Smeared Phase-Space Distributions

• Smeared phase-space distributions are defined by convolving reference functions like the Wigner function with smoothing kernels to enforce positivity and measurable resolution.
• They regularize rapid oscillations and non-positive features, allowing clearer interpretation of quantum and nuclear systems in experimental settings.
• Applications span quantum tomography, nuclear physics, and quantum optics, where they facilitate state reconstruction and comparison between theory and measurement.

A smeared phase-space distribution is a positive or quasi-probabilistic function on quantum or classical phase space obtained from a reference distribution—such as the Wigner function or nuclear Wigner distribution—by convolution with a smoothing kernel. This process arises both in quantum tomography, information theory, and nuclear many-body physics, with prominent examples being the Husimi (Q) distribution, Gaussian smoothed Wigner functions, Born–Jordan distributions, and the quark phase-space distributions in nuclei. Smeared distributions regularize or average highly oscillatory (possibly non-positive) features of their parent distributions, enforce physical measurement resolution, and facilitate the interpretation of phase-space densities as occupation numbers or probabilistic descriptors of physical states. They underpin rigorous uncertainty relationships, provide information-theoretic quantifiers, and enable direct connection between experimentally accessible observables and theoretical models across domains.

1. Definitions and Mathematical Construction

A smeared phase-space distribution is typically constructed via the convolution

$$F_{\text{sm}}(\Omega) = \int \! d\Omega' \; K(\Omega-\Omega') \;F(\Omega')$$

where $F(\Omega)$ is the original phase-space quasi-probability (such as the Wigner function), and $K$ is a smoothing kernel (often Gaussian, but may be more general, e.g. sinc kernel for Born–Jordan). The smearing operation can be expressed equivalently via Fourier or symplectic transforms, as in Cohen-class distributions (Koczor et al., 2018), or through explicit convolution in position/momentum (or their multidimensional generalizations) as in nuclear, optical, or oscillator models (Nikolakopoulos et al., 27 Jun 2025, Ojha et al., 2024, Barbier et al., 2021, 1311.0666).

Prototypical Examples:

• Husimi Distribution (Q function): Gaussian smoothing of the Wigner function, yielding

$$Q(x,p) = \frac{1}{\pi}\int \! dx'dp' \,e^{-(x-x')^2 - (p-p')^2} W(x',p')$$

(Ojha et al., 2024, Barbier et al., 2021)

• Gaussian-Smoothed Wigner Function:

$$W_\sigma(x, p) = \int dx' dp' \; \frac{1}{2\pi \sigma_x \sigma_p} \exp\left[-\frac{(x-x')^2}{2\sigma_x^2} - \frac{(p-p')^2}{2\sigma_p^2}\right] W(x', p')$$

with positivity guaranteed for $\sigma_x \sigma_p \geq \hbar/2$ (1311.0666).

• Quark Phase-Space Distributions:

$$f_q(r, p) = \int d^3p' \; W(r, p-p') \; f_{q/N}(p')$$

where $W(r, p)$ is the nuclear Wigner distribution and $f_{q/N}$ is the intrinsic quark momentum distribution (Nikolakopoulos et al., 27 Jun 2025).

2. Principal Types and Physical Interpretation

The most studied smeared phase-space distributions include:

• Husimi Q function: A minimal-uncertainty (σ_x σ_p = ℏ/2) smearing producing a strictly positive function, representing the overlap with coherent states (Ojha et al., 2024, Barbier et al., 2021).
• Gaussian-smoothed Wigner functions: Arbitrary-resolution coarse-grainings that interpolate between the Wigner function and the Q function. For detector models, the smearing width encodes measurement resolution and minimal uncertainty constraints (1311.0666).
• Cohen-class distributions: A family of phase-space representations parametrized by a kernel in the characteristic-function (Fourier) domain. The Born–Jordan distribution is a key example, obtained via a sinc kernel (Koczor et al., 2018).
• Detector-agnostic distributions: In photonic systems, smearing encapsulates unknown detector response, and the resultant distribution is reconstructed directly from click statistics, independent of detector characterization (Sperling et al., 2019).
• Nuclear and sub-nucleonic smearing: In many-body theory, convolution of the nuclear Wigner distribution with intrinsic partonic distributions incorporates correlations and substructure, allowing physical interpretation of smeared phase-space occupation numbers (Nikolakopoulos et al., 27 Jun 2025).

The smearing process regularizes negative or rapidly oscillating features endemic to unsmeared (e.g., Wigner) distributions, enforces positivity for suitable kernels, and matches the finite resolution of realistic measurements.

3. Information-Theoretic and Statistical Properties

Smeared distributions play a central role in quantum and information-theoretic analysis:

• Entropy Measures: Shannon, Wehrl, and Rényi entropies can be associated with both Wigner and smeared distributions. For the Husimi (Q) function, the Wehrl entropy $S_Q$ is strictly real and well-defined, reflecting the additional uncertainty introduced by coarse-graining. Smeared distributions consistently yield higher entropy and phase-space uncertainty relative to their unsmeared counterparts (Ojha et al., 2024).
• Information Loss: The Kullback–Leibler divergence

$$D(W \parallel Q) = \int dx dp \; W(x,p) \ln\!\frac{W(x,p)}{Q(x,p)}$$

quantifies the information lost in Gaussian smearing. It is strictly positive (when both distributions are positive) and increases with state complexity and degree of smearing (Ojha et al., 2024).

