Entanglement monogamy via multivariate trace inequalities

TL;DR

This paper introduces new matrix trace inequalities and variational formulas for relative entropies, providing simplified proofs and generalizations of entanglement monogamy inequalities in quantum information theory.

Contribution

It develops matrix-analysis-based proofs for entanglement monogamy, strengthening and generalizing existing inequalities using multivariate trace inequalities.

Findings

Proves faithfulness of squashed entanglement via matrix inequalities.

Relates conditional entanglement of mutual information to separably measured relative entropy.

Extends results to states with positive partial transpose and multipartite systems.

Abstract

Entropy is a fundamental concept in quantum information theory that allows to quantify entanglement and investigate its properties, for example its monogamy over multipartite systems. Here, we derive variational formulas for relative entropies based on restricted measurements of multipartite quantum systems. By combining these with multivariate matrix trace inequalities, we recover and sometimes strengthen various existing entanglement monogamy inequalities. In particular, we give direct, matrix-analysis-based proofs for the faithfulness of squashed entanglement by relating it to the relative entropy of entanglement measured with one-way local operations and classical communication, as well as for the faithfulness of conditional entanglement of mutual information by relating it to the separably measured relative entropy of entanglement. We discuss variations of these results using the relative entropy to states with positive partial transpose, and multipartite setups. Our results simplify and generalize previous derivations in the literature that employed operational arguments about the asymptotic achievability of information-theoretic tasks.