Fig. 13

A particular solution to the inverse problem of finding a set of points in \(\mathcal {P}\) that follows a path with entanglement measure \(\mu = \mu _0 = {\text {const}}\), see Figure 9(b), is obtained after making \(b_{\ell } = 1/\sqrt{5}\) in (1). The pure state \(\vert \psi \rangle\) is parameterized by the phase \(0 \le \omega \le \pi\) and is projected onto the path \(\vec {\lambda }_{\psi (\omega )}\) in \(\mathcal {P}\) . (a) \(\vec {\lambda }_{\psi (\omega )}\) starts and finishes at \(\left( {\mu _{0}}/{2}, \varepsilon , \varepsilon \right)\) and \(\left( {\mu _{0}}/{2}, 2/5, 2/5 \right)\), respectively. Here \(\mu _{0} = 1 - {\sqrt{13}}/{5}\) and \(\varepsilon = \tfrac{1}{2} \left( 1 - {\sqrt{17}}/{5} \right)\). In the interval \([0, \omega _0]\), with \(\omega _0 = \pi /3\), the path overflows the \(\mu _0\)-region. (b) Measures \(\mu\), \(\xi\), \(F_{123}\) and \(\mathcal {C}_{GME}\) as functions of \(\omega\). Only \(\mu\) and \(\mathcal {C}_{GME}\) maintain constant in the interval \([\omega _0, \pi ]\), with values \(\mu = 1 - {\sqrt{13}}/{5}\) and \(\mathcal {C}_{GME} = {12}/{25}\).