# Why is the quantum Fisher information useful in quantum metrology?

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• jamie.j1989
In summary, the conversation discusses the use of quantum Fisher information (QFI) as a measure of phase sensitivity in interferometers. Two equations are presented, with equation (2) providing a general measure of sensitivity without dependence on a specific measurement, while equation (3) requires a defined measurement process to estimate sensitivity. The advantage of (3) is the ability to optimize the measurement process, while (2) provides an upper bound and can identify suboptimal measurements.
jamie.j1989
TL;DR Summary
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?

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.

## 1. What is quantum Fisher information?

Quantum Fisher information is a measure of the sensitivity of a quantum state to small changes in a parameter. It is a fundamental quantity in quantum metrology, which is the study of how to make precise measurements using quantum systems.

## 2. How is quantum Fisher information useful in quantum metrology?

Quantum Fisher information is useful in quantum metrology because it tells us the maximum possible precision that can be achieved in measuring a parameter using a quantum state. It also helps us design optimal measurement strategies to achieve this precision.

## 3. Can quantum Fisher information be measured?

Yes, quantum Fisher information can be measured using a variety of techniques, such as quantum state tomography or interferometric methods. These measurements can be used to validate theoretical predictions and assess the performance of quantum metrology protocols.

## 4. How does quantum Fisher information relate to other measures of quantum information?

Quantum Fisher information is closely related to other measures of quantum information, such as quantum entanglement and quantum coherence. It provides a complementary perspective on the usefulness of quantum systems for information processing tasks.

## 5. What are the practical applications of quantum Fisher information?

Quantum Fisher information has many practical applications, including in quantum sensing, quantum imaging, and quantum communication. It also has potential applications in quantum computing and quantum cryptography.

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