Schur complement inequalities for covariance matrices and monogamy of quantum correlations

TL;DR

This paper establishes new inequalities for quantum covariance matrices using Schur complements, leading to insights on the monogamy of quantum correlations and a hierarchy of correlation measures for Gaussian states.

Contribution

It introduces operator-based inequalities for quantum covariance matrices and applies them to prove monogamy relations and a novel hierarchy of correlation measures.

Findings

Proved general monogamy constraints for quantum correlations.

Established a hierarchy of correlation measures based on the log-determinant.

Extended inequalities to continuous variable quantum systems with multiple modes.

Abstract

We derive fundamental constraints for the Schur complement of positive matrices, which provide an operator strengthening to recently established information inequalities for quantum covariance matrices, including strong subadditivity. This allows us to prove general results on the monogamy of entanglement and steering quantifiers in continuous variable systems with an arbitrary number of modes per party. A powerful hierarchical relation for correlation measures based on the log-determinant of covariance matrices is further established for all Gaussian states, which has no counterpart among quantities based on the conventional von Neumann entropy.