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The Cabello Nonlocality Argument is Stronger Control than the Hardy Nonlocality Argument for Detecting Post-Quantum Correlations in the Bipartite Systems

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Abstract

In this paper, we study the Hardy nonlocality argument (HNA) and the Cabello nonlocality argument (CNA) under the Information Causality (IC), Macroscopic Locality (ML) and Local Orthogonality (LO) principles with respected to Local Randomness. We show that, in the context of all possibilities of local randomness, the gap between the quantum mechanics and the above principles, in the CNA is larger than the HNA. Therefore, the CNA is stronger control than the HNA for detecting post-quantum nosignalling correlations in the bipartite systems.

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References

  1. Tsirelson, B.S.: Quantum generalizations of Bell's inequality. Lett. Math. Phys. 4, 93–100 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  2. Bell, J.S.: On the Einstein Podolsky Rosen paradox. Physics. 1, 195–200 (1964)

    Article  MathSciNet  Google Scholar 

  3. Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed Experiment to Test Local Hidden-Variable Theories. Phys. Rev. Lett. 23, 880–884 (1969)

    Article  ADS  MATH  Google Scholar 

  4. Popescu, S., Rohrlich, D.: Quantum nonlocality as an axiom. Found. Phys. 24, 379–385 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  5. Hardy, L.: Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories. Phys. Rev. Lett. 68, 2981–2984 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Hardy, L.: Nonlocality for two particles without inequalities for almost all entangled states. Phys. Rev. Lett. 71, 1665–1668 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Mermin, N.D.: Quantum mysteries refined. Am. J. Phys. 62, 880–887 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Cabello, A.: Phys.Rev. A. 65, 032108 (2002)

    Article  ADS  Google Scholar 

  9. Liang, L.-m., Li, C.-z.: Phys. Lett. A. 335, 371–373 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  10. Choudhary, S.K., Ghosh, S., Kar, G., Kunkri, S., Rahaman, R., Roy, A.: Quantum Information and Computation. 10(9 & 10), 0859–0871 (2010)

    MathSciNet  Google Scholar 

  11. Kar, G.: Hardy's nonlocality for mixed states. Phys. Lett. A. 228, 119–120 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Rabelo, R., Law, Y.Z., Scarani, V.: Device-Independent Bounds for Hardy’s Experiment. Phys. Rev. Lett. 109, 180401 (2012)

    Article  ADS  Google Scholar 

  13. Kunkri, S., Choudhary, S.K., Ahanj, A., Joag, P.: Nonlocality without inequality for almost all two-qubit entangled states based on Cabello’s nonlocality argument. Phys. Rev. A. 73, 022346 (2006)

    Article  ADS  Google Scholar 

  14. van Dam, W.: Ph.D. thesis. In: University of Oxford (2000)

    Google Scholar 

  15. Brassard, G., Buhrman, H., Linden, N., Methot, A.A., Tapp, A., Unger, F.: Limit on Nonlocality in Any World in Which Communication Complexity Is Not Trivial. Phys. Rev. Lett. 96, 250401 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Pawlowski, M., Paterek, T., Kaszlikowski, D., Scarani, V., Winter, A., Zukowski, M.: Information causality as a physical principle. Nature. 461, 1101–1104 (2009)

    Article  ADS  Google Scholar 

  17. Navascués, M., Wunderlich, H.: Proc. Roy. Soc. Lond. A. 466, 881–890 (2009)

    Article  ADS  Google Scholar 

  18. Fritz, T., Sainz, A.B., Augusiak, R., Brask, J.B., Chaves, R., Leverrier, A., Acin, A.: Nat. Commun. 4(2263), (2013)

  19. Cabello, A.: Exclusivity principle and the quantum bound of the Bell inequality. Phys. Rev. A. 90, 062125 (2014)

    Article  ADS  Google Scholar 

  20. Oppenheim, J., Wehner, S.: The Uncertainty Principle Determines the Nonlocality of Quantum Mechanics. Science. 330, 1072–1074 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Banik, M., Gazi, M.R., Ghosh, S., Kar, G.: Phys. Rev. A. 87, 052125 (2013)

