Extremal properties of the variance and the quantum Fisher information

@article{Tth2013ExtremalPO,
  title={Extremal properties of the variance and the quantum Fisher information},
  author={G{\'e}za T{\'o}th and D{\'e}nes Petz},
  journal={Physical Review A},
  year={2013},
  volume={87},
  pages={032324}
}
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… 

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