Four-qubit pure states as fermionic states

Abstract

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 two- (or three)-qubit orbits and the two- (or three)-fermion orbits with four (or six) 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 mapping of orbits from qubits to fermions is injective under SLOCC, and a similar result holds under LU for generic orbits. As a by-product, we obtain a complete answer to the problem of SLOCC equivalence of pure four-qubit states. © 2013 American Physical Society.