Two computable sets of multipartite entanglement measures

We present two sets of computable entanglement measures for multipartite systems where each subsystem can have different degrees of freedom (so-called qudits). One set, called “separability” measure, reveals which of the subsystems are separable or entangled. For that we have to extend the concept of $k$ separability for multipartite systems to a unambiguous separability concept which we call ${\gamma }_k$ separability. The second set of entanglement measures reveals the “kind” of entanglement, i.e., if it is bipartite, tripartite, …, $n$-partite entangled and is denoted as the “physical” measure. We show how lower bounds on both sets of measures can be obtained by the observation that any entropy may be rewritten via operational expressions known as $m$ concurrences. Moreover, for different classes of bipartite or multipartite qudit systems we compute the bounds explicitly and discover that they are often tight or equivalent to positive partial transposition.

Figure 1

Here the convexity of ${\gamma }_k$ separability is visualized, i.e., any convex mixture of two ${\gamma }_k$-separable states, e.g., ${\gamma }_{k_1}$ and ${\gamma }_{k_2}$, is either ${\gamma }_{k_1}$, ${\gamma }_{k_2}$, or ${\gamma }_k$ separable with $k_1,k_2<k$.

Figure 2

(Color online) The graphs show the set of the physical measure of the mixture of the GHZ state and the $\text{EPR}\otimes \text{EPR}$ state, Eq. (40). Solid (red) curve shows the four-partite entanglement ${\mathcal{E}}_4={\mathcal{E}}_{1234}$, dashed (green) curve shows the three-partite entanglement ${\mathcal{E}}_3={\mathcal{E}}_{123}+{\mathcal{E}}_{124}+{\mathcal{E}}_{134}+{\mathcal{E}}_{234}$, and dotted (blue) curve shows the two-partite entanglement ${\mathcal{E}}_2={\mathcal{E}}_{12}+{\mathcal{E}}_{13}+{\mathcal{E}}_{14}+{\mathcal{E}}_{23}+{\mathcal{E}}_{24}+{\mathcal{E}}_{34}$ in dependence of $\alpha$. The amount of the total entanglement is for the GHZ state and the $\text{EPR}\otimes \text{EPR}$ state 4, however, in the first case it due to four-partite entanglement whereas in the other case the bipartite entanglement maximizes. The separability measurement reveals that the state is ${\gamma }_1=\left\{1234\right\}$ separable ($E_{1234}=4$, all others zero) except for $\alpha =\frac{\pi }{2}$ then the state is ${\gamma }_2=\left\{12\mid 34\right\}$ separable ($E_{tot}=4$, $E_{12}=E_{34}=2$ all others zero).

Figure 3

(Color online) Here two slices through the class of “line” states [46], Eqs. (48, 49), are shown. Green triangle visualizes the parameter space for which the states are positive, blue triangle/ellipse the parameter space for which the states are positive under partial transpose (PPT). Colored areas denote the regions where the bounds on the physical measure are nonzero (labelling from inner to outer regions: purple: $0.2\le \mathbf{C}>0$; blue: $0.4\ge \mathbf{C}>0.2$; green: $0.6\ge \mathbf{C}>0.4$; yellow: $0.8\ge \mathbf{C}>0.6$; red: $1\ge \mathbf{C}>0.8$). Note that not all states negative under partial transpose are detected. In (a) for $\alpha <0$ or $\beta <0$ the bound is equivalent to the boundary by PPT, however, as was shown in Ref. [46] a small region of bound entangled states exists in this case. Only for the class of states visualized in (b) for $\alpha \ge \frac{1}{4}$ the bounds are tight.