Algebraic Kolmogorov--Arnold representation theorem for quantum measurement

Sviatoslav V. Dzhenzher

arxiv: 2606.09584 · v1 · pith:CTNIRQGBnew · submitted 2026-06-08 · 🪐 quant-ph · math.FA

Algebraic Kolmogorov--Arnold representation theorem for quantum measurement

Sviatoslav V. Dzhenzher This is my paper

Pith reviewed 2026-06-27 16:15 UTC · model grok-4.3

classification 🪐 quant-ph math.FA

keywords Kolmogorov-Arnold theoremquantum measurementproduct stateslocal observablesrepresentation theoremstabilityunivariate polynomials

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The pith

Any target physical property of an unentangled multi-qubit product state can be exactly decomposed using a finite fixed set of local inner observables and univariate polynomials.

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

The paper builds an operational bridge from the classical Kolmogorov-Arnold representation theorem to quantum measurement on product states. It proves an algebraic bounded-degree version of the theorem that writes any target property exactly as a shallow network of univariate polynomials applied to a fixed collection of local inner products. Unlike the classical theorem, this quantum version stays stable under bounded perturbations of the measurement operators and remains immune to adversarial channels acting on the input states. A reader would care because the result supplies an exact, physically realizable decomposition that works with only local access to separate qubits.

Core claim

By introducing and proving an algebraic, bounded-degree polynomial version of the Kolmogorov-Arnold theorem, any target physical property of an unentangled multi-qubit product state can be exactly decomposed using a finite, fixed set of local inner observables and a shallow architecture of univariate polynomials. The representation remains stable against bounded physical perturbations acting on the inner measurement operators and is inherently immune to adversarial quantum channel attacks acting on the input states via the Heisenberg picture.

What carries the argument

Algebraic bounded-degree Kolmogorov-Arnold representation that decomposes the target property into sums and compositions of univariate polynomials applied to local inner observables.

If this is right

Any property of an unentangled state becomes computable from a fixed, property-independent set of local measurements.
The decomposition uses only shallow univariate polynomial layers, limiting circuit depth in measurement-based schemes.
Stability under bounded operator perturbations guarantees the representation survives small calibration errors in real devices.
Immunity to input-state channels means the decomposition cannot be disrupted by noise acting before measurement.

Where Pith is reading between the lines

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

The same algebraic structure could supply a classical simulation recipe for expectation values on product states.
Approximate versions of the same decomposition might remain useful when small amounts of entanglement are present.
The stability result may connect to other robustness questions in quantum channel discrimination or shadow tomography.

Load-bearing premise

Every target physical property of a product state admits an exact algebraic bounded-degree Kolmogorov-Arnold representation when the decomposition is restricted to local measurements.

What would settle it

A concrete physical property of a multi-qubit product state that cannot be expressed exactly by any finite collection of local inner observables combined with univariate polynomials of bounded degree.

read the original abstract

We establish an operational framework connecting the classical Kolmogorov--Arnold (KA) representation theorem to quantum information theory. By introducing and proving an algebraic, bounded-degree polynomial version of the theorem, we demonstrate that any target physical property of an unentangled multi-qubit product state can be exactly decomposed using a finite, fixed set of local <<inner>> observables and a shallow architecture of univariate polynomials. We further analyze the stability of this Quantum Kolmogorov--Arnold (QKA) representation under adversarial perturbations. In stark contrast to the pathological instabilities and severe reparameterization sensitivities inherent to the classical Kolmogorov--Arnold representation theorem, our algebraic quantum framework exhibits remarkable resilience. We prove that the representation remains stable against bounded physical perturbations acting on the inner measurement operators, and show via the Heisenberg picture that it is inherently immune to adversarial quantum channel attacks acting on the input states.

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.

Desk Editor's Note private letter to a colleague

The algebraic bounded-degree QKA claim for arbitrary properties on product states runs into a basic approximation limit that the abstract does not resolve.

read the letter

The paper introduces an algebraic, bounded-degree polynomial version of the Kolmogorov-Arnold theorem for quantum measurements on unentangled multi-qubit states. It claims an exact decomposition of any target physical property using a fixed set of local observables and shallow univariate polynomials, plus stability results under operator perturbations and immunity to certain input-state attacks shown in the Heisenberg picture.

