Quantum Fisher Information vs Classical Fisher Information 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?
 A: 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.
