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Entanglement is a resource in quantum information theory when statemanipulation is restricted to Local Operations assisted by ClassicalCommunication (LOCC). It is therefore of paramount importance to decide whichLOCC transformations are possible and, particularly, which states are maximallyuseful under this restriction. While the bipartite maximally entangled state iswell known (it is the only state that cannot be obtained from any other and, atthe same time, it can be transformed to any other by LOCC), no such stateexists in the multipartite case. In order to cope with this fact, we introducehere the notion of the Maximally Entangled Set (MES) of n-partite states. Thisis the set of states which are maximally useful under LOCC manipulation, i.e.any state outside of this set can be obtained via LOCC from one of the stateswithin the set and no state in the set can be obtained from any other state viaLOCC. We determine the MES for states of three and four qubits and provide asimple characterization for them. In both cases, infinitely many states arerequired. However, while the MES is of measure zero for 3-qubit states, almostall 4-qubit states are in the MES. This is because, in contrast to the 3-qubitcase, deterministic LOCC transformations are almost never possible among fullyentangled four-partite states. We determine the measure-zero subset of the MESof LOCC convertible states. This is the only relevant class of states forentanglement manipulation.

The maximally entangled set of multipartite quantum states