Maximally entangled three-qubit states via geometric measure of entanglement

Bipartite maximally entangled states have the property that the largest Schmidt coefficient reaches its lower bound. However, for multipartite states, the standard Schmidt decomposition generally does not exist. We use a generalized Schmidt decomposition and the geometric measure of entanglement to characterize three-qubit pure states and derive a single-parameter family of maximally entangled three-qubit states. The paradigmatic Greenberger-Horne-Zeilinger (GHZ) and $W$ states emerge as extreme members in this family of maximally entangled states. This family of states possesses different trends of entanglement behavior: in going from GHZ to $W$ states, the geometric measure, the relative entropy of entanglement, and the bipartite entanglement all increase monotonically whereas the three-tangle and bipartition negativity both decrease monotonically.

Figure 1

(Color online) Plots of $t$ dependencies of the largest $g_1\left(t\right)$ (solid line) and next $g_2\left(t\right)$ (dashed line) roots of the degeneracy condition for $\gamma =\pi /6$. $g_1\left(t\right)$ is the largest Schmidt coefficient and has a minimum at $t=0$ which gives the GHZ state.

Figure 2

(Color online) Plots of $t$ dependencies of the largest $g_1\left(t\right)$ (solid line) and next $g_2\left(t\right)$ (dashed line) roots of the degeneracy condition for $\gamma =2\pi /5$. $g_1\left(t\right)$ has a minimum while $g_2\left(t\right)$ has a maximum at $t=0.31943$. These roots never intersect unless $\gamma =\pm \pi /2$.

Figure 3

(Color online) Plots of $t$ dependencies of the largest $g_1\left(t\right)$ (solid line) and next $g_2\left(t\right)$ (dashed line) roots of the degeneracy condition for $\gamma =\pi /2$. The two curves touch at (1/3,2/3).

Figure 4

(Color online) The gauge phase $\gamma$ dependence of the largest Schmidt coefficient $g$ as well as $h$ of maximally entangled states. $g$ is constant and equal to $1/\sqrt{2}$ within $0\le \gamma \le \pi /4$, then it decreases monotonically and becomes 2/3 at $\gamma =\pi /2$. Similarly, $h$ is constant and equal to $1/\sqrt{2}$ within $0\le \gamma \le \pi /4$, then it decreases monotonically and becomes $\sqrt{2}/3$ at $\gamma =\pi /2$. The parameter $t$ is obtained from the normalization $g^2+3t^2+h^2=1$.

Figure 5

(Color online) The maximal overlap $g$ vs the gauge phase $\gamma$ for the family of maximally entangled three-qubit states (red solid curve) as well as randomly generated states (dots). This shows that the family of states we have derived is indeed maximally entangled (with minimal overlap $g$).

Figure 6

(Color online) The phase dependence of three-tangle $\tau$ (red solid curve) and the residual bipartite entanglement $E_r$ (blue dashed curve) vs $\gamma$ for the family of maximally entangled three-qubit states.

Figure 7

(Color online) The phase dependence of the negativity $N$ (red solid curve) and the relative entropy of entanglement $E_R$ (blue dashed curve) vs $\gamma$ for the family of maximally entangled three-qubit states.