Necessary and sufficient conditions for positive semidefinite quantum mutual information matrices

Original · EN

For any n-partite state ρₐ₁ₐ₂··· ₐₙ, we define its quantum mutual information matrix as an n by n matrix whose (i,j)-entry is given by quantum mutual information I(ρₐᵢₐⱼ). 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 genuine n-partite mutual information which can be easily calculated. This definition is symmetric, nonnegative, bounded and more accurate for measuring multipartite states.