Testing the Monogamy Relations via Rank-2 Mixtures

We introduce two tangle-based four-party entanglement measures $t_1$ and $t_2$, and two negativity-based measures $n_1$ and $n_2$, which are derived from the monogamy relations. These measures are computed for three four-qubit maximally entangled and W states explicitly. We also compute these measures for the rank-$2$ mixture $\rho_4 = p \ket{\mbox{GHZ}_4} \bra{\mbox{GHZ}_4} + (1 - p) \ket{\mbox{W}_4} \bra{\mbox{W}_4}$ by finding the corresponding optimal decompositions. It turns out that $t_1 (\rho_4)$ is trivial and the corresponding optimal decomposition is equal to the spectral decomposition. Probably, this triviality is a sign of the fact that the corresponding monogamy inequality is not sufficiently tight. We fail to compute $t_2 (\rho_4)$ due to the difficulty for the calculation of the residual entanglement. The negativity-based measures $n_1 (\rho_4)$ and $n_2 (\rho_4)$ are explicitly computed and the corresponding optimal decompositions are also derived explicitly.