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The famous Cramer-Rao bound is $$\Delta\theta\ge\frac{1}{\sqrt {F[\rho,H]}}$$ But what happens if the denominator vanishes, i.e., $F[\rho,H]=0$ ($F[\rho,H]$ here stands for the quantum fisher information in the unitary process)? For example, $F[\rho,H]$ can be zero in this paper below eq$(26)$.

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  • $\begingroup$ Well, it could only mean that variance of the unknown parameter $\theta$ is infinite. $\endgroup$
    – kludg
    Apr 19, 2021 at 5:46
  • $\begingroup$ But what does infinite variance mean, I mean, the physical meaning instead of the mathematical meaning? $\endgroup$
    – narip
    Apr 19, 2021 at 6:43
  • $\begingroup$ I think you need to explain what is $\theta$, physically. The question, as it is currently formulated, is a formal mathematical question IMO. $\endgroup$
    – kludg
    Apr 19, 2021 at 6:51
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    $\begingroup$ infinite variance means that that you cannot estimate the parameter $\theta$ from measuring the state $\rho=\rho(\theta)$ with the measurement corresponding to $H$. Vanishing Fisher information in this context means that the state effectively does not depend on the parameter $\theta$ (hence why $\theta$ cannot be estimated from the state) $\endgroup$
    – glS
    Apr 19, 2021 at 9:39

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