A computable multipartite multimode Gaussian correlation measure and the monogamy relation for continuous-variable systems

In this paper, a computable multipartite multimode Gaussian quantum correlation measure ${\mathcal M}^{(k)}$ is proposed for any $k$-partite continuous-variable (CV) systems with $k\geq 2$. ${\mathcal M}^{(k)}$ depends only on the covariance matrix of CV states, is invariant under any permutation of subsystems, is a quantification without ancilla problem, nonincreasing under $k$-partite local Gaussian channels (particularly, invariant under $k$-partite local Gaussian unitary operations), vanishes on $k$-partite product states. For a $k$-partite Gaussian state $\rho$, ${\mathcal M}^{(k)}(\rho)=0$ if and only if $\rho$ is a $k$-partite product state. Thus, for the bipartite case, ${\mathcal M}={\mathcal M}^{(2)}$ is an accessible replacement of the Gaussian quantum discord and Gaussian geometric discord. Moreover, ${\mathcal M}^{(k)}$ satisfies the unification condition, hierarchy condition that a multipartite quantum correlation measure should obey. ${\mathcal M}^{(k)}$ is not bipartite like monogamous, but, ${\mathcal M}^{(k)}$ is complete monogamous and tight complete monogamous.