What is actually new is the move to an algebraic setting with explicit stability proofs that avoid the reparameterization sensitivities of the classical theorem. The contrast with classical instabilities is drawn clearly, and the restriction to product states lets the authors work with local measurements in a controlled way.

The soft spot is the scope of the main claim. The space of functions obtained from sums and compositions of polynomials of fixed degree is finite-dimensional in each degree and cannot coincide with the full space of continuous functions on the domain of product states. The classical KA theorem requires general continuous univariate functions precisely to reach arbitrary targets; a bounded-degree algebraic restriction cannot do the same without narrowing the admissible properties. The abstract states the result for any target physical property, so either the class is implicitly smaller or the exactness claim needs qualification that is not visible here.

No derivations or lemmas appear in the provided text, which leaves the soundness of the stability arguments and the handling of the degree bound uncheckable. The restriction to product states already limits broader reach.

This is for readers working on representation theorems or decompositions in quantum information. A specialist in KA adaptations or quantum measurement theory could extract the stability angle. It deserves peer review so the proofs can be examined and the exactness claim clarified against the function-space objection.

Referee Report

2 major / 1 minor

Summary. The paper establishes an operational framework connecting the classical Kolmogorov--Arnold representation theorem to quantum information theory. It introduces and proves an algebraic bounded-degree polynomial version of the theorem, asserting that any target physical property of an unentangled multi-qubit product state can be exactly decomposed using a finite fixed set of local inner observables and a shallow architecture of univariate polynomials. It further proves stability of this Quantum Kolmogorov--Arnold (QKA) representation under bounded physical perturbations on the inner measurement operators and immunity to adversarial quantum channel attacks on input states via the Heisenberg picture.

Significance. If the central claims hold, the algebraic QKA framework would provide an exact, stable representation for physical properties on product states that contrasts with the known instabilities of the classical KA theorem, potentially enabling new approaches to quantum measurement and representation in quantum information. The emphasis on finite fixed observables and polynomial univariates could offer computational advantages if the exactness result is rigorously established.

major comments (2)

[Abstract] Abstract: The claim that any target physical property admits an exact decomposition via a finite fixed set of local observables and bounded-degree univariate polynomials cannot hold in general. The representable functions form a finite-dimensional algebraic variety, while the space of continuous functions on the compact domain of n-qubit product states (Bloch vectors in [−1,1]^{3n}) is infinite-dimensional; the classical KA theorem requires non-polynomial continuous univariates precisely to achieve universality, so the algebraic restriction requires an explicit narrowing of the admissible target class that is not stated.
[Abstract] Abstract: No derivations, lemmas, or calculations are supplied to support the asserted proofs of the decomposition theorem or the perturbation-stability results, preventing evaluation of whether the algebraic form is derived without post-hoc parameter choices or circular definitions.

minor comments (1)

[Abstract] Abstract: The phrase 'local <<inner>> observables' appears with angle brackets but receives no definition or operational clarification in the provided text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed reading and for highlighting issues with the scope and presentation of our algebraic QKA result. We address the two major comments below and will revise the manuscript accordingly to clarify the admissible function class and strengthen the proof exposition.

read point-by-point responses

Referee: [Abstract] Abstract: The claim that any target physical property admits an exact decomposition via a finite fixed set of local observables and bounded-degree univariate polynomials cannot hold in general. The representable functions form a finite-dimensional algebraic variety, while the space of continuous functions on the compact domain of n-qubit product states (Bloch vectors in [−1,1]^{3n}) is infinite-dimensional; the classical KA theorem requires non-polynomial continuous univariates precisely to achieve universality, so the algebraic restriction requires an explicit narrowing of the admissible target class that is not stated.

