Generalized CHSH nonlocality criteria for continuous variable bipartite Gaussian states: testing in a double-cavity optomechanical device.

We derive a general condition for nonlocality using the Banaszek-Wódkiewicz phase-space Wigner representation of Bell's function in continuous-variable Gaussian states, showing rigorously that while entanglement is necessary, it is not sufficient for nonlocality, whereas nonlocality always implies entanglement. Furthermore, to demonstrate the applicability of the framework, we analyze the two-mode squeezed output generated by a double-cavity optomechanical system coupled to a common mechanical resonator-an experimentally accessible hybrid electro/optomechanical interface. We show that maximal entanglement does not necessarily lead to Bell violation, whereas nonlocality may arise in states with lower entanglement, highlighting the crucial role of mixedness. The developed formalism is universally applicable to all bipartite Gaussian states and provides a general tool for investigating foundational and operational aspects of nonlocality in continuous-variable quantum platforms.