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We examine important properties of the difference between the variance andthe quantum Fisher information over four, i.e., $(\Delta A)^2-F_{\rmQ}[\varrho,A]/4.$ We find that it is equal to a generalized variance defined inPetz [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 isproportional to the linear entropy. As expected, our relation shows that forstates that are close to being pure, the quantum Fisher information over fouris close to the variance. We also obtain the variance and the quantum Fisherinformation averaged over all Hermitian operators, and examine its relation tothe von Neumann entropy. Apart from the usual quantum Fisher information, wealso consider the Kubo-Mori-Bogoliubov quantum Fisher information.

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