Characterizing translation-invariant Bell inequalities using tropical algebra and graph polytopes

Nonlocality is one of the key features of quantum physics, which is revealed through the violation of a Bell inequality. In large multipartite systems, nonlocality characterization quickly becomes a challenging task. A common practice is to make use of symmetries, low-order correlators, or exploiting local geometries, to restrict the class of inequalities. In this paper, we characterize translation-invariant (TI) Bell inequalities with finite-range correlators in one-dimensional geometries. We introduce a novel methodology based on tropical algebra tensor networks and highlight its connection to graph theory. Surprisingly, we find that the TI Bell polytope has a number of extremal points that can be uniformly upper-bounded with respect to the system size. We give an efficient method to list all vertices of the polytope for a particular system size, and characterize the tightness of a given TI Bell inequality. The connections highlighted in our work allow us to re-interpret concepts developed in the fields of tropical algebra and graph theory in the context of Bell nonlocality, and vice-versa. This work extends a parallel article [M. Hu \textit{et al.}, arXiv: 2208.02798 (2022)] on the same subject.