Insights into Quantum Contextuality and Bell Nonclassicality: A Study on Random Pure Two-Qubit Systems

We explore the relationship between Kochen-Specker quantum contextuality and Bell-nonclassicality for ensembles of two-qubit pure states. We present a comparative analysis showing that the violation of a noncontextuality inequality on a given quantum state reverberates on the Bell-nonclassicality of the considered state. In particular, we use suitable inequalities that are experimentally testable to detect quantum contextuality and nonlocality for systems in a Hilbert space of dimension $d=4$. While contextuality can be assessed on different degrees of freedom of the same particle, the violation of local realism requires parties spatially separated.