  For any $n$-partite state $\rho_{A_{1}A_{2}\cdot\cdot\cdot A_{n}}$, we define
its quantum mutual information matrix as an $n$ by $n$ matrix whose
$(i,j)$-entry is given by quantum mutual information $I(\rho_{A_{i}A_{j}})$.
Although each entry of quantum mutual information matrix, like its classical
counterpart, is also used to measure bipartite correlations, the similarity
ends here: quantum mutual information matrices are not always positive
semidefinite even for collections of up to 3-partite states. In this work, we
obtain necessary and sufficient conditions for the positive semidefinite
quantum mutual information matrix. We further define the \emph{genuine}
$n$-partite mutual information which can be easily calculated. This definition
is symmetric, nonnegative, bounded and more accurate for measuring multipartite
states.
