Non-locality and entanglement in multi-qubit systems from a unified framework

Non-classical probability is the underlying feature of quantum mechanics. The emergence of Bell-CHSH non-locality [We stress that CHSH inequality (Clauser et al. in Phys. Rev. Lett. 23(15):880, 1969) is more general and reduces to Bell inequality (Bell in Physics 1:195, 1964) under the assumption of perfect correlations between the observables] for bipartite systems and linear entanglement inequalities for two-qubit systems has been shown in Adhikary et al. (Eur Phys J D 74:68, 2020), purely as violations of classical probability rules. In this paper, we improve upon that work by showing that violation of any nonlocality inequality implies violation of classical probability rules, manifested through negative probabilities, without recourse to any underlying theory. Moving on to entanglement, we employ parent pseudoprojections to show how any number of linear and nonlinear entanglement witnesses for multiqubit systems can be obtained as violations of classical probability rules. They include the ones that have been derived earlier by employing different methods. It provides a perspective complementary to the current understanding in terms of the algebraic approaches.