Quantifying Tripartite Entanglement of Three-Qubit Generalized Werner States

Multipartite entanglement is a key concept in quantum mechanics for which, despite the experimental progress in entangling three or more quantum devices, there is still no general quantitative theory that exists. In order to characterize the robustness of multipartite entanglement, one often employs generalized Werner states, that is, mixtures of a Greenberger-Horne-Zeilinger (GHZ) state and the completely unpolarized state. While two-qubit Werner states have been instrumental for various important advancements in quantum information, as of now there is no quantitative account for such states of more than two qubits. By using the GHZ symmetry introduced recently, we find exact results for tripartite entanglement in three-qubit generalized Werner states and, moreover, the entire family of full-rank mixed states that share the same symmetries.

Figure 1

The convex set of GHZ-symmetric density matrices ${\rho }^{\mathrm{S}}$ for two qubits. The Bell states ${\Phi }^{+}$ and ${\Phi }^{-}$ form the upper corners of the triangle, while the equal mixture of ${\Psi }^{+}$ and ${\Psi }^{-}$ defines the lower corner. The completely mixed state $\frac{1}{4}{\mathbb{1}}_4$ is located at the origin. The border between separable and entangled states is given by the separability line $y^{\mathrm{sep}}=\pm (\frac{1}{2\sqrt{2}}-\sqrt{2}x)$. The solid black line shows an example for the directions characterized by $u$ along which we parametrize the concurrence in Eq. (7). The parameter $z$ gives the position on that line. Further, the red dotted line illustrates the convex combination for the decomposition (9) of the arbitrary state ${\rho }^{\mathrm{S}}(x_0,y_0)$ indicated by a red dot.

Figure 2

The set of GHZ-symmetric three-qubit states [19]. The states ${\mathrm{GHZ}}_{+}$ and ${\mathrm{GHZ}}_{-}$ form the upper corners of the triangle, the lower corner is the symmetrization of $|001⟩$, and the mixed state $\frac{1}{8}{\mathbb{1}}_8$ is located at the origin. The generalized Werner states lie on the line $y=\frac{\sqrt{3}}{2}x$ (magenta solid line). The border between $W$-class (yellow area) and GHZ-class states (brown area) is given by the $\mathrm{GHZ}/W$ line (red solid line), Eq. (14). The directions along which we parametrize the states using $q$ and $z$ in Eq. (15) are indicated by the black solid line. A state ${\rho }^{\mathrm{S}}(x_0,y_0)$ with nonvanishing three-tangle (19) can be decomposed according to Eq. (18), illustrated by the red dotted line.

Figure 3

The three-tangle for GHZ-symmetric three-qubit states. In the triangle of the states we have used the same colors for the entanglement classes as in Fig. 2. The red solid line in the $xy$ plane represents again the $\mathrm{GHZ}/W$ line, and the magenta solid line the three-tangle of the generalized Werner states. The thin black lines indicate convex characteristic curves ${\tau }^{*}(q,z)$ for different $q$ values [Eq. (17)].