  Basic properties of von Neumann entropy such as the triangle inequality and
what we call MONO-SSA are studied for CAR systems.
  We show that both inequalities hold for any even state. We construct a
certain class of noneven states giving counter examples of those inequalities.
  It is not always possible to extend a set of prepared states on disjoint
regions to some joint state on the whole region for CAR systems.
  However, for every even state, we have its `symmetric purification' by which
the validity of those inequalities is shown.
  Some (realized) noneven states have peculiar state correlations among
subsystems and induce the failure of those inequalities.
