Three-outcome multipartite Bell inequalities: applications to dimension witnessing and spin-nematic squeezing in many-body systems

We present a three-outcome Bell inequality, which we show to be naturally suited to explore nonlocal correlations in many-body spin-1 systems or SU(3) models. In the specific, we demonstrate from this inequality an experimentally practical Bell correlation test based on the measurement of collective spin components. Next, we analyze such an inequality as a dimension witness, namely, criteria whose violation signals the impossibility of reproducing the inferred correlations by Hilbert spaces of a certain dimension. In particular, our approach enables the certification of genuine three-level correlations in systems of arbitrary size, which cannot be reproduced by an ensemble of qubits. Then, we demonstrate how metrological resources, in the form of spin-nematic squeezed states as produced in spin-1 Bose Einstein condensates involving thousands of atoms, maximally violate such a witness. Moreover, we propose that the robustness of the violation can be used to certify the number of high-dimensional parties in the system. In this direction, we demonstrate that, in the thermodynamic limit, the squeezed correlations described above require the system to be composed entirely of qutrits. Our results shed light on the fundamental role of high-dimensional quantum resources in many-body systems and their application for metrological purposes.