B. Bell, “On the Problem of Hidden Variables in Quantum Mechanics,” Rev. Mod. Phys. 38, 447 (1966).
[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]

Bourennane, M.

K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, M. Pawlowski, and M. Bourennane, “Contextuality offers device-independent security,” arXiv: 1006.0468 (2010).

Brunner, N.

N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86, 419–478 (2014).
[Crossref]

Budroni, C.

A. Cabello, M. Kleinmann, and C. Budroni, “Necessary and Sufficient Condition for Quantum State-Independent Contextuality,” Phys. Rev. Lett. 114, 250402 (2015).
[Crossref] [PubMed]

M. Kleinmann, C. Budroni, J.-Å. Larsson, O. Gühne, and A. Cabello, “Optimal Inequalities for State-Independent Contextuality,” Phys. Rev. Lett. 109, 250402 (2012).
[Crossref]

Cabello, A.

A. Cabello, M. Kleinmann, and J. R. Portillo, “Quantum State-Independent Contextuality Requires 13 Rays,” J. Phys. A: Math. Theor. 49, 38LT01 (2016).
[Crossref]

A. Cabello, M. Gu, O. Gühne, J. Larsson, and K. Wiesner, “Thermodynamical cost of some interpretations of quantum theory,” Phys. Rev. A 94, 052127 (2016).
[Crossref]

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

A. Cabello, “Simple Explanation of the Quantum Limits of Genuine n-Body Nonlocality,” Phys. Rev. Lett. 114, 220402 (2015).
[Crossref] [PubMed]

A. Cabello, M. Kleinmann, and C. Budroni, “Necessary and Sufficient Condition for Quantum State-Independent Contextuality,” Phys. Rev. Lett. 114, 250402 (2015).
[Crossref] [PubMed]

A. Cabello, S. Severini, and A. Winter, “Graph-Theoretic Approach to Quantum Correlations,” Phys. Rev. Lett. 112, 040401 (2014).
[Crossref] [PubMed]

A. Cabello, “Simple Explanation of the Quantum Violation of a Fundamental Inequality,” Phys. Rev. Lett. 110, 060402 (2013).
[Crossref] [PubMed]

M. Kleinmann, C. Budroni, J.-Å. Larsson, O. Gühne, and A. Cabello, “Optimal Inequalities for State-Independent Contextuality,” Phys. Rev. Lett. 109, 250402 (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. García-Alcain, “Bell-Kochen-Specker theorem: A proof with 18 vectors,” Phys. Lett. A 212, 183–187 (1996).
[Crossref]

A. Tavakoli and A. Cabello, “Cost of classically simulating quantum sequential measurements on entangled systems,” arXiv:1705.07456 (2017).

A. Cabello, M. Gu, O. Gühne, and Z.-P. Xu, “Optimal simulation of state-independent quantum contextuality,” arXiv:1709.07372 (2017).

Cao, D.-Y.

Y.-F. Huang, M. Li, D.-Y. Cao, C. Zhang, Y.-S. Zhang, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Experimental test of state-independent quantum contextuality of an indivisible quantum system,” Phys. Rev. A 87, 052133 (2013).
[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,” Nature Mat. 13, 151–156 (2014).
[Crossref]

Cavalcanti, D.

N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86, 419–478 (2014).
[Crossref]

Chang, X.-Y.

C. Zu, Y.-X. Wang, D.-L. Deng, X.-Y. Chang, K. Liu, P.-Y. Hou, H.-X. Yang, and L.-M. Duan, “State-Independent Experimental Test of Quantum Contextuality in An Indivisible System,” Phys. Rev. Lett. 109, 150401 (2012).
[Crossref] [PubMed]

Chiribella, G.

G. Chiribella and X. Yuan, “Measurement sharpness cuts nonlocality and contextuality in every physical theory,” arXiv: 1404.3348 (2014).

Deng, D.-L.

X. Zhang, M. Um, J. Zhang, S. An, Y. Wang, D.-L. Deng, C. Shen, L.-M. Duan, and K. Kim, “State-Independent Experimental Test of Quantum Contextuality with a Single Trapped Ion,” Phys. Rev. Lett. 110, 070401 (2013).
[Crossref] [PubMed]

C. Zu, Y.-X. Wang, D.-L. Deng, X.-Y. Chang, K. Liu, P.-Y. Hou, H.-X. Yang, and L.-M. Duan, “State-Independent Experimental Test of Quantum Contextuality in An Indivisible System,” Phys. Rev. Lett. 109, 150401 (2012).
[Crossref] [PubMed]

Duan, L.-M.

X. Zhang, M. Um, J. Zhang, S. An, Y. Wang, D.-L. Deng, C. Shen, L.-M. Duan, and K. Kim, “State-Independent Experimental Test of Quantum Contextuality with a Single Trapped Ion,” Phys. Rev. Lett. 110, 070401 (2013).
[Crossref] [PubMed]

C. Zu, Y.-X. Wang, D.-L. Deng, X.-Y. Chang, K. Liu, P.-Y. Hou, H.-X. Yang, and L.-M. Duan, “State-Independent Experimental Test of Quantum Contextuality in An Indivisible System,” Phys. Rev. Lett. 109, 150401 (2012).
[Crossref] [PubMed]

Emerson, J.

