We examine important properties of the difference between the variance
and the quantum Fisher information over four, i.e., (ΔA)2−FQ[ϱ,A]/4.
We find that it is equal to a generalized variance defined in Petz [J.
Phys. A 35, 929 (2002)] and Gibilisco, Hiai, and Petz [IEEE Trans.
Inf. Theory 55, 439 (2009)]. We present an upper bound on this
quantity that is proportional to the linear entropy. As expected, our
relations show that for states that are close to being pure, the
quantum Fisher information over four is close to the variance. We also
obtain the variance and the quantum Fisher averaged over all Hermitian
operators, and examine their relation to the von Neumann entropy.
 G. Toth, arxiv:1701.07461.