Length filtration of the separable states

We investigate the separable states ρ of an arbitrary multi-partite quantum system with Hilbert space H 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, S , is the increasing chain ∅ ⊊ S 1 ′ ⊆ S 2 ′ ⊆ ⋯ , where S i ′ = { ρ ∈ S : L ( ρ ) ≤ i } . We define the maximum length, L max = max ρ ∈ S L ( ρ ) , 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 Dim S . We show that in general d ≤ L c ≤ L crit ≤ L max ≤ d 2 . 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 S as its range.