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

Estebaranz, J. M.

A. Cabello, J. M. Estebaranz, and G. García-Alcain, “Bell-Kochen-Specker theorem: A proof with 18 vectors,” Phys. Lett. A 212, 183–187 (1996).
[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,” Nature Mat. 13, 151–156 (2014).
[Crossref]

García-Alcain, G.

A. Cabello, J. M. Estebaranz, and G. García-Alcain, “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]

Gu, M.

A. Cabello, M. Gu, O. Gühne, J. Larsson, and K. Wiesner, “Thermodynamical cost of some interpretations of quantum theory,” Phys. Rev. A 94, 052127 (2016).
[Crossref]

A. Cabello, M. Gu, O. Gühne, and Z.-P. Xu, “Optimal simulation of state-independent quantum contextuality,” arXiv:1709.07372 (2017).

Gühne, O.

A. Cabello, M. Gu, O. Gühne, J. Larsson, and K. Wiesner, “Thermodynamical cost of some interpretations of quantum theory,” Phys. Rev. A 94, 052127 (2016).
[Crossref]

M. Kleinmann, C. Budroni, J.-Å. Larsson, O. Gühne, and A. Cabello, “Optimal Inequalities for State-Independent Contextuality,” Phys. Rev. Lett. 109, 250402 (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, M. Gu, O. Gühne, and Z.-P. Xu, “Optimal simulation of state-independent quantum contextuality,” arXiv:1709.07372 (2017).

Guo, G.-C.

Y.-F. Huang, M. Li, D.-Y. Cao, C. Zhang, Y.-S. Zhang, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Experimental test of state-independent quantum contextuality of an indivisible quantum system,” Phys. Rev. A 87, 052133 (2013).
[Crossref]

Horodecki, K.

K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, M. Pawlowski, and M. Bourennane, “Contextuality offers device-independent security,” arXiv: 1006.0468 (2010).

Horodecki, M.

K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, M. Pawlowski, and M. Bourennane, “Contextuality offers device-independent security,” arXiv: 1006.0468 (2010).

Horodecki, P.

R. Ramanathan and P. Horodecki, “Necessary and sufficient condition for state-independent contextual measurement scenarios,” Phys. Rev. Lett. 112, 040404 (2014).
[Crossref] [PubMed]

K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, M. Pawlowski, and M. Bourennane, “Contextuality offers device-independent security,” arXiv: 1006.0468 (2010).

Horodecki, R.

K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, M. Pawlowski, and M. Bourennane, “Contextuality offers device-independent security,” arXiv: 1006.0468 (2010).

Hou, P.-Y.

C. Zu, Y.-X. Wang, D.-L. Deng, X.-Y. Chang, K. Liu, P.-Y. Hou, H.-X. Yang, and L.-M. Duan, “State-Independent Experimental Test of Quantum Contextuality in An Indivisible System,” Phys. Rev. Lett. 109, 150401 (2012).
[Crossref] [PubMed]

Howard, M.

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

Huang, Y.-F.

Y.-F. Huang, M. Li, D.-Y. Cao, C. Zhang, Y.-S. Zhang, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Experimental test of state-independent quantum contextuality of an indivisible quantum system,” Phys. Rev. A 87, 052133 (2013).
[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,” Nature Mat. 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,” Nature Mat. 13, 151–156 (2014).
[Crossref]

Kernaghan, M.

M. Kernaghan and A. Peres, “Kochen-Specker theorem for eight-dimensional space,” Phys. Lett. A 198, 1–5 (1995).
[Crossref]

M. Kernaghan, “Bell-Kochen-Specker theorem for 20 vectors,” J. Phys. A: Math. Gen. 27, L829–L830 (1994).
[Crossref]

Kim, J.

C. Monroe and J. Kim, “Scaling the ion trap quantum processor,” Science 339, 1164–1169 (2013).
[Crossref] [PubMed]

Kim, K.

X. Zhang, M. Um, J. Zhang, S. An, Y. Wang, D.-L. Deng, C. Shen, L.-M. Duan, and K. Kim, “State-Independent Experimental Test of Quantum Contextuality with a Single Trapped Ion,” Phys. Rev. Lett. 110, 070401 (2013).
[Crossref] [PubMed]

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.

A. Cabello, M. Kleinmann, and J. R. Portillo, “Quantum State-Independent Contextuality Requires 13 Rays,” J. Phys. A: Math. Theor. 49, 38LT01 (2016).
[Crossref]

A. Cabello, M. Kleinmann, and C. Budroni, “Necessary and Sufficient Condition for Quantum State-Independent Contextuality,” Phys. Rev. Lett. 114, 250402 (2015).
[Crossref] [PubMed]

M. Kleinmann, C. Budroni, J.-Å. Larsson, O. Gühne, and A. Cabello, “Optimal Inequalities for State-Independent Contextuality,” Phys. Rev. Lett. 109, 250402 (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]

Kochen, S.

