Canonical form of three-fermion pure-states with six single particle states

Abstract

We construct a canonical form for pure states in $\bwe^3(\bC^6)$, the three-fermion system with six single particle states, under local unitary (LU) transformations, i.e., the unitary group $\Un(6)$. We also construct a minimal set of generators of the algebra of polynomial $\Un(6)$-invariants on $\bwe^3(\bC^6)$. It turns out that this algebra is isomorphic to the algebra of polynomial LU-invariants of three-qubits which are additionally invariant under qubit permutations. As a consequence of this surprising fact, we deduce that there is a one-to-one correspondence between the $\Un(6)$-orbits of pure three-fermion states in $\bwe^3(\bC^6)$ and the LU orbits of pure three-qubit states when qubit permutations are allowed. As an important byproduct, we obtain a new canonical form for pure three-qubit states under LU transformations $\Un(2)\times\Un(2)\times\Un(2)$ (no qubit permutations allowed).