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

For continuous-variable systems, we introduce a measure of entanglement, the continuous variable 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 continuous-variable analogs of both the GHZ and the W states of three qubits: in continuous-variable 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.