Phase-Space Entropy at Acquisition

• The paper establishes that phase-space entropy quantifies structural information loss at the acquisition stage by applying Shannon-type entropy to joint space–momentum representations.
• Results indicate that acquisition protocols with higher ΔS_B correlate with poorer downstream performance, highlighting the impact of sampling, aliasing, and instrument constraints.
• The methodology spans spectrogram analysis and Wannier basis projections, enabling cross-domain evaluation from quantum tomography to plasma turbulence.

Phase-space entropy at acquisition quantifies the structural disorder or information content of a system at the earliest interface between physical measurement and any downstream processing. Across quantum, classical, and signal processing domains, phase-space entropy is defined by exploiting a joint representation over position and momentum (or frequency), systematically measuring how acquisition protocols reorder, mix, or suppress space–frequency structure. This concept subsumes a range of phenomena: from the role of entropy increase in quantum evolution, to the detection of aliasing in signal acquisition, and the identification of irreversibility in turbulent plasma heating. By measuring entropy immediately upon data acquisition, one obtains a universal, modality-independent indicator of information preservation that is sensitive to structure disruption well before any learning or reconstruction task begins.

1. Formal Definitions of Phase-Space Entropy at Acquisition

Phase-space entropy at acquisition is generally an application of a Shannon–type entropy functional to a joint space–momentum (or time–frequency) representation derived directly from the measured or acquired field.

Classical and signal-processing settings:

Given an empirical field $I(x)$ over spatial domain $x\in\mathbb R^d$, one constructs a smoothed phase-space (Husimi or spectrogram) density by convoluting the Wigner distribution $W_I(x;k)$ with an instrument-matched kernel $\Phi$:

$\rho_I(x;k) = (\Phi * W_I)(x;k).$

The local spectral density is then band-normalized, and the position-resolved phase-space entropy is

$$s_\mathcal{B}(x) = -\int_{k\in\mathcal{B}} N\rho_I(x;k) \log N\rho_I(x;k)\,dk,$$

with $\mathcal{B}$ the Nyquist band, and the total band-entropy

$$S_\mathcal{B}[I] = \int s_\mathcal{B}(x) \, w(x) dx.$$

The acquisition-level entropy change is then

$$\Delta S_\mathcal{B} = S_\mathcal{B}[I_{acq}] - S_\mathcal{B}[I_{ref}]$$

where $I_{acq}$ is the measured (sampled/masked) field, and $I_{ref}$ is a high-fidelity reference (Wang et al., 22 Dec 2025).

Quantum settings:

A widely accepted structure is to define, for a pure state $|\psi\rangle$, a phase-space entropy via the overlap with a basis of Planck-scale localized Wannier functions $|w_j\rangle$:

$$p_j = |\langle w_j|\psi \rangle|^2, \quad S_w(\psi) = -\sum_j p_j \ln p_j,$$

with $j$ ranging over a tiling of phase space into Planck cells. This entropy is directly computable from raw state acquisition and is additive over composite systems (Han et al., 2014).

Alternative definitions, e.g., the Wigner entropy $$H[W_\rho] = -\iint W_\rho(x,p) \ln W_\rho(x,p) dx dp$$, apply to states with non-negative Wigner functions and quantify phase-space disorder at the acquisition level (Herstraeten et al., 2024).

2. Theoretical Properties and Acquisition-Induced Effects

Phase-space entropy at acquisition is sensitive to joint structure rather than marginal (pixelwise or spectral) statistics and provides modality-agnostic, coordinate-invariant quantification of acquisition-induced disorder.

• Coordinate and invariance properties: Multidimensional continuous entropy functionals, when properly weighted with the correct phase-space measure, are invariant under coordinate changes provided an appropriate Jacobian factor $A(x)$ is included:

$S[\rho] = -\int dx \, \rho(x) \ln [A(x) \rho(x)]$

with the correct $A(x)$ derivable from the microphysics of acquisition or collision laws (Maynar et al., 2011).

• Quantum lower bounds: Entropic uncertainty principles impose universal lower bounds. For example, the Wigner entropy for nonnegative Wigner states satisfies

$H[W_\rho] \geq \ln(\pi e \hbar),$

a phase-space analog of minimal quantum uncertainty (Herstraeten et al., 2024). In the Wannier basis, entropy is always strictly positive and strictly decreases upon taking subsystems (Han et al., 2014).

• Sampling and aliasing: The phase-space entropy framework recovers classical consequences of periodic (coherent) sampling—such as aliasing and information loss—by showing that periodic subsampling always increases the measured $\Delta S_\mathcal{B}$, while random sampling leaves it essentially unchanged on average; this dichotomy is evident via Jensen's inequality (Wang et al., 22 Dec 2025).

