Generalized Wehrl entropies and Euclidean Landau levels

Abstract

We are concerned with an information-theoretic measure of uncertainty for quantum systems. Precisely, the Wehrl entropy of the phase-space probability Q (m) ρ = z, m|ρ|z, m which is known as Husimi function, whereρ is a density operator and |z, m are coherent states attached to an Euclidean mth Landau level. We obtain the Husimi function Q (m) β of the thermal density operator ρ β 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) β 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)β , m = 0, 1, 2, .... For m ≥ 1, we establish an exact expression for the Wehrl entropy of the density operatorρ β and we discuss the behavior of this entropy with respect to the temperature parameter T = 1/β.

Husmi Q-functions attached to hyperbolic Landau levels

“…, from which we derive the mean and the variance parameters. Finally, we recover most of the results [4] of the flat case as the radius parameter R goes to infinity. As a tool we establish a summation formula for the special Kampé de Fériet function F 1:2:2 2:0:0 .…”

“…on the number of zeros of the Husimi density and their location in connection with properties of harmonic and anaharmonic oscillators can be found in [32], particularly [4] in the Euclidean setting.…”

“…For this distribution we write down the characteristic function from which we derive the main value and the variance. As in the Euclidean setting [4] this distribution may be useful for tackling the hyperbolic version of Ginibre-type processes [5]. Next, from Q for the thermal density operator (heat semi-group) ρβ of the Hamiltonian of the isotonic oscillator.…”