The algebra of Grassmann canonical anticommutation relations and its applications to fermionic systems

We present an approach to a noncommutativelike phase space which allows to analyze quasifree states on the algebra of canonical anti-commutation relations (CAR) in analogy to quasifree states on the algebra of canonical commutation relations (CCR). The used mathematical tools are based on a new algebraic structure the "Grassmann algebra of canonical anticommutation 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 quasifree fermionic states which is needed for the study of entanglement distillation within fermionic systems. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3282845]