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 ${\rho }_4=p|{\text{GHZ}}_4\rangle \langle {\text{GHZ}}_4|+(1-p)|W_4\rangle \langle W_4|$ by finding the corresponding optimal decompositions. It turns out that $t_1\left({\rho }_4\right)$ 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 $t_2\left({\rho }_4\right)$ due to the difficulty in the calculation of the residual entanglement. The negativity-based measures $n_1\left({\rho }_4\right)$ and $n_2\left({\rho }_4\right)$ are explicitly computed and the corresponding optimal decompositions are also derived explicitly.

Figure 1

The $p$ dependence of (a) ${\mathcal{C}}_{I|J}$ and (b) ${\mathcal{C}}_{I|J}^{\left(3\right)}$. As (a) shows, ${\mathcal{C}}_{I|J}$ is independent of the phase factor $\phi$ unlike ${\mathcal{C}}_{I|J}^{\left(3\right)}$. This makes $t_1\left({\rho }_4\right)$ trivial.

Figure 2

The $p$ dependence of (a) $t_1\left[Z_4(p,\phi )\right]$ and (b) $\tau \left[Z_3(p,\phi )\right]$. Contrary to $\tau \left[Z_3(p,\phi )\right],t_1\left[Z_4(p,\phi )\right]$ is independent of the phase factor $\phi$.

Figure 3

The $p$ dependence of ${\mathcal{N}}_{I\left|\right(JKL)},N_{I\left|\right|J|K}$, and ${\mathcal{N}}_{I|J}$ for $\left|Z\right(p,\phi \rangle$. It is shown that all negativities are independent of the phase factor $\phi$.

Figure 4

The $p$ dependence of (a) $n_1\left[Z(p,\phi )\right]$ (dashed and dotted) and $n_1\left({\rho }_4\right)$ (solid) at ${\nu }_1={\nu }_{1*}$ [red (lower) solid] and ${\nu }_1=\infty$ [blue (upper) solid], and (b) $n_2\left[Z(p,\phi )\right]$ (dashed and dotted) and $n_2\left({\rho }_4\right)$ (solid) at ${\nu }_2={\nu }_{2*}$ [red (lower) solid] and ${\nu }_2=\infty$ [blue (upper) solid].

Figure 5

(a) The $p$ dependence of ${\tau }_3(p,0),{\tau }_3(p,\pi )$, and ${\tau }_3(p,\pi \pm {\phi }_0)$ with ${\phi }_0=1.27672$. The dashed curve is a $p$ dependence of ${\tau }_3^{max}$. The nontrivial zeros given in Table 3 are plotted as black dots. (b) The $p$ dependence of ${\tau }_3(p,\phi )$ with varying $\phi$ from 0 to $\pi$ with a step 0.05. These curves have been referred to as the characteristic curves. The red (lowest) solid line is a minimum of the characteristic curves. This red curve seems to indicate that ${\tau }_3\left(\mathrm{\Pi }\right)$ is zero at $p_1\le p\le p_3$. However, we cannot find the corresponding optimal decompositions.