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We investigate the structural aspects of genuine multipartite entanglement inGaussian states of continuous variable systems. Generalizing the results of[Adesso & Illuminati, Phys. Rev. Lett. 99, 150501 (2007)], we analyze whetherthe entanglement shared by blocks of modes distributes according to a strongmonogamy law. This property, once established, allows to quantify genuineN-partite entanglement in terms of the "residual contangle" not encoded into2,...,K,...,(N-1)-partite quantum correlations. The explicit expression of thisentanglement measure is derived, by a recursive formula, for a subclass ofGaussian states. These are fully symmetric (permutation-invariant) statesmulti-partitioned into blocks, each consisting of an arbitrary number of modes.We compute the genuine multipartite entanglement shared by the blocks("molecules") and investigate its scaling properties with the number and sizeof the molecules, the total number of modes, the global mixedness of the state,and the squeezed resources. To achieve the exact computation we prove a generalresult of symplectic analysis: Correlations among K molecules in N-modemulti-symmetric Gaussian states can be transformed by a local unitary operationinto correlations shared by K single modes, one per molecule, in effectivenon-symmetric states. Due to this theorem, the above results extend to thesubclass of non-symmetric Gaussian states that are obtained by unitarilylocalizing the multipartite molecular entanglement of symmetric states. Thesefindings provide evidence that distributed Gaussian entanglement is stronglymonogamous even beyond specific symmetry constraints, and that the residualcontangle is a bona fide measure of genuine multipartite entanglement forpermutation-invariant Gaussian states under any multi-partition of the modes.

physical review. a, atomic, molecular, and optical physics/physical review, a, atomic, molecular, and optical physics

Genuine multipartite entanglement of symmmetric Gaussian states: Strong  monogamy, unitary localization, scaling behavior, and molecular sharing  structure