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

Larsson, J.

A. Cabello, M. Gu, O. Gühne, J. Larsson, and K. Wiesner, “Thermodynamical cost of some interpretations of quantum theory,” Phys. Rev. A 94, 052127 (2016).
[Crossref]

Larsson, J.-Å.

M. Kleinmann, C. Budroni, J.-Å. Larsson, O. Gühne, and A. Cabello, “Optimal Inequalities for State-Independent Contextuality,” Phys. Rev. Lett. 109, 250402 (2012).
[Crossref]

Li, C.-F.

Y.-F. Huang, M. Li, D.-Y. Cao, C. Zhang, Y.-S. Zhang, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Experimental test of state-independent quantum contextuality of an indivisible quantum system,” Phys. Rev. A 87, 052133 (2013).
[Crossref]

Li, M.

Y.-F. Huang, M. Li, D.-Y. Cao, C. Zhang, Y.-S. Zhang, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Experimental test of state-independent quantum contextuality of an indivisible quantum system,” Phys. Rev. A 87, 052133 (2013).
[Crossref]

Liu, B.-H.

Y.-F. Huang, M. Li, D.-Y. Cao, C. Zhang, Y.-S. Zhang, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Experimental test of state-independent quantum contextuality of an indivisible quantum system,” Phys. Rev. A 87, 052133 (2013).
[Crossref]

Liu, K.

C. Zu, Y.-X. Wang, D.-L. Deng, X.-Y. Chang, K. Liu, P.-Y. Hou, H.-X. Yang, and L.-M. Duan, “State-Independent Experimental Test of Quantum Contextuality in An Indivisible System,” Phys. Rev. Lett. 109, 150401 (2012).
[Crossref] [PubMed]

Monroe, C.

C. Monroe and J. Kim, “Scaling the ion trap quantum processor,” Science 339, 1164–1169 (2013).
[Crossref] [PubMed]

Oh, C. H.

S. Yu and C. H. Oh, “State-Independent Proof of Kochen-Specker Theorem with 13 Rays,” Phys. Rev. Lett. 108, 030402 (2012).
[Crossref] [PubMed]

Ohshima, T.

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

Pawlowski, M.

K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, M. Pawlowski, and M. Bourennane, “Contextuality offers device-independent security,” arXiv: 1006.0468 (2010).

Peres, A.

M. Kernaghan and A. Peres, “Kochen-Specker theorem for eight-dimensional space,” Phys. Lett. A 198, 1–5 (1995).
[Crossref]

Pironio, S.

N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86, 419–478 (2014).
[Crossref]

Portillo, J. R.

A. Cabello, M. Kleinmann, and J. R. Portillo, “Quantum State-Independent Contextuality Requires 13 Rays,” J. Phys. A: Math. Theor. 49, 38LT01 (2016).
[Crossref]

Ramanathan, R.

R. Ramanathan and P. Horodecki, “Necessary and sufficient condition for state-independent contextual measurement scenarios,” Phys. Rev. Lett. 112, 040404 (2014).
[Crossref] [PubMed]

Raussendorf, R.

R. Raussendorf, “Contextuality in measurement-based quantum computation,” Phys. Rev. A 88, 022322 (2013).
[Crossref]

Roos, C. F.

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]

Scarani, V.

N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86, 419–478 (2014).
[Crossref]

Seevinck, M. P.

M. P. Seevinck, “E. Specker: ‘The logic of non-simultaneously decidable propositions’ (1960),” arXiv: 1103.4537 (2011).

Severini, S.

A. Cabello, S. Severini, and A. Winter, “Graph-Theoretic Approach to Quantum Correlations,” Phys. Rev. Lett. 112, 040401 (2014).
[Crossref] [PubMed]

Shen, C.

X. Zhang, M. Um, J. Zhang, S. An, Y. Wang, D.-L. Deng, C. Shen, L.-M. Duan, and K. Kim, “State-Independent Experimental Test of Quantum Contextuality with a Single Trapped Ion,” Phys. Rev. Lett. 110, 070401 (2013).
[Crossref] [PubMed]

Specker, E. P.

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

E. P. Specker, “Die Logik nicht gleichzeitig entscheidbarer Aussagen,” Dialectica 14, 239–246 (1960).
[Crossref]

Stavrias, N.

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

Tavakoli, A.

A. Tavakoli and A. Cabello, “Cost of classically simulating quantum sequential measurements on entangled systems,” arXiv:1705.07456 (2017).

Um, M.

X. Zhang, M. Um, J. Zhang, S. An, Y. Wang, D.-L. Deng, C. Shen, L.-M. Duan, and K. Kim, “State-Independent Experimental Test of Quantum Contextuality with a Single Trapped Ion,” Phys. Rev. Lett. 110, 070401 (2013).
[Crossref] [PubMed]

Umeda, T.

