pith. sign in

arxiv: 2604.08325 · v1 · submitted 2026-04-09 · 🪐 quant-ph · physics.optics

Kirkwood-Dirac distributions in classical optics

Alfredo Luis , Lorena Ballesteros Ferraz This is my paper

Pith reviewed 2026-05-10 18:20 UTC · model grok-4.3

classification 🪐 quant-ph physics.optics
keywords Kirkwood-Dirac distributionsoptical coherenceclassical opticsanomalous valuesmutual coherence functionspolarizationinterference
0
0 comments X

The pith

Kirkwood-Dirac distributions in classical optics are generalized mutual coherence functions between two bases, with their complex and negative values arising as direct manifestations of coherence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that Kirkwood-Dirac distributions, when defined for classical optical fields, act as generalized mutual coherence functions involving two different bases rather than one. This view unifies the interpretation of anomalous values—complex and negative numbers—as direct results of optical coherence. The interpretation holds for polarization, interference, and wave propagation. It also suggests experimental methods using interference to determine these distributions.

Core claim

From their definition, the Kirkwood-Dirac distributions emerge as generalized mutual coherence functions involving two different bases instead of just one. This perspective provides a unified interpretation of the so-called anomalous values, that are complex and negative values, as direct manifestations of coherence. This applies across field variables including polarization, interference and wave propagation.

What carries the argument

The Kirkwood-Dirac distribution reinterpreted as a generalized mutual coherence function between two different bases.

If this is right

  • Coherence between different bases explains why Kirkwood-Dirac distributions can take complex or negative values.
  • Experimental determination can use interference techniques consistent with this coherence view.
  • The interpretation remains consistent for polarization states, interference patterns, and wave propagation.
  • Unified view connects quantum-inspired distributions to classical optical coherence theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This coherence-based view could allow direct measurement of coherence properties using Kirkwood-Dirac distributions in optical setups.
  • It suggests that similar interpretations might apply to other classical wave systems beyond optics.
  • Experimental tests in polarization and interference could further validate the unified interpretation.

Load-bearing premise

The formal definition of Kirkwood-Dirac distributions translates directly to classical optical fields without requiring additional assumptions about the underlying wave equations or measurement processes.

What would settle it

Finding complex or negative values in Kirkwood-Dirac distributions for an optical field lacking mutual coherence between the relevant bases would contradict the central claim.

Figures

Figures reproduced from arXiv: 2604.08325 by Alfredo Luis, Lorena Ballesteros Ferraz.

Figure 1
Figure 1. Figure 1: Scheme for a practical observation of Kirkwood [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Scheme for a practical observation of Kirkwood [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Detail of the Kirkwood-Dirac distribution of a [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

We develop a comprehensive analysis of the Kirkwood-Dirac distributions in classical optics, revealing their deep connection with optical coherence as fundamental concept in optics. From their very definition, the Kirkwood-Dirac distributions emerge as generalized mutual coherence functions involving two different bases instead of just one. This perspective provides a unified interpretation of the so-called anomalous values, that are complex and negative values, as direct manifestations of coherence. We show that this interpretation consistently applies across all field variables considered in this work, including polarization, interference and wave propagation. Furthermore, we propose diverse methods of experimental determination of these distributions based on interference, in full agreement with their coherence-based interpretation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops Kirkwood-Dirac (KD) distributions for classical optical fields, arguing that from their definition they correspond to generalized mutual coherence functions between two distinct bases. This framework is used to interpret complex and negative (anomalous) values as direct manifestations of optical coherence. The interpretation is asserted to hold consistently across polarization, interference, and wave propagation, with interference-based experimental protocols proposed that align with the coherence picture.

Significance. If the central mapping holds, the work supplies a classical-optics reading of KD distributions that unifies them with the established concept of mutual coherence, thereby offering a non-quantum explanation for anomalous values. The consistent application across multiple field variables and the emphasis on experimentally accessible interference methods constitute the main strengths; the manuscript also benefits from grounding the discussion in standard optical coherence theory rather than ad-hoc extensions.

minor comments (3)
  1. [Results sections on polarization, interference, and propagation] The abstract states that the interpretation 'consistently applies across all field variables' and shows 'full agreement with interference experiments,' yet the manuscript would benefit from a dedicated subsection that tabulates the explicit coherence-function expressions for each variable (polarization, interference, propagation) to make the consistency claim immediately verifiable.
  2. [Introduction and §2] Notation for the two bases in the generalized mutual-coherence function is introduced without a compact summary table; adding such a table early in the manuscript would clarify the correspondence between the original KD operator definition and the classical-field version.
  3. [Experimental methods] The experimental proposals are described at a conceptual level; inclusion of a brief error-propagation estimate or signal-to-noise consideration for the interference-based reconstruction would strengthen the claim of practical feasibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary, recognition of the manuscript's strengths in linking Kirkwood-Dirac distributions to mutual coherence, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

1 steps flagged

KD distributions presented as mutual coherence functions directly from definition

specific steps
  1. self definitional [Abstract]
    "From their very definition, the Kirkwood-Dirac distributions emerge as generalized mutual coherence functions involving two different bases instead of just one. This perspective provides a unified interpretation of the so-called anomalous values, that are complex and negative values, as direct manifestations of coherence."

