Algebraic Kolmogorov--Arnold representation theorem for quantum measurement

• URL: http://arxiv.org/abs/2606.09584v1
• Date: Mon, 08 Jun 2026 14:57:41 GMT
• Title: Algebraic Kolmogorov--Arnold representation theorem for quantum measurement
• Abstract summary: We connect the classical Kolmogorov-Arnold-Arnold representation theorem to quantum information theory.<n>We prove that any target physical property of an unentangled multi-qubit product state can be exactly using a finite, fixed set of local inner observables.<n>We prove that the representation remains stable against bounded physical perturbations acting on the inner measurement operators.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

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