\phi \rangle$}}$ nor the other way round.

The previous result is based on the fact that, unlike the GHZ state, not all entangled states of three qubits can be expressed as a linear combination of only two product states. Remarkably enough, the inequivalence under SLOCC of the states ${\mbox{$| GHZ \rangle$}}$ and ${\mbox{$| W \rangle$}}$ can alternatively be shown from the fact that the 3-tangle (residual tangle), a measure of tripartite correlations introduced by Coffman et. al. [@Wo99], does not increase on average under LOCC, as we will prove here.

We will then move to the second main goal of this work, namely the analysis of the state ${\mbox{$| W \rangle$}}$. It can not be obtained from a state ${\mbox{$| GHZ \rangle$}}$ by means of LOCC and thus one could expect, in principle, that it has some interesting, characteristic properties. Recall that in several aspects the GHZ state can be regarded as the maximally entangled state of three qubits. However, if one of the three qubits is traced out, the remaining state is completely unentangled. Thus, the entanglement properties of the state ${\mbox{$| GHZ \rangle$}}$ are very fragile under particle losses. We will prove that, oppositely, the entanglement of ${\mbox{$| 