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?


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.

  • $\begingroup$ Thank you :). That is enough. $\endgroup$ Jul 12 '20 at 21:09

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