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We introduce a permutationally invariant multipartite Bell inequality for many-body three-level systems and use it to explore a possible connection between the presence of nonlocality and quantum chaos. For this, we interpret the Bell operator resulting from fixing the measurement settings as a system Hamiltonian and explore the energy level spacing distribution for different measurement choices and SU(3) irreducible representations. Interestingly, we find that in every irreducible representation displaying nonlocality, the measurement settings yielding maximal violation result in a Bell operator whose energy level spacing distribution signals the absence of quantum chaos. This situation changes when the measurement settings slightly depart from the optimal ones, as we generally observe a level spacing distribution characteristic of chaotic systems. Our work suggests an intriguing connection between Bell nonlocality, quantum chaos and random matrix theory. These subjects belonged to the beloved areas of the late Fritz Haake, and we dedicate this paper to his memory.
Aloy et al. (2024) studied this question.