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

Veitch, V.

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

Wallman, J.

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

Wang, Y.

X. Zhang, M. Um, J. Zhang, S. An, Y. Wang, D.-L. Deng, C. Shen, L.-M. Duan, and K. Kim, “State-Independent Experimental Test of Quantum Contextuality with a Single Trapped Ion,” Phys. Rev. Lett. 110, 070401 (2013).
[Crossref] [PubMed]

Wang, Y.-X.

C. Zu, Y.-X. Wang, D.-L. Deng, X.-Y. Chang, K. Liu, P.-Y. Hou, H.-X. Yang, and L.-M. Duan, “State-Independent Experimental Test of Quantum Contextuality in An Indivisible System,” Phys. Rev. Lett. 109, 150401 (2012).
[Crossref] [PubMed]

Wehner, S.

N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86, 419–478 (2014).
[Crossref]

Wiesner, K.

A. Cabello, M. Gu, O. Gühne, J. Larsson, and K. Wiesner, “Thermodynamical cost of some interpretations of quantum theory,” Phys. Rev. A 94, 052127 (2016).
[Crossref]

Winter, A.

A. Cabello, S. Severini, and A. Winter, “Graph-Theoretic Approach to Quantum Correlations,” Phys. Rev. Lett. 112, 040401 (2014).
[Crossref] [PubMed]

Xu, Z.-P.

A. Cabello, M. Gu, O. Gühne, and Z.-P. Xu, “Optimal simulation of state-independent quantum contextuality,” arXiv:1709.07372 (2017).

Yan, B.

B. Yan, “Quantum Correlations are Tightly Bound by the Exclusivity Principle,” Phys. Rev. Lett. 110, 260406 (2013).
[Crossref] [PubMed]

Yang, H.-X.

C. Zu, Y.-X. Wang, D.-L. Deng, X.-Y. Chang, K. Liu, P.-Y. Hou, H.-X. Yang, and L.-M. Duan, “State-Independent Experimental Test of Quantum Contextuality in An Indivisible System,” Phys. Rev. Lett. 109, 150401 (2012).
[Crossref] [PubMed]

Yu, S.

S. Yu and C. H. Oh, “State-Independent Proof of Kochen-Specker Theorem with 13 Rays,” Phys. Rev. Lett. 108, 030402 (2012).
[Crossref] [PubMed]

Yuan, X.

G. Chiribella and X. Yuan, “Measurement sharpness cuts nonlocality and contextuality in every physical theory,” arXiv: 1404.3348 (2014).

Zähringer, F.

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]

Zhang, C.

Y.-F. Huang, M. Li, D.-Y. Cao, C. Zhang, Y.-S. Zhang, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Experimental test of state-independent quantum contextuality of an indivisible quantum system,” Phys. Rev. A 87, 052133 (2013).
[Crossref]

Zhang, J.

X. Zhang, M. Um, J. Zhang, S. An, Y. Wang, D.-L. Deng, C. Shen, L.-M. Duan, and K. Kim, “State-Independent Experimental Test of Quantum Contextuality with a Single Trapped Ion,” Phys. Rev. Lett. 110, 070401 (2013).
[Crossref] [PubMed]

Zhang, X.

X. Zhang, M. Um, J. Zhang, S. An, Y. Wang, D.-L. Deng, C. Shen, L.-M. Duan, and K. Kim, “State-Independent Experimental Test of Quantum Contextuality with a Single Trapped Ion,” Phys. Rev. Lett. 110, 070401 (2013).
[Crossref] [PubMed]

Zhang, Y.-S.

Y.-F. Huang, M. Li, D.-Y. Cao, C. Zhang, Y.-S. Zhang, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Experimental test of state-independent quantum contextuality of an indivisible quantum system,” Phys. Rev. A 87, 052133 (2013).
[Crossref]

Zu, C.

C. Zu, Y.-X. Wang, D.-L. Deng, X.-Y. Chang, K. Liu, P.-Y. Hou, H.-X. Yang, and L.-M. Duan, “State-Independent Experimental Test of Quantum Contextuality in An Indivisible System,” Phys. Rev. Lett. 109, 150401 (2012).
[Crossref] [PubMed]

Dialectica (1)

E. P. Specker, “Die Logik nicht gleichzeitig entscheidbarer Aussagen,” Dialectica 14, 239–246 (1960).
[Crossref]

J. Math. Mech. (1)

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

J. Phys. A: Math. Gen. (1)

M. Kernaghan, “Bell-Kochen-Specker theorem for 20 vectors,” J. Phys. A: Math. Gen. 27, L829–L830 (1994).
[Crossref]

J. Phys. A: Math. Theor. (1)

A. Cabello, M. Kleinmann, and J. R. Portillo, “Quantum State-Independent Contextuality Requires 13 Rays,” J. Phys. A: Math. Theor. 49, 38LT01 (2016).
[Crossref]

Nature (2)

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

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]

Nature Mat. (1)

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

Phys. Lett. A (2)

