  We demonstrate the existence of general constraints on distributed quantum
correlations, which impose a trade-off on bipartite and multipartite
entanglement at once. For all N-mode Gaussian states under permutation
invariance, we establish exactly a monogamy inequality, stronger than the
traditional one, that by recursion defines a proper measure of genuine
N-partite entanglement. Strong monogamy holds as well for subsystems of
arbitrary size, and the emerging multipartite entanglement measure is found to
be scale invariant. We unveil its operational connection with the optimal
fidelity of continuous variable teleportation networks.