• Uncertainty Principles: Smeared (Husimi) distributions satisfy phase-space uncertainty inequalities that are saturated or closely approached by the Wigner marginals. Smoothing increases the joint entropy, consistent with tightening classicality (Ojha et al., 2024).
• Mutual Information and Correlations: Correlation measures are strictly positive for positive-definite smeared distributions but may be complex for Wigner functions. Smearing suppresses negative (nonclassical) mutual information components (Ojha et al., 2024).

4. Applications: Nuclear, Quantum Optics, and Measurement Theory

Nuclear and Quark Distributions: In nuclear many-body physics, smeared phase-space distributions are employed to model the convolution of Fermi-level nucleonic motion with intrinsic quark distributions, modeling the appearance of quarkyonic phases. This leads to phenomena such as:

• Suppression of low-momentum nucleons: Smearing creates a "hole" in the momentum distribution at low p, a counterintuitive feature relative to standard mean-field theory but natural for Pauli-saturated quark distributions (Nikolakopoulos et al., 27 Jun 2025).
• Experimental signals: Specific observables, such as inclusive and exclusive electron scattering at low missing momentum, are predicted to be sensitive to these features; deviations from standard predictions may falsify quark-saturated models (Nikolakopoulos et al., 27 Jun 2025).

Quantum Optics and Detector Modeling: Smeared distributions accurately characterize the effects of imperfect measurement in homodyne and photon counting experiments:

• Direct tomographic reconstruction: Gaussian-smoothed Wigner functions correspond to the joint probability distribution for simultaneous finite-resolution measurements of canonical observables. In eight-port homodyne detection, the measured distribution maps directly to the smeared Wigner function, with widths determined by detector efficiency (1311.0666).
• Detector-agnostic phase-space tomography: Techniques enabling direct sampling of smeared distributions (via, e.g., click statistics of unknown detectors) yield nonclassicality certification even in the presence of arbitrary detector imperfections. Negativity in the reconstructed distribution is a robust signature of nonclassicality (Sperling et al., 2019).

Coarse-Graining and Quantum-Classical Transition: Smeared distributions interpolate between fully quantum quasi-probabilities (Wigner) and classical probability distributions, enabling controlled investigation of quantum-to-classical and measurement-induced transitions (Barbier et al., 2021, Koczor et al., 2018).

5. Resolution, Positivity, and Physical Interpretation

The smearing kernel's width controls the trade-off between phase-space resolution and positivity:

• Parameter Dependence: For Gaussian smearing, $\sigma_x \sigma_p = \hbar/2$ is the positivity threshold (Husimi limit). Larger widths further regularize, smaller widths retain more negativity (1311.0666, Ojha et al., 2024).
• Physical Meaning: In quantum optics, the Q function gives the overlap probability with coherent states; in nuclear physics, convolution with partonic distributions reflects the substructure of composite particles; in time–frequency analysis, smeared distributions decompose signal content with controlled precision.
• Coarse-graining and classicality: As smearing increases, the distribution approaches a purely classical probability density in the appropriate limit (Ojha et al., 2024, Barbier et al., 2021).

6. Experimental and Theoretical Implications

Smeared phase-space distributions provide powerful frameworks for theory–experiment comparison and the extraction of physical observables under realistic conditions:

• Precision Analysis: All moments and expectation values of suitably ordered operators can be computed directly from smeared distributions via known ordering prescriptions, avoiding the need for deconvolution—even with imperfect detector data (1311.0666).
• Robustness and Limitations: Excessive smearing leads to loss of fine-scale information, complicating inversion and making high-order moments sensitive to noise. For moderate smearing and suitable efficiency, smeared distributions are near-optimal for reconstructing low-order observables (1311.0666, Sperling et al., 2019).
• Physical Tests: Deviations between smeared-model predictions (e.g., depletion of low-momentum states or suppressed interference contrast) and experiment constitute direct tests of underlying physical principles. In quantum optics, detector-agnostic negativity is a strong comparator to traditional correlation tests (Sperling et al., 2019).

7. Generalizations and Cohen-Class Framework

The Cohen-class formalism encompasses all phase-space distributions constructed as convolutions of the Wigner function with prescribed kernels:

• Parity-Operator Representation: Every Cohen-class distribution can be written as the expectation value of a displaced parity operator parameterized by the filter kernel (Koczor et al., 2018).
• Born–Jordan Distribution: The Born–Jordan kernel is a sinc function in the characteristic domain, and the corresponding distribution smoothes sharp oscillatory features in the Wigner function, with quantitative control provided by operator norm bounds and explicit matrix element expressions (Koczor et al., 2018).
• Spectral Properties: The Born–Jordan parity operator admits a generalized spectral decomposition in terms of squeezing eigenstates, reflecting the deeper structure of smeared phase-space representations (Koczor et al., 2018).

This general framework supports both theoretical analysis of quantum-classical hybrids and the design of optimized experimental protocols for state reconstruction, nonclassicality certification, and signal analysis.

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References:

(Nikolakopoulos et al., 27 Jun 2025, Ojha et al., 2024, Barbier et al., 2021, 1311.0666, Koczor et al., 2018, Sperling et al., 2019)