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

M. Kernaghan and A. Peres, “Kochen-Specker theorem for eight-dimensional space,” Phys. Lett. A 198, 1–5 (1995).
[Crossref]

Phys. Rev. A (4)

R. Raussendorf, “Contextuality in measurement-based quantum computation,” Phys. Rev. A 88, 022322 (2013).
[Crossref]

Y.-F. Huang, M. Li, D.-Y. Cao, C. Zhang, Y.-S. Zhang, B.-H. Liu, C.-F. Li, and G.-C. Guo, “Experimental test of state-independent quantum contextuality of an indivisible quantum system,” Phys. Rev. A 87, 052133 (2013).
[Crossref]

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

A. Cabello, M. Gu, O. Gühne, J. Larsson, and K. Wiesner, “Thermodynamical cost of some interpretations of quantum theory,” Phys. Rev. A 94, 052127 (2016).
[Crossref]

Phys. Rev. Lett. (10)

A. Cabello, S. Severini, and A. Winter, “Graph-Theoretic Approach to Quantum Correlations,” Phys. Rev. Lett. 112, 040401 (2014).
[Crossref] [PubMed]

X. Zhang, M. Um, J. Zhang, S. An, Y. Wang, D.-L. Deng, C. Shen, L.-M. Duan, and K. Kim, “State-Independent Experimental Test of Quantum Contextuality with a Single Trapped Ion,” Phys. Rev. Lett. 110, 070401 (2013).
[Crossref] [PubMed]

S. Yu and C. H. Oh, “State-Independent Proof of Kochen-Specker Theorem with 13 Rays,” Phys. Rev. Lett. 108, 030402 (2012).
[Crossref] [PubMed]

C. Zu, Y.-X. Wang, D.-L. Deng, X.-Y. Chang, K. Liu, P.-Y. Hou, H.-X. Yang, and L.-M. Duan, “State-Independent Experimental Test of Quantum Contextuality in An Indivisible System,” Phys. Rev. Lett. 109, 150401 (2012).
[Crossref] [PubMed]

M. Kleinmann, C. Budroni, J.-Å. Larsson, O. Gühne, and A. Cabello, “Optimal Inequalities for State-Independent Contextuality,” Phys. Rev. Lett. 109, 250402 (2012).
[Crossref]

R. Ramanathan and P. Horodecki, “Necessary and sufficient condition for state-independent contextual measurement scenarios,” Phys. Rev. Lett. 112, 040404 (2014).
[Crossref] [PubMed]

A. Cabello, M. Kleinmann, and C. Budroni, “Necessary and Sufficient Condition for Quantum State-Independent Contextuality,” Phys. Rev. Lett. 114, 250402 (2015).
[Crossref] [PubMed]

A. Cabello, “Simple Explanation of the Quantum Violation of a Fundamental Inequality,” Phys. Rev. Lett. 110, 060402 (2013).
[Crossref] [PubMed]

B. Yan, “Quantum Correlations are Tightly Bound by the Exclusivity Principle,” Phys. Rev. Lett. 110, 260406 (2013).
[Crossref] [PubMed]

A. Cabello, “Simple Explanation of the Quantum Limits of Genuine n-Body Nonlocality,” Phys. Rev. Lett. 114, 220402 (2015).
[Crossref] [PubMed]

Rev. Mod. Phys. (2)

J. B. Bell, “On the Problem of Hidden Variables in Quantum Mechanics,” Rev. Mod. Phys. 38, 447 (1966).
[Crossref]

N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86, 419–478 (2014).
[Crossref]

Science (1)

C. Monroe and J. Kim, “Scaling the ion trap quantum processor,” Science 339, 1164–1169 (2013).
[Crossref] [PubMed]

Other (5)

A. Tavakoli and A. Cabello, “Cost of classically simulating quantum sequential measurements on entangled systems,” arXiv:1705.07456 (2017).

A. Cabello, M. Gu, O. Gühne, and Z.-P. Xu, “Optimal simulation of state-independent quantum contextuality,” arXiv:1709.07372 (2017).

K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, M. Pawlowski, and M. Bourennane, “Contextuality offers device-independent security,” arXiv: 1006.0468 (2010).

M. P. Seevinck, “E. Specker: ‘The logic of non-simultaneously decidable propositions’ (1960),” arXiv: 1103.4537 (2011).

G. Chiribella and X. Yuan, “Measurement sharpness cuts nonlocality and contextuality in every physical theory,” arXiv: 1404.3348 (2014).

