We introduce a classification of mixed three-qubit states, in which we define the classes of separable, biseparable, $W$, and Greenberger-Horne-Zeilinger states. These classes are successively embedded into each other. We show that contrary to pure $W$-type states, the mixed $W$ class is not of measure zero. We construct witness operators that detect the class of a mixed state. We discuss the conjecture that all entangled states with positive partial transpose (PPTES) belong to the $W$ class. Finally, we present a new family of PPTES “edge” states with maximal ranks.

• Received 8 March 2001

DOI:https://doi.org/10.1103/PhysRevLett.87.040401