D-nontrilocality of sparse probability tensors and the triangle network

Abstract Bell’s inequalities are linear and apply for cases of two entangled bodies. In this work, we consider the case of entanglement among three bodies as previously discussed in [Renou, et al Phys. Rev. Lett., 123, 140 401 (2019)] and based on triangle network. By discussing the question whether a sparse probability tensor (SPT) can be represented by a discrete trilocal hidden variable model (D-triLHVM), we show that every SPT having a D-triLHVM satisfies a set of concrete equalities and a nonlinear inequality, which can be used to detect whether a D-triLHVM can describe the network completely. As an application, we re-explore the D-nontrilocality of the correlations studied by Renou et al and that of the triangle network with shared entangled pure states. We also leave open questions about the closednees of the set of all D-trilocal probability tensors and the description with a continuous trilocal hidden variable model.