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arxiv: 2508.16601 · v4 · submitted 2025-08-10 · 📡 eess.SP · cs.IT· math.IT

On the Unification of Deterministic and Stochastic Electromagnetic Information Theory via Symplectic Geometry

Marco Donald Migliore This is my paper

Pith reviewed 2026-05-18 23:24 UTC · model grok-4.3

classification 📡 eess.SP cs.ITmath.IT
keywords electromagnetic information theorysymplectic geometryspatial degrees of freedomradiometric etendueLiouville theoremGromov non-squeezing theoremspatial information flowsphase-space volume
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The pith

For spatially incoherent sources, deterministic and stochastic electromagnetic information theories agree on eigenvalues and degrees of freedom because both reduce to the same symplectic invariant of the source-observer geometry.

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

This paper shows that deterministic and stochastic versions of electromagnetic information theory agree on the eigenvalues and spatial degrees of freedom when sources lack spatial coherence. The agreement is not accidental but follows from the fact that radiometric etendue, Hamiltonian phase-space volume, and degrees of freedom are all expressions of the same symplectic invariant fixed by the source and observer positions. Liouville's theorem then ensures this quantity remains constant as waves propagate without loss, while Gromov's theorem limits how small the phase-space cells can be, setting a bound on resolution. The resulting structure produces curves of high mutual information known as spatial information flows that match previously known optimal sampling paths for symmetric convex sources.

Core claim

Both deterministic and stochastic formulations of electromagnetic information theory produce identical eigenvalues and spatial degrees of freedom for incoherent sources. This identity follows necessarily from the fact that the radiometric étendue, the Hamiltonian phase-space volume, and the number of degrees of freedom are identical symplectic invariants determined by the source-observer configuration. Consequently, Liouville's theorem conserves the degrees of freedom under lossless propagation, and Gromov's non-squeezing theorem imposes a fundamental limit on resolution via the minimum phase-space cell size. Spatial information flows appear as level sets of high mutual information, aligning

What carries the argument

Symplectic invariants of the source-observer configuration that equate radiometric étendue to Hamiltonian phase-space volume and spatial degrees of freedom.

If this is right

  • Liouville's theorem guarantees conservation of the number of degrees of freedom under lossless propagation.
  • Gromov's Non-Squeezing Theorem establishes an irreducible minimum phase-space cell that bounds resolving power.
  • Spatial Information Flows form as level-set curves of high mutual information.
  • For convex sources with rotational symmetry these flows coincide with the optimal sampling curves identified by Bucci et al.
  • Spatial information in electromagnetic fields is governed by the geometry of the source-observer configuration.

Where Pith is reading between the lines

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

  • Antenna or sensor layouts could be designed by calculating phase-space volumes rather than optimizing information measures directly.
  • The geometric approach may extend to other wave systems that preserve symplectic structure during propagation.
  • Controlled experiments could compare predicted spatial information flow curves against measured mutual information in incoherent source setups.

Load-bearing premise

The sources are spatially incoherent and electromagnetic propagation can be modeled as a Hamiltonian system in which phase-space volume corresponds directly to information measures.

What would settle it

Compute the eigenvalues and number of spatial degrees of freedom from both deterministic and stochastic models for a specific incoherent source-observer pair and verify whether they match exactly or whether the number changes after lossless propagation.

Figures

Figures reproduced from arXiv: 2508.16601 by Marco Donald Migliore.

