Four qubits can be entangled in nine different ways

We consider a single copy of a pure four-partite state of qubits and investigate its behavior under the action of stochastic local quantum operations assisted by classical communication (SLOCC). This leads to a complete classification of all different classes of pure states of four qubits. It is shown that there exist nine families of states corresponding to nine different ways of entangling four qubits. The states in the generic family give rise to Greenberger-Horne-Zeilinger-like entanglement. The other ones contain essentially two-or three-qubit entanglement distributed among the four parties. The concept of concurrence and 3-tangle is generalized to the case of mixed states of four qubits, giving rise to a seven-parameter family of entanglement monotones. Finally, the SLOCC operations maximizing all these entanglement monotones are derived, yielding the optimal single-copy distillation protocol.