Entanglement of Three-Qubit Greenberger-Horne-Zeilinger–Symmetric States

The first characterization of mixed-state entanglement was achieved for two-qubit states in Werner’s seminal work [Phys. Rev. A 40, 4277 (1989)]. A physically important extension concerns mixtures of a pure entangled state [such as the Greenberger-Horne-Zeilinger (GHZ) state] and the unpolarized state. These mixed states serve as benchmark for the robustness of multipartite entanglement. They share the symmetries of the GHZ state. We call such states GHZ symmetric. Here we give a complete description of the entanglement in the family of three-qubit GHZ-symmetric states and, in particular, of the three-qubit generalized Werner states. Our method relies on the appropriate parametrization of the states and on the invariance of entanglement properties under general local operations. An application is the definition of a symmetrization witness for the entanglement class of arbitrary three-qubit states.

Figure 1

The convex set of GHZ-symmetric density matrices ${\rho }^{\mathrm{S}}$. The upper corners of the triangle are the standard GHZ state $|{\mathrm{GHZ}}_{+}⟩$, and $|{\mathrm{GHZ}}_{-}⟩$. Note that these are the only pure states. Applying ${\sigma }_z$ to any one of the qubits changes the sign of $x$. Therefore for properties invariant under local unitaries, we have a mirror symmetry about the $y$ axis. At the center of the upper horizontal line there is the separable state $\frac{1}{2}(|000⟩⟨000|+|111⟩⟨111|)$. The points for the pure states $|001⟩$, $|+++⟩$, $|+⟩|{\varphi }^{+}⟩$, $|W_{+--}⟩$ indicate the positions of the corresponding symmetrized mixed state (here we have used the definitions $|\pm ⟩\equiv \frac{1}{\sqrt{2}}(|0⟩\pm |1⟩)$, $|{\varphi }^{+}⟩\equiv \frac{1}{\sqrt{2}}(|00⟩+|11⟩)$, and $|W_{+--}⟩\equiv \frac{1}{\sqrt{3}}(|+--⟩+|-+-⟩+|--+⟩)$). The solid magenta line (Werner line) represents the generalized Werner states ${\rho }_{\mathrm{WS}}(p)$ with $p_{\mathrm{sep}}=\frac{1}{5}$ and $p_{\mathrm{bisep}}=\frac{3}{7}$ (see text). The two lower dashed lines are first guesses for the boundaries of fully separable (“sep”) and biseparable (“bisep”) states from the known values of $p_{\mathrm{sep}}$ and $p_{\mathrm{bisep}}$, respectively. The upper dashed line represents a first guess for the boundary between $W$ and GHZ states as $|W_{+--}⟩$ is the $W$ state with the largest overlap to $|{\mathrm{GHZ}}_{+}⟩$ (cf. Ref. 4). The intersection with the Werner line occurs at $p=9/13$.

Figure 2

The SLOCC classes of three-qubit GHZ-symmetric states ${\rho }^{\mathrm{S}}$. The dark blue region shows the separable states (“sep”) with the light blue lines $x=\pm (\frac{1}{4}-\frac{1}{\sqrt{3}}y{)}^{3/2}$. Green areas represent the biseparable states (“bisep”). The $W$ states “$W$”(yellow) and the GHZ states “GHZ” (grey) are separated by the curve Eq. (13) (red line). The Werner line (magenta) crosses that curve at $p_W\approx 0.6955$. Some geometrical aspects are noteworthy. The curve (13) nearly (within a few per cent) describes a circle about the point ${\rho }^{\mathrm{S}}(001)$. The radius has a minimum in the vicinity of the Werner line. Further, it is intriguing to note that each SLOCC class shares exactly one fourth of the lower edge of the triangle.