We investigate the separable states ρ of an arbitrary multi-partite quantum system with Hilbert space 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, , is the increasing chain , where . We define the maximum length, , 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 . 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 as its range.
- Received May 16, 2016.
- Accepted October 3, 2016.
- © 2016 The Author(s)
Published by the Royal Society. All rights reserved.