have to prove that the state ${\mbox{$| W \rangle$}}$ cannot be expressed as a linear combination of just two product vectors. In order to complete our classification we also have to show that any pure state of three qubits with maximal local ranks can be reversibly converted into either the state ${\mbox{$| GHZ \rangle$}}$ or the state ${\mbox{$| W \rangle$}}$. We start with an obvious lemma regarding product decompositions:

[**Lemma:**]{} Let $\sum_{i=1}^l {\mbox{$| e_i \rangle$}}{\mbox{$| f_i \rangle$}}$ be a product decomposition for the state ${\mbox{$| \eta \rangle$}}\in\H_{E}\otimes\H_{F}$. Then the set of states $\{{\mbox{$| e_i \rangle$}}\}_{i=1}^l$ span the range of $\rho_E\equiv$Tr$_F {{\mbox{$| \eta \rangle$}}\!{\mbox{$\langle \eta |$}}}$.

[**Proof:**]{} We have that $\rho_E = \sum_{i,j=1}^l {\mbox{$\langle f_i  | f_j \rangle$}}
{\mbox{$| e_j \rangle$}}{\mbox{$\langle e_i |$}}$. On the other hand ${\mbox{$| \nu \rangle$}}$ is in the range of $\rho_E$ iff a state ${\mbox{$| \mu \rangle$}}$ exists such that ${\mbox{$| \nu \rangle$}}=\rho_E{\mbox{$| \mu \rangle$}}$, that is ${\mbox{$| \nu \rangle$}} = \sum_{i,j=1}^l {\mbox{$\langle f_i  | f_j \rangle$}}{\mbox{$\langle e_i  | \mu \rangle$}} {\mbox{$| e_j \rangle$}}$. $\Box$

In particular, $r(\rho_A)=2$ implies that at least two product terms are needed to expand ${\mbox{$| \psi \rangle$}}\in \C^2\otimes\C^2\otimes\C^2$. Let us suppose that a product decomposition with only two terms is possible, 