State space structure and entanglement of rotationally invariant spin systems

We investigate the structure of SO(3)-invariant quantum systems which are composed of two particles with spins j(1) and j(2). The states of the composite spin system are represented by means of two complete sets of rotationally invariant operators, namely by the projections P-J onto the eigenspaces of the total angular momentum J, and by certain invariant operators Q(K) which are built out of spherical tensor operators of rank K. It is shown that these representations are connected by an orthogonal matrix whose elements are expressible in terms of Wigner's 6-j symbols. The operation of the partial time reversal of the combined spin system is demonstrated to be diagonal in the Q(K)-representation. These results are employed to obtain a complete characterization of spin systems with j(1) = 1 and arbitrary j(2) >= 1. We prove that the Peres-Horodecki criterion of positive partial transposition (PPT) is necessary and sufficient for separability if j(2) is an integer, while for half-integer spins j(2) there always exist entangled PPT states (bound entanglement). We construct an optimal entanglement witness for the case of half-integer spins and design a protocol for the detection of entangled PPT states through measurements of the total angular momentum.