  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.
