Continuous and discrete local hidden variable theories are equivalent

Abstract

In quantum theory, Bell locality of quantum states (generally, of correlations, or boxes) is characterized by local hidden variable models (LHVMs) given by integrals and sums. We call these models continuous and discrete LHVMs (C-LHVMs and D-LHVMs), respectively. Theoretically, this seems leading to two types of local hidden variable theories (LHVTs) given by C-LHVMs and D-LHVMs, respectively. In this paper, we show the equivalence of these two types of LHVTs by mathematically proving that a correlation tensor (equivalently, a box) has a C-LHVM if and only if it has a D-LHVM. As application, we check the Bell locality of some correlation tensors that are defined by integrals.