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We present Bell-type tests of nonclassicality and non-Gaussianity forsingle-mode fields employing a generalized quasiprobability function. Ournonclassicality tests are based on the observation that two orthogonalquadratures in phase space (position and momentum) behave as independentrealistic variables for a coherent state. Taking four (three) points at thevertices of a rectangle (right triangle) in phase space, our tests detect everypure nonclassical Gaussian state and a range of mixed Gaussian states. Thesetests also set an upper bound for all Gaussian states and their mixtures, whichthereby provide criteria for genuine quantum non-Gaussianity. We optimize thenon-Gaussianity tests by employing a squeezing transformation in phase spacethat converts a rectangle (right triangle) to a parallelogram (triangle), whichenlarges the set of non-Gaussian states detectable in our formulation. Weaddress fundamental and practical limits of our generalized phase-space testsby looking into their relation with decoherence under a lossy Gaussian channeland their robustness against finite data and nonoptimal choice of phase-spacepoints. Furthermore, we demonstrate that our parallelogram test can identifyuseful resources for nonlocality testing in phase space.

Demonstrating nonclassicality and non-Gaussianity of single-mode fields: Bell-type tests using generalized phase-space distributions