interpret the inequivalence under SLOCC of states whose reduced density matrices differ in rank also in terms of the impossibility of creating any of the bipartite measures. For instance, a state in the $A-BC$ class has $S_A=0$, and thus cannot be transformed with any finite probability into a state of the $AB-C$ class, because this would have $S_A>0$. We conclude that the monotonicity of these measures readily split the set of pure states of three qubits into five subsets which are inequivalent under SLOCC, namely unentagled states $A-B-C$, the three classes $A-BC$, $AB-C$ and $C-AB$ containing only bipartite entanglement, and a fifth subset of entangled states with $S_A,S_B,S_C \neq 0$ (i.e. r$(\rho_A)=$r$(\rho_B)=$r$(\rho_C)=2$). Bipartite measures cannot, however, determine the inequivalence of the GHZ and W classes.

Is there any known measure of tripartite entanglement which can distinguish between these two classes? The 3-tangle does. Indeed, it can be computed from the product decompositions (\[GHZclass\]) and (\[Wclass\]) (see [@Wo99] for details), and reads (\_[GHZ]{})=(2Ks\_s\_s\_s\_c\_)\^2 \[tangle\] =0\[tauGHZ\] for any state in the GHZ class, while it vanishes for any state in the W class. In the appendix B we prove that the 3-tangle is an entanglement monotone, a very desirable property for any 