Necessary and sufficient conditions for positive semidefinite quantum mutual information matrices

TL;DR

This paper establishes the exact conditions under which quantum mutual information matrices are positive semidefinite and introduces a new, symmetric measure for multipartite quantum correlations.

Contribution

It provides necessary and sufficient conditions for positive semidefiniteness and defines a new genuine multipartite mutual information measure.

Findings

Quantum mutual information matrices are not always positive semidefinite.

Necessary and sufficient conditions for positive semidefiniteness are derived.

A new symmetric, nonnegative, and bounded multipartite mutual information measure is proposed.

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.