Detecting multimode entanglement by symplectic uncertainty relations

A hierarchy of multimode uncertainty relations on the second moments of n pairs of canonical operators is derived in terms of quantities invariant under linear canonical (i.e. symplectic) transformations. Conditions for the separability of multimode continuous variable states are derived from the uncertainty relations, generalizing the inequalities obtained in [Phys. Rev. Lett. 96, 110402 (2006)] to states with some transposed symplectic eigenvalues equal to 1. Finally, to illustrate the methodology proposed for the detection of continuous variable entanglement, the separability of multimode noisy GHZ-like states is analysed in detail with the presented techniques, deriving a necessary and sufficient condition for the separability of such states under an `even' bipartition of the modes.