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


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.

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  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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys. 86, 419–478 (2014)

    ADS  Article  Google Scholar 

  6. 6.

    Cirel’son, B.: Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4, 93 (1980). doi:10.1007/BF00417500

    ADS  MathSciNet  Article  Google 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)

    ADS  Article  Google 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

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Dada, A., et al.: Experimental high-dimensional two-photon entanglement and violations of generalized bell inequalities. Nat. Phys. 7, 677 (2011)

    Article  Google 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)

    ADS  MathSciNet  Article  Google 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)

    Article  Google 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

    ADS  Article  Google 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)

  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

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Popescu, S., Rohrlich, D.: Which states violate bell’s inequality maximally? Phys. Lett. A 169(6), 411 (1992)

    ADS  MathSciNet  Article  Google 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

    MathSciNet  MATH  Google 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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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.

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Correspondence to Radha Pyari Sandhir.

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Sandhir, R.P., Adhikary, S. & Ravishankar, V. CGLMP and Bell–CHSH formulations of non-locality: a comparative study. Quantum Inf Process 16, 263 (2017).

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  • Bell inequality
  • Non-locality
  • Higher-dimensional systems