Quantum vs Classical estimation using Fisher Information

Question

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?

Answer 1

The key difference between the QFI and CFI is that CFI depends on the measurement you use.

Is it possible to have a different estimated value of a parameter

Neither of these are estimators, they give you bounds on your estimate so I'm not 100% what you mean here. You would use say, maximum likelihood estimation if you want to actually estimate some parameter. Both QFI/CFI have nothing to do with the actual estimate you get from an estimation procedure.

If it is only one parameter you are interested in then there exists a measurement such that the CFI and QFI are equal. This is the optimal measurement to use. If it is a multi-parameter problem then this is not necessarily the case.

$10^{30}$ 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.