3. Methodologies for Evaluation at Acquisition

Evaluation of phase-space entropy at the point of acquisition involves projection, transformation or sampling operators tailored to the physical modality:

• Instrument-matched spectrogram analysis: For imaging, MRI, and MIMO wireless systems, local short-time Fourier transforms (STFT) with Gaussian windows are applied at the instrument's resolution scale, generating a local spectral density per acquisition window. The resulting normalized spectrogram is used for entropy assessment (Wang et al., 22 Dec 2025).
• Wannier-basis projections: In quantum state acquisition, precomputed Wannier bases enable direct projection of the measured state onto phase-space cells, producing a fully operational, non-ensemble-averaged entropy evaluation just after tomography or read-out (Han et al., 2014).
• Maximum-entropy phase-space tomography: For severely underdetermined, high-dimensional inference (e.g., 6D beams in accelerators), normalizing flows parameterize phase-space densities, and the entropy functional is optimized subject to projection constraints to yield the maximally conservative estimate at acquisition (Hoover et al., 2024).
• Quadratic (Casimir) invariants in plasma physics: In kinetic plasma models, quadratic invariants such as $$C_2 = (1/L)\iint dx\,dv\, \frac12 \langle f^2 \rangle$$ serve as proxies for phase-space disorder. Acquisition (application of an external field) launches a cascade transferring this invariant toward fine scales and increasing entropy, directly reflecting the irreversibility of the measurement process (Nastac et al., 2023).

4. Empirical Impact Across Physical and Information Systems

Phase-space entropy at acquisition enables prediction and ranking of downstream task difficulty in diverse measurement scenarios, independently of further training or reconstruction.

Summary of empirical findings (Wang et al., 22 Dec 2025):

Domain	Higher $\lvert \Delta S_\mathcal{B} \rvert$	Lower $\lvert \Delta S_\mathcal{B} \rvert$
Vision (mini-ImageNet)	Periodic patch masks — higher classification error	Uniform random masks, higher accuracy
MRI	Periodic or structured k-space undersampling — poor PSNR/SSIM	Poisson-disc masks — better reconstruction
Massive MIMO	Periodic pilot deactivation — higher NMSE	Random pilot deactivation — lower NMSE

These results show that $\lvert \Delta S_\mathcal{B} \rvert$ consistently ranks acquisition protocols in accordance with actual learning or reconstruction performance, with greater entropy drift indicating more severe structure loss. Notably, these rankings are accessible before any downstream training.

5. Connections to Fundamental Limits and Thermodynamics

Phase-space entropy at acquisition interfaces with uncertainty principles and informs operational constraints across quantum, classical, and stochastic regimes:

• Uncertainty bounds: Minimal achievable phase-space entropy is determined by quantum mechanics; no acquisition protocol can undercut the bound $H[W_\rho]\ge\ln(\pi e\hbar)$ for Wigner-positive states, setting a hard limit on spatial and spectral resolution (Herstraeten et al., 2024).
• Dissipative anomalies in turbulence: In kinetic plasma systems, phase-space entropy increase is tied to entropy cascades and dissipation anomalies that are robust even in the limit of vanishing collisionality, setting the scale and irreversibility of acquired disorder (Nastac et al., 2023).
• Quantum entropy growth: Under generic unitary evolution, phase-space entropy for pure states not only grows but typically plateaus close to a maximum, in accord with quantum H-theorems generalizing von Neumann's original framework (Han et al., 2014, Geiger et al., 2021).

6. Practical Guidelines, Implementation, and Limitations

Operational recommendations (Wang et al., 22 Dec 2025):

• Select a high-fidelity reference and calibrate instrument-specific spectrogram/STFT parameters.
• Evaluate candidate acquisition or sampling policies by computing $\lvert \Delta S_\mathcal{B} \rvert$ on calibration data.
• Rank acquisition protocols by $\lvert \Delta S_\mathcal{B} \rvert$ for pre-training selection, as this correlates with downstream learnability.
• Attend to window scales, ensuring they match the artifact or feature size to avoid over- or underestimating entropy changes.

Limitations and considerations:

• $\Delta S_\mathcal{B}$ is a relative rather than absolute measure; a high-quality reference is critical.
• For task-specific sufficiency, band-weighted variants may offer improved alignment; $\lvert \Delta S_\mathcal{B} \rvert$ is most effective for broad-based structure preservation.
• In quantum metrology, measurement resolution and statistical noise are fundamentally limited by the lower bounds set by uncertainty principles; post-processing cannot circumvent these bounds.

7. Outlook and Cross-Domain Relevance

Phase-space entropy at acquisition provides a universal, physically principled measure for evaluating information content and structure disruption at the point of measurement. Its implications span quantum state tomography, signal sampling, plasma turbulence, and learning-based data acquisition. As acquisition strategies diversify, phase-space entropy offers a single, robust tool for benchmarking, calibration, and policy selection in increasingly complex, high-dimensional, and cross-modal measurement environments (Wang et al., 22 Dec 2025, Herstraeten et al., 2024, Han et al., 2014, Hoover et al., 2024).