A simplex of probability measures associated with classical states of the harmonic oscillator (Q1074951)

For the one dimensional quantum harmonic oscillator a classical state W is a convex combination of the coherent states parametrized by \(z\in {\mathbb{C}}:\) \[ W=\int_{{\mathbb{C}}}d\nu (z)\quad | z><z|. \] Here \(\nu\) is a probability measure on \({\mathbb{C}}\). The paper proves the uniqueness of \(\nu\) (thereby establishing the geometric fact that the classical states form a ''simplex''). The key is that in mixing Poisson distributions the correspondence between mixing measure and mixture is a homeomorphism. The authors overlooked that this is known in point process theory in much wider generality.