CGLMP and Bell–CHSH formulations of non-locality: a comparative study

  • Radha Pyari SandhirEmail author
  • Soumik Adhikary
  • V. Ravishankar


The CGLMP prescription of non-locality is known to identify certain states that satisfy the Bell–CHSH inequality to be non-local. The demonstration is, however, restricted to a specific family of states. In this paper, we address the converse question: can there be states that satisfy the CGLMP inequality but violate Bell–CHSH? We find the answer to be in the affirmative. Examining coupled \(4 \times 4\) level systems, we find that there exist a large number of such states. As a direct consequence, states that violate the CGLMP inequality do not form a superset over the ones that violate Bell–CHSH.


Bell inequality Non-locality Higher-dimensional systems 



Radha thanks the Department of Science and Technology (DST), India, for funding her research under the WOS-A Women’s Scientist Scheme. Soumik thanks Council of Scientific and Industrial Research (CSIR), India, for funding his research. The authors would also like to thank the referees for their valuable comments and suggestions.


  1. 1.
    Acín, A., Durt, T., Gisin, N., Latorre, J.I.: Quantum nonlocality in two three-level systems. Phys. Rev. A 65(052), 325 (2002)Google Scholar
  2. 2.
    Avis, D., Imai, H., Ito, T., Sasaki, Y.: Deriving tight bell inequalities for 2 parties with many 2-valued observables from facets of cut polytopes. (2004). arXiv:quant-ph/0404014
  3. 3.
    Bell, J.S.: On the Einstein podolsky rosen paradox. Physics 1, 195 (1964)Google Scholar
  4. 4.
    Braunstein, S., Mann, A., Revzen, M.: Maximal violation of bell inequalities for mixed states. Phys. Rev. Lett. 68, 3259 (1992). doi: 10.1103/PhysRevLett.68.3259 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys. 86, 419–478 (2014)ADSCrossRefGoogle Scholar
  6. 6.
    Cirel’son, B.: Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4, 93 (1980). doi: 10.1007/BF00417500 ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23(15), 880 (1969)ADSCrossRefGoogle Scholar
  8. 8.
    Collins, D., Gisin, N., Linden, N., Massar, S., Popescu, S.: Bell inequalities for arbitrarily high-dimensional systems. Phys. Rev. Lett. 88(040), 404 (2002). doi: 10.1103/PhysRevLett.88.040404 MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dada, A., et al.: Experimental high-dimensional two-photon entanglement and violations of generalized bell inequalities. Nat. Phys. 7, 677 (2011)CrossRefGoogle Scholar
  10. 10.
    Durt, T., Kaszlikowski, D., Zukowski, M.: Violations of local realism with quantum systems described by N -dimensional hilbert spaces up to \(n=16\). Phys. Rev. A 64(024), 101 (2001). doi: 10.1103/PhysRevA.64.024101 Google Scholar
  11. 11.
    Fonseca, E.A., Parisio, F.: Measure of nonlocality which is maximal for maximally entangled qutrits. Phys. Rev. A 92(030), 101 (2015). doi: 10.1103/PhysRevA.92.030101 Google Scholar
  12. 12.
    Fu, L.: General correlation functions of the Clauser-Horne-Shimony-Holt inequality for arbitrarily high-dimensional systems. Phys. Rev. Lett. 92(130), 404 (2004). doi: 10.1103/PhysRevLett.92.130404 Google Scholar
  13. 13.
    Gisin, N.: Bell’s inequality holds for all non-product states. Phys. Lett. A 154(5), 201–202 (1991)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Howell, J., Lamas-Linares, A., Bouwmeester, D.: Experimental violation of a spin-1 bell inequality using maximally entangled four-photon states. Phys. Rev. Lett. 88(3), 030,401 (2002)CrossRefGoogle Scholar
  15. 15.
    Kaszlikowski, D., Gnaciński, P., Zukowski, M., Miklaszewski, W., Zeilinger, A.: Violations of local realism by two entangled \(\mathit{N}\)-dimensional systems are stronger than for two qubits. Phys. Rev. Lett. 85, 4418 (2000). doi: 10.1103/PhysRevLett.85.4418 ADSCrossRefGoogle Scholar
  16. 16.
    Lo, H.P., Li, C.M., Yabushita, A., Chen, Y.N., Luo, C.W., Kobayashi, T.: Experimental violation of bell inequalities for multi-dimensional systems. Sci. Rep. 6(22), 088 (2016)Google Scholar
  17. 17.
    Masanes, L.: Tight bell inequality for d-outcome measurements correlations. Quantum Inf. Comput. 3(4), 345 (2003)Google Scholar
  18. 18.
    Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7(4), 308 (1965). doi: 10.1093/comjnl/7.4.308 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Popescu, S., Rohrlich, D.: Which states violate bell’s inequality maximally? Phys. Lett. A 169(6), 411 (1992)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Scarani, V.: The device-independent outlook on quantum physics. Acta Phys. Slovaca 62, 347 (2013)Google Scholar
  21. 21.
    Thew, R.T., Acín, A., Zbinden, H., Gisin, N.: Bell-type test of energy-time entangled qutrits. Phys. Rev. Lett. 93(010), 503 (2004). doi: 10.1103/PhysRevLett.93.010503 MathSciNetzbMATHGoogle Scholar
  22. 22.
    Vaziri, A., Weihs, G., Zeilinger, A.: Experimental two-photon, three-dimensional entanglement for quantum communication. Phys. Rev. Lett. 89(240), 401 (2002). doi: 10.1103/PhysRevLett.89.240401 Google Scholar

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Authors and Affiliations

  1. 1.Department of Physics and Computer ScienceDayalbagh Educational InstituteAgraIndia
  2. 2.Department of PhysicsIndian Institute of Technology DelhiNew DelhiIndia

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