  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.