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Figures (12)

Fig. 1
Fig. 1

Graph of the exclusivity relations between the measurements for the Yu-Oh scenario. Vertices represent vectors vi and edges represent exclusivity relations. The blue vertices represent the vectors in the Yu-Oh set. The red vertices represent the extra vectors used for assuring that each projector is always measured using a complete basis. The states are represented as v1 = (1, 0, 0), v2 = (0, 1, 0), v3 = (0, 0, 1), v 4 = 1 2 ( 0 , 1 , 1 ), v 5 = 1 2 ( 1 , 0 , 1 ), v 6 = 1 2 ( 1 , 1 , 0 ), v 7 = 1 2 ( 0 , 1 , 1 ), v 8 = 1 2 ( 1 , 0 , 1 ), v 9 = 1 2 ( 1 , 1 , 0 ), v 10 = 1 3 ( 1 , 1 , 1 ), v 11 = 1 3 ( 1 , 1 , 1 ), v 12 = 1 3 ( 1 , 1 , 1 ), v 13 = 1 3 ( 1 , 1 , 1 ), v 14 = 1 6 ( 1 , 2 , 1 ), v 15 = 1 6 ( 1 , 1 , 2 ), v 16 = 1 6 ( 2 , 1 , 1 ), v 17 = 1 6 ( 2 , 1 , 1 ), respectively.

Fig. 2
Fig. 2

Experimental setup to test the noncontextuality inequality. (a). The preparation of a single photon source. A single photon is generated by using a 632.8 nm laser to excite an intrinsic defect in a 4H-SiC sample which is mounted on a piezoelectric XYZ stage (PZT). The dichroic mirror (DM) is used to separated the pump laser and the fluorescence, which is further filtered by an interference filter (IF) centered at 720 nm with a bandwidth of 13 nm. (b). State preparation of a single photon qutrit. The polarization of the photon is set to be horizontal by the polarization beam splitter 1 (PBS1). After passing the first three haft-wave plates (HWP1, HWP2 and HWP3) and two beam displacers (BD1 and BD2), the photon is split into three paths. By adjusting the angle settings of HWPs, we can prepare different initial qutrit states. (c). The setup for state measurement. The angle of HWP4 is set to 45° and the photon in paths 1 and 3 are combined by the BD3. Different measurement bases are implemented by rotating HWP5, HWP6, and HWP7. Several phase plates (PCs) are inserted into paths to compensate relative phases. The photon is then detected by a single-photon avalanche detector (SPAD).

Fig. 3
Fig. 3

Influence between adjacent measurements with the input state |φ1〉(= |i1〉). The numbers along the x axis represent the settings of (vi, vj) as (v1, v2) (number 1), (v1, v3) (number 2), (v1, v4) (number 3), (v1, v7) (number 4), (v2, v3) (number 5), (v2, v5) (number 6), (v2, v8) (number 7), (v3, v6) (number 8), (v3, v9) (number 9), (v4, v7) (number 10), (v4, v10) (number 11), (v4, v13) (number 12), (v5, v13) (number 13), (v5, v8) (number 14), (v5, v11) (number 15), (v6, v9) (number 16), (v6, v13) (number 17), (v6, v12) (number 18), (v7, v12) (number 19), (v7, v11) (number 20), (v8, v12) (number 21), (v8, v10) (number 22), (v9, v10) (number 23), (v9, v11) (number 24). The error bars are estimated from counting statistics.

Fig. 4
Fig. 4

The experimental results of Iopt (shown by the red squares) with different input states. For all states, we obtain a violation of the inequality, which demonstrates its state-independent character. The dashed line in the left-hand side specifies the upper bound imposed by noncontextual hidden variable (HV) models, while the dashed line in the right-hand side corresponds to the quantum mechanical prediction. The error bars are estimated from the counting statistics.

Fig. 5
Fig. 5

The second-order photon correlation function (g2(τ)) as a function of the delay time τ. g2 = 0.179 with a fitting to 0.026 clearly confirms the character of single-photon emission.

Fig. 6
Fig. 6

The setup to detect P|φ (vi = 0, vj = 1)

Fig. 7
Fig. 7

Influence between adjacent measurements with the input state |φ2〉 (= |i2〉). The numbers along the x axis represent the settings of (vi, vj) as (v1, v2) (number 1), (v1, v3) (number 2), (v1, v4) (number 3), (v1, v7) (number 4), (v2, v3) (number 5), (v2, v5) (number 6), (v2, v8) (number 7), (v3, v6) (number 8), (v3, v9) (number 9), (v4, v7) (number 10), (v4, v10) (number 11), (v4, v13) (number 12), (v5, v13) (number 13), (v5, v8) (number 14), (v5, v11) (number 15), (v6, v9) (number 16), (v6, v13) (number 17), (v6, v12) (number 18), (v7, v12) (number 19), (v7, v11) (number 20), (v8, v12) (number 21), (v8, v10) (number 22), (v9, v10) (number 23), (v9, v11) (number 24). The error bars are estimated from counting statistics.

