Length filtration of the separable states

We investigate the separable states p of an arbitrary multi-partite quantum system with Hilbert space H of dimension d. The length L(p) of p is defined as the smallest number of pure product states having p as their mixture. The length filtration of the set of separable states, S, is the increasing chain θ S₁ S₂ C, where S_i = {p ϵ S : L(p) ≤ i}. We define the maximum length, Lmax =maxpS L(p), critical length, Lcrit, and yet another special length, Lc, 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 Dim S. We show that in general d ≤ Lc ≤ Lcrit ≤ Lmax ≤ d2. We conjecture that the equality Lcrit =Lc holds for all finite-dimensional multi-partite quantum systems. Our main result is that Lcrit =Lc 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 S as its range.