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Phys. A 35, 929 (2002)] and Gibilisco, Hiai, and Petz [IEEE Trans. Inf. Theory 55, 439 (2009)]. For the rank-2 case, we present an upper bound that is proportional to the linear entropy. We also present similar bounds for states with a higher rank. 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. 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