Conditions for multipartite continuous-variable entanglement

Necessary and sufficient observable conditions for the non-negativity of all partial transpositions of multimode quantum states are derived. The result is a hierarchy of inequalities for minors in terms of moments of the given state. The violation of any inequality is a sufficient condition for entanglement. Full entanglement can be certified for a manifold of multimode quantum states. A MATHEMATICA package is given for a systematic test of the hierarchy of conditions.

Figure 1

The minors $d_1$ and $d_2$ are given for different noise levels: $\overline{n}=0$ (solid line), $\overline{n}=0.01$ (dashed line), and $\overline{n}=0.05$ (dot-dashed line).