Read the conditions behind a quantum precision bound

For $m$ independent identical preparations, classical information adds. At a regular parameter value, an estimator that is locally unbiased has variance at least the reciprocal of that total classical information, which is itself bounded below by the reciprocal quantum information. Local unbiasedness requires both the correct expectation at the reference point and unit derivative of that expectation there. Getting the mean right at just one isolated point is insufficient: a constant estimator equal to that point has zero variance but no local response.

The usual proof differentiates expectations and uses Cauchy–Schwarz on the estimator and score. It therefore needs smoothness, suitable interchange of differentiation and integration or summation, and regular support behavior. A parameter-dependent boundary can invalidate the elementary argument. When information vanishes or an inverse is singular, first diagnose identifiability rather than quoting an infinite or zero error mechanically. For multiple unknown parameters, inverse information matrices and measurement compatibility introduce further conditions beyond this single-parameter treatment.

An ideal midpoint sine fringe illustrates exact local saturation. Let $K$ count successes with $p=(1+\sin\phi)/2$ and define $\widehat\phi=2K/m-1$. Its expectation is $\sin\phi$, so it is locally unbiased at zero but not globally unbiased as an angle estimator. Its variance at zero is $1/m$, equal to the phase information bound. The construction uses a linear local calibration and retains a finite-sample equality at that reference point; it does not establish an unbiased estimator over a full period.

A cosine or sine fringe repeats when phase advances by a full turn. High Fisher information can sharpen every repeated likelihood peak without selecting the correct peak. A known narrow parameter interval, extra interrogation times, or additional calibration information is needed to resolve aliases. The same issue appears in a more severe form for entangled fringes that oscillate several times faster. Local distinguishability and global identifiability answer different questions, and an experimental uncertainty statement must say which one is controlled.

An SLD measurement can extract quantum Fisher information at a chosen parameter value, yet an estimator based on its finite record can still fail to reach the variance bound. Conversely, a statistically efficient estimator cannot recover information that the detector never collected. Adaptive protocols often use some preparations to obtain a rough estimate and then tune the readout near its local optimum. The tuning data and feedback time are resources, and the rough estimate must select the correct neighborhood with sufficiently high reliability.

With a correctly specified identifiable regular model, likelihood-based procedures often approach the bound in a large-data regime. The rate of that approach and the effect of outliers require checking for the actual protocol. Bias can intentionally reduce mean-squared error relative to an unbiased-variance benchmark, especially with prior information or restricted parameter ranges. Such a result does not violate quantum mechanics. Compare the same loss function and the same assumptions before describing a claimed bound violation; uncertainty from model mismatch remains a separate concern.