# Quantum vs Classical estimation using Fisher Information

Is it possible to have a different estimated value of a parameter by Classical Fisher Information (CFI) compared to Quantum Fisher Information (QFI)?

Say, for example, I am plotting, $$\mathcal{F_{\nu}}$$, Fisher Information for parameter $$\nu$$ with respect to $$\nu$$. When estimating $$\nu$$ using CFI I get a peak in plot at, say, $$\nu = 1.03$$ and when estimating using QFI the peak is at $$\nu = 1.06$$. Is it ever possible?

Using Cramer-Rao Bound, I think that only the estimation precision should be different. For example, estimation using QFI could be 6 orders of magnitude more precise compared to CFI if QFI $$\sim 10^{30}$$ whereas CFI $$\sim 10^{24}$$ because the variance is now 6 orders of magnitude less for QFI compared to CFI using the Cramer-Rao Bound. Whatever this be, I think the parameter's estimated value should be same.

Can anyone please give arguments in support or against the above?

$$10^{30}$$ is a pretty big QFI but not completely outside the realms of possibility! If you want to supply some more details of the system you're interested in I could give a more detailed response.