Quantum statistical information contained in a semi-classical   Fisher--Husimi measure
  We study here the difference between quantum statistical treatments and semi-classical ones, using as the main research tool a semi-classical, shift-invariant Fisher information measure built up with Husimi distributions. Its semi-classical character notwithstanding, this measure also contains information of a purely quantal nature.   Such a tool allows us to refine the celebrated Lieb bound for Wehrl entropies and to discover thermodynamic-like relations that involve the degree of delocalization. Fisher-related thermal uncertainty relations are developed and the degree of purity of canonical distributions, regarded as mixed states, is connected to this Fisher measure as well. 
F. Pennini, A. Plastino and G.L. Ferri