Figure 1
Figure 1. Figure 1: Geometry of the problem in case of non stochastic source; [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The red curves show the ξ hyperbolas for a linear source of length ℓ = 8λ (drawn as a green line). The gray rectangles are "receiving domains". with recent advances in NDoF theory [10] that use a characteristic mode expansion to relate the NDoF to the total shadow area of the source. It is worth noting that the preceding discussion is based on radiometry, which studies the flux of energy (i.e., power in ha… view at source ↗
Figure 3
Figure 3. Figure 3: From a ’spatial information’ point of view, linear source sees the scene in Fig. 2 in a [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Real part of the Wigner distribution along the hyperbola passing through [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Spectral degree of coherence amplitude, |µ(r1, r2)| (left), and mutual information, IMI (r1, r2) (right), for a linear source of length ℓ = 8λ. The point r1, indicated as ¯r in the main text, is marked with a white circle, while r2 ranges across the entire domain of the figure. The subset Qξ of hyperbolas of constant ξ is drawn as dashed curves [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Spectral degree of coherence amplitude, |µ(r1, r2)| (left), and mutual information, IMI (r1, r2) (right) for a linear source of length ℓ = 8λ. The point r1 is marked with a white circle, while r2 ranges across the entire domain of the figure. The subset Qξ of hyperbolas of constant ξ is drawn as dashed curves. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
read the original abstract

This paper unifies deterministic and stochastic Electromagnetic Information Theory (EIT) through symplectic geometry. For spatially incoherent sources, both formulations yield identical eigenvalues and spatial Degrees of Freedom (NDF). This equivalence is shown to be a structural necessity: the radiometric \'etendue, the Hamiltonian phase-space volume, and the NDF are the same symplectic invariant of the source--observer configuration. Liouville's theorem guarantees conservation of the NDF under lossless propagation; Gromov's Non-Squeezing Theorem establishes an irreducible minimum phase-space cell, setting a fundamental geometric bound on resolving power. The physical manifestation of this symplectic structure is the formation of \textit{Spatial Information Flows} (SIFs) -- level-set curves of high mutual information which, for convex sources with rotational symmetry, coincide with the optimal sampling curves of Bucci et al. Spatial information in electromagnetic fields is therefore governed by the geometry of the source--observer configuration, providing the foundation for a geometric theory of electromagnetic information.

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

2 major / 2 minor

Summary. The manuscript claims to unify deterministic and stochastic Electromagnetic Information Theory (EIT) via symplectic geometry. For spatially incoherent sources, both formulations are asserted to yield identical eigenvalues and spatial Degrees of Freedom (NDF) because the radiometric étendue, Hamiltonian phase-space volume, and NDF are the same symplectic invariant of the source-observer configuration. This equivalence is presented as a structural necessity following from Liouville's theorem (conservation of NDF under lossless propagation) and Gromov's non-squeezing theorem (irreducible minimum phase-space cell). The paper introduces Spatial Information Flows (SIFs) as level-set curves of high mutual information that coincide with the optimal sampling curves of Bucci et al. for convex, rotationally symmetric sources.

Significance. If the claimed identification of NDF with the symplectic invariant holds with explicit derivations, the work would supply a geometric, largely parameter-free foundation for electromagnetic information measures and fundamental resolution bounds. The unification of deterministic and stochastic views, together with the link to Liouville and Gromov invariants, could influence antenna theory, imaging, and wave-based communication by emphasizing configuration geometry over specific operator details.

major comments (2)
  1. [Abstract] Abstract: the central claim that the eigenvalues of the deterministic propagation operator and the stochastic mutual-coherence operator both equal the symplectic invariant (radiometric étendue / Hamiltonian phase-space volume) is load-bearing for the unification, yet the manuscript supplies no explicit mapping from the continuous symplectic measure to the discrete spectrum (singular values above threshold or trace of coherence kernel).
  2. [Main text (equivalence argument)] The step converting the phase-space volume into the number of significant eigenvalues is not shown; without it the asserted 'structural necessity' reduces to an identification by definition rather than an independent derivation, undermining the equivalence result for spatially incoherent sources.
minor comments (2)
  1. [SIFs discussion] The precise mathematical definition of Spatial Information Flows (SIFs) as level-set curves and the conditions under which they coincide with Bucci et al.'s curves should be stated more formally, including any symmetry assumptions.
  2. [Notation] Notation for the symplectic invariants (étendue vs. phase-space volume) could be unified earlier to avoid potential reader confusion between radiometric and Hamiltonian formulations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript unifying deterministic and stochastic Electromagnetic Information Theory via symplectic geometry. We address each major comment point by point below, indicating revisions where appropriate to strengthen the explicit derivations.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the eigenvalues of the deterministic propagation operator and the stochastic mutual-coherence operator both equal the symplectic invariant (radiometric étendue / Hamiltonian phase-space volume) is load-bearing for the unification, yet the manuscript supplies no explicit mapping from the continuous symplectic measure to the discrete spectrum (singular values above threshold or trace of coherence kernel).

