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

TL;DR

This paper establishes a canonical form for three-fermion pure states with six single particle states under local unitary transformations, revealing a deep connection to three-qubit states and their invariants.

Contribution

It constructs a canonical form for three-fermion states in six dimensions and links their invariants to those of three-qubit states, including a new LU canonical form for three-qubit states.

Findings

Algebra of invariants is isomorphic to that of three-qubit states with permutation invariance.

One-to-one correspondence between fermion and qubit state orbits under LU transformations.

New canonical form for three-qubit states under LU without permutations.

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).