Interrelation of nonclassicality conditions through stabiliser group homomorphism

In this paper, we show that coherence witness for a single qubit itself yields conditions for nonlocality and entanglement inequalities for multiqubit systems. It also yields a condition for quantum discord in two--qubit systems. It is shown by employing homomorphism among the stabiliser group of a single qubit and those of multi--qubit states. Interestingly, globally commuting homomorphic images of single qubit stabilisers do not allow for consistent assignments of outcomes of local observables. As an application, we show that CHSH inequality can be straightforwardly generalised to nonlocality inequalities for multiqubit GHZ states. It also reconfirms the fact that quantumness prevails even in the large $N$--limit, if coherence is sustained. The mapping provides a way to construct many nonlocality inequalities, given a {\it seed} inequality. This study gives us a motivation to gain a better control over multiple degrees of freedom and multi-party systems. It is because in multi-party systems, the same nonclassical feature, {\it viz.}, coherence may appear in many avatars.