Mermin and Svetlichny inequalities for non-projective measurement observables

TL;DR

This paper derives comprehensive criteria for violating Mermin and Svetlichny inequalities in three-qubit states using both projective and non-projective measurements, revealing conditions for maximal violation and the role of measurement bias.

Contribution

It generalizes existing criteria to include arbitrary measurement observables and provides maximal bounds for violations in three-qubit states, including those with maximally mixed marginals.

Findings

Criteria for violation are valid for both projective and arbitrary measurements.

Maximal bounds of inequalities are obtained for various measurement types.

Violation can occur only with biased measurements in certain regimes.

Abstract

The necessary and sufficient criteria for violating the Mermin and Svetlichny inequalities by arbitrary three-qubit states are presented. Several attempts have been made, earlier, to find such criteria, however, those extant criteria are neither tight for most of the instances, nor fully general. We generalize the existing criteria for Mermin and Svetlichny inequalities which are valid for the local projective measurement observables as well as for the arbitrary ones. We obtain the maximal achievable bounds of the Mermin and Svetlichny operators with unbiased measurement observables for arbitrary three-qubit states and with arbitrary observables for three-qubit states having maximally mixed marginals. We find that for certain ranges of measurement strengths, it is possible to violate Mermin and Svetlichny inequalities only by biased measurement observables. The necessary and sufficient criteria of violating any one of the six possible Mermin and Svetlichny inequalities are also derived.