Simple algorithm for computing the geometric measure of entanglement

We present an easy implementable algorithm for approximating the geometric measure of entanglement from above. The algorithm can be applied to any multipartite mixed state. It involves only the solution of an eigenproblem and finding a singular value decomposition; no further numerical techniques are needed. To provide examples, the algorithm was applied to the isotropic states of three qubits and the three-qubit $XX$ model with external magnetic field.

Figure 1

Approximation of the geometric measure of entanglement ${\tilde{E}}_G$ for isotropic states of three qubits given in (38) as a function of $p$ (solid line) compared to the two-qubit case (dashed line).

Figure 2

Approximation of the geometric measure of entanglement ${\tilde{E}}_G$ plotted as function of the temperature $T$ for $\rho =\frac{e^{-\frac{H}{kT}}}{Z}$ with $H$ given in (39). The parameter $J$ is set to $\frac{1}{2}$ and $k=1$.

Figure 3

Approximation of the geometric measure of entanglement ${\tilde{E}}_G$ for fixed values of $T$ plotted as a function of the magnetic field $B$. The parameter $J$ is set to $\frac{1}{2}$ and $k=1$.