way entangled states can be expressed minimally as a linear combination of product states. For instance, as we shall prove later on, the GHZ and W states have a different number of terms in their minimal product decompositions (\[GHZ\]) and (\[W\]), namely 2 and 3 product terms respectively, and this readily implies that there is no way to convert one state into the other by means of an ILO $A\otimes B\otimes C$. Indeed, let us consider, e.g., the most general pure state that can be obtained reversibly from a ${\mbox{$| GHZ \rangle$}}$. It reads ABC=( +), \[new\] where ${\mbox{$| A0 \rangle$}}$ and ${\mbox{$| A1 \rangle$}}$ are linearly independent vectors (since $A$ is invertible) and similarly for the other two qubits. That is, the minimal number of terms in a product decomposition for the state (\[new\]) is also 2. Actually, we have that also for a general multipartite system,

[**Observation:**]{} The minimal number of product terms for any given state remains unchanged under SLOCC.

This simple observation tells us already that in three qubits there are at least two inequivalent kinds of genuine tripartite entanglement under SLOCC, that of ${\mbox{$| GHZ \rangle$}}$ and that of ${\mbox{$| W \rangle$}}$.

However, we still 