Wehrl entropies and Euclidean Landau levels

We are concerned with an information-theoretic measure of uncertainty for quantum systems. Precisely, the Wehrl entropy of the phase-space probability $Q^{(m)}_{\hat{\rho}}=\left\langle z,m|\hat{\rho}|z,m\right\rangle$ which is known as Husimi function, where $\hat{\rho}$ is a density operator and $$ are coherent states attached to an Euclidean $m$th Landau level. We obtain the Husimi function $Q^{(m)}_{\beta}$ of the thermal density operator $\hat{\rho}_{\beta}$ of the harmonic oscillator, which leads by duality, to the Laguerre probability distribution of the mixed light. We discuss some basic properties of $Q^{(m)}_{\beta}$ such as its characteristic function and its limiting logarithmic moment generating function from which we derive the rate function of the sequence of probability distributions $Q^{(m)}_{\beta},\ m=0,1,2,...$. For $m\geq1$, we establish an exact expression for the Wehrl entropy of the density operator $$ and we discuss the behavior of this entropy with respect to the temperature parameter $T=1/\beta$