Entanglement in SO(3)-invariant bipartite quantum systems

The structure of the state spaces of bipartite N circle times N quantum systems which are invariant under product representations of the group SO(3) of three-dimensional proper rotations is analyzed. The subsystems represent particles of arbitrary spin j which transform according to an irreducible representation of the rotation group. A positive map v is introduced which describes the time reversal symmetry of the local states and which is unitarily equivalent to the transposition of matrices. It is shown that the partial time reversal transformation v(2)= I circle times v acting on the composite system can be expressed in terms of the invariant 6-j symbols introduced by Wigner into the quantum theory of angular momentum. This fact enables a complete geometrical construction of the manifold of states with positive partial transposition and of the sets of separable and entangled states of 4 circle times 4 systems. The separable states are shown to form a three-dimensional prism and a three- dimensional manifold of bound entangled states is identified. A positive map is obtained which yields, together with the time reversal, a necessary and sufficient condition for the separability of states of 4 circle times 4 systems. The relations to the reduction criterion and to the recently proposed cross norm criterion for separability are discussed.