Separability in2×Ncomposite quantum systems

We analyze the separability properties of density operators supported on $C^2\otimes C^N$ whose partial transposes are positive operators. We show that if the rank of $\rho$ equals N then it is separable, and that bound entangled states have ranks larger than N. We also give a separability criterion for a generic density operator such that the sum of its rank and the one of its partial transpose does not exceed $3N.$ If it exceeds this number, we show that one can subtract product vectors until it decreases to $3N,$ while keeping the positivity of $\rho$ and its partial transpose. This automatically gives us a sufficient criterion for separability for general density operators. We also prove that all density operators that remain invariant after partial transposition with respect to the first system are separable.