Fig. 8
Fig. 8

Influence between adjacent measurements with the input state |φ3〉 (= |i3〉). The numbers along the x axis represent the settings of (vi, vj) as (v1, v2) (number 1), (v1, v3) (number 2), (v1, v4) (number 3), (v1, v7) (number 4), (v2, v3) (number 5), (v2, v5) (number 6), (v2, v8) (number 7), (v3, v6) (number 8), (v3, v9) (number 9), (v4, v7) (number 10), (v4, v10) (number 11), (v4, v13) (number 12), (v5, v13) (number 13), (v5, v8) (number 14), (v5, v11) (number 15), (v6, v9) (number 16), (v6, v13) (number 17), (v6, v12) (number 18), (v7, v12) (number 19), (v7, v11) (number 20), (v8, v12) (number 21), (v8, v10) (number 22), (v9, v10) (number 23), (v9, v11) (number 24). The error bars are estimated from counting statistics.

Fig. 9
Fig. 9

Influence between adjacent measurements with the input state | φ 4 ( = 1 2 ( | i 1 + | i 2 ) ). The numbers along the x axis represent the settings of (vi, vj) as (v1, v2) (number 1), (v1, v3) (number 2), (v1, v4) (number 3), (v1, v7) (number 4), (v2, v3) (number 5), (v2, v5) (number 6), (v2, v8) (number 7), (v3, v6) (number 8), (v3, v9) (number 9), (v4, v7) (number 10), (v4, v10) (number 11), (v4, v13) (number 12), (v5, v13) (number 13), (v5, v8) (number 14), (v5, v11) (number 15), (v6, v9) (number 16), (v6, v13) (number 17), (v6, v12) (number 18), (v7, v12) (number 19), (v7, v11) (number 20), (v8, v12) (number 21), (v8, v10) (number 22), (v9, v10) (number 23), (v9, v11) (number 24). The error bars are estimated from counting statistics.

Fig. 10
Fig. 10

Influence between adjacent measurements with the input state | φ 5 ( = 1 2 ( | i 1 + | i 3 ) ). The numbers along the x axis represent the settings of (vi, vj) as (v1, v2) (number 1), (v1, v3) (number 2), (v1, v4) (number 3), (v1, v7) (number 4), (v2, v3) (number 5), (v2, v5) (number 6), (v2, v8) (number 7), (v3, v6) (number 8), (v3, v9) (number 9), (v4, v7) (number 10), (v4, v10) (number 11), (v4, v13) (number 12), (v5, v13) (number 13), (v5, v8) (number 14), (v5, v11) (number 15), (v6, v9) (number 16), (v6, v13) (number 17), (v6, v12) (number 18), (v7, v12) (number 19), (v7, v11) (number 20), (v8, v12) (number 21), (v8, v10) (number 22), (v9, v10) (number 23), (v9, v11) (number 24). The error bars are estimated from counting statistics.

Fig. 11
Fig. 11

Influence between adjacent measurements with the input state | φ 6 ( = 1 2 ( | i 2 + | i 3 ) ). The numbers along the x axis represent the settings of (vi, vj) as (v1, v2) (number 1), (v1, v3) (number 2), (v1, v4) (number 3), (v1, v7) (number 4), (v2, v3) (number 5), (v2, v5) (number 6), (v2, v8) (number 7), (v3, v6) (number 8), (v3, v9) (number 9), (v4, v7) (number 10), (v4, v10) (number 11), (v4, v13) (number 12), (v5, v13) (number 13), (v5, v8) (number 14), (v5, v11) (number 15), (v6, v9) (number 16), (v6, v13) (number 17), (v6, v12) (number 18), (v7, v12) (number 19), (v7, v11) (number 20), (v8, v12) (number 21), (v8, v10) (number 22), (v9, v10) (number 23), (v9, v11) (number 24). The error bars are estimated from counting statistics.

Fig. 12
Fig. 12

Influence between adjacent measurements with the input state | φ 7 ( = 1 3 ( | i 1 + | i 2 + | i 3 ) ). The numbers along the x axis represent the settings of (vi, vj) as (v1, v2) (number 1), (v1, v3) (number 2), (v1, v4) (number 3), (v1, v7) (number 4), (v2, v3) (number 5), (v2, v5) (number 6), (v2, v8) (number 7), (v3, v6) (number 8), (v3, v9) (number 9), (v4, v7) (number 10), (v4, v10) (number 11), (v4, v13) (number 12), (v5, v13) (number 13), (v5, v8) (number 14), (v5, v11) (number 15), (v6, v9) (number 16),(v6, v13) (number 17), (v6, v12) (number 18), (v7, v12) (number 19), (v7, v11) (number 20), (v8, v12) (number 21), (v8, v10) (number 22), (v9, v10) (number 23), (v9, v11) (number 24). The error bars are estimated from counting statistics.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