    Authors: We acknowledge that the abstract and main text would benefit from a clearer, self-contained mapping between the continuous symplectic phase-space volume and the discrete spectrum. In the revised manuscript we will add an explicit derivation in a new subsection of the main text (and a corresponding sentence in the abstract) showing how the Liouville measure, combined with the trace of the mutual-coherence kernel and the singular-value decomposition of the propagation operator, yields the same set of eigenvalues above threshold for spatially incoherent sources. This step uses the fact that both operators share the same symplectic invariant of the source-observer geometry. revision: yes

  2. Referee: [Main text (equivalence argument)] The step converting the phase-space volume into the number of significant eigenvalues is not shown; without it the asserted 'structural necessity' reduces to an identification by definition rather than an independent derivation, undermining the equivalence result for spatially incoherent sources.

    Authors: The referee correctly identifies that the conversion step must be shown explicitly rather than asserted via the shared invariant alone. While the manuscript already invokes Liouville’s theorem for conservation and Gromov’s non-squeezing theorem for the irreducible cell size, we agree that an independent derivation of the eigenvalue count from the phase-space volume is needed. In the revision we will insert a dedicated paragraph (with supporting equations) that discretizes the symplectic volume into the number of significant eigenvalues by integrating the level sets of the mutual-information density and applying the trace-class property of the coherence operator, thereby establishing the equivalence as a derived geometric result rather than a definitional identification. revision: yes

Circularity Check

1 steps flagged

NDF identified with symplectic phase-space volume by definitional assertion of shared invariant

specific steps
  1. self definitional [Abstract]
    "This equivalence is shown to be a structural necessity: the radiometric étendue, the Hamiltonian phase-space volume, and the NDF are the same symplectic invariant of the source--observer configuration."

    The paper claims to derive the identity between deterministic/stochastic eigenvalues and the geometric invariant, yet the quoted statement defines NDF, étendue, and phase-space volume as identical by construction. No separate operator-theoretic step is shown converting the continuous symplectic measure into the discrete spectrum of the integral operator, making the 'structural necessity' equivalent to the input identification rather than a derived result.

full rationale

The paper's central unification rests on the claim that NDF equals the radiometric étendue / Hamiltonian phase-space volume as the same symplectic invariant for incoherent sources. This is presented as a structural necessity following from Liouville and Gromov theorems. However, standard EIT defines NDF via singular values of the propagation operator (or trace of coherence kernel) above threshold; the text equates the discrete spectrum to the continuous volume without exhibiting the explicit isomorphism or eigenvalue derivation, reducing the equivalence to a re-identification of quantities rather than an independent derivation from the operators.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on two standard mathematical theorems and introduces one new conceptual entity without independent falsifiable evidence supplied in the abstract.

axioms (2)
  • standard math Liouville's theorem guarantees conservation of the NDF under lossless propagation
    Invoked to establish that the number of degrees of freedom remains invariant during propagation.
  • standard math Gromov's Non-Squeezing Theorem establishes an irreducible minimum phase-space cell, setting a fundamental geometric bound on resolving power
    Used to derive a lower bound on the phase-space cell size that limits information resolution.
invented entities (1)
  • Spatial Information Flows (SIFs) no independent evidence
    purpose: Level-set curves of high mutual information that coincide with optimal sampling curves for convex rotationally symmetric sources
    New entity introduced as the physical manifestation of the underlying symplectic structure.

