Extremal properties of the variance and the quantum Fisher information

TL;DR

This paper explores the extremal properties of variance and quantum Fisher information, establishing analytical results for specific cases and providing numerical evidence and conjectures for the general case.

Contribution

It proves that variance is its own concave roof and shows quantum Fisher information is four times the convex roof of variance for certain states, with conjectures for the general case.

Findings

Variance is its own concave roof.

Quantum Fisher information equals four times the convex roof of variance for rank-2 states with zero diagonal elements.

Numerical evidence suggests the quantum Fisher information is maximal among generalized versions.

Abstract

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 Fisher information considered in [D. Petz, J. Phys. A: Math. Gen. 35, 929 (2002); P. Gibilisco, F. Hiai, and D. Petz, IEEE Trans. Inf. Theory 55, 439 (2009)], after appropriate normalization, the quantum Fisher information is the largest. Hence, we conjecture that the quantum Fisher information is four times the convex roof of the variance even for the general case.