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 prove analytically that the quantum Fisher information is four times the convex roof of the variance. Strong numerical evidence suggests that this statement is true even for operators with nonzero diagonal elements or density matrices with a rank larger than 2. We also find that within the different types of generalized quantum Fisher information considered in Petz [J. Phys. A1361-644710.1088/0305-4470/35/4/305 35, 929 (2002)] and Gibilisco, Hiai, and Petz [IEEE Trans. Inf. TheoryIETTAW0018-944810.1109/TIT.2008.2008142 55, 439 (2009)], after appropriate normalization, the quantum Fisher information is the largest. Hence, we conjecture that the quantum Fisher information is four times the convex roof of the variance even for the general case.