Entanglement in SO(3)-invariant bipartite quantum systems

The structure of the state spaces of bipartite $N\otimes N$ 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 $\vartheta$ 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 ${\vartheta }_2=I\otimes \vartheta$ acting on the composite system can be expressed in terms of the invariant $6\text{−}j$ 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 $4\otimes 4$ 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 $4\otimes 4$ systems. The relations to the reduction criterion and to the recently proposed cross norm criterion for separability are discussed.