Three-tangle of assistance

In this report, we calculate the three tangle of assistance and give the corresponding best possible pure state decomposition of rank-two mixed states for the three-qubit Greenberger-Horne-Zeilinger (GHZ) state and $W$ state. As the three tangle can be transformed into bipartite concurrence with assistance, we achieve a lower bound of the assisted concurrence with local operations and classical communication (LOCC) constraints via these pure state decompositions. It is smaller than the total assisted concurrence, which shows that local assistance alone is not sufficient for maximizing bipartite entanglement. The increase of three tangle is attributed to the genuine multipartite entanglement in the purified state. A genuine four-qubit entanglement measure is calculated for the purified four-qubit state and compared to the three tangle of assistance.

Figure 1

Comparison between the total assisted concurrence $C_a^2({\rho }_{\mathit{AB}})$ and the assisted concurrence $C_{a,1}^2({\rho }_{\mathit{AB}})$. The dashed line represents the total assisted concurrence $C_a^2({\rho }_{\mathit{AB}})$ and the solid line represents the assisted concurrence $C_{a,1}^2({\rho }_{\mathit{AB}})$.

Figure 2

Comparison between the increase of three tangle under assistance ${\Delta }_3={\tau }_a({\rho }_{\mathit{ABC}})-\tau ({\rho }_{\mathit{ABC}})$ and its possible origins ${\Delta }_4={\mathcal{C}}_{C|\mathit{ABD}}^2(|\Psi \rangle )-[{\mathcal{C}}_{\mathit{AC}}^2({\rho }_{\mathit{AC}})+{\mathcal{C}}_{\mathit{CB}}^2({\rho }_{\mathit{BC}})+{\mathcal{C}}_{\mathit{CD}}^2({\rho }_{\mathit{CD}})+\tau ({\rho }_{\mathit{ACD}})+\tau ({\rho }_{\mathit{BCD}})+\tau ({\rho }_{\mathit{ABC}})]$. The dashed line represents ${\Delta }_4$ and the solid line represents ${\Delta }_3$.

Figure 3

Comparison between the genuine four-qubit entanglement measure $\left|{\mathcal{F}}_1\right|{}^{2/3}$ and the increase of three tangle under assistance ${\Delta }_3={\tau }_a({\rho }_{\mathit{ABC}})-\tau ({\rho }_{\mathit{ABC}})$. The dashed line represents $\left|{\mathcal{F}}_1\right|{}^{2/3}$ and the solid line represents ${\Delta }_3$.