Extremal properties of the variance and the quantum Fisher information

We show that the variance is its own concave roof. For rank-2 density matrices and operators with zero diagonal elements in the eigenbasis of the density matrix, we show analytically that the quantum Fisher information is 4 times the convex roof of the variance. Strong numerical evidence suggests that the quantum Fisher information is very close to the convex roof even for operators with nonzero diagonal elements or density matrices with a rank larger than 2. Hence, we conjecture that the quantum Fisher information is 4 times the convex roof of the variance even for the general case.