Classical Fisher information
The slope of a probability distribution relative to its noise determines information about a parameter.
$$F_C(\theta)=\sum_x p(x\mid\theta)\left[\partial_\theta\ln p(x\mid\theta)\right]^2$$
Physical intuition
The slope of a probability distribution relative to its noise determines information about a parameter.
Deep dive: physical mechanism
Compute the score function for each outcome and average its square. A flat likelihood carries little information even if the underlying probe is highly coherent.
What is measured experimentally?
The expected estimator variance is bounded by the Cramér–Rao relation under regularity conditions.
Assumptions, noise and limitations
The bound may not be attained for small samples or biased estimators.
Research and simulation exercise
Compare Fisher information for Ramsey measurements at different operating points.
Connect the ideas
Explore the chapter overview, learning path, and references.