Identifiability as Uniqueness Before Noise Enters

The highlighted sentence—“Identifiability concerns uniqueness from noiseless observations”—is a compact definition of a structural question. It says that before asking how noisy, efficient, biased, or experimentally calibrated a receiver is, one should first ask a simpler mathematical question: if the model were observed exactly, would the unknown quantity be determined uniquely?

In the parent document, this distinction matters because the Rydberg receiver is not merely estimating a signal in the presence of ordinary measurement noise. It is also trying to separate the incident microwave signal from nuisance quantities: local-oscillator amplitudes at the atoms, detuning, phase increments, and possible phase offsets. Some of these nuisance parameters are not known externally. The paper therefore studies when the observations themselves contain enough structure to determine the signal.

The word “noiseless” is important. It does not mean that real Rydberg spectra are noiseless. Rather, it defines an ideal mathematical test: suppose the measured spectral separations were exactly equal to their model-predicted values. Would two different parameter settings produce the same data? If yes, the parameter is not identifiable from that protocol, no matter how precise the measurement apparatus becomes. If no, the parameter is identifiable, although it may still be hard to estimate accurately under noise.

The Model Map Behind the Definition

A convenient way to state identifiability is through a model map. Let $\eta$ denote all unknown parameters in the model, including both the desired signal and nuisance parameters. In the parent document’s ideal squared-separation model, the signal is written

$$ s = a e^{i\theta}, \qquad a>0,\qquad P=a^2, $$

where $a$ is the signal amplitude, $\theta$ is the signal phase, and $P$ is signal power. At LO setting $j$ and phase sample $k$, the total microwave coupling has the form

$$ \Omega_{jk}=L_j e^{i\phi_{jk}} + s, $$

where $L_j>0$ is the LO amplitude at the atoms and $\phi_{jk}$ is the LO phase used for that spectral acquisition. The ideal squared-separation observation can be expressed, up to the model conventions of the paper, as a sinusoidal quantity of the form

$$ m_{jk} = D + L_j^2 + P + 2L_j\sqrt{P}\cos(\phi_{jk}-\theta), $$

where $D=\delta^2$ is the squared detuning. For a collection of measurements, these noiseless values form a vector

$$ m(\eta)=\bigl(m_{11},m_{12},\ldots\bigr). $$

Identifiability asks whether the equation

$$ m(\eta)=m(\eta') $$

can hold for two distinct parameter points $\eta\neq \eta'$. If equality of the noiseless observation vectors forces the desired signal parameters to agree, then those signal parameters are identifiable. If equality forces all parameters to agree, then the full parameter vector is identifiable. These are not always the same question: a signal phase might be identifiable even when a nuisance background is not, or the signal power might be identifiable only after combining two LO settings.

This is the sense in which identifiability is about uniqueness. It is not about variance, finite-sample performance, or experimental traceability. It is about whether the inverse problem has a unique answer in the ideal model.

Why Noiseless Uniqueness Is a Separate Question

The parent document explicitly separates identifiability from statistical efficiency and physical calibration. That separation is mathematically necessary.

A protocol can fail identifiability in a very strong way. Suppose two different signals and nuisance-parameter combinations generate exactly the same ideal observation vector. Then even an infinite number of perfectly repeatable measurements of the same design would not distinguish them. No estimator can recover the true parameter uniquely because the data do not contain the required information. This kind of nonuniqueness is sometimes described as structural non-identifiability in the broader identifiability literature [Bellman and Åström 1970; Rothenberg 1971].

By contrast, if the inverse is unique but nearly singular, the parameter is identifiable but poorly conditioned. Small perturbations in the data may cause large perturbations in the recovered parameter. In differential terms, if the Jacobian of the map $m(\eta)$ with respect to $\eta$ has directions that are very small, then nearby parameter values produce nearly indistinguishable observations. The noiseless inverse may exist, but noise can make estimation unstable. This is the bridge from identifiability to Fisher information: when the observation noise model is specified, the sensitivity of the mean vector contributes to the Fisher information matrix, and singular or nearly singular information matrices signal directions that are unidentifiable or weakly estimable under that model [Kay 1993; Stoica and Marzetta 2001].

The parent paper’s statement that “identifiability concerns uniqueness from noiseless observations” is therefore not a casual definition. It fixes the logical order of the analysis. First ask whether the mapping from parameters to ideal observations is one-to-one, at least for the parameter of interest. Only afterward ask how much uncertainty remains when observations are noisy.

A Concrete Reading in the Rydberg Receiver Problem

At a fixed LO setting, the ideal observation as a function of LO phase is a sampled sinusoid,

$$ m_{jk}=A_j+B_j\cos(\phi_{jk}-\theta), $$

with

$$ A_j=D+L_j^2+P,\qquad B_j=2L_j\sqrt{P}. $$

If $A_j$, $B_j$, and $\theta$ were the only unknowns, sufficiently rich phase samples could identify the sinusoid’s phase modulo $2\pi$, subject to standard degeneracies such as zero modulation depth. This is why the parent document relates part of the problem to phase-shifting interferometry [Bruning et al. 1974; Carré 1966; Surrel 1993].

