Introduces a covariance-dependent observability geometry for dynamical-decoupling spectroscopy using the noise-whitened Fisher metric F=JTΣ−1JF = J^T\Sigma^{-1}JF=JTΣ−1J. The work shows that correlated observational noise induces smooth but localized geometric reorganization, including substantial soft-mode rotation and anisotropy redistribution in reduced inverse-map geometry.
Introduces a covariance-dependent observability geometry for dynamical-decoupling spectroscopy using the noise-whitened Fisher metric F=JTΣ−1JF = J^T\Sigma^{-1}JF=JTΣ−1J. The work shows that correlated observational noise induces smooth but localized geometric reorganization, including substantial soft-mode rotation and anisotropy redistribution in reduced inverse-map geometry.
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.
Inverse Geometry and Identifiability of Protocol Ratios in Dynamical-Decoupling Spectroscopy This work develops an inverse identifiability framework for protocol ratios in dynamical-decoupling spectroscopy. Building on reduced observation geometry, it asks the inverse question: given an observed protocol ratio, what spectral information is uniquely reconstructible, what is only weakly constrained, and what is fundamentally unrecoverable? The analysis is formulated in terms of the Jacobian of the reduced observation map and the associated Fisher information matrix. The paper shows that the inverse problem separates into three qualitatively distinct sectors. In the regular sector, the ratio map is locally invertible and reconstruction accuracy is controlled by a Cramér–Rao bound. For the pure power-law family S(ω)=Aω^α, the ratio R(α)=2/(1+2^{-α}) is strictly monotone, admits a closed-form inverse, and yields an explicit conditioning profile through the slope ∂_αR. In the degenerate sector, exemplified by the white-noise family S(ω)=A, the protocol ratio is exactly pinned to R=1 for all amplitudes. The Fisher information for the amplitude direction vanishes identically, so amplitude reconstruction from ratio data alone is impossible. This gives an exact no-go theorem for identifiability from protocol ratios in the white-noise case. In the branched sector, non-injectivity of the inverse map produces intrinsic ambiguity. Using the mixed family S(ω; α1, α2, w)=wAω^{α1}+(1−w)Aω^{α2}, the paper shows both global non-injectivity and local fold branching. Along an explicit symmetric slice, the ratio develops a fold at Rc=4/3, so nearby observed values admit two distinct inverse branches. These results establish protocol ratios as inverse-geometric probes with sharply defined domains of identifiability and intrinsic limits of spectral reconstruction. The framework clarifies the difference between unique inference, exact information collapse, and multi-valued inversion, and provides a basis for future extensions to multi-ratio, mixed-spectrum, and frequency-resolved inverse design. V2:This work develops an inverse identifiability framework for protocol ratios in dynamical-decoupling spectroscopy. Interpreting the reduced observation map through Fisher information, we show that the inverse problem separates into three sectors: regular (locally invertible with Cramér–Rao bounds), degenerate (exact collapse with vanishing Fisher information), and branched (non-injective with multiple solutions). Using power-law and mixed spectral families, we derive explicit inversion formulas, a white-noise no-go theorem, and fold-induced branching. The framework establishes protocol ratios as inverse-geometric probes with sharply defined limits of spectral reconstruction.
This paper studies the covariance geometry of filter-integrated observables in dynamical-decoupling spectroscopy. We show that the covariance admits a non-commuting additive decomposition,Ctotal=λCflat+CdevC^{\rm total}=\lambda C^{\rm flat}+C^{\rm dev}Ctotal=λCflat+Cdev,where CflatC^{\rm flat}Cflat represents flat filter-overlap geometry and CdevC^{\rm dev}Cdev encodes spectral deformation induced by power-law noise weights. Numerical analysis of CPMG filters reveals persistent but diluted non-commutativity under protocol enlargement, selective intermediate-mode hybridization, and horizon-precursor growth of covariance ill-conditioning.
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.
This paper extends the recoverable‑null and observable‑horizon framework of reduced dynamical‑decoupling (DD) spectroscopy to colored noise floors. Earlier work (Papers Q6–Q7) established that a flat floor B produces a recoverable near‑null: the reduced Jacobian remains full‑rank at every finite floor ratio ρ=B/A, and the weakest reconstruction direction becomes unobservable only when its singular value falls below the measurement noise, defining an observable horizon. Here we replace the flat floor by a shape‑dependent floor ρ Sfloor(ω;ν) and compute the horizon map ρ∗(α;ν) for three canonical spectral families: a flat floor, a 1/f-like floor, and a Lorentzian floor parameterised by a knee frequency ωc. The main findings are: Shape‑dependent horizon deformation: each floor family produces a qualitatively distinct deformation of the horizon map. A flat floor yields a smooth baseline; a 1/f-like floor introduces a sharp but finite minimum at α=1; a Lorentzian floor shifts the horizon‑gain region across the α-axis as ωc varies. 1/f resonance collapse: when the floor exponent matches the signal exponent (α=1), the ln(ωk)-amplified component of the α-column of the leading Jacobian J(1) is suppressed by the prefactor 1−21−α, which vanishes at α=1. This produces a strong recoverable‑null degeneracy and a sharp horizon minimum. Lorentzian knee as a design parameter: the knee frequency ωc determines which α sector benefits from the floor. An intermediate knee near the geometric mean ωminωmax achieves the broadest horizon gain (up to ∼3.5× near α≈0.9). Together with Papers Q6 and Q7, these results complete the three‑part analysis of recoverable‑null geometry under coarse‑grained floors in the DD series. Floor spectral shape emerges as a geometrically active design variable, providing a full toolkit for horizon optimisation in reduced DD spectroscopy under realistic colored noise conditions. v2: Clarified the distinction between recoverable near-nulls and exact structural nulls in reduced DD spectroscopy under colored noise floors. Added the exact 1/f resonance result at α=1, where the observable map loses the logρ direction identically, together with revised horizon geometry, Lorentzian knee-frequency analysis, normalization conventions, convergence checks, and improved figure annotations for structural rank loss.
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.