pith-pipeline@v0.9.0 · 5699 in / 1427 out tokens · 43488 ms · 2026-05-18T23:24:42.375077+00:00 · methodology

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Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

  1. [1]

    Communication theory and physics,

    D. Gabor, “Communication theory and physics,”Transactions of the IRE Professional Group on Information Theory, vol. 1, no. 1, pp. 48–59, 1953

    work page 1953

  2. [2]

    A mathematical theory of communication,

    C. E. Shannon, “A mathematical theory of communication,”The Bell System Technical Journal, vol. 27, no. 3, pp. 379–423, 1948

    work page 1948

  3. [3]

    Spatial channels for communicating with waves between volumes,

    D. A. Miller, “Spatial channels for communicating with waves between volumes,”Optics letters, vol. 23, no. 21, pp. 1645–1647, 1998

    work page 1998

  4. [4]

    Degrees of freedom in multiple-antenna channels: A signal space approach,

    A. S. Poon, R. W. Brodersen, and D. N. Tse, “Degrees of freedom in multiple-antenna channels: A signal space approach,”IEEE Transactions on Information Theory, vol. 51, no. 2, pp. 523–536, 2005

    work page 2005

  5. [5]

    On electromagnetics and information theory,

    M. D. Migliore, “On electromagnetics and information theory,”IEEE transactions on antennas and propagation, vol. 56, no. 10, pp. 3188–3200, 2008

    work page 2008

  6. [6]

    New aspects of electromagnetic information theory for wireless and antenna systems,

    F. K. Gruber and E. A. Marengo, “New aspects of electromagnetic information theory for wireless and antenna systems,”IEEE Transactions on Antennas and Propagation, vol. 56, no. 11, pp. 3470–3484, 2008

    work page 2008

  7. [7]

    Ontheemdegreesoffreedominscatteringenvironments,

    R.Janaswamy, “Ontheemdegreesoffreedominscatteringenvironments,” IEEE transactions on antennas and propagation, vol. 59, no. 10, pp. 3872–3881, 2011

    work page 2011

  8. [8]

    The information carried by scattered waves: Near-field and nonasymptotic regimes,

    M. Franceschetti, M. D. Migliore, P. Minero, and F. Schettino, “The information carried by scattered waves: Near-field and nonasymptotic regimes,”IEEE Transactions on Antennas and Propagation, vol. 63, no. 7, pp. 3144–3157, 2015

    work page 2015

  9. [9]

    Franceschetti,Wave theory of information

    M. Franceschetti,Wave theory of information. Cambridge University Press, 2017

    work page 2017

  10. [10]

    Degrees of freedom for radiating systems,

    M. Gustafsson, “Degrees of freedom for radiating systems,”IEEE Transactions on Antennas and Propagation, 2025. 16

    work page 2025

  11. [11]

    A phase-space approach for propagating field–field correlation functions,

    G. Gradoni, S. C. Creagh, G. Tanner, C. Smartt, and D. W. Thomas, “A phase-space approach for propagating field–field correlation functions,”New Journal of Physics, vol. 17, no. 9, p. 093027, 2015

    work page 2015

  12. [12]

    Stochastic electromagnetic field propagation—measurement and modelling,

    G. Gradoni, J. Russer, M. H. Baharuddin, M. Haider, P. Russer, C. Smartt, S. C. Creagh, G. Tanner, and D. W. Thomas, “Stochastic electromagnetic field propagation—measurement and modelling,”Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 376, no. 2134, p. 20170455, 2018

    work page 2018

  13. [13]

    Capacity for electromagnetic information theory,

    Z. Wan, J. Zhu, Z. Zhang, and L. Dai, “Capacity for electromagnetic information theory,” arXiv preprint arXiv:2111.00496, 2021

    work page arXiv 2021

  14. [14]

