title: Length filtration of the separable states.
abstract: We investigate the separable statesρof an arbitrary multi-partite quantum system with Hilbert space [Formula: see text] of dimensiond. The lengthL(ρ) ofρis defined as the smallest number of pure product states havingρas 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,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 [Formula: see text]. We show that in generald≤Lc≤Lcrit≤Lmax≤d2. We conjecture that the equalityLcrit=Lcholds for all finite-dimensional multi-partite quantum systems. Our main result is thatLcrit=Lcfor 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.
authors.affiliation: School of Mathematics and Systems Science, Beihang University, Beijing 100191, People's Republic of China; International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, People's Republic of China.
authors.forename: Lin
authors.identifier: 0000-0002-8367-3651
authors.initials: L
authors.lastname: Chen
authors.affiliation: Department of Pure Mathematics and Institute for Quantum Computing , University of Waterloo , Waterloo, Ontario, N2L 3G1, Canada.
authors.forename: Dragomir Ž
authors.initials: DŽ
authors.lastname: Ðoković
keywords: Jacobian matrix; length; separable states