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

Abstract

Based on results well known in the mathematics literature but not yet common knowledge in the physics literature, 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 stochastic local operations and classical communication (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.