Length filtration of the separable states

We investigate the separable states ρ of an arbitrary multi-partite quantum system with Hilbert space [Formula: see text] of dimension d. The length L(ρ) of ρ is defined as the smallest number of pure product states having ρ as their mixture. The length filtration of the set of separable states, [Formula: see text], is the increasing chain [Formula: see text], where [Formula: see text]. We define the maximum length, [Formula: see text], critical length, L_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 filtration whose dimension is equal to [Formula: see text]. We show that in general d≤L_c≤L_crit≤L_max≤d². We conjecture that the equality L_crit=L_c holds for all finite-dimensional multi-partite quantum systems. Our main result is that L_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 [Formula: see text] as its range.