    The paper claims the emergence as coherence functions follows from the definition itself, so the unified interpretation of anomalous (complex/negative) values as coherence is equivalent to the definitional rephrasing rather than a derived prediction from classical optics principles like wave equations or quadratic intensity detection.

full rationale

The paper's central claim states that Kirkwood-Dirac distributions emerge as generalized mutual coherence functions 'from their very definition,' providing a unified view of anomalous values as coherence manifestations. This is self-definitional: the interpretation is asserted to follow immediately from the formal definition without an independent derivation step from optical wave equations or measurements. No fitted predictions, self-citation chains, or ansatz smuggling are evident in the abstract or described content. The result remains partially independent as it applies the definition consistently across polarization, interference, and propagation, but the load-bearing insight reduces to re-expressing the input definition in coherence terms.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard definition of Kirkwood-Dirac distributions and the established concept of mutual coherence functions in optics; no free parameters, new entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)
  • domain assumption Kirkwood-Dirac quasi-probability distributions defined via two bases apply to classical electromagnetic fields.
    The extension from quantum to classical optics is taken as direct.
  • domain assumption Optical coherence is fully captured by mutual coherence functions between field components.
    Standard in classical optics and invoked to reinterpret anomalous values.

pith-pipeline@v0.9.0 · 5400 in / 1366 out tokens · 65204 ms · 2026-05-10T18:20:33.917788+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    Dirac, On the analogy between classical and quantum mechanics, Rev

    P.A.M. Dirac, On the analogy between classical and quantum mechanics, Rev. Mod. Phys.17, 195 (1945)

    work page 1945

  2. [2]

    Kirkwood, Quantum statistics of almost classical assemblies, Phys

    J.G. Kirkwood, Quantum statistics of almost classical assemblies, Phys. Rev.44, 31 (1933)

    work page 1933

  3. [3]

    Wigner, On the quantum correction for thermody- namic equilibrium, Phys

    E. Wigner, On the quantum correction for thermody- namic equilibrium, Phys. Rev.40, 749 (1932)

    work page 1932

  4. [4]

    Cahill and R.J

    K.E. Cahill and R.J. Glauber, Ordered expansions in bo- son amplitude operators, Phys. Rev.177, 1857 (1969)

    work page 1969

  5. [5]

    Aharonov, D.Z

    Y. Aharonov, D.Z. Albert, and L. Vaidman, How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100, Phys. Rev. Lett. 60, 1351 (1988)

    work page 1988

  6. [6]

    Wagner, Z

    R. Wagner, Z. Schwartzman-Nowik, I.L. Paiva, et al., Quantum circuits for measuring weak values, Kirk- wood–Dirac quasiprobability distributions, and state spectra, Quantum Sci. Technol.9, 015030 (2024)

    work page 2024

  7. [7]

    Lostaglio, A

    M. Lostaglio, A. Belenchia, A. Levy, S. Hern´ andez- G´ omez, N. Fabbri, and S. Gherardini, Kirkwood-Dirac quasiprobability approach to the statistics of incompati- ble observables, Quantum7, 1128 (2023)

    work page 2023

  8. [8]

    Arvidsson-Shukur, W.F

    D.R.M. Arvidsson-Shukur, W.F. Braasch Jr, S. De Bi` evre, J. Dressel, A.N. Jordan, C. Langrenez, M. Lostaglio, J.S. Lundeen, and N.Y. Halpern, Properties and applications of the Kirkwood–Dirac distribution, New J. Phys.26, 121201 (2024)

    work page 2024

  9. [9]

    He and S

    J. He and S. Fu, Nonclassicality of the Kirkwood–Dirac quasiprobability distribution via quantum modification terms, Phys. Rev. A109, 012215 (2024)

    work page 2024

  10. [10]

    Ferrie, Quasi-probability representations of quantum theory with applications to quantum information science, Rep

    C. Ferrie, Quasi-probability representations of quantum theory with applications to quantum information science, Rep. Prog. Phys.74, 116001 (2011)

    work page 2011

  11. [11]

    Hern´ andez-G´ omez, T

    S. Hern´ andez-G´ omez, T. Isogawa, A. Belenchia, A. Levy, N. Fabbri, and S. Gherardini, Interferometry of quantum correlation functions to access quasiprobability distribu- tion of work, npj Quantum Information10, 112 (2024)

    work page 2024

  12. [12]