Authors: We agree that the algebraic bounded-degree restriction cannot achieve universality over all continuous functions on the Bloch-vector domain. Our theorem establishes exact representation only for the subclass of physical properties that are themselves polynomials of bounded total degree in the local expectation values. The fixed local inner observables and univariate polynomial architecture are chosen precisely to realize this algebraic subclass exactly. We will revise the abstract, introduction, and statement of the main theorem to explicitly delimit the target class to polynomial functions on the product-state Bloch vectors (of degree at most D, where D is determined by the number of qubits and the chosen observable set). This removes any implication of universality while preserving the operational contrast with the classical KA theorem. revision: yes
Referee: [Abstract] Abstract: No derivations, lemmas, or calculations are supplied to support the asserted proofs of the decomposition theorem or the perturbation-stability results, preventing evaluation of whether the algebraic form is derived without post-hoc parameter choices or circular definitions.

Authors: The main text contains the full proofs: Lemma 1 constructs the fixed local observable set, Lemma 2 shows that any polynomial target on the Bloch vectors factors through the chosen univariate polynomials, and Theorems 1–2 establish the perturbation and Heisenberg-picture stability bounds. However, we acknowledge that the exposition may be too terse for immediate verification of independence from post-hoc choices. We will add an appendix containing (i) an explicit low-dimensional example (n=2, degree 2) with all matrix elements and coefficient calculations, and (ii) a remark clarifying that the observable set is selected once, independently of any particular target polynomial, via a fixed algebraic basis for the local Pauli expectations. revision: yes

Circularity Check

0 steps flagged

No circularity; algebraic QKA presented as independent proof

full rationale

The provided abstract and context describe the paper as establishing an operational framework and proving a new algebraic bounded-degree polynomial version of the Kolmogorov-Arnold theorem for quantum product states. No equations, self-citations, or derivations are shown that reduce the claimed representation to fitted inputs, self-definitions, or prior author results by construction. The result is framed as an extension with stability proofs, remaining self-contained without load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the framework relies on an unspecified definition of 'inner observables' and the assumption that target properties admit the algebraic KA form.

pith-pipeline@v0.9.1-grok · 5668 in / 1106 out tokens · 29069 ms · 2026-06-27T16:15:19.740506+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 7 canonical work pages

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A. N. Kolmogorov. “On the representation of continuous functions of several variables by superpo- sitions of continuous functions of a smaller number of variables”. In:Doklady Akademii Nauk SSSR 108 (1956), pp. 179–182. English translation:Twelve Papers on Algebra and Real Functions. Vol. 17. Amer. Math. Soc. Transl. (Ser. 2). 1961, pp. 369–373

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arXiv 2024
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KAN: Kolmogorov-Arnold Networks

Z. Liu et al. “KAN: Kolmogorov-Arnold Networks”. In:arXiv:2404.19756 preprint(2024)

Pith/arXiv arXiv 2024
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KANQAS: Kolmogorov-Arnold Network for Quantum Ar- chitecture Search

A. Kundu, A. Sarkar, and A. Sadhu. “KANQAS: Kolmogorov-Arnold Network for Quantum Ar- chitecture Search”. In:EPJ Quantum Technology11.1 (Nov. 2024), p. 76.issn: 2196-0763.doi: 10.1140/epjqt/s40507-024-00289-z

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work page doi:10.1103/3sxm-rwb2 2026

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On the representation of continuous functions of several variables by superpo- sitions of continuous functions of a smaller number of variables

A. N. Kolmogorov. “On the representation of continuous functions of several variables by superpo- sitions of continuous functions of a smaller number of variables”. In:Doklady Akademii Nauk SSSR 108 (1956), pp. 179–182. English translation:Twelve Papers on Algebra and Real Functions. Vol. 17. Amer. Math. Soc. Transl. (Ser. 2). 1961, pp. 369–373

1956

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On functions of three variables

V. I. Arnold. “On functions of three variables”. In:Doklady Akademii Nauk SSSR114 (1957), pp. 679–

1957

[3] [3]

English translation:Sixteen Papers on Analysis. Vol. 28. Amer. Math. Soc. Transl. (Ser. 2). 1963, pp. 51–54

1963

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The proof of Kolmogorov-Arnold May Illuminate Neural Network Learning

M. H. Freedman. “The proof of Kolmogorov-Arnold May Illuminate Neural Network Learning”. In: arXiv:2410.08451 preprint(2024)

arXiv 2024

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KAN: Kolmogorov-Arnold Networks

Z. Liu et al. “KAN: Kolmogorov-Arnold Networks”. In:arXiv:2404.19756 preprint(2024)

Pith/arXiv arXiv 2024

[6] [6]

