Experimentally implementable criteria revealing substructures of genuine multipartite entanglement

We present a general framework that reveals substructures of genuine multipartite entanglement. Via simple inequalities it is possible to discriminate different sets of multipartite qubit states. These inequalities are beneficial regarding experimental examinations as only local measurements are required. Furthermore, the number of observables scales favorably with system size. In exemplary cases we demonstrate the noise resistance and discuss implementations.

Figure 1

Here the nested convex structure of $k$ separability in multipartite systems is illustrated.

Figure 2

Here the structure of (a) three- and (b) five-qubit mixed states is illustrated. The three entanglement classes $\mathcal{C}(\{|{\Psi }_{(2)}^1\rangle \})$, $\mathcal{C}(\{|{\Psi }_{(n)}^1\rangle \})$, and $\mathcal{C}(\{|{\Psi }_{(n-1)}^1\rangle \})$ of course form convex sets together with the partially separable states, which can be excluded by our inequalities I(2), $\mathrm{I}(n)$, and I($n-1)$ and inequality II from Ref. [24]. The partially separable states are subsumed as PS. The different sets are drawn completely disjoint, although it might still be possible that there is a nonvanishing overlap. Numerical analysis of tripartite systems suggests otherwise, but even if there was, it would not change any of the presented results. Also note that a two-dimensional picture will always fail to incorporate all essential properties of multipartite entanglement. This illustration should facilitate the understanding of how the inequalities work.

Figure 3

Here the entanglement classes for the four-qubit state $\rho =\alpha |{\varphi }_1\rangle \langle {\varphi }_1|+\beta |{\varphi }_2\rangle \langle {\varphi }_2|+\frac{1-\alpha -\beta }{16}1$ are illustrated, with $|{\varphi }_1\rangle =\frac{1}{\sqrt{2}}(|0000\rangle +|1111\rangle )\in \mathcal{C}\{|{\Psi }_{(2)}^1\rangle \}$ and $|{\varphi }_2\rangle =\frac{1}{2}(|1000\rangle +|0100\rangle +|0010\rangle +|0001\rangle )\in \mathcal{C}\{|{\Psi }_{(4)}^1\rangle \}$. For the parameter region I (red) the state is not in $\mathcal{C}(\{|{\Psi }_{(4)}^1\rangle \})$, for region II (green) it is not in $\mathcal{C}(\{|{\Psi }_{(2)}^1\rangle \})$, and for region III (gray) it is genuinely multipartite entangled, as detected by the criteria introduced in Ref. [24]. The region labeled PPT (blue) contains all states which are positive under partial transposition for the two distinct bipartitions.