CHSH inequalities with appropriate response function for POVM and their quantum violation

  • Asmita Kumari
  • A. K. PanEmail author


In the derivation of local bound of a Bell’s inequality, the response functions corresponding to the different outcomes of measurements are fixed by the relevant hidden variables irrespective of the fact whether the measurement is unsharp. In the context of a recent result by Spekkens that tells even in a ontological theory the unsharp observable cannot be assigned a deterministic response function, we derive a modified local bound of CHSH inequality in unsharp measurement scenario. We consider response function for a given POVM which is determined by the response functions of the relevant projectors appearing in its spectral representation. In this scenario, the local bound of CHSH inequality is found to be dependent on the unsharpness parameter. This then enables us to show that the quantum violation of CHSH inequality for unbiased spin-POVMs occurs whenever there is violation for their sharp counterpart. For the case of biased POVMs, it is shown that the quantum violation of CHSH inequality can be obtained for ranges of sharpness parameter for which no violation obtained using standard local bound of CHSH inequality.


Bell’s inequality Ontological model Response function of POVM 



Authors gratefully acknowledge the discussion with Prof G. Kar (ISI Kolkata). AKP acknowledges the support from Ramanujan Fellowship research Grant (SB/S2/RJN-083/2014).


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Authors and Affiliations

  1. 1.National Institute Technology PatnaPatnaIndia

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