Separability criterion and inseparable mixed states with positive partial transposition

It is shown that any separable state on the Hilbert space H = H-1 x H-2 can be written as a convex combination of N pure product states with N less than or equal to (dim H)(2). Then a new separability criterion for mixed states in terms of the range of the density matrix is obtained. It is used in the construction of inseparable mixed states with positive partial transposition in the case of 3 x 3 and 2 x 4 systems, The states represent an entanglement which is hidden in a more subtle way than known so far. (C) 1997 Published by Elsevier Science B.V.