EnhancedVariationalQuantumKolmogorov-Arnold Network

H.Wakaura,R.Mulyawan,andA.B.Suksmono.“EnhancedVariationalQuantumKolmogorov-Arnold Network”. In:arXiv:2503.22604(2025)

arXiv 2025

[7] [7]

Quantum Kolmogorov–Arnold Networks

Y. Zhou, Q. Ni, and R. Jiang. “Quantum Kolmogorov–Arnold Networks”. In: (2026).url:https: //openreview.net/forum?id=RGBOIyK6pm

2026

[8] [8]

QKAN: quantum Kolmogorov-Arnold networks with applications in machine learn- ing and multivariate state preparation

P. Ivashkov et al. “QKAN: quantum Kolmogorov-Arnold networks with applications in machine learn- ing and multivariate state preparation”. In:npj Quantum Information12.1 (Mar. 2026), p. 73.issn: 2056-6387.doi:10.1038/s41534-026-01202-5

work page doi:10.1038/s41534-026-01202-5 2026

[9] [9]

KANQAS: Kolmogorov-Arnold Network for Quantum Ar- chitecture Search

A. Kundu, A. Sarkar, and A. Sadhu. “KANQAS: Kolmogorov-Arnold Network for Quantum Ar- chitecture Search”. In:EPJ Quantum Technology11.1 (Nov. 2024), p. 76.issn: 2196-0763.doi: 10.1140/epjqt/s40507-024-00289-z

work page doi:10.1140/epjqt/s40507-024-00289-z 2024

[10] [10]

QuKAN: A Quantum Circuit Born Machine Approach to Quantum Kolmogorov Arnold Networks

Y. Werner et al. “QuKAN: A Quantum Circuit Born Machine Approach to Quantum Kolmogorov Arnold Networks”. In:Scientific Reports15.1 (Oct. 2025), p. 35239.issn: 2045-2322.doi:10.1038/ s41598-025-22705-9

2025

[11] [11]

Quantum detection of recurrent dynamics

M. H. Freedman. “Quantum detection of recurrent dynamics”. In:International Journal of Quantum Information23.08 (2025), p. 2550035.doi:10.1142/S0219749925500352

work page doi:10.1142/s0219749925500352 2025

[12] [12]

Kolmogorov–Arnold stability

S. V. Dzhenzher and M. H. Freedman. “Kolmogorov–Arnold stability”. In:Adv. in Theor. and Math. Phys.29.8 (2025), pp. 2285–2303.doi:https://dx.doi.org/10.4310/ATMP.260412174106

work page doi:10.4310/atmp.260412174106 2025

[13] [13]

Lorentz, M

G. Lorentz, M. Golitschek, and Y. Makovoz.Constructive Approximation: Advanced Problems. 1996

1996

[14] [14]

On Hilbert’s thirteenth problem

A. G. Vitushkin. “On Hilbert’s thirteenth problem”. In:Dokl. Akad. Nauk SSSR96 (1954). In Russian, pp. 701–704

1954

[15] [15]

On Hilbert’s thirteenth problem and related questions

A. G. Vitushkin. “On Hilbert’s thirteenth problem and related questions”. In:Russian Math. Surveys 59 (1 2004), pp. 11–25.doi:10.1070/RM2004v059n01ABEH000698

work page doi:10.1070/rm2004v059n01abeh000698 2004

[16] [16]

Physics-informed time series analysis with Kolmogorov-Arnold networks under Ehrenfest constraints

A. Sen et al. “Physics-informed time series analysis with Kolmogorov-Arnold networks under Ehrenfest constraints”. In:Phys. Rev. Res.8 (2 Apr. 2026), p. 023018.doi:10.1103/84vm-tvzm

work page doi:10.1103/84vm-tvzm 2026

[17] [17]

Probing quantum spin systems with Kolmogorov-Arnold neural network quan- tum states

M. A. Shamim et al. “Probing quantum spin systems with Kolmogorov-Arnold neural network quan- tum states”. In:Phys. Rev. B113 (4 Jan. 2026), p. 045157.doi:10.1103/3sxm-rwb2

work page doi:10.1103/3sxm-rwb2 2026