Length filtration of the separable states

We investigate the separable states $\r$ of an arbitrary multipartite quantum system with Hilbert space $\cH$ of dimensionin $d$. The length $L(\r)$ of $\r$ is defined as the smallest number of pure product states having $\r$ as their mixture. The length filtration of the set of separable states, $\cS$, is the increasing chain $\emptyset\subset\cS'_1\subseteq\cS'_2\subseteq\cdots$, where $\cS'_i=\{\r\in\cS:L(\r)\le i\}$. We define the maximum length, $L_{\rm max}=\max_{\r\in\cS} L(\r)$, critical length, $L_{\rm crit}$, and yet another special length, $L_c$, which was defined by a simple formula in one of our previous papers. The critical length indicates the first term in the length filtrartion whose dimension is equal to $\dim\cS$. We show that in general $d\le L_c\le L_{\rm crit}\le L_{\rm max}\le d^2$. We conjecture that the equality $L_{\rm crit}=L_c$ holds for all finite-dimensional multipartite quantum systems. Our main result is that $L_{\rm crit}=L_c$ for the bipartite systems having a single qubit as one of the parties. This is accomplished by computing the rank of the Jacobian matrix of a suitable map having $\cS$ as its range.