But the Rydberg receiver problem contains additional structure. The sinusoid’s background $A_j$ and modulation depth $B_j$ are not arbitrary independent parameters; they are tied to physical quantities $D$, $L_j$, and $P$. A single LO setting generally does not separate these quantities uniquely. For example, the background contains the sum $D+L_j^2+P$, so changing one term may be compensated by changing another unless additional structure is available. The parent document’s two-setting inversion uses different LO amplitudes to break this ambiguity: the way $A_j$ and $B_j$ change across settings carries information about $P$ and the LO amplitudes.

This illustrates the highlighted sentence well. The identifiability question is not, “Will the estimator have a small error bar?” It is, “Do the ideal spectra determine the signal and nuisance parameters uniquely, under the assumed acquisition design?” If two different values of $P$ can be paired with different $L_j$ and $D$ to produce the same ideal spectra, then $P$ is not identifiable under that design. If the two-setting structure eliminates that ambiguity, then $P$ may be identifiable, even though the actual estimator may still have high variance when the LO settings are poorly chosen.

Global, Local, and Generic Identifiability

The highlighted sentence gives the central idea, but practical theoretical papers usually refine it. “Uniqueness” can mean several related things.

Global identifiability means that the noiseless observations determine the parameter uniquely over the full allowed parameter space, except for intended conventions such as phase modulo $2\pi$. In symbols, if

$$ m(\eta)=m(\eta') $$

then $\eta=\eta'$ or, for target identifiability, the target components agree.

Local identifiability is weaker. It means that no distinct parameter point in a sufficiently small neighborhood gives the same observations. A model can be locally identifiable but not globally identifiable if there are distant mirror solutions or periodic aliases. Phase problems often have this flavor because $\theta$ and $\theta+2\pi$ represent the same physical phase, and because undersampled phase steps can introduce additional aliases.

Generic identifiability is weaker still. It means uniqueness holds for almost all parameter values, but may fail on an exceptional set. The parent document’s language about “generic identification” versus “identification at every signal phase” belongs to this distinction. A protocol might work except when the signal phase, offset, or step size lies in a special configuration that collapses the sinusoidal information. Such a protocol is generically identifiable but not uniformly identifiable over all allowed parameter values.

These distinctions are especially important in the unknown-phase-increment designs discussed by the parent document. If the phase increment itself is unknown, the observations must identify not only a sinusoid but also the sampling geometry. A design may have enough samples for most increments but fail at special increments where the samples become symmetrically ambiguous or linearly dependent in the recurrence used to infer the phase law. The highlighted sentence tells us what kind of failure this is: a noiseless nonuniqueness, not merely a noisy-estimation problem.

What Identifiability Does Not Guarantee

The sentence also warns against overinterpreting identifiability. If a parameter is identifiable, then the ideal model has a unique inverse. But this does not guarantee that an estimator is efficient, unbiased under real experimental conditions, or robust.

Statistical efficiency requires a probability model for the observations, such as homoscedastic Gaussian noise in the squared separations. Under such a model, one may compute Fisher information and Cramér–Rao bounds, as the parent document does. These quantities describe lower bounds on estimator variance under regularity conditions [Kay 1993; Van Trees et al. 2013]. They are not part of the bare identifiability definition, although singular Fisher information often reveals local non-identifiability.

Physical calibration is another layer. The ideal model might be identifiable and statistically well-conditioned while still being physically wrong. For example, if the actual field at the atoms differs from the assumed LO model, or if detuning contributes in an LO-power-dependent way not included in the fitted model, then the noiseless inverse of the wrong model may return a unique but biased signal. The parent document explicitly emphasizes this possibility when it discusses observational ambiguities and calibration errors that can produce internally consistent but physically incorrect amplitudes.

Thus, the highlighted passage should be read as a boundary marker. Identifiability is the first structural test: uniqueness in the ideal observation model. It is necessary for reliable self-calibration, but it is not sufficient for accurate experimental reception. A complete receiver analysis must also examine conditioning, noise statistics, estimator variance, systematic model error, and absolute calibration.

References

Bellman, R., and Åström, K. J. (1970). “On structural identifiability.” Mathematical Biosciences, 7(3–4), 329–339.

Bruning, J. H., Herriott, D. R., Gallagher, J. E., Rosenfeld, D. P., White, A. D., and Brangaccio, D. J. (1974). “Digital wavefront measuring interferometer for testing optical surfaces and lenses.” Applied Optics, 13(11), 2693–2703.

Carré, P. (1966). “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures.” Metrologia, 2(1), 13–23.

Kay, S. M. (1993). Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory. Prentice Hall.

Rothenberg, T. J. (1971). “Identification in parametric models.” Econometrica, 39(3), 577–591.

Stoica, P., and Marzetta, T. L. (2001). “Parameter estimation problems with singular information matrices.” IEEE Transactions on Signal Processing, 49(1), 87–90.

Surrel, Y. (1993). “Phase stepping: A new self-calibrating algorithm.” Applied Optics, 32(19), 3598–3600.

Van Trees, H. L., Bell, K. L., and Tian, Z. (2013). Detection, Estimation, and Modulation Theory, Part I (2nd ed.). Wiley.