Continuous variable tangle, monogamy inequality, and entanglement sharing in Gaussian states of continuous variable systems

For continuous-variable ( CV) systems, we introduce a measure of entanglement, the CV tangle (contangle), with the purpose of quantifying the distributed ( shared) entanglement in multimode, multipartite Gaussian states. This is achieved by a proper convex-roof extension of the squared logarithmic negativity. We prove that the contangle satisfies the Coffman - Kundu - Wootters monogamy inequality in all three-mode Gaussian states, and in all fully symmetric N-mode Gaussian states, for arbitrary N. For three-mode pure states, we prove that the residual entanglement is a genuine tripartite entanglement monotone under Gaussian local operations and classical communication. We show that pure, symmetric three-mode Gaussian states allow a promiscuous entanglement sharing, having both maximum tripartite residual entanglement and maximum couplewise entanglement between any pair of modes. These states are thus simultaneous CV analogues of both the GHZ and the W states of three qubits: in CV systems monogamy does not prevent promiscuity, and the inequivalence between different classes of maximally entangled states, holding for systems of three or more qubits, is removed.