Necessary and sufficient conditions for positive semidefinite quantum mutual information matrices

Abstract

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.