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

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.