Quantum Fisher information

The quantum Fisher information equals the expectation value of ${\displaystyle L_{\varrho }^{2}}$, where ${\displaystyle L_{\varrho }}$ is the Symmetric Logarithmic Derivative.

The quantum Fisher information equals four times the variance for pure states

${\displaystyle F_{\rm {Q}}[\vert \Psi \rangle ,H]=4(\Delta H)_{\Psi }^{2}}$.

For mixed states, when the probabilities are parameter independent, i.e., when ${\displaystyle p(\theta )=p}$, the quantum Fisher information is convex:

${\displaystyle F_{\rm {Q}}[p\varrho _{1}(\theta )+(1-p)\varrho _{2}(\theta ),H]\leq pF_{\rm {Q}}[\varrho _{1}(\theta ),H]+(1-p)F_{\rm {Q}}[\varrho _{2}(\theta ),H].}$

The quantum Fisher information is the largest function that is convex and that equals four times the variance for pure states. That is, it equals four times the convex roof of the variance [6][7]

${\displaystyle F_{\rm {Q}}[\varrho ,H]=4\inf _{\{p_{k},\vert \Psi _{k}\rangle \}}\sum _{k}p_{k}(\Delta H)_{\Psi _{k}}^{2},}$

where the infimum is over all decompositions of the density matrix

${\displaystyle \varrho =\sum _{k}p_{k}\vert \Psi _{k}\rangle \langle \Psi _{k}\vert .}$

Note that ${\displaystyle \vert \Psi _{k}\rangle }$ are not necessarily orthogonal to each other.

When the probabilities are ${\displaystyle \theta }$-dependent, an extended-convexity relation has been proved:[8]

${\displaystyle F_{\rm {Q}}[\sum _{i}p_{i}(\theta )\varrho _{i}(\theta )]\leq \sum _{i}p_{i}(\theta )F_{\rm {Q}}[\varrho _{i}(\theta )]+F_{\rm {C}}[\{p_{i}(\theta )\}],}$

where ${\displaystyle F_{\rm {C}}[\{p_{i}(\theta )\}]=\sum _{i}{\frac {\partial _{\theta }p_{i}(\theta )^{2}}{p_{i}(\theta )}}}$ is the classical Fisher information associated to the probabilities contributing to the convex decomposition. The first term, in the right hand side of the above inequality, can be considered as the average quantum Fisher information of the density matrices in the convex decomposition.

We need to understand the behavior of quantum Fisher information in composite system in order to study quantum metrology of many-particle systems.[9] For product states,

${\displaystyle F_{\rm {Q}}[\varrho _{1}\otimes \varrho _{2},H_{1}\otimes {\rm {Identity}}+{\rm {Identity}}\otimes H_{2}]=F_{\rm {Q}}[\varrho _{1},H_{1}]+F_{\rm {Q}}[\varrho _{2},H_{2}]}$

holds.

For the reduced state, we have

${\displaystyle F_{\rm {Q}}[\varrho _{12},H_{1}\otimes {\rm {Identity}}_{2}]\geq F_{\rm {Q}}[\varrho _{1},H_{1}],}$

where ${\displaystyle \varrho _{1}={\rm {Tr}}_{2}(\varrho _{12})}$.

There are strong links between quantum metrology and quantum information science. For a multiparticle system of ${\displaystyle N}$ spin-1/2 particles [10]

${\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq N}$

holds for separable states, where

${\displaystyle J_{z}=\sum _{n=1}^{N}j_{z}^{(n)},}$

and ${\displaystyle j_{z}^{(n)}}$ is a single particle angular momentum component. The maximum for general quantum states is given by

${\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq N^{2}.}$ Hence, quantum entanglement is needed to reach the maximum precision in quantum metrology.

Moreover, for quantum states with an entanglement depth ${\displaystyle k}$,

${\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq kN+N_{R}^{2}}$

holds, where ${\displaystyle N_{R}}$ is the remainder from dividing ${\displaystyle N}$ by ${\displaystyle k}$. Hence, a higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation.[11][12]

The Wigner–Yanase skew information is defined as [13]

${\displaystyle I(\varrho ,H)={\rm {Tr}}(H^{2}\varrho )-{\rm {Tr}}(H{\sqrt {\varrho }}H{\sqrt {\varrho }}).}$

It follows that ${\displaystyle I(\varrho ,H)}$ is convex in ${\displaystyle \varrho .}$

For the quantum Fisher information and the Wigner–Yanase skew information, the inequality

${\displaystyle F_{\rm {Q}}[\varrho ,H]\geq 4I(\varrho ,H)}$

holds, where there is an equality for pure states.