Separability of n-particle mixed states:: necessary and sufficient conditions in terms of linear maps

We provide necessary and sufficient conditions for separability of mixed states of rt-particle systems. The conditions are formulated in terms of maps which are positive on product states of n - 1 particles. The method of construction of the maps on the basis of unextendible product bases is provided. The three-qubit problem is discussed in more details involving Bloch sphere parameters of qubit state. (C) 2001 Elsevier Science B.V. All rights reserved.