Area of overlap and interference in phase space versus Wigner pseudoprobabilities

In the semiclassical approximation, the quantum-mechanical scalar product between two quantum states is governed by (1) the areas of overlap in phase space of these states and (2) interference between the probability amplitudes contributed by these areas. We compare and contrast this principle with Wigner’s concept of pseudoprobabilities in phase space and illustrate the essential points of both treatments by the oscillations in the photon distribution of a highly squeezed state.