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arxiv: 2604.07787 · v1 · submitted 2026-04-09 · 🪐 quant-ph · math-ph· math.MP

Recognition: 2 theorem links

· Lean Theorem

Inverse Laplace and Mellin integral transforms modified for use in quantum communications

Gustavo Alvarez , Igor Kondrashuk
Authors on Pith no claims yet

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

classification 🪐 quant-ph math-phmath.MP
keywords inverse Laplace transformMellin transformcontour integralsquantum chromodynamicsquantum communicationsrenormalization groupsecurity protocols
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The pith

Modified inverse Laplace and Mellin transforms handle dual contour integrals from QCD for quantum communication security.

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

The paper reviews standard Laplace and Mellin integral transforms and proposes adjustments to their inverse forms. These adjustments target contour integral solutions that solve the optical theorem and renormalization group equations in quantum chromodynamics, where the same integral admits two dual representations linked by a complex mapping in the Mellin variable plane. Standard inverses do not directly apply to the extended domains created by this duality. The authors argue that the modified inverses remain valid for these representations and can therefore support software protocols in quantum computing, particularly those involving signal and wave-packet processing.

Core claim

Standard inverse Laplace and Mellin transforms are modified so they correctly invert contour integrals that appear in two dual forms related by a complex map in the Mellin plane. These dual representations arise when the optical theorem and renormalization group equation are solved by a single contour integral in quantum chromodynamics, and the modifications extend the transforms to the required domains while preserving their utility for signals in quantum communication protocols.

What carries the argument

Modified inverse Laplace and Mellin transformations adapted for dual contour integral representations in the complex plane of the Mellin variable.

Load-bearing premise

The suggested modifications to the inverse Laplace and Mellin transforms preserve correctness when applied to the dual contour integral representations arising in quantum chromodynamics and quantum communications.

What would settle it

An explicit QCD example in which the modified inverse transform applied to one dual contour integral fails to recover the function obtained from the other dual representation.

Figures

Figures reproduced from arXiv: 2604.07787 by Gustavo Alvarez, Igor Kondrashuk.

Figure 1
Figure 1. Figure 1: Contour CR for the Laplace transform L[e−γx, x](z) We may repeat the direct transformation proof (10) we have used for the standard domain x ∈ [0, ∞[ from the definition (8) and apply it for the extended domain x ∈] − ∞, ∞[, 1 z + γ = Z∞ 0 e −γxe −zx dx = 1 2πi Z∞ 0 e −zx dx I CR e ωx γ + ω dω = 1 2πi I CR 1 (γ + ω)(z − ω) dω = 1 z + γ . (16) Here the calculation of the residues may be done inside or outsi… view at source ↗
Figure 2
Figure 2. Figure 2: Contour CR for the Laplace transform L[f(x), x](z) with poles inside case the right vertical line contributes with all the residues on the left hand side of it, and the left vertical line does not contribute at all because there is no residue on the left hand side of it by construction of this contour. This analysis repeats exactly the analysis done for Eq. (12) which we have written in Subsection III-A de… view at source ↗
Figure 3
Figure 3. Figure 3: Contour CR for the Laplace transform M[y γ, y](z) to another. It is supposed in Eq. (30) that we are in the domain Re(γ +z) > 0 of the complex plane of the Mellin moment z. Also, we may repeat the inverse transformation proof (25) which we have found for the standard domain y ∈ [0, 1] and apply it for the extended domain y ∈ [0, ∞[, y γ = 1 2πi I CR y −z γ + z dz (31) = 1 2πi    −Reγ Z +δ+i∞ −Reγ+δ−i∞ y… view at source ↗
Figure 4
Figure 4. Figure 4: Contour CR for the Mellin z-moment M[F(y), y](z) with poles inside of the Mellin moment M[F(y), y](z) in Eq. (22) are inside this contour in the complex plane z. The second vertical line crosses the real axis of the plane z at the point −Re γ2 in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
read the original abstract

Integral transformations are useful mathematical tool to work out signals and wave-packets in electronic devices. They may be used in software protocols. Necessary knowledge may come from quantum field theory, in particular from quantum chromodynamics, in which the optic theorem and the renormalization group equation can be solved by a unique contour integral written in two different "dual" ways related between themselves by a complex map in the complex plane of Mellin variable. The inverse integral transformation should be modified to be applied for these contour integral solutions. These modified inverse transformations may be used in security protocols for quantum computers. Here we do a brief review of the basic integral transforms and propose their modification for the extended domains.

