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

We construct a canonical form for pure states in ${\text{bwe}}^3({\text{bC}}^6)$, the three-fermion system with six single particle states, under local unitary (LU) transformations, i.e., the unitary group $\text{Un}(6)$. We also construct a minimal set of generators of the algebra of polynomial $\text{Un}(6)$-invariants on ${\text{bwe}}^3({\text{bC}}^6)$. 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 $\text{Un}(6)$-orbits of pure three-fermion states in ${\text{bwe}}^3({\text{bC}}^6)$ 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 $\text{Un}(2)\times \text{Un}(2)\times \text{Un}(2)$ (no qubit permutations allowed).