Comparison of different definitions of the geometric measure of entanglement

Several inequivalent definitions of the geometric measure of entanglement (GM) have been introduced and studied in the past. Here we review several known and new definitions, with the qualifying criterion being that for pure states the measure is a linear or logarithmic function of the maximal fidelity with product states. The entanglement axioms and properties of the measures are studied, and qualitative and quantitative comparisons are made between all definitions. Streltsov et al. [New J. Phys. 12, 123004 (2010)] proved the equivalence of two linear definitions of GM, whereas we show that the corresponding logarithmic definitions are distinct. Certain classes of states such as “maximally correlated states” and isotropic states are particularly valuable for this analysis. A little-known GM definition is found to be the first one to be both normalized and weakly monotonous, thus being a prime candidate for future studies of multipartite entanglement. We also find that a large class of graph states, which includes all cluster states, have a “universal” closest separable state that minimizes the quantum relative entropy, the Bures distance, and the trace distance.

Figure 1

The quantitative hierarchy of the different measures is shown, with the six distinct extensions of GM in white boxes. For a given state $\rho \in \mathcal{S}\left(\mathcal{H}\right)$ the value of the measures increases monotonically from bottom to top along the vertical lines, and measures that are not vertically connected are not in an inequality relationship to each other. The quantities $\widetilde{E_{\text{T}}}$ and $E_{\text{R}}+S$ are not extensions of GM, but they provide lower and upper bounds, respectively.

Figure 2

Illustration of the partitioning of state space $\mathcal{S}\left(\mathcal{H}\right)=A\cup B\cup C\cup D$ into four pairwise disjoint sets. The pure states ($A\cup B$) lie at the boundary, and the separable states ($A\cup C$) form a closed, convex subset of $\mathcal{S}\left(\mathcal{H}\right)$. The set a state $\rho$ belongs to is uniquely determined by whether the inequalities in (35) are strict or equalities. By means of ${\Lambda }_{\text{f}}^2\left(\rho \right)$ and ${\Lambda }_{\text{m}}^2\left(\rho \right)$, the same partitioning can be facilitated by the logarithmic quantities $G_{\text{l}}^{\text{f}}$ and $G_{\text{l}}^{\text{m}}$. When taking $G_{\text{l}}^{\text{c}}$ into account, as well, the set of genuinely mixed entangled states $D=D_1\cup D_2\cup D_3$ is further subdivided into three pairwise disjoint sets.