Separability and entanglement in 2x2xN composite quantum systems

Abstract

We investigate separability and entanglement of mixed states in ${\cal C}^2\otimes{\cal C}^2\otimes{\cal C}^N$ three party quantum systems. We show that all states with positive partial transposes that have rank $\le N$ are separable. For the 3 qubit case (N=2) we prove that all states $ρ$ that have positive partial transposes and rank 3 are separable. We provide also constructive separability checks for the states $ρ$ that have the sum of the rank of $ρ$ and the ranks of partial transposes with respect to all subsystems smaller than 15N-1.