  Entanglement is a resource in quantum information theory when state
manipulation is restricted to Local Operations assisted by Classical
Communication (LOCC). It is therefore of paramount importance to decide which
LOCC transformations are possible and, particularly, which states are maximally
useful under this restriction. While the bipartite maximally entangled state is
well known (it is the only state that cannot be obtained from any other and, at
the same time, it can be transformed to any other by LOCC), no such state
exists in the multipartite case. In order to cope with this fact, we introduce
here the notion of the Maximally Entangled Set (MES) of n-partite states. This
is 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 states
within the set and no state in the set can be obtained from any other state via
LOCC. We determine the MES for states of three and four qubits and provide a
simple characterization for them. In both cases, infinitely many states are
required. However, while the MES is of measure zero for 3-qubit states, almost
all 4-qubit states are in the MES. This is because, in contrast to the 3-qubit
case, deterministic LOCC transformations are almost never possible among fully
entangled four-partite states. We determine the measure-zero subset of the MES
of LOCC convertible states. This is the only relevant class of states for
entanglement manipulation.