    Article  ADS  Google Scholar 

  22. D. Rohrlich, “PR-box correlations have no classical limit, in Quantum Theory", A Two Time Success Story, Yakir Aharonov Festschrift; D.C. Struppa, J.M. Tollaksen, Eds.; Springer: Milan, Italy, 2013; pp. 205–211

  23. D. Rohrlich,arXiv: 1407.8530

  24. D. Rohrlich,arXiv: 1408.3125

  25. Gisin, N.: Quantum correlations in Newtonian space and time: faster than light communication or nonlocality. In: (the Frontiers Collection) Quantum [Un]Speakables, vol. II, pp. 321–330. Half a Century of Bell’s Theorem; R. Bertlmann, A. Zeilinger, Eds.; Springer, Berlin, Germany (2017)

    Chapter  Google Scholar 

  26. Ahanj, A., Kunkri, S., Rai, A., Rahaman, R., Joag, P.S.: Bound on Hardy’s nonlocality from the principle of information causality. Phys. Rev. A. 81, 032103 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  27. Das, S., Banik, M., Gazi, M.R., Rai, A., Kunkri, S.: Phys. Rev. A. 88, 062101 (2013)

    Article  ADS  Google Scholar 

  28. Gallego, R., Wurflinger, L.E., Acin, A., Navascues, M.: Quantum Correlations Require Multipartite Information Principles. Phys. Rev. Lett. 107, 210403 (2011)

    Article  ADS  Google Scholar 

  29. Das, S., Banik, M., Rai, A., Gazi, M.D.R., Kunkri, S.: Hardy's nonlocality argument as a witness for postquantum correlations. Phys. Rev. A. 87, 012112 (2013)

    Article  ADS  Google Scholar 

  30. Gazi, M.D.R., Rai, A., Kunkri, S., Rahaman, R.: Local randomness in Hardy's correlations: implications from the information causality principle. J. Phys. A Math. Theor. 43, 452001 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zoka, G., Ahanj, A.: Local randomness in Cabello’s non-locality argument from the information causality principle. Quantum Stud.: Math. Found. 3(2), 135–145 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Barrett, J., Linden, N., Massar, S., Pironio, S., Popescu, S., Roberts, D.: Nonlocal correlations as an information-theoretic resource. Phys. Rev. A. 71, 022101 (2005)

    Article  ADS  Google Scholar 

  33. Allcock, J., Brunner, N., Pawlowski, M., Scarani, V.: Phys. Rev. A. 80(R), 040103 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  34. W. van Dam, e-print arXiv:quant-ph/0501159

  35. S. Wolf and J. Wullschleger, e-print arXiv:quant-ph/0502030v1 (2005)

    Google Scholar 

  36. Allcock, J., Brunner, N., Pawlowski, M., Scarani, V.: Phys. Rev. A. 80(R), 040103 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  37. Navascues, M., Pironio, S., Acin, A.: Bounding the Set of Quantum Correlations. Phys. Rev. Lett. 98, 010401 (2007)

    Article  ADS  Google Scholar 

  38. Navascues, M., Pironio, S., Acin, A.: A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New J. Phys. 10, 073013 (2008)

    Article  ADS  Google Scholar 

  39. Cereceda, J.L.: Found. Phys. Lett. 13, 427–442 (2000)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Khayyam University, Mashhad, Iran

    Ali Ahanj

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  1. Ali Ahanj
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Appendices

