I am currently studying about Fisher Information and have a rather simple doubt which I can't figure out.

Is it possible to have Quantum Fisher Information (QFI) more than Classical Fisher Information (CFI) for a system? What would be the implications, if so?

I am thinking that this is possible when we have entanglement whereupon the QFI being more than CFI would imply that the estimation on the parameter is better as a result of the Classical and Quantum Cramer-Rao Bounds. If so, then what about the case of calculating QFI without entanglement and comparing it with CFI?


1 Answer 1


The quantum Fisher information is always greater than or equal to the classical Fisher information, by definition. The QFI with respect to a parameter $\theta$ is defined by $$\mathcal{H}_\theta[\hat{\rho}_\theta] = \max_{\{\hat{M}_\alpha\}} \mathcal{F}_\theta[\hat{\rho}_\theta,\{\hat{M}_\alpha\}],$$ where $\mathcal{F}_\theta[\hat{\rho},\{\hat{M}_\alpha\}] = \int{\rm d}\alpha\, p(\alpha|\theta)[\partial_\theta \ln p(\alpha|\theta)]^2$ is the classical Fisher information for a measurement of the state $\hat{\rho}_\theta$, which yields outcome $\alpha$ with probability $p(\alpha|\theta) = {\rm Tr}[\hat{M}_\alpha \hat{\rho}_\theta]$, with $\{\hat{M}_\alpha\}$ a positive operator-valued measure satisfying $\int{\rm d}\alpha\, \hat{M}_\alpha = 1$. The maximum defining the QFI is taken over all possible measurements. See Braunstein & Caves, PRL 72, 3439 (1994) for details.

When the CFI for a given measurement is less than the QFI, it just means that one is not using the optimal measurement to estimate the parameter. This may be related to entanglement, e.g. in phase estimation with NOON states, but is not necessarily. In general, even for a simple two-level system, there are infinitely many measurements one can make that do not saturate the quantum Cramer-Rao bound and thus have a smaller Fisher information than the QFI.

  • $\begingroup$ In the second paragraph do you mean `When the QFI is not larger than the CFI..."? $\endgroup$ May 14, 2020 at 13:53
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    $\begingroup$ @Nitin No that is not what I mean. The QFI is either larger than the CFI, or equal to it. Thus when the QFI is not larger than the CFI, the two are equal, meaning that one is considering the optimal measurement that saturates the quantum Cramer-Rao bound. I have edited for clarity. $\endgroup$ May 14, 2020 at 14:42
  • $\begingroup$ Thank you. That clarified my doubt perfectly. :) $\endgroup$ May 14, 2020 at 15:16
  • $\begingroup$ Is there any way to calculate the Quantum Fisher Information without a brute force search of all the states we could possible measure? $\endgroup$
    – John
    Jul 29, 2021 at 19:16

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