We develop a minimal framework for dynamic noise spectroscopy in time-dependent environments, where observables probe window-averaged rather than instantaneous spectra. Using a windowed filter-function formulation and Fisher-based identifiability, we show that observability is confined to a finite band in the joint space of time and measurement window. This band arises from a scale competition between Fisher amplification (∝Twin\propto \sqrt{T_{\rm win}}∝Twin) and envelope-induced suppression of the forward-model sensitivity (C(Twin)C(T_{\rm win})C(Twin)). The resulting “observable horizon” is not a geometric rank effect but a consequence of this competition. A minimal two-parameter model demonstrates the effect, and numerical results confirm the analytic scaling of the optimal window Twin∗=Tc/(2γ−1)T_{\rm win}^* = T_c/(2\gamma - 1)Twin∗=Tc/(2γ−1). The framework provides a quantitative criterion for when dynamic spectral information is accessible.
We develop a minimal framework for dynamic noise spectroscopy in time-dependent environments, where observables probe window-averaged rather than instantaneous spectra. Using a windowed filter-function formulation and Fisher-based identifiability, we show that observability is confined to a finite band in the joint space of time and measurement window. This band arises from a scale competition between Fisher amplification (∝Twin\propto \sqrt{T_{\rm win}}∝Twin) and envelope-induced suppression of the forward-model sensitivity (C(Twin)C(T_{\rm win})C(Twin)). The resulting “observable horizon” is not a geometric rank effect but a consequence of this competition. A minimal two-parameter model demonstrates the effect, and numerical results confirm the analytic scaling of the optimal window Twin∗=Tc/(2γ−1)T_{\rm win}^* = T_c/(2\gamma - 1)Twin∗=Tc/(2γ−1). The framework provides a quantitative criterion for when dynamic spectral information is accessible.
Dynamical-decoupling spectroscopy using solid-state spin qubits provides a powerful probe of environmental noise across wide frequency ranges. Here we identify a previously unrecognized observable fixed-point behavior in decoherence dynamics arising from additive decoherence channels sampled under narrow-band spectral filtering. We show that the ratio of decoherence times between two experimentally distinguishable configurations converges to a parameter-independent plateau value R ≈ 4/3 over an intermediate range of pulse numbers. This fixed point is independent of microscopic noise models and admits a natural geometric interpretation in terms of decoherence channel structure in the influence-functional description of open quantum systems. A central consequence is a strong experimental prediction: the existence of the plateau requires a band-limited spectral distortion, such that reconstructed noise spectra satisfy S_XY(ω)/S_CPMG(ω) ≈ 4/3 within a finite frequency window while approaching unity outside it. Numerical simulations demonstrate robustness of the effect across realistic noise environments without parameter fine tuning. We further propose an experimentally accessible protocol based solely on standard dynamical-decoupling measurements on a single qubit, enabling direct falsification of the predicted behavior. More broadly, these results indicate that measurement filtering can generate emergent invariant structures in open quantum-system dynamics, suggesting a new organizing principle for interpreting decoherence spectroscopy across quantum sensing platforms. Version 2 (v2) update: This version clarifies the logical structure of the paper by separating the existence of the measurement-induced plateau from the specific benchmark value R=4/3R=4/3R=4/3. The plateau mechanism is presented as a consequence of narrow-band spectral cancellation, while the value R=4/3R=4/3R=4/3 is treated more conservatively as a minimal effective-weight assignment consistent with pulse-sequence symmetry and the numerical results. The revised manuscript also makes the falsifiability structure more explicit and narrows the scope of the numerical claims: the simulations support the existence and robustness of the plateau, but are not presented as a unique microscopic derivation of the effective channel weights. Note (updated): This record corresponds to an earlier exploratory version of the idea. Parts of the theoretical interpretation presented here were not fully established and have been refined in subsequent work. In particular, the universal interpretation of ratio plateaus should be treated with caution. Readers are encouraged to consult the later work (see related records) for a more robust and conservative formulation of the effective description.