Appendix 1

The 16 Local matrices for bipartite system with binary input-output:

$$ {\displaystyle \begin{array}{l}{P}^{0000}=\left(\begin{array}{llll}1& 0& 0& 0\\ {}1& 0& 0& 0\\ {}1& 0& 0& 0\\ {}1& 0& 0& 0\end{array}\right),{P}^{0001}=\left(\begin{array}{llll}0& 1& 0& 0\\ {}0& 1& 0& 0\\ {}0& 1& 0& 0\\ {}0& 1& 0& 0\end{array}\right)\\ {}{P}^{0010}=\left(\begin{array}{llll}1& 0& 0& 0\\ {}0& 1& 0& 0\\ {}1& 0& 0& 0\\ {}0& 1& 0& 0\end{array}\right),{P}^{0011}=\left(\begin{array}{llll}0& 1& 0& 0\\ {}1& 0& 0& 0\\ {}0& 1& 0& 0\\ {}1& 0& 0& 0\end{array}\right)\\ {}{P}^{0100}=\left(\begin{array}{llll}0& 0& 1& 0\\ {}0& 0& 1& 0\\ {}0& 0& 1& 0\\ {}0& 0& 1& 0\end{array}\right),{P}^{0101}=\left(\begin{array}{llll}0& 0& 0& 1\\ {}0& 0& 0& 1\\ {}0& 0& 0& 1\\ {}0& 0& 0& 1\end{array}\right)\\ {}{P}^{0110}=\left(\begin{array}{llll}0& 0& 1& 0\\ {}0& 0& 0& 1\\ {}0& 0& 1& 0\\ {}0& 0& 0& 1\end{array}\right),{P}^{0111}=\left(\begin{array}{llll}0& 0& 0& 1\\ {}0& 0& 1& 0\\ {}0& 0& 0& 1\\ {}0& 0& 1& 0\end{array}\right)\\ {}{P}^{1000}=\left(\begin{array}{llll}1& 0& 0& 0\\ {}1& 0& 0& 0\\ {}0& 0& 1& 0\\ {}0& 0& 1& 0\end{array}\right),{P}^{1001}=\left(\begin{array}{llll}0& 1& 0& 0\\ {}0& 1& 0& 0\\ {}0& 0& 0& 1\\ {}0& 0& 0& 1\end{array}\right)\\ {}{P}^{1010}=\left(\begin{array}{llll}1& 0& 0& 0\\ {}0& 1& 0& 0\\ {}0& 0& 1& 0\\ {}0& 0& 0& 1\end{array}\right),{P}^{1011}=\left(\begin{array}{llll}0& 1& 0& 0\\ {}1& 0& 0& 0\\ {}0& 0& 0& 1\\ {}0& 0& 1& 0\end{array}\right)\\ {}{P}^{1100}=\left(\begin{array}{llll}0& 0& 1& 0\\ {}0& 0& 1& 0\\ {}1& 0& 0& 0\\ {}1& 0& 0& 0\end{array}\right),{P}^{1101}=\left(\begin{array}{llll}0& 0& 0& 1\\ {}0& 0& 0& 1\\ {}0& 1& 0& 0\\ {}0& 1& 0& 0\end{array}\right)\\ {}{P}^{1110}=\left(\begin{array}{llll}0& 0& 1& 0\\ {}0& 0& 0& 1\\ {}1& 0& 0& 0\\ {}0& 1& 0& 0\end{array}\right),{P}^{1111}=\left(\begin{array}{llll}0& 0& 0& 1\\ {}0& 0& 1& 0\\ {}0& 1& 0& 0\\ {}1& 0& 0& 0\end{array}\right)\end{array}} $$

The 8 Non-Local matrices for bipartite system with binary input-output:

