Remarks on the relation between quantum and classical entropy as proposed by A. Wehrl (Q791116)

There are several methods of constructing a classical probability distribution starting from a quantum-mechanical density matrix, for instance the method of Wigner. Wehrl proposes to consider the quantity \((z| \omega | z), \omega =density\) matrix, \(| z| =coherent\) state centered around the point \(z=\{q,p\}\) of phase space. The main theorems of this article are the following ones: \[ \int_{N}(z| \omega | z)dz\leq \lambda_ 1+...+\lambda_ m+(s- m)\lambda_{m+1}, \] where \(\lambda_ 1\geq \lambda_ 2..\). are the eigenvalues of \(\omega\) and the Liouville measure of N is smaller than s. \[ \int_{N}(z| \omega | z)dz\leq s e^{-s^ n/ne}\quad for\quad 0\leq s^ n\leq e. \] Let \(F(\cdot)\) be a continuous, concave function defined on \([0,\infty)\) with \(F(0)=0\). Then \(F^{c1}(\omega):=\int F((z| \omega | z))dz\geq e^{1/n}\int^{1}_{0}F[(1-t^ n)e^{-t^ n/n}]dt.\)