Genuine Collapse, Apparent Flattening, and Observable Horizons in Inverse Spectral Reconstruction This paper studies two distinct mechanisms by which inverse spectral reconstruction can fail. In one case, the forward observation map loses rank and information is destroyed at the geometric level; this is termed genuine collapse. In the other, the map remains full-rank but its weakest reconstruction direction is compressed below practical observational reach; this is termed apparent flattening. The paper introduces the corresponding notion of an observable horizon, defined by the competition between geometric compression and finite measurement noise. The analysis is carried out for the spectral family S(ω) = Aω^α + B, using scale-independent ratio observables motivated by dynamical-decoupling spectroscopy. Writing ρ = B/A, the paper derives the Jacobian of the ratio map with respect to (α, ρ), relates its weakest singular direction to the Fisher information matrix, and shows that practical identifiability is governed by the smallest singular value of the reduced observation map. For the minimal two-frequency ratio model, the paper obtains an exact closed-form formula for the singular value s(J) and proves a classification theorem for its large-ρ decay. Away from a distinguished fragility locus α = α*, the singular value decays generically as O(ρ^−1). On the fragility locus, the leading response cancels and the decay accelerates to O(ρ^−2), causing observability to deteriorate more rapidly. The paper also proves that, in this minimal model, no genuine-collapse locus exists at any finite parameter value: all finite-parameter information loss is therefore of the apparent-flattening type. This leads to an observable-horizon map in the (α, ρ) plane. Lowering observational noise expands the visible region and pushes the horizon outward, but it does not move the fragility locus or the asymptotic geometric boundary. The resulting picture distinguishes a movable observational wall from an immovable geometric wall and provides a concrete criterion for probe-frequency selection: choose frequencies so that the fragility value α* lies outside the target spectral-exponent range. Overall, the paper gives a geometric and statistical characterization of information loss in floor-dominated inverse spectral reconstruction. It clarifies the difference between exact geometric impossibility and noise-limited practical impossibility, and formulates observable horizons as the natural interface between them. V2:This version refines the interpretation and presentation of the single-ratio model without altering the main analytical results. Clarified the Fisher structure: the single-ratio model is rank-1, and observability is formulated via the scalar response norm s(J)s(J)s(J) rather than a two-parameter Fisher minimum eigenvalue. Introduced the scalar observable horizon explicitly as a local detectability (sensitivity) threshold, removing ambiguity with estimation-based notions. Strengthened the framing as a scalar sensitivity prototype for multi-ratio Fisher geometry (treated in the companion paper). Corrected minor issues, including numerical consistency (αmono\alpha_{\rm mono}αmono) and full alignment between figures, code, and text. Improved wording to avoid overstatements and clarify scope (e.g., scalar-response interpretation). The analytical results, including the fragility locus α∗\alpha_*α∗ and the O(ρ−1)O(\rho^{-1})O(ρ−1) vs. O(ρ−2)O(\rho^{-2})O(ρ−2) classification, remain unchanged.
We present a geometric formulation of parameter identifiability in dynamical-decoupling noise spectroscopy based on the projected Fisher information. The key result is an exact kernel equivalence,kerIθ,eff=ker(PobsKθ)\ker \mathcal{I}_{\theta,\mathrm{eff}} = \ker(P_{\mathrm{obs}}K_\theta)kerIθ,eff=ker(PobsKθ),which identifies unidentifiable parameter directions as those invisible to the observable protocol subspace. We show that symmetry-controlled protocol design induces rank transitions in the observable projector, defining an observable horizon where new identifiable directions can emerge. Numerical results demonstrate that the lifted Fisher mode corresponds uniquely to the release of a symmetry-protected null direction. The framework separates geometric identifiability (rank structure) from quantitative sensitivity, providing a model-independent link between control symmetry and estimation limits.
The dynamics of quantum systems are unavoidably influenced by their environment, but in turn observing a quantum system (probe) can allow one to measure its environment: Measurements and controlled manipulation of the probe such as dynamical decoupling sequences as an extension of the Ramsey interference measurement allow to spectrally resolve a noise field coupled to the probe. Here, we introduce fast and robust estimation strategies for the characterization of the spectral properties of classical and quantum dephasing environments. These strategies are based on filter function orthogonalization, optimal control filters maximizing the relevant Fisher Information and multi-qubit entanglement. We investigate and quantify the robustness of the schemes under different types of noise such as finite-precision measurements, dephasing of the probe, spectral leakage and slow temporal fluctuations of the spectrum.
This work formulates dynamical-decoupling (DD) spectroscopy as a reduced observation problem. Rather than treating measured coherence responses as direct probes of the full environmental noise spectrum, the paper introduces a protocol-dependent map from parametrized spectral families to a small set of observables and analyzes its local Jacobian structure. The main result is a geometric framework in which spectral directions are classified into observable, weakly observable, and compressed sectors according to the singular-value structure and effective rank of the reduced map. Within this framework, plateau behavior and fixed protocol ratios arise when the reduced image becomes effectively low-dimensional, while abrupt transitions are interpreted as switching between response sectors with different local observability structure. A running power-law example is used to illustrate how amplitude and slope deformations are transmitted differently into protocol observables, and how protocol ratios can become locally insensitive without implying full spectral universality. The analysis emphasizes that DD protocols access environmental structure only through compressed, protocol-shaped images of the underlying spectrum. This provides a unifying interpretation of several familiar DD phenomena, including plateau regions, fixed-ratio structures, and sharp protocol-level transitions, in terms of reduced observation geometry rather than direct spectral reconstruction. v2 :This work formulates dynamical-decoupling spectroscopy as a reduced observation map from parameterized noise spectra to protocol-level observables. Using the Jacobian of this map, we classify spectral directions into observable, weakly observable, and compressed sectors via an effective-rank criterion. Plateau behavior and fixed protocol ratios are shown to arise from geometric compression of the reduced image, combined with a proportional-locking condition, rather than from full spectral universality. Abrupt transitions are interpreted as switching between response sectors with different local observability structures. The framework provides a unified geometric interpretation of sensitivity, compression, and invariance in DD-based noise spectroscopy.
This work develops a geometric framework for protocol-to-filter maps in dynamical decoupling noise spectroscopy. By analyzing the singular spectrum of the local Jacobian in a normalized feature space, it shows that increasing protocol-control complexity does not necessarily increase the number of experimentally accessible filter directions. CPMG exhibits moderate anisotropy with a weak but finite secondary direction, while the XY-8 deformation studied here shows symmetry-protected strong anisotropy: the deformation direction vanishes at first order at λ=0\lambda=0λ=0 and remains strongly suppressed over a finite range. A two-component toy model confirms that the framework detects genuinely two-dimensional accessibility when independent spectral directions are present.