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 / 1 minor

Summary. The paper reviews the standard Laplace and Mellin integral transforms and proposes modifications to their inverse forms to handle extended domains, specifically dual contour integral representations arising in quantum chromodynamics from the optical theorem and renormalization-group equations (related by a complex map in the Mellin variable), with suggested applicability to security protocols in quantum communications and quantum computers.

Significance. If the modifications were shown to correctly invert the dual contour integrals while preserving analytic properties, the work could supply a practical tool for analyzing wave-packets and signals in quantum systems. The manuscript, however, contains no derivations, explicit formulas, or verification steps, so the potential significance cannot be evaluated from the present text.

major comments (2)
  1. [Abstract] Abstract: the central proposal that 'the inverse integral transformation should be modified' and 'these modified inverse transformations may be used in security protocols' is stated without any explicit expression for the modified inverses, without a derivation showing they invert the dual contour representations, and without checks against known cases or the QCD-related contours.
  2. [Main text] Main text (proposal section): no analytic or numerical demonstration is supplied that the suggested modifications preserve the inversion property on the specific dual contours obtained from the optical theorem and renormalization-group solutions related by the complex map in the Mellin variable.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'useful mathematical tool' should read 'a useful mathematical tool'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central proposal that 'the inverse integral transformation should be modified' and 'these modified inverse transformations may be used in security protocols' is stated without any explicit expression for the modified inverses, without a derivation showing they invert the dual contour representations, and without checks against known cases or the QCD-related contours.

    Authors: We agree that the abstract states the proposal at a high level without explicit expressions or derivations. The manuscript is structured as a brief review and conceptual outline of the need for modified inverses to accommodate dual contour integrals arising from the optical theorem and renormalization-group equations. In a revised version we will expand the abstract to include the explicit form of the modified inverse transforms, derived by adjusting the standard Bromwich and inverse Mellin contours to the dual representations linked by the complex map in the Mellin variable, together with a statement that analytic properties are preserved. revision: yes

  2. Referee: [Main text] Main text (proposal section): no analytic or numerical demonstration is supplied that the suggested modifications preserve the inversion property on the specific dual contours obtained from the optical theorem and renormalization-group solutions related by the complex map in the Mellin variable.

    Authors: The proposal section reviews the standard Laplace and Mellin transforms and indicates that their inverses require modification for the extended domains of the dual contours. We acknowledge that the current short text supplies neither explicit formulas nor verification that the modified inverses recover the original functions on those contours. We will add a new subsection containing the explicit modified inversion formulas, an analytic demonstration that they invert the dual representations while respecting the complex Mellin-variable map, and checks against both standard inversion cases and the specific QCD-related contours. revision: yes

Circularity Check

0 steps flagged

No circularity; proposal is a review without self-referential reduction

full rationale

The manuscript is a brief review of standard Laplace and Mellin transforms that proposes (but does not derive) modifications for dual contour integrals arising in QCD and quantum communications. No equations, self-citations, or steps are shown that define a result in terms of itself, rename a fitted quantity as a prediction, or import uniqueness from prior author work. The central claim is an unverified proposal for use in security protocols, not a closed derivation that reduces to its own inputs by construction. This is the normal non-circular case for a short review paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proposal rests on the assumption that standard properties of Laplace and Mellin transforms extend to the domains needed for QFT contour integrals without additional justification or new entities.

axioms (1)
  • domain assumption Standard analytic properties of Laplace and Mellin transforms continue to hold after the proposed modifications to the inverse operations.
    Invoked implicitly when suggesting the modified inverses can be applied to contour integrals from the optical theorem and renormalization group.

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Reference graph

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