    Spatial characterization of electromagnetic random channels,

    A. Pizzo, L. Sanguinetti, and T. L. Marzetta, “Spatial characterization of electromagnetic random channels,”IEEE Open Journal of the Communications Society, vol. 3, pp. 847–866, 2022

    work page 2022

  15. [15]

    Mutual information for electromagnetic information theory based on random fields,

    Z. Wan, J. Zhu, Z. Zhang, L. Dai, and C.-B. Chae, “Mutual information for electromagnetic information theory based on random fields,”IEEE Transactions on Communications, vol. 71, no. 4, pp. 1982–1996, 2023

    work page 1982

  16. [16]

    On the degrees of freedom of scattered fields,

    O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,”IEEE transactions on Antennas and Propagation, vol. 37, no. 7, pp. 918–926, 1989

    work page 1989

  17. [17]

    On the spatial bandwidth of scattered fields,

    ——, “On the spatial bandwidth of scattered fields,”IEEE transactions on antennas and propagation, vol. 35, no. 12, pp. 1445–1455, 1987

    work page 1987

  18. [18]

    Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples,

    O. M. Bucci, C. Gennarelli, and C. Savarese, “Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples,”IEEE Transactions on Antennas and Propagation, vol. 46, no. 3, pp. 351–359, 1998

    work page 1998

  19. [19]

    On the role of the number of degrees of freedom of the field in mimo channels,

    M. D. Migliore, “On the role of the number of degrees of freedom of the field in mimo channels,”IEEE Transactions on Antennas and Propagation, vol. 54, no. 2, pp. 620–628, 2006

    work page 2006

  20. [20]

    Mandel and E

    L. Mandel and E. Wolf,Optical coherence and quantum optics. Cambridge university press, 1995

    work page 1995

  21. [21]

    Kolmogorov and S

    A. Kolmogorov and S. Fomin,Elements of the theory of functions and functional analysis. Dover Publications, 1999

    work page 1999

  22. [22]

    The world beneath the physical layer: An introduction to the deep physical layer,

    M. D. Migliore, “The world beneath the physical layer: An introduction to the deep physical layer,”IEEE Access, vol. 9, pp. 77106–77126, 2021

    work page 2021

  23. [23]

    Tse and P

    D. Tse and P. Viswanath,Fundamentals of Wireless Communication. New York, NY: Cambridge University Press, 2005

    work page 2005

  24. [24]

    R. G. Gallager,Information theory and reliable communication. Springer, 1968, vol. 588

    work page 1968

  25. [25]

    Spatial coherence properties of planar antennas,

    M. A. Plonus, “Spatial coherence properties of planar antennas,”IEEE transactions on antennas and propagation, vol. 39, no. 7, pp. 892–897, 1991

    work page 1991

  26. [26]

    A wigner function approach for describing the radiation of complex sources,

    G. Gradoni, S. C. Creagh, and G. Tanner, “A wigner function approach for describing the radiation of complex sources,” in2014 IEEE International Symposium on Electromagnetic Compatibility (EMC). IEEE, 2014, pp. 882–887

    work page 2014

  27. [27]

    Wigner distribution in optics,

    M. J. Bastiaanset al., “Wigner distribution in optics,” pp. 1–44, 2009. 17

    work page 2009

  28. [28]

    Heisenberg proof of the balian-low theorem,

    G. Battle, “Heisenberg proof of the balian-low theorem,”Letters in Mathematical Physics, vol. 15, no. 2, pp. 175–177, 1988

    work page 1988

  29. [29]

    Uncertainty principles for orthonormal sequences,

    P. Jaming and A. M. Powell, “Uncertainty principles for orthonormal sequences,”Journal of Functional Analysis, vol. 243, no. 2, pp. 611–630, 2007

    work page 2007

  30. [30]

    On landau’s eigenvalue theorem and information cut-sets,

    M. Franceschetti, “On landau’s eigenvalue theorem and information cut-sets,”IEEE Trans- actions on Information Theory, vol. 61, no. 9, pp. 5042–5051, 2015

    work page 2015

  31. [31]

    R. E. Collin,Antennas and radiowave propagation. McGraw-Hill Book Company, 1985

    work page 1985

  32. [32]

    R. F. Harrington, Time harmonic electromagnetic fields. New York City, New York, McGraw-Hill, 1961

    work page 1961

  33. [33]

    Characterization of mimo antennas using spherical vector waves,

    M. Gustafsson and S. Nordebo, “Characterization of mimo antennas using spherical vector waves,”IEEE Transactions on Antennas and Propagation, vol. 54, no. 9, pp. 2679–2682, 2006

    work page 2006

  34. [34]

    Degrees of freedom and characteristic modes a perspective and opportunities: Estimates for radiating and arbitrarily shaped objects,

    M. Gustafsson and J. Lundgren, “Degrees of freedom and characteristic modes a perspective and opportunities: Estimates for radiating and arbitrarily shaped objects,”IEEE Antennas and Propagation Magazine, 2024

    work page 2024

  35. [35]

    Degrees of freedom and sampling representation of elec- tromagnetic fields: Concepts and applications,

    O. M. Bucci and M. D. Migliore, “Degrees of freedom and sampling representation of elec- tromagnetic fields: Concepts and applications,”IEEE Antennas and Propagation Magazine, 2024

    work page 2024

  36. [36]

    R. J. Koshel,Illumination Engineering: design with nonimaging optics. John Wiley & Sons, 2012

    work page 2012

  37. [37]

    A radiometric generalization of the van Cittert-Zernike theorem for fields generated by sources of arbitrary state of coherence,

    E. Wolf and W. H. Carter, “A radiometric generalization of the van Cittert-Zernike theorem for fields generated by sources of arbitrary state of coherence,”Optics Communications, vol. 16, no. 3, pp. 297–302, 1976

    work page 1976

  38. [38]

    The thermodynamics of optical étendue,

    T. Markvart, “The thermodynamics of optical étendue,”Journal of Optics A: pure and applied optics, vol. 10, no. 1, p. 015008, 2007

    work page 2007

  39. [39]

    Horse (electromagnetics) is more important than horseman (information) for wireless transmission,

    M. D. Migliore, “Horse (electromagnetics) is more important than horseman (information) for wireless transmission,”IEEE Transactions on Antennas and Propagation, vol. 67, no. 4, pp. 2046–2055, 2018

    work page 2046

  40. [40]

    Classical and quantum processing in the deep physical layer,

    ——, “Classical and quantum processing in the deep physical layer,”IEEE Access, 2023

    work page 2023

  41. [41]

    Information flows at the deep physical layer level,

    ——, “Information flows at the deep physical layer level,” inA Glimpse Beyond 5G in Wireless Networks. Springer, 2022, pp. 59–87

    work page 2022

  42. [42]

    Electromagnetic inverse scattering: Retrievable information and measurement strategies,

    O. Bucci and T. Isernia, “Electromagnetic inverse scattering: Retrievable information and measurement strategies,”Radio science, vol. 32, no. 6, pp. 2123–2137, 1997

    work page 1997

  43. [43]

    Sampling approach for singular system computa- tion of a radiation operator,

    R. Solimene, M. A. Maisto, and R. Pierri, “Sampling approach for singular system computa- tion of a radiation operator,”Journal of the Optical Society of America A, vol. 36, no. 3, pp. 353–361, 2019

    work page 2019

  44. [44]

    Electromagnetic signal and information theory,

    M. Di Renzo and M. D. Migliore, “Electromagnetic signal and information theory,”IEEE BITS the Information Theory Magazine, 2024. 18

    work page 2024