  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.
