Build a parameter-estimation experiment

Suppose the measurand is a constant angular-frequency offset $\omega$, measured in radians per second relative to a specified reference oscillator. During a known interrogation time $t$, a two-level probe acquires relative phase $\phi=\omega t$. For the Hamiltonian $H=\hbar\omega\sigma_z/2$, evolution is a rotation about the Bloch z axis. An initially equatorial state changes its relative phase, whereas a z eigenstate changes only an unobservable overall phase. A large interaction energy alone therefore does not guarantee a useful sensor.

The physical parameter must be defined with its reference conditions. Frequency offset differs from absolute transition frequency; an instantaneous magnetic field differs from its average during a sensing interval. Calibration supplies the relation between the parameter and phase, including sign conventions and coupling coefficients. Uncertain timing or coupling is then part of the inference problem. Treating either as exact without justification can produce a precise estimate of the wrong quantity, even when the quantum probabilities are calculated correctly.

A measurement is specified by positive operators whose sum is the identity. Its outcome probabilities follow from the Born rule. For an equatorial probe, measuring z populations immediately after free precession gives equal probabilities independently of the accumulated phase. A second coherent pulse can rotate an equatorial component onto the population axis. The same encoded state can consequently be informative or useless depending on the selected readout. Recording a probability requires repeated preparations; it is not a deterministic property of one observed binary outcome.

Choose a readout convention giving $p_1=(1+\cos\phi)/2$. For $m$ independent identical preparations, the success count follows a binomial law. A likelihood compares candidate parameter values against this law rather than equating each outcome with an angle. Inverting the observed fraction is a simple estimator only after choosing a phase branch. Since cosine is periodic and even, the data alone cannot distinguish opposite phases or phases differing by a full turn. Prior range information or additional readout phases must supply that missing distinction.

A protocol description should give the probes used simultaneously, the number of independent repetitions, each interrogation duration, and the preparation and detection time. Parallel probes and repeated uses of one probe can produce the same number of binary records while consuming different laboratory resources. Entangled probes also require specifying which collective measurement is available. Comparisons are meaningful only after fixing the resource that the experiment actually constrains, such as total elapsed time, particle exposure, or energy delivered to a specimen.

Statistical spread describes variation over hypothetical repetitions of the complete experiment under the stated probability model. It does not include an unknown reference offset merely because many outcomes were collected. Slow drift can also invalidate the independence assumption behind a binomial likelihood. A useful design therefore includes reference checks and an acquisition schedule, then states whether the reported uncertainty is local, conditional on calibration, or intended to cover additional effects. Quantum preparation is one part of that measurement model, not a substitute for it.