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

Abstract

We construct a canonical form for pure states in \documentclass[12pt]{minimal}\begin{document}$\mathop {\wedge }\nolimits ^3({\mathbb {C}}^6)$\end{document}∧3(C6), the three-fermion system with six single particle states, under local unitary (LU) transformations, i.e., the unitary group U(6). We also construct a minimal set of generators of the algebra of polynomial U(6)-invariants on \documentclass[12pt]{minimal}\begin{document}$\mathop {\wedge }\nolimits ^3({\mathbb {C}}^6)$\end{document}∧3(C6). 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 U(6)-orbits of pure three-fermion states in \documentclass[12pt]{minimal}\begin{document}$\mathop {\wedge }\nolimits ^3({\mathbb {C}}^6)$\end{document}∧3(C6) 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 U(2) × U(2) × U(2) (no qubit permutations allowed).