$$ {\displaystyle \begin{array}{l}{P}^{000}=\frac{1}{2}\left(\begin{array}{llll}1& 0& 0& 1\\ {}1& 0& 0& 1\\ {}1& 0& 0& 1\\ {}0& 1& 1& 0\end{array}\right),{P}^{001}=\frac{1}{2}\left(\begin{array}{llll}0& 1& 1& 0\\ {}0& 1& 1& 0\\ {}0& 1& 1& 0\\ {}1& 0& 0& 1\end{array}\right)\\ {}{P}^{010}=\frac{1}{2}\left(\begin{array}{llll}1& 0& 0& 1\\ {}0& 1& 1& 0\\ {}1& 0& 0& 1\\ {}1& 0& 0& 1\end{array}\right),{P}^{011}=\frac{1}{2}\left(\begin{array}{llll}0& 1& 1& 0\\ {}1& 0& 0& 1\\ {}0& 1& 1& 0\\ {}0& 1& 1& 0\end{array}\right)\\ {}{P}^{100}=\frac{1}{2}\left(\begin{array}{llll}1& 0& 0& 1\\ {}1& 0& 0& 1\\ {}0& 1& 1& 0\\ {}1& 0& 0& 1\end{array}\right),{P}^{101}=\frac{1}{2}\left(\begin{array}{llll}0& 1& 1& 0\\ {}0& 1& 1& 0\\ {}1& 0& 0& 1\\ {}0& 1& 1& 0\end{array}\right)\\ {}{P}^{110}=\frac{1}{2}\left(\begin{array}{llll}1& 0& 0& 1\\ {}0& 1& 1& 0\\ {}0& 1& 1& 0\\ {}0& 1& 1& 0\end{array}\right),{P}^{111}=\frac{1}{2}\left(\begin{array}{llll}0& 1& 1& 0\\ {}1& 0& 0& 1\\ {}1& 0& 0& 1\\ {}1& 0& 0& 1\end{array}\right)\end{array}} $$

Appendix 2

The following inequalities are sufficient for obtaining the upper bound of the HNA under LO principle [27]. The inequalities can be shown, in terms of variables ek, mi and ni (k ∈ {1, 2, 3, 4} and i ∈ {1, 2}).

$$ {e}_3^2+2{e}_2{g}_2-{g}_2^2-{e}_2^2\le 0 $$
(32)
$$ {e}_3^2+2{e}_1{g}_2-{e}_1^2-{g}_2^2\le 0 $$
(33)
$$ {e}_3^2+\left({e}_3-{e}_1\right)\left(1-{e}_2-{f}_2\right)\le 0 $$
(34)
$$ {e}_3^2+\left({e}_3-{e}_2\right)\left(1-{f}_2-{g}_2\right)\le 0 $$
(35)
$$ {e}_2\left({e}_3+{f}_2-{e}_2\right)+\left({e}_3-{f}_2\right){g}_2\le 0 $$
(36)
$$ {e}_1\left({f}_2-{e}_3\right)+{e}_3\left({e}_3+{g}_2\right)-{f}_2{g}_2\le 0 $$
(37)
$$ \left({g}_2-{e}_2\right)\left({f}_2+{g}_2-1\right)+2{e}_1{e}_3-{e}_1^2\le 0 $$
(38)
$$ {e}_3\left(1+{f}_2-{g}_2\right)+{e}_2\left({f}_2+{g}_2-1\right)-{f}_2^2\le 0 $$
(39)
$$ {e}_3^2+\left({f}_2-{e}_3\right)\left({f}_2+{g}_2-1\right)-{\left({e}_1-{e}_2\right)}^2\le 0 $$
(40)
$$ {e}_1\left({e}_3-{f}_2-{g}_2\right)+{e}_2\left(-1+{e}_1-{e}_3+{f}_2+{g}_2\right)+{e}_3\le 0 $$
(41)

The relation between the 8 independent parameter and the coefficients ci is obtained by comparing the matrix (5) and the Hardy correlation matrix (15). So, we have:

$$ {\displaystyle \begin{array}{l}{e}_1={c}_3+\frac{c_6}{2}\\ {}{e}_2={c}_{3,4}+\frac{c_6}{2}\\ {}{e}_3=\frac{c_6}{2}\\ {}{e}_4=0\\ {}{f}_1={e}_2\\ {}{f}_2={c}_5+\frac{c_6}{2}\\ {}{g}_1={e}_1\\ {}{g}_2={c}_2+{e}_2\end{array}} $$

