three qubits can be locally transformed into a GHZ with finite probability of success (see also [@Co00]). The W-class is of zero measure compared to the GHZ-class. This does not mean, however, that it is irrelevant. In a similar way as separable mixed states are not of zero measure with respect to entangled states, even though product states are, it is in principle conceivable that mixed states having only W-class entanglement are also not of zero measure in the set of mixed states.

Relating SLOCC–classes by means of non–invertible operators
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In this subsection, we investigate the hierarchical relation of the 6 SLOCC-equivalence classes under non–invertible operators, i.e. under general LOCC.

A non–invertible local operator transforms ${\mbox{$| \psi \rangle$}}$ into ${\mbox{$| \phi \rangle$}}$ according to (\[phipsi\]), but with at least one of the local operators $A$, $B$ and $C$ having rank 1. This means that the local ranks of the pure states can be diminished. For instance, if the initial state ${\mbox{$| \psi \rangle$}}$ belongs either to the GHZ or W class, then a non-invertible operator will diminish at least one of the local ranks. That is, ${\mbox{$| \phi \rangle$}}$ belongs necessarily to one of the bipartite classes $\kappa-\mu\nu~ (\kappa\not=\mu\not=\nu 