I = i = 1 N w i P ^ i α ( G ) ,
I = 3 i = 1 9 P ^ i + 2 i = 13 10 P ^ i 11 ,
I opt = 3 i = 1 9 P | φ ( v i = 1 ) ( 1 1 2 ( i , j ) E P | v i ( v i = 1 ) ) + 2 i = 10 13 P | φ ( v i = 1 ) ( 1 1 2 ( i , j ) E P | v i ( v j = 1 ) ) 11 ,
P | φ ( v i = 1 ) = N | φ ( v i ) N | φ ( v i ) + N | φ ( v i ) + N | φ ( v i ) ,
P ^ 1 = [ 1 0 0 0 0 0 0 0 0 ] , P ^ 2 = [ 0 0 0 0 1 0 0 0 0 ] , P ^ 3 = [ 0 0 0 0 0 0 0 0 1 ] , P ^ 4 = 1 2 [ 0 0 0 0 1 1 0 1 1 ] , P ^ 5 = 1 2 [ 1 0 1 0 0 0 1 0 1 ] , P ^ 6 = 1 2 [ 1 1 0 1 1 0 0 0 0 ] , P ^ 7 = 1 2 [ 0 0 0 0 1 1 0 1 1 ] , P ^ 8 = 1 2 [ 1 0 1 0 0 0 1 0 1 ] , P ^ 9 = 1 2 [ 1 1 0 1 1 0 0 0 0 ] , P ^ 10 = 1 3 [ 1 1 1 1 1 1 1 1 1 ] , P ^ 11 = 1 3 [ 1 1 1 1 1 1 1 1 1 ] , P ^ 12 = 1 3 [ 1 1 1 1 1 1 1 1 1 ] , P ^ 13 = 1 3 [ 1 1 1 1 1 1 1 1 1 ] .
I QM ( w ) = i = 1 13 w i P ^ i = w 0 𝟙 .
w 1 = w 2 = w 3 , w 4 = w 5 = w 6 = w 7 = w 8 = w 9 , w 10 = w 11 = w 12 = w 13 .
α ( w ) = max { w 1 + 4 w 10 , w 1 + w 10 + 2 w 4 , w 10 + 3 w 4 } .
α = max { w 1 + 4 , w 1 + 2 w 4 + 1 , 1 + 3 w 4 } .
r = I Q M α = w 1 + 2 w 4 + 4 / 3 w 1 + 4 = { ( w 1 + 13 / 3 ) / ( w 1 + 4 ) w 1 3 / 2 , ( 5 w 1 / 3 + 10 / 3 ) / ( w 1 + 4 ) , w 1 3 / 2 .
I = 3 i = 1 9 P ^ i + 2 i = 10 10 P ^ i 11 ,
I = 2 i = 1 9 P ^ i + i = 10 13 P ^ i 7 ,
I = 3 i = 1 , 4 , 7 , 10 , 11 , 12 , 13 P ^ i + 6 i = 2 , 3 , 5 , 6 , 8 , 9 P ^ i 18 ,
I = 5 i = 1 9 P ^ i + 3 i = 10 13 P ^ i 18 ,
I opt = 3 i = 1 9 P | φ ( v i = 1 ) ( 1 1 2 ( i , j ) E P | v i ( v j = 1 ) ) + 2 i = 10 13 P | φ ( v i = 1 ) ( 1 1 2 ( i , j ) E P | v i ( v j = 1 ) ) 11 ,
P | φ ( v i = 1 ) 1 2 ( i , j ) E P | φ ( v i = 1 , v j = 1 ) .
P | φ ( v i = 1 , v j = 1 ) = P | φ ( v i = 1 ) P | v i ( v j = 1 ) ,
ε ( , 0 | , v j ) = | P | φ ( v j = 0 ) P | φ ( v i = 0 , v j = 0 ) P | φ ( v i = 1 , v j = 0 ) | ,
ε ( , 1 | , v j ) = | P | φ ( v j = 1 ) P | φ ( v i = 0 , v j = 1 ) P | φ ( v i = 1 , v j = 0 ) | ,
ε ( 0 , | v i , ) = | P | φ ( v i = 0 ) P | φ ( v i = 0 , v j = 0 ) P | φ ( v i = 0 , v j = 1 ) | ,
ε ( 1 , | v i , ) = | P | φ ( v i = 1 ) P | φ ( v i = 1 , v j = 0 ) P | φ ( v i = 1 , v j = 1 ) | ,
P | φ ( v i = 0 ) = 1 P | φ ( v i = 1 ) , P | φ ( v i = 1 , v j = 1 ) = P | φ ( v i = 1 ) P | v i ( v j = 1 ) , P | φ ( v i = 0 , v j = 1 ) = P | φ ( v i = 0 ) P | v i ( v j = 1 ) , P | φ ( v i = 1 , v j = 0 ) = P | φ ( v i = 1 ) P | φ ( v i = 1 , v j = 1 ) , P | φ ( v i = 0 , v j = 0 ) = P | φ ( v i = 0 ) P | φ ( v i = 0 , v j = 1 ) .
ε ( , 0 | , v j ) ε ( 0 , | v i , ) = 0 , ε ( , 1 | , v j ) ε ( 1 , | v i , ) = 0 .
SD ( I opt ) = 3 i = 1 9 SD ( P | φ ( v i = 1 ) ( 1 + 1 2 ( i , j ) E P | v i ( v j = 1 ) ) ) + 2 i = 10 13 SD ( P | φ ( v i = 1 ) ( 1 + 1 2 ( i , j ) E P | v i ( v j = 1 ) ) ) .