Fig. 2

The parameters \(\alpha\) and \(\beta\) completely define the state \(\vert \psi _{3a} \rangle\) introduced in Eq. (6). The constraints on these parameters give rise to the convex set \(\mathcal {M}\), a right triangle with legs of one unit on the plane \(\mathbb {R}^2\). The extreme points, \((\alpha , \beta ) = (0,0)\), (0, 1) and (1, 0), characterize \(\vert \psi _{3a} \rangle\) as a fully separable state. The middle points, (1/2, 1/2), (1/2, 0) and (0, 1/2), produce the bi-separable states \(\vert 0 \rangle _i \vert \Psi ^+ \rangle _{jk}\), where \(\vert \Psi ^+ \rangle _{jk}\) stands for the Bell state \(\vert \Psi ^+ \rangle\) shared by the jth and kth qubits. The convex combinations of extreme and middle points yield four different convex subsets (regions) of \(\mathcal {M}\) that permit a classification of entanglement for \(\vert \psi _{3a} \rangle\).