Appendix 3

The following inequalities are sufficient for obtaining the upper bound of CNA under LO principle [27]. The inequalities can be represented, in terms of variables ek, mi and ni (k ∈ {1, 2, 3, 4} and i ∈ {1, 2}):

$$ {e}_3\left(1+{e}_1-{f}_1-{g}_2\right)+\left(1+{e}_2+{e}_3-{f}_1-{f}_2-{g}_2\right){e}_2-{e}_1+\left({e}_1+{f}_2+{g}_2-1\right){g}_2\le 0 $$
$$ {\displaystyle \begin{array}{c}{e}_3\left(1+{e}_1-{f}_1-{g}_2\right)+\left(1+{e}_2+{e}_3-{f}_1-{f}_2-{g}_2\right){e}_2\\ {}-{e}_1+\left({e}_1+{f}_2+{g}_2-1\right){g}_2\le 0\end{array}} $$
(42)
$$ {\displaystyle \begin{array}{c}{\left({e}_2\right)}^2+\left(1-{f}_1-{g}_2\right){e}_3+\left(1+{e}_1+{e}_3-2{f}_1-{f}_2-{g}_2\right){e}_2\\ {}-{e}_1^2+\left({e}_3+{f}_1-{f}_2\right){e}_1+\left({f}_1+{f}_2\right)\left({f}_2+{g}_2\right)-\left({f}_1+{f}_2\right)\le 0\end{array}} $$
(43)
$$ {\displaystyle \begin{array}{c}\left({e}_2+{e}_3\right)\left({e}_3-{f}_1\right)+2{e}_2{g}_2-{g}_2^2\le 0\\ {}{e}_2^2+\left({e}_2+2{f}_2-{f}_1\right){e}_3-{e}_1{e}_2+\left(1-{f}_2\right){e}_1-\\ {}{f}_2^2+\left({f}_2-1\right){f}_1\le 0\end{array}} $$
(44)
$$ {\displaystyle \begin{array}{c}{e}_2^2+{e}_3\left(1+{f}_2-{f}_1-{g}_2\right)+{e}_3{e}_2+{e}_2\left(1-{f}_1-{f}_2-{g}_2\right)\\ {}+{e}_1\left({g}_2-{f}_1\right)+{f}_1^2+{f}_2\left({f}_1+{g}_2-1\right)-{f}_1\le 0\end{array}} $$
(45)
$$ {\displaystyle \begin{array}{c}{e}_3^2+{e}_3\left(-{f}_1\right)+{e}_2{e}_3+{e}_2\left(2-2{f}_1-{f}_2\right)+\\ {}{e}_1\left(1-{f}_2-{g}_2\right)+{f}_1^2+{f}_1\left(2{g}_2+{f}_2-2\right)\\ {}+\left({f}_2-1\right){g}_2\le 0\end{array}} $$
(46)
$$ {\displaystyle \begin{array}{c}{e}_3^2+\left({e}_2-{f}_1\right){e}_3+{e}_2\left(1-2{f}_1+{g}_2\right)+\\ {}{f}_1\left({f}_1+{g}_2\right)-{f}_1-{g}_2^2\le 0\end{array}} $$
(47)
$$ \left({e}_2+{e}_1\right)\left({e}_3+{e}_2+1-{e}_1-{g}_2\right)+{f}_1\left({f}_1+2{g}_2-2{e}_2-2\right)\le 0 $$
(48)

The relation between 8 independent parameters and coefficients ci is obtained by comparing the matrix (5) and the Cabelo correlation matrix(18). So, we have:

$$ {\displaystyle \begin{array}{l}{e}_1={c}_{3,8,10}+\frac{c_6+{c}_{11}}{2}\\ {}{e}_2={c}_{3,4}+\frac{c_6}{2}\\ {}{e}_3={c}_8+\frac{c_6}{2}\\ {}{e}_4=0\\ {}{f}_1={e}_1+{c}_{4,7,9}\\ {}{f}_2={e}_3+{c}_{5,7}+\frac{c_11}{2}\\ {}{g}_1={e}_1\\ {}{g}_2={e}_2+{c}_2+\frac{c_{11}}{2}\end{array}} $$

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Ahanj, A. The Cabello Nonlocality Argument is Stronger Control than the Hardy Nonlocality Argument for Detecting Post-Quantum Correlations in the Bipartite Systems. Int J Theor Phys 58, 1441–1455 (2019). https://doi.org/10.1007/s10773-019-04032-8

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