we will be able to make an exhaustive classification of three-qubit entanglement. Later on we will rephrase some of these results in terms of well-known measures of entanglement. In particular, we will see that the existence of two inequivalent kinds of true tripartite entanglement under SLOCC is very much related to the fact that the 3-tangle, a measure of tripartite entanglement introduced in [@Wo99], is an entanglement monotone (see appendix B).

At the end of the section also a practical way to identify the class an arbitrary state belongs to will be discussed.

Non-entangled states and bipartite entanglement.
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If at least one of the local ranks r$(\rho_A)$, r$(\rho_B)$ or r$(\rho_C)$ is 1, then the pure state of the three qubits factors out as the tensor product of two pure states, and this implies that at least one of the qubits is uncorrelated with the other two. SLOCC distinguish states with this feature depending on which qubits are uncorrelated from the rest.

[**Class A-B-C (product states)**]{}\
This class corresponds to states with $r(\rho_A)=r(\rho_B)=r(\rho_C)=1$. They can be taken, after using some convenient LU, into the form = , where we have already relaxed the notation for ${\mbox{$| 0 \rangle$}}\otimes{\mbox{$| 0 \rangle$}}\otimes{\mbox{$| 