Best separable approximation of multipartite diagonal symmetric states

The structural study of entanglement in multipartite systems is hindered by the lack of necessary and sufficient operational criteria able to discriminate among the various entanglement properties of a given mixed state. Here, we pursue a different route to the study of multipartite entanglement based on the closeness of a multipartite state to the set of separable ones. In particular, we analyze multipartite diagonal symmetric $N$-qubit states and provide the analytical expression for their best separable approximation (BSA, [Phys. Rev. Lett. 80, 2261 (1998)]), that is, their unique convex decomposition into a separable part and an entangled part, with maximal weight of the separable part.

Figure 1

Feasible region for the BSA of full-$\mathrm{NPT}$ $N$-qubit symmetric states as a function of $p_0$ and $p_N$, for $N=2,3,$ and 4. For each $N$, the region covers the shaded area below the curve given by the boundary of $\lambda ={(\sqrt[N]{p_0}+\sqrt[N]{p_N})}^N\le 1$. The dashed region corresponds to unphysical states, and the area between the dashed and shaded regions corresponds to nonfull $\mathrm{NPT}$ entangled states, for each $N$.