In Quantum Metrology, the aim is to estimate some unknown parameters(I will talk about one parameter estimation in this post, while multiparameter is also available) as precise as possible. Without quantum resources, we can only realize the Standard Quantum Limit(SQL), normally referred to as the form $$var(\hat{\theta})=1/n$$, where $$\hat{\theta}$$ is the estimator(function of random variables to estimate the unknown parameter $$\theta$$) and $$n$$ is the number of resources, e.g. number of particles. While with quantum resources, we may reach the so-called Heisenberg's Limit(HL), normally referred to as the form $$var(\hat{\theta})=1/n^2$$, a square enhancement in the precision. Here is a frequently used state example, stating that the GHZ state $$1/\sqrt{2}(|0\rangle^{\otimes n}+|1\rangle^{\otimes n})$$ after the evolution described by the unitary operator $$U = e^{-i\theta\sigma_z/2}\otimes e^{-i\theta\sigma_z/2}\otimes...$$ will become $$1/\sqrt{2}(|0\rangle^{\otimes n}+e^{in\theta}|1\rangle^{\otimes n})\tag{1}$$ ignoring the global phase. And we can estimate the value of the parameter by measuring this parameterized quantum state.

My question is, the HL will show its advantage only when the scale is $$1/n^2$$, while from eq.(1), we can easily see that there's a $$2\pi$$ period in an exponential function, no matter how small $$\theta$$ is, we cannot always enhance our precision by HL scale when $$n$$ pass some specific value. So, how to understand this paradox, did I miss something?

• Actually, this is the drawback of quantum metrology under the framework of locally unbiased estimators. Another framework is global estimation, e.g. Bayesian framework and minimax framework. A related paper discussing such a problem is this one. Commented May 12 at 8:37

You are correct, but the SQL is a local limit, when you already have a very good idea what the value of $$\theta$$ is, so there is no contradiction.
Let's work through it. You measure some relative phase $$\Theta$$, and you infer that $$\Theta=n\theta-2\pi k,\quad k\in \mathbb{N}.$$ You work out that $$\theta=\frac{\Theta +2\pi k}{n},$$ where $$\Theta$$ and $$n$$ are known ($$n$$ is known because you set up your experiment) and $$k$$ is unknown. You now must determine what values of $$k$$ are possible, then you are finished.
Because the SQL is local, we know a priori that $$\theta_{\text{low}}\leq \theta\leq \theta_{\text{high}}$$ for some values of $$\theta_{\text{low}}$$ and $$\theta_{\text{high}}$$, so we deduce that $$\frac{n\theta_{\text{low}}-\Theta}{2\pi}\leq k\leq \frac{n\theta_{\text{high}}-\Theta}{2\pi}.$$ So as long as our initial knowledge has sufficiently small $$\theta_{\text{high}}-\theta_{\text{low}}$$, there will only be one possible value of $$k$$, and we will fully determine the value of $$\theta$$. If our initial knowledge is not precise enough, you are correct that using too large a value of $$n$$ will yield multiple possible results.
• @narip I mean exactly what I said, that the possible values of $\theta$ are restricted to a small region around the "true" value of $\theta$. See e.g. doi.org/10.1007/s00220-019-03433-4. This is why the SQL is an asymptotic limit, as you can only achieve that precision in the limit of doing lots of measurements. Commented Nov 15, 2021 at 15:05
• @narip any text on Fisher information (regardless of quantum) will explain that it can be attained in the asymptotic limit where the sample size is very large (and thus you have very good prior information about the parameter). As for only learning about a parameter up to $2\pi k/n$, this is true for all phase estimation. For example en.wikipedia.org/wiki/Quantum_phase_estimation_algorithm assumes the phase to be close to 0, setting $\theta_{\mathrm{low}}=0$ and $\theta_{\mathrm{high}}=1/2^{n+1}$ Commented Apr 10, 2023 at 13:08