Low Rank States with Bound Entanglement in a System of three qubits

We have studied mixed quantum states in the system of three qubits with the property that all their partial transposes are positive; these are called PPT states. We classify a PPT state by the ranks of the state itself and its three single partial transposes. We have studied especially the rank 4444 and rank 5555 entangled PPT states.We find two distinct classes of rank four states, identified by a real valued quadratic expression invariant under local $\sln$ transformations, mathematically equivalent to continuous Lorentz transformations. This quadratic Lorentz invariant is non-zero for one class of states (type 1) and zero for the other class (type 2). We present analytical constructions of states of both types, general enough to reproduce all the rank four PPT states we have found numerically.There are six product vectors in a generic five dimensional subspace. The product vectors are used to define SL invariants of the state and its partial transposes. We find four distinct classes of states based which invariants are conserved or not under partial transposition.We find that a state is in the equivalence class of a symmetric state when its invariants are conserved under a partial transposition, and we have shown how to transform the state into its symmetric form through Lorentz transformations.The dimensions and geometry of extremal rank 5555 PPT states has also been examined. We find that states of different types lie on surfaces of rank 5555 different dimensions. And the surfaces touch the simplex of separable states in different way