Optimal representations of quantum states by gaussians in phase space

A two-step optimization is proposed to represent an arbitrary quantum state to a desired accuracy with the least number of gaussians in phase space. The Husimi distribution of the quantum state provides the information to determine the modulus of the weight for the gaussians. Then, the phase information contained in the Wigner distribution is used to obtain the full complex weights by considering the relative phases for pairs of gaussians, the chords. The method is exemplified with several excited states n of the harmonic and the Morse oscillators. A semiclassical interpretation of the number of gaussians needed as a function of the quantum number n is given. The representation can also be used to characterize Wigner and Husimi distributions directly which do not originate in a quantum state.