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

The algebra of Grassmann canonical anti-commutation relations (GAR) and its applications to fermionic systems