Computable multipartite multimode Gaussian quantum correlation measure and the monogamy relations for continuous-variable systems

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In this paper, definitions of the unification condition, the hierarchy condition, and three kinds of monogamy relations for multipartite quantum correlation measures are given and discussed. A computable multipartite multimode Gaussian quantum correlation measure ${\mathcal{M}}^{(k)}$ is proposed for any $k$-partite multimode continuous-variable systems with $k\ensuremath{\ge}2$. The value of ${\mathcal{M}}^{(k)}$ only depends on the covariance matrices of continuous-variable states, is invariant under any permutation of subsystems, has no ancilla problem, is nonincreasing under $k$-partite local Gaussian channels (particularly, invariant under $k$-partite locally Gaussian unitary operations), and vanishes on $k$-partite product states. For a $k$-partite Gaussian state $\ensuremath{\rho}, {\mathcal{M}}^{(k)}(\ensuremath{\rho})=0$ if and only if $\ensuremath{\rho}$ is a $k$-partite product state. Moreover, ${\mathcal{M}}^{(k)}$ satisfies the unification condition and the hierarchy condition that a multipartite quantum correlation measure should obey. We also show that ${\mathcal{M}}^{(k)}$ is not strongly monogamous but completely monogamous and tightly monogamous.