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

Abstract

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.

25 pages

References in corpus (4)

• Density Operators for Fermions
• Determinant Bounds and the Matsubara UV Problem of Many-Fermion Systems
• Clustering of fermionic truncated expectation values via functional integration
• A rigorous treatment of the perturbation theory for many-electron systems

Cited by in corpus (6)

• Comment on "Fermionic entanglement ambiguity in noninertial frames"
• Robust fermionic-mode entanglement of a nanoelectronic system in non-Markovian environments
• Capacities of Grassmann channels
• Dynamics of decoherence of an entangled pair of qubits locally connected to a one-dimensional disordered spin chain
• Generation of concurrence between two qubits locally coupled to a one dimensional spin chain
• Decoherence of Bell states by local interactions with a suddenly quenched spin environment