Proving genuine multiparticle entanglement from separable nearest-neighbor marginals

We address the question of whether or not global entanglement of a quantum state can be inferred from local properties. Specifically, we are interested in genuinely multiparticle entangled states whose two-body marginals are all separable, but where the entanglement can be proven using knowledge of a subset of the marginals only. Using an iteration of semidefinite programs we prove that for any possible marginal configuration up to six particles multiqubit states with the desired properties can be found. We then present a method to construct states with the same properties for more particles in higher dimensions.

Figure 1

If global entanglement should be concluded from a set of two-body marginals $\left\{{\rho }_{ij}\right\}$ then the set of marginals has to obey two conditions: First, any particle $k$ must be covered by the set and secondly, the graph arising from the interpretation of the ${\rho }_{ij}$ as edges must be connected. Among the marginal sets with these conditions, the minimal sets are of special interest, and these are necessarily treelike configurations. The figure shows all minimal tree configurations of marginals for five particles, up to permutations of the particles [19].

Figure 2

Illustration of biseparable states and PPT mixtures for a three-particle system; see text for further details. The figure is taken from Ref. [22].

Figure 3

Constructing a state with the desired properties for a simple six-party configuration. Here, it is assumed that the marginals ${\rho }_{AB}$, ${\rho }_{BC}$, ${\rho }_{CD}$, ${\rho }_{DE}$, and ${\rho }_{EF}$ are known. See text for further details.

Figure 4

Covering a two-dimensional lattice with linear numerically found states. See text for further details.

Figure 5

Constructing a state with the desired properties for an arbitrary configuration. See the text for further details.