Entanglement classification of three fermions with up to nine single-particle states

Based on results well known in the mathematics literature but have not made their debut to the physics literature yet we conduct a study on three-fermionic systems with six, seven, eight and nine single-particle states. Via introducing special polynomial invariants playing the role of entanglement measures the structure of the SLOCC entanglement classes is investigated. The SLOCC classes of the six- and seven-dimensional cases can elegantly be described by special subconfigurations of the Fano plane. Some special embedded systems containing distinguishable constituents are arising naturally in our formalism, namely, three-qubits and three-qutrits. In particular the three fundamental invariants $I_6$, $I_9$, and $I_{12}$ of the three-qutrits system are shown to arise as special cases of the four fundamental invariants of three-fermions with nine single-particle states.