Numerical evaluation of convex-roof entanglement measures with applications to spin rings

We present two ready-to-use numerical algorithms to evaluate convex-roof extensions of arbitrary pure-state entanglement monotones. Their implementation leaves the user merely with the task of calculating derivatives of the respective pure-state measure. We provide numerical tests of the algorithms and demonstrate their good convergence properties. We further employ them in order to investigate the entanglement in particular few-spins systems at finite temperature. Namely, we consider ferromagnetic Heisenberg exchange-coupled spin-$\frac{1}{2}$ rings subject to an inhomogeneous in-plane field geometry obeying full rotational symmetry around the axis perpendicular to the ring through its center. We demonstrate that highly entangled states can be obtained in these systems at sufficiently low temperatures and by tuning the strength of a magnetic field configuration to an optimal value which is identified numerically.

Figure 1

(Color online) Convergence plots of the algorithms used to evaluate the entanglement of formation on ten random full-rank two-qubit states (each plot was done using the same ten states) showing the difference between the numerical data and the analytical result as a function of the iteration number. Top: the modified steepest-descent algorithm from Ref. [29]. Middle: the generalized conjugate-gradient method from Sec. 3A. Bottom: quasi-Newton on the parametrized search space, Sec. 3B. Each curve in the main plot is averaged over ten randomly chosen initial points. The typical behavior of the algorithms for a single fixed density matrix, but with varying initial points of the iteration, is displayed in the insets.

Figure 2

(Color online) Convergence of the generalized conjugate-gradient (top) and the parametrized quasi-Newton (bottom) algorithms for the tangle of GHZ-$W$ states Eq. (38). The curves for the values $\eta =\frac{1}{5}{\eta }_0$ (solid line), $\eta =\left(1-{10}^{-4}\right){\eta }_0$ (dashed line), $\eta =\left(1+{10}^{-4}\right){\eta }_0$ (dotted line), and $\eta =\frac{7}{5}{\eta }_0$ (dashed-dotted line) have each been obtained by averaging 100 successful runs starting with random initial points. The typical behavior of single runs is shown in the insets, where the 100 successful tries that yield one curve in the main plots are displayed.

Figure 3

(Color online) (a) Schematic depiction of the system described by the Hamiltonian (44) for $N=3$. Three spins ${\mathbf{S}}_i$ are situated at the corners of an equilateral triangle and are ferromagnetically exchange coupled with coupling strength $J$. Local radial in-plane magnetic fields ${\mathbf{b}}_i$ (shown as green arrows in the $xy$ plane) point radially outwards. As discussed in the text, any other in-plane field geometry obeying the same radial symmetry (such as, e.g., a “chiral” field looping around the triangle) leads to equivalent results. (b) Classical energy surface $E_c$ of the system shown in the top panel. The “mean” angles $\overline{\vartheta }$ and $\overline{\phi }$ (introduced in the top panel) are well suited to characterize the state of the system since fluctuations around these angles are small for $b⪡J$ and sum to zero. The superimposed white line shows the perturbatively calculated energy barrier at $\overline{\phi }=\pi /2$ [see Eq. (2) in Ref. [22]], whereas the crosses are due to a corresponding numerical minimization of the energy.

Figure 4

(Color online) Meyer-Wallach entanglement measure for the system described by the Hamiltonian Eq. (44) at different temperatures for several numbers of particles. Concretely, the cases $N=2$ (top left), $N=3$ (top right), $N=4$ (bottom left), and $N=5$ (bottom right) are studied at temperatures $T={10}^{-4}J/k_B$ (dashed line), $T={10}^{-3}J/k_B$ (dashed-dotted line), $T={10}^{-2}J/k_B$ (dashed-dot-dotted line), and $T=5\times {10}^{-2}J/k_B$ (dotted line).

Figure 5

(Color online) Difference between zero-temperature and finite-temperature maximally achievable Meyer-Wallach entanglement measure as a function of temperature for systems with $N=2,3,4,5,7,10$ spins (from bottom to top). Inset: value of the magnetic field strength $b$ as a function of system size at which the ground state yields the Meyer-Wallach value 0.5 (full width at half maximum).