We classify multipartite entangled states in the Hilbert space $\mathcal{H}=C^2\otimes C^2\otimes C^n(n>~4),$ for example, the four-qubit system distributed over three parties, under local filtering operations. We show that there exist nine essentially different classes of states, giving rise to a five-graded partially ordered structure, including the celebrated Greenberger-Horne-Zeilinger and W classes of three qubits. In particular, all $2\times 2\times n$ states can be deterministically prepared from one maximally entangled state, and some applications such as entanglement swapping are discussed.

• Received 9 July 2003

DOI:https://doi.org/10.1103/PhysRevA.69.012101