Highly Entangled Ground States in Tripartite Qubit Systems

We investigate the creation of highly entangled ground states in a system of three exchange-coupled qubits arranged in a ring geometry. Suitable magnetic field configurations yielding approximate Greenberger-Horne-Zeilinger and exact $W$ ground states are identified. The entanglement in the system is studied at finite temperature in terms of the mixed-state tangle $\tau$. By generalizing a conjugate gradient optimization algorithm originally developed to evaluate the entanglement of formation, we demonstrate that $\tau$ can be calculated efficiently and with high precision. We identify the parameter regime for which the equilibrium entanglement of the tripartite system reaches its maximum.

Figure 1

The tangle $\tau$ of the system with isotropic positive (ferromagnetic) coupling $J$ and radial magnetic field as a function of $b/J$ for different temperatures $T={10}^{-4}J/k_B$ (dashed line), ${10}^{-3}J/k_B$ (dash-dotted line), ${10}^{-2}J/k_B$ (dash-dot-dotted line), and $5\times {10}^{-2}J/k_B$ (dotted line). Note the twofold influence of the temperature on $\tau$: Although higher temperatures reduce the maximally achievable entanglement, a stabilizing effect is observed as well. A maximum in the tangle is more robust against fluctuations in $b$ at higher temperatures due to the less rapid drop-off of $\tau$ as $b/J$ is reduced. Conversely, $\tau$ of the approximate GHZ GS $|\mathrm{GS}⟩$ ($T=0$, solid red line) shows a discontinuity at $b=0$, where $\tau (\rho )=0$. For $b>0$, we find the simple algebraic expression ${\tau }_p(|\mathrm{GS}⟩)=(3-8b/J)/C+2/\sqrt{C}$, where $C=9+4b(4b/J-3)/J$. Inset: Energy splitting $\Delta E_{0,1}$ of the ground-state doublet as a function of $b/J$.

Figure 2

Top: Maximally achievable tangle ${\tau }_{\max }$ in the anisotropic system (GHZ GS) with homogeneous in-plane magnetic field and $J_z>0$ as a function of temperature for six anisotropy ratios $J_{xy}/J_z$ (see legend). The curves end at $\tau (\rho )={10}^{-5}$. Bottom: The corresponding optimal values $(b/J_z{)}_{\mathrm{opt}}$ of the scaled magnetic field strength $b/J_z$.