\in\{A,B,C\})$ or else is a product state $A-B-C$.

Thus we have that the classes GHZ and W are also inequivalent even under most general LOCC, whereas e.g. a measurement of the projector $P=|+\rangle\langle+|$ with $|+\rangle=1/\sqrt{2}(|0\rangle+|1\rangle)$ in party $A$ maps states within the classes $W$ (\[Wclass\]) and $GHZ$ (\[GHZclass\]) to states within the class $A-BC$. In a similar way, non–invertible local operators (local, standard measurements) can convert states within one of the classes $\kappa-\mu\nu$ to states within the class $A-B-C$. Note that in all cases described above, the inverse transformations, e.g. from the class $A-B-C$ to one of the classes $\kappa-\mu\nu$ are impossible as they would imply an increase of the rank of at least one of the reduced density operators $\rho_A,\rho_B,\rho_C$. These results are summarized in Fig. \[Fig1\].

Measures of entanglement and classes under SLOCC
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Several measures have been introduced so far in the literature in order to quantify entanglement. Although this section is mainly concerned with qualitative aspects of multipartite quantum correlations, we would like to relate some of these measures, namely some bipartite ones and the tripartite 3-tangle [@Wo99](see appendix B), to our classification. Remarkably, the existence of two kinds of genuine tripartite entanglement in a three-qubit 