0 \rangle$}}$.

[**Classes A-BC, AB-C and C-AB\
(bipartite entanglement)**]{}\
These three classes of states contain only bipartite entanglement between two of the qubits, one of the reduced density matrices having rank 1 and the other two having rank 2. For example, the states in class $A-BC$ possess entanglement between the systems $B$ and $C$ ($r(\rho_B)=r(\rho_C)=2$) and are product with respect to system $A$ ($r(\rho_A)=1$). LU allow us to write uniquely states of the class $A-BC$ as |\_[A-BC]{}=|0(c\_|0|0+s\_|1|1),     c\_ s\_&gt;0, and similary for $|\psi_{B-AC}\rangle$ and $|\psi_{C-AB}\rangle$. We choose the maximally entangled state |0(|0|0+|1|1) \[repreA-BC\] as representative of the class $A-BC$. Any other state within this class can be obtained from (\[repreA-BC\]) with certainty by means of LOCC.

The proof that these four marginal classes are inequivalent under SLOCC is very simple. We only need to recall that the local ranks are invariant under ILO (see appendix A). In what follows we will analyze the more interesting case of $r(\rho_{\kappa})=2,~\kappa = A,B,C$. To see that there are two inequivalent classes fulfilling this condition we will have to study possible product decompositions of pure states.

True three-qubit entanglement.
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There turns out to be a close connection between convertibility under SLOCC and the 