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PreprintCC BY-NC-SA 4.0arXiv

Algebraic Kolmogorov--Arnold representation theorem for quantum measurement

  • Sviatoslav V. Dzhenzher
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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 analyse the stability of this Quantum Kolmogorov--Arnold (QKA) representation under adversarial perturbations. In stark contrast to the pathological instabilities and severe reparameterisation 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.

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Cite

BibTeXRIS
BibTeX
@misc{dzhenzher2026algebraic,
  title = {{Algebraic Kolmogorov--Arnold representation theorem for quantum measurement}},
  author = {Sviatoslav V. Dzhenzher},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.09584},
  eprint = {2606.09584},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.09584},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Algebraic Kolmogorov--Arnold representation theorem for quantum measurement
AU  - Sviatoslav V. Dzhenzher
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2606.09584
UR  - https://arxiv.org/abs/2606.09584
AB  - 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 analyse the stability of this Quantum Kolmogorov--Arnold (QKA) representation under adversarial perturbations. In stark contrast to the pathological instabilities and severe reparameterisation 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.
ER  -