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 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 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.
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.
This work develops a microscopic account of sequence-dependent decoherencetime ratios in dynamical-decoupling (DD) spectroscopy. We show that a nontrivial ratio between coherence times measured under XY-8 and CPMG does not arise generically from standard Gaussian pure-dephasing theory: in the white-noise limit, Parseval’s theorem enforces R ≡ τXY/τCPMG = 1, so any deviation from unity must originate from spectral structure rather than total filter power alone. We then consider a minimal noncommuting-noise extension in which sequence-dependent channel redistribution shifts one transverse contribution sampled by XY-8 from ωeff to ωeff/2, while CPMG continues to sample the relevant channels at ωeff. In the narrow-band regime this yields the microscopic ratio law R = 2S(ωeff)/[S(ωeff) + S(ωeff/2)]. For power-law spectra S(ω) ωα, the ratio becomes R = 2/(1 + 2 α), giving R = 1 for white∝⁻ noise (α = 0), R = 4/3 for an Ohmic bath (α = 1), and R = 2/3 for 1/f noise (α = −1). The value R = 4/3 is therefore not universal but emerges as the special signature of an Ohmic spectrum. These results identify the XY-8/CPMG coherence-time ratio as a spectral-slope-sensitive observable and suggest a direct diagnostic strategy: measuring R in the narrow-band regime directly constrains the local bath exponent governing the sampled frequency window. This establishes a distinct theoretical direction from the effective benchmark treatment of Paper A and from the time-discontinuous channel-accessibility mechanism of Paper B, while remaining directly testable on NV-center platforms. V2: This version clarifies the phenomenological status of the XY-8/CPMG ratio law and reorganizes the manuscript so that the sequence-dependent spectral-slope interpretation is self-contained. Parts of the earlier Paper A framework have been re-examined and absorbed into the present paper in a revised form: rather than treating R=4/3R=4/3R=4/3 as a protocol-universal value, the ratio is now interpreted as a bath-dependent observable determined by the local spectral structure, with R=4/3R=4/3R=4/3 identified as the ideal Ohmic point. The revised version also makes the assumptions explicit (narrow-band regime, equal-weight transverse channels, and common exponential-decay form), strengthens the white-noise no-go benchmark, refines the inverse-problem discussion, and clarifies that the half-frequency XY-8 contribution is introduced as a minimal phenomenological assumption illustrated by a toy toggling-frame model rather than a complete microscopic derivation.