Separability in 2 x N composite quantum systems

We analyze the separability properties of density operators supported on C(2)xC(N) whose partial transposes an 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.