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In the derivation of local bound of a Bell's inequality, the responsefunctions corresponding to the different outcomes of measurements are fixed bythe relevant hidden variables irrespective of the fact if the measurement isunsharp. In the context of a recent result by Spekkens that tells even in anontological model the unsharp observable cannot be assigned a deterministicresponse function, we derive a modified local bound of CHSH inequality inunsharp measurement scenario. We consider response function for a given POVMwhich is determined by the response functions of the relevant projectorsappearing in its spectral representation. In this scenario, the local bound ofCHSH inequality is found to be dependent on the unsharpness parameter. Thisthen enables us to show that the quantum violation of CHSH inequality forunbiased spin-POVMs occurs whenever there is violation for their sharpcounterpart. For the case of biased POVMs, it is shown that the quantumviolation of CHSH inequality can be obtained for ranges of sharpness parameterfor which no violation obtained using standard local bound of CHSH inequality.

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