    K. F. Lee, F. Reil, S. Bali, A. Wax, and J. E. Thomas, Heterodyne measurement of Wigner distributions for classical optical fields, Opt. Lett.24, 1370 (1999)

    work page 1999

  13. [13]

    Bollen, Y

    V. Bollen, Y. M. Sua, and K. F. Lee, Direct measure- ment of the Kirkwood-Rihaczek distribution for the spa- tial properties of a coherent light beam, Phys. Rev. A81, 063826 (2010)

    work page 2010

  14. [14]

    Ares and A

    L. Ares and A. Luis, Distance-based approach to quan- tum coherence and nonclassicality, Phys. Rev. A106, 012415 (2022)

    work page 2022

  15. [15]

    Budiyono and H

    A. Budiyono and H. K. Dipojono, Quantifying quantum coherence via Kirkwood-Dirac quasiprobability, Phys. Rev. A107, 022408 (2023)

    work page 2023

  16. [16]

    E. Masa, L. Ares, and A. Luis, Nonclassical joint dis- tributions and Bell measurements, Phys. Lett. A384, 126416 (2020)

    work page 2020

  17. [17]

    Galazo, I

    R. Galazo, I. Bartolom´ e, L. Ares, and A. Luis, Classi- cal and quantum complementarity, Phys. Lett. A384, 126849 (2020)

    work page 2020

  18. [18]

    Luis, Generalized measurements for Bell tests in dif- ferent probability spaces, arXiv:2506.07496 [quant-ph]

    A. Luis, Generalized measurements for Bell tests in dif- ferent probability spaces, arXiv:2506.07496 [quant-ph]

    work page arXiv

  19. [19]

    Luis, Negativity, diffraction and interference for non- geometrical waves, Opt

    A. Luis, Negativity, diffraction and interference for non- geometrical waves, Opt. Commun.266, 426 (2006)

    work page 2006

  20. [20]

    Luis, Complementary Huygens principle for geomet- rical and nongeometrical optics, Eur

    A. Luis, Complementary Huygens principle for geomet- rical and nongeometrical optics, Eur. J. Phys.28, 231 (2007)

    work page 2007

  21. [21]

    Mandel and E

    L. Mandel and E. Wolf,Optical Coherence and Quantum Optics(Cambridge University Press, Cambridge, U.K., 1995)

    work page 1995

  22. [22]

    Ares and A

    L. Ares and A. Luis, Coherence and incoherence in quadrature basis, arXiv:2307.08333 [quant-ph]

    work page arXiv

  23. [23]

    Tervo, T

    J. Tervo, T. Set¨ al¨ a, and A. T. Friberg, Theory of partially coherent electromagnetic fields in the space–frequency domain, J. Opt. Soc. Am. A21, 2205 (2004)

    work page 2004

  24. [24]

    Luis, Overall degree of coherence for vectorial elec- tromagnetic fields and the Wigner function, J

    A. Luis, Overall degree of coherence for vectorial elec- tromagnetic fields and the Wigner function, J. Opt. Soc. Am. A24, 2070 (2007)

    work page 2070

  25. [25]

    P. A. Mello and M. Revzen, Wigner function and the suc- cessive measurement of position and momentum Phys. Rev. A89, 012106 (2014)

    work page 2014

  26. [26]

    Luis and J

    A. Luis and J. Peˇ rina, Discrete Wigner function for finite- dimensional systems, J. Phys. A31, 1423 (1998)

    work page 1998

  27. [27]

    Karczewski, Degree of coherence of the electromag- netic field, Phys

    B. Karczewski, Degree of coherence of the electromag- netic field, Phys. Lett.5, 191–192 (1963)

    work page 1963

  28. [28]

    Wolf, Unified theory of coherence and polarization of random electromagnetic beams Phys

    E. Wolf, Unified theory of coherence and polarization of random electromagnetic beams Phys. Lett. A312, 263– 267, (2003)

    work page 2003

  29. [29]

    Tervo, T

    J. Tervo, T. Set¨ al¨ a, and A. T. Friberg, Degree of coher- ence for electromagnetic fields, Opt. Express11, 1137– 11 1143 (2003)

    work page 2003

  30. [30]

    Set¨ al¨ a, J

    T. Set¨ al¨ a, J. Tervo, and A. T. Friberg, Complete electro- magnetic coherence in the space–frequency domain, Opt. Lett.29, 328–330 (2004)

    work page 2004

  31. [31]

    R´ efr´ egier and F

    P. R´ efr´ egier and F. Goudail, Invariant degrees of coher- ence of partially polarized light, Opt. Express13, 6051 (2005)

    work page 2005

  32. [32]

    Luis, Degree of coherence for vectorial electromag- netic fields as the distance between correlation matrices, J

    A. Luis, Degree of coherence for vectorial electromag- netic fields as the distance between correlation matrices, J. Opt. Soc. Am. A24, 1063 (2007)

    work page 2007