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

TL;DR

This paper derives a modified local bound for the CHSH inequality considering unsharp measurements (POVMs), showing how quantum violations depend on measurement sharpness and revealing new violation regimes.

Contribution

It introduces a response function framework for POVMs, deriving a unsharpness-dependent local bound and analyzing quantum violations in this context.

Findings

Quantum violation occurs for unbiased spin-POVMs whenever the sharp case is violated.

For biased POVMs, violations can occur even when standard bounds show none.

The local bound depends on the unsharpness parameter, affecting violation conditions.

Abstract

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 if the measurement is unsharp. In the context of a recent result by Spekkens that tells even in an ontological model 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.