Fig. 11

(a) When the parameter \(\alpha\) sweeps the domain \([0, \alpha _{**} ] \equiv [0, \pi /2]\), the state \(\vert \psi _{\alpha } \rangle\) introduced in (22) describes a path \(\vec {\lambda }_{\psi _{\alpha }}\) on the lower tetrahedron of \(\mathcal {P}\). The trajectory goes first from the vertex \(\vec {S}\) to the midpoint of \(\overline{B_{2}B_{3}}\) (at \(\alpha = \alpha _* = \pi /4\)), and then to the vertex \(\vec {B}_1\) (at \(\alpha = \alpha _{**}\)) on the triangle \(\triangle _{B_1B_2B_3}\). In this last journey, \(\vec {\lambda }_{\psi _{\alpha }}\) passes through the geometric center of \(\triangle _{B_1B_2B_3}\), at \(\alpha = \alpha _0 = \arccos ({1}/{\sqrt{3}})\), which houses state \(\vert W \rangle\). (b) Measures \(\xi\) and \(\mu\) for state \(\vert \psi _{\alpha } \rangle\), the former gives \(\xi _0 = 2/3\) for \(\alpha \in [\alpha _*, \alpha _{**}]\) whereas the latter cancels at the ends of the \(\alpha\)-domain. In contrast with \(\xi\), the measure \(\mu\) also distinguishes among the points living on \(\triangle _{B_1B_2B_3}\). That is, points \(\vec {\lambda }_{\psi _{\alpha }}\) closer to the center of \(\triangle _{B_1B_2B_3}\) will correspond to more \(\mu\)-entangled states \(\vert \psi _{\alpha } \rangle\).