Uncertainty relations with variances and the quantum Fisher information based on convex decompositions of density matrices

We discuss recent findings relating the quantum Fisher information to convex roofs of variances. We present several improvements on the Robertson-Schrodinger uncertainty relation. In all these improvements, we consider a decomposition of the density matrix into a mixture of pure states, and use the fact that the Robertson-Schrodinger uncertainty relation is valid for all these components. By considering a convex roof of the bound, we obtain an alternative derivation of the relation in F. Frowis, R. Schmied, and N. Gisin [Phys. Rev. A 92, 012102 (2015)], and we can also list a number of conditions that are needed to saturate the relations. We also gain new insights about the Cramer-Rao bound. By considering a concave roof of the bound in the Robertson-Schrodinger uncertainty relation, we obtain other type of improvements. We consider similar techniques for uncertainty relations with three variances. Finally, we present further uncertainty relations that provide lower bounds on the metrological usefulness of bipartite quantum states based on the variances of canonical position and momentum operators for two-mode continuous variable systems and angular momentum operators for spin systems.