Contextuality, superlocality and supernoncontextuality

Contextuality is a fundamental manifestation of nonclassicality, indicating that for certain quantum correlations, sets of jointly measurable variables cannot be pre-assigned values independently of the measurement context. In this work, we characterize nonclassical quantum correlation beyond contextuality, in terms of supernoncontextuality, namely the higher-than-quantum hidden-variable(HV) dimensionality required to reproduce the given noncontextual quantum correlations. Thus supernoncontextuality is the contextuality analogue of superlocality. Specifically, we study the quantum system of two-qubit states in a scenario composed of five contexts that demonstrate contextuality in a state-dependent fashion. For this purpose, we use the framework of boxes, whose behavior is described by a set of probabilities satisfying the no-disturbance conditions. We observe that superlocal states must have sufficiently a high HV dimension in order to lead to a contextual box. On the other hand, a noncontextual superlocal box can be supernoncontextual, but superlocality is not a necessary condition. For sublocal (i.e., not superlocal) boxes, supernoncontextuality arises owing to a different type of simultaneous correlation in two MUBs present in any nonproduct state and correlations due to nonlocal measurements. From an operational perspective, the superlocal states that can or cannot be used to demonstrate contextuality can be distinguished by quantum correlations in two and three mutually unbiased bases (MUBs) of the discordant correlations in the states.