A transformational property of the Husimi function and its relation to the Wigner function and symplectic tomograms

We consider the Husimi $Q$-functions, which are quantum quasiprobability distributions in the phase space, and investigate their transformation properties under a scale transformation $(q,p)\to(\lambda q,\lambda p)$. We prove a theorem that under this transformation, the Husimi function of a physical state is transformed into a function that is also a Husimi function of some physical state. Therefore, the scale transformation defines a positive map of density operators. We investigate the relation of Husimi functions to Wigner functions and symplectic tomograms and establish how they transform under the scale transformation. As an example, we consider the harmonic oscillator and show how its states transform under the scale transformation.