arXiv subjects

Sviatoslav V. Dzhenzher

Publications and source records attributed to Sviatoslav V. Dzhenzher.

7 recordsLinked to original sources

arXiv2026

Algebraic Kolmogorov--Arnold representation theorem for quantum measurement

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.

quant-ph↗

arXiv2026

Quantum channels on duals of von Neumann algebras in the Schrödinger picture

The theory of quantum channels is traditionally studied either on finite-dimensional state spaces or within the Heisenberg picture as completely positive maps on C^*-algebras. In this paper, we consider quantum channels as completely positive maps on the duals of general von Neumann algebras in the Schrodinger picture. We investigate the construction of such channels through Pettis integrals using representations of topological groups.

math.OA↗

arXiv2026

Kolmogorov--Arnold stability for discontinuous functions

Here we investigate the stability of the Kolmogorov--Arnold representation theorem (KART) under adversarial reparameterisations of the hidden layer for multivariate discontinuous and unbounded functions. Our results provide a rigorous mathematical foundation for the structural robustness of modern deep learning architectures, such as Kolmogorov--Arnold Networks (KANs), under adversarial configurations.

cs.LG↗

arXiv2026

Quantum Kolmogorov--Arnold representation theorem for continuous unitary-valued maps

The classical Kolmogorov--Arnold representation theorem states that any continuous multivariate function can be exactly decomposed into a finite composition of univariate continuous functions and addition operations. This foundational result has recently inspired the development of Kolmogorov--Arnold Networks (KANs) in classical machine learning, as well as their extensions into the quantum domain (QKANs). In this paper, we establish two quantum analogues of the Kolmogorov--Arnold representation theorem for continuous unitary-valued maps of several variables within an open $