  We present an approach to a non-commutative-like phase space which allows to
analyze quasi-free states on the CAR algebra in analogy to quasi-free states on
the CCR algebra. The used mathematical tools are based on a new algebraic
structure the "Grassmann algebra of canonical anti-commutation relations" (GAR
algebra) which is given by the twisted tensor product of a Grassmann and a CAR
algebra. As a new application, the corresponding theory provides an elegant
tool for calculating the fidelity of two quasi-free fermionic states which is
needed for the study of entanglement distillation within fermionic systems.
