Entangled Three-Qubit States without Concurrence and Three-Tangle

We provide a complete analysis of mixed three-qubit states composed of a Greenberger-Horne-Zeilinger state and a $W$ state orthogonal to the former. We present optimal decompositions and convex roofs for the three-tangle. Further, we provide an analytical method to decide whether or not an arbitrary rank-2 state of three qubits has vanishing three-tangle. These results highlight intriguing differences compared to the properties of two-qubit mixed states, and may serve as a quantitative reference for future studies of entanglement in multipartite mixed states. By studying the Coffman-Kundu-Wootters inequality we find that, while the amounts of inequivalent entanglement types strictly add up for pure states, this “monogamy” can be lifted for mixed states by virtue of vanishing tangle measures.

Figure 1

The three-tangle of the states $|Z(p,\phi )⟩$ in Eq. (8) for various values of $\phi =\gamma ·\frac{2\pi }{3}$ (from top to bottom: $\gamma =1/2$, $1/3$, $1/5$, $1/10$, 0). The three-tangle is periodic in $\phi$ with a period of $2\pi /3$. Further, ${\tau }_3(Z)$ vanishes trivially for zero GHZ component, but also for $\phi =0$ and nonzero GHZ amplitude $p_0=\frac{4\sqrt[3]{2}}{3+4\sqrt[3]{2}}\approx 0.627$. This is in striking contrast to the two-qubit case where, in a superposition of a Bell state with an orthogonal unentangled state, the concurrence equals the weight of the Bell state [25], that is the entanglement remains “untouched”.

Figure 2

Bloch sphere for the two-dimensional space spanned by the GHZ state and the $W$ state. The simplex $S_0$ contains all states with zero three-tangle. The leaves between $p_0$ and $p_1$ represent sets of constant three-tangle, and in the simplex $S_1$ the three-tangle is affine (for further explanation, see text).

Figure 3

Plot of the one-tangle (solid line), the sum of squared concurrences $C({\rho }_{AB}{)}^2+C({\rho }_{AC}{)}^2$ (dashed line) and three-tangle (dotted line) in $\rho (p)$ as a function of $p$. Clearly, the CKW inequality (13) is satisfied.