Correspondence rules and properties of smoothed phase space distribution functions

According to the link between phase space distribution functions and correspondence rules via an arbitrary weighting function, we show that a necessary and sufficient condition for obtaining a direct correspondence between a real phase space distribution function and the density operator of a pure state imposes the weighting function to be unimodular. The same condition is also shown to a particular case of interest for deriving a general formula from which previous known results such as orthonormality of phase space eigenfunctions, non-negativity of smoothed Wigner distributions or phase space interpretation of the scalar product are recovered as special cases.