Nonclassical quadrature distributions

The characterization of nonclassicality of quantum states in terms of measurable quadrature distributions is studied, on the basis of recently derived necessary and sufficient conditions for the failure of the Glauber-Sudarshan P-function to be a probability density. When the noise-subtracted quadrature distribution, that is the marginal of the P-function, is not a probability density, the same is true for the Glauber-Sudarshan P-function. Inversely, for a complete characterization of a nonclassical P-function one needs phase-sensitive criteria that connect quadrature distributions for different phases. Even for phase-insensitive quantum states, phase-sensitive nonclassicality conditions are needed, as is illustrated for one-photon added thermal states