State what scales in a quantum limit

Take $N$ spin-one-half probes with dimensionless generator $J_z=\frac12\sum_j\sigma_z^{(j)}$, and encode one common phase through $\exp(-i\phi J_z)$. For independent equatorial inputs, each single-probe generator variance is $1/4$. Independent variances add, giving collective variance $N/4$ and quantum Fisher information $N$. Convexity extends the information ceiling to separable mixtures for this local linear encoding. With $m$ independent complete preparations, the regular locally unbiased phase standard-deviation bound is $1/\sqrt{mN}$.

This is often called shot-noise scaling even when the probes are atoms rather than detected photons. The name refers to independent contributions, not to a universal source of all laboratory noise. Technical noise can make precision worse, and correlations can change the scaling. A comparison must keep the generator's single-probe spectral range fixed. Increasing the coupling strength with probe number changes the physical encoding and cannot be credited solely to preparing a better state under the original resource model.

The collective generator has extreme eigenvalues $-N/2$ and $N/2$. An equal superposition of their eigenstates, the GHZ state, has variance $N^2/4$, so its phase quantum information is $N^2$. A suitable collective coherence readout can access a fringe whose argument is $N\phi$. Repeating the entire preparation $m$ times gives the local standard-deviation benchmark $1/(N\sqrt m)$. The gain arises from a larger spread of generator eigenvalues, not from eliminating quantum fluctuations.

The faster fringe also repeats more often. A single enhanced fringe cannot choose the correct phase interval across a full turn, and a local precision number does not account for that ambiguity. Practical estimation can combine different group sizes or start with a coarse measurement. Those preparations consume resources. Moreover, creating and reading the GHZ coherence must be feasible at the required size. Treat the scaling derivation as an idealized information ceiling under explicit operations, then compare a complete protocol that actually obtains unambiguous estimates.

If the total number of particle interrogations is $Q=mN$, the ideal GHZ total phase information is $QN$ rather than merely $Q$. Increasing group size at fixed Q therefore improves the local bound within the ideal model, while reducing the number of independent preparations. That change can undermine asymptotic inference or branch identification. At fixed N, repeating the experiment improves standard deviation only as the inverse square root of m for both strategies. Never substitute a total particle count for N in one formula while leaving m implicit in another.

A single probe making several coherent passes can accumulate the same phase amplification as several entangled probes. Counting only detected particles would hide those extra interactions and longer exposure. Frequency estimation additionally depends on interrogation duration, and the number of repetitions available in a fixed elapsed time falls as cycles get longer. Loss and dephasing may grow with either group size or duration. Nonlinear generators can produce other exponents, but those exponents refer to different interactions and need their own energy, time, and coupling budgets.