Validity and failure of some entropy inequalities for CAR systems
  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. 
Hajime Moriya