Standard quantum limit
Independent probes usually average down uncorrelated noise with shot-number scaling.
$$\Delta\theta\propto N^{-1/2}$$
Physical intuition
Independent probes usually average down uncorrelated noise with shot-number scaling.
Deep dive: physical mechanism
For N independent identical probes, information adds linearly, producing a precision scaling of inverse square root N under fixed per-probe resources.
What is measured experimentally?
Measure the slope of log precision against log probe count.
Assumptions, noise and limitations
Resource accounting matters: time, photon number, bandwidth and detector efficiency must be specified.
Research and simulation exercise
Generate binomial data for increasing N and fit the scaling exponent.
Connect the ideas
Explore the chapter overview, learning path, and references.