Four-qubit pure states as fermionic states

The embedding of the n-qubit space into the n-fermion space with 2n modes is a widely used method in studying various aspects of these systems. This simple mapping raises a crucial question: does the embedding preserve the entanglement structure? It is known that the answer is affirmative for n=2 and n=3. That is, under either local unitary (LU) operations or with respect to stochastic local operations and classical communication (SLOCC), there is a one-to-one correspondence between the 2- (or 3)-qubit orbits and the 2- (or 3)-fermion orbits with 4 (or 6) modes. However these results do not generalize as the mapping from the n-qubit orbits to the n-fermion orbits with 2n modes is no longer surjective for n>3. Here we consider the case of n=4. We show that surprisingly, the orbit mapping from qubits to fermions remains injective under SLOCC, and a similar result holds under LU for generic orbits. As a byproduct, we obtain a complete answer to the problem of SLOCC equivalence of pure 4-qubit states.