Why is the quantum Fisher information useful in quantum metrology?

[jamie.j1989]

TL;DR

What advantages does the phase sensitivity estimation obtained from the QFI give us compared to say, the phase sensitivity estimation obtained via the calculus of error propagation?

I'm getting interested in quantum-enhanced metrology and have come across the quantum Fisher information (QFI) as a measure of how much a quantum state $|\Psi(\theta)\rangle$ changes with respect to some variable, for example, the phase accumulated during an interferometer, $\theta$. This is interesting as it provides a means to estimate the phase sensitivity of the interferometer given by
$$\Delta\theta=1/\sqrt{F_Q},\qquad\qquad\qquad(1)$$
where $F_Q$ is the QFI and for pure states can be written as
$$F_Q=4\left(\langle\Psi'|\Psi'\rangle-\left|\langle\Psi'|\Psi\rangle\right|^2\right),\qquad\qquad\qquad(2)$$
where $|\Psi'\rangle=\tfrac{d}{d\theta}|\Psi\rangle$ and $|\Psi\rangle$ being the output state. Now, I'm also aware of other phase sensitivity estimations such as the formula derived via the calculus of error propagation
$$\Delta\theta=\frac{\langle\Delta O\rangle}{\left|\frac{d\langle O\rangle}{d\theta}\right|},\qquad\qquad\qquad(3)$$
where $\langle\Delta O\rangle$ is the standard deviation, and $O$ some Hermitian operator normally describing some measurement such as, the population difference between the two output arms of the interferometer.

My question is, why or when would I prefer one method over the other? My current understanding draws me to the form of both equations, (2) has no dependence on the measurement process whilst (3) does. This implies that a measurement procedure explained with $O$ might not be appropriate to obtain the degree of sensitivity given by (1), so comparing the two can tell you whether your measurement procedure is optimal?

 

[azntoon]

My current understanding of the two equations is that (2) gives a measure of the sensitivity of a quantum system to a parameter change without any reference to any particular measurement, whereas (3) requires a specific measurement to be defined in order to estimate the sensitivity. The advantage of (3) is that it can be used to optimize the measurement process by tuning the parameters of the measurement to maximize the sensitivity. The advantage of (2) is that it provides an upper bound on the sensitivity achievable with any measurement, and can be used to identify cases where the sensitivity of a given measurement is suboptimal.