Extremal properties of the variance and the quantum Fisher information

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 Petz [J. Phys. A 35, 929 (2002)] and Gibilisco, Hiai, and 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.

Figure 1

Basic parameter estimation task in quantum metrology. The small parameter $\theta$ must be estimated by making measurements on the output state ${\varrho }_{\mathrm{output}}.$

Figure 2

A single-qubit mixed state $\varrho$ is decomposed as Eq. (23) into the mixture of pure states $|{\Psi }_{{\varphi }_1}\rangle$ and $|{\Psi }_{{\varphi }_1+\pi }\rangle$ defined in the text. Quantum states are represented by points in the $(\langle {\sigma }_x\rangle ,\langle {\sigma }_y\rangle ,\langle {\sigma }_z\rangle )$ space. The sphere with radius 1 is the Bloch sphere. Density matrices ${\varrho }^{\prime }$ for which the expectation value of $A$ is the same as for $\varrho$ fulfill the linear condition $\mathrm{Tr}\left[A(\varrho -{\varrho }^{\prime })\right]=0$ and are on a plane. The elliptic curve is the cross section between this plane and the surface of the Bloch sphere. Points corresponding to $\varrho ,$ $|{\Psi }_{{\varphi }_1}\rangle$ and $|{\Psi }_{{\varphi }_1+\pi }\rangle$ are all on this plane, thus the operator $A$ has the same expectation value in these three states.

Figure 3

The rank-3 mixed state ${\varrho }_0$ is decomposed as Eq. (26) into the mixture of two rank-2 states, ${\varrho }_{-}$ and ${\varrho }_{+}.$ The coordinate axes are the eigenvalues of the density matrix, while we assume that all density matrices have the same eigenstates as $\varrho .$ Points corresponding to states ${\varrho }^{\prime }$ for which $\mathrm{Tr}\left({\varrho }^{\prime }A\right)=\mathrm{Tr}\left({\varrho }_{0}A\right)$ are on the dashed line. For ${\varrho }_{+}$ we have ${\lambda }_2=0,$ while for ${\varrho }_{-}$ we have ${\lambda }_1=0.$