Geometry of two-body correlations in three-qubit states

We study restrictions of two-body correlations in three-qubit states, using three local-unitarily invariant coordinates based on the Bloch vector lengths of the marginal states. First, we find tight nonlinear bounds satisfied by all pure states and extend this result by including the three-body correlations. Second, we consider mixed states and conjecture a tight non-linear bound for all three-qubit states. Finally, within the created framework we give criteria to detect different types of multipartite entanglement as well as characterize the rank of the quantum state.