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

TL;DR

This paper introduces the GAR algebra, a new algebraic framework for analyzing quasi-free fermionic states, enabling efficient fidelity calculations crucial for entanglement studies in fermionic quantum systems.

Contribution

It develops the GAR algebra structure, combining Grassmann and CAR algebras, to facilitate analysis of fermionic states and their fidelities, advancing mathematical tools in quantum physics.

Findings

Introduces the GAR algebra as a new mathematical framework.

Provides an elegant method for calculating fidelity of fermionic states.

Enables better analysis of entanglement in 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.