## Lower bounds on the quantum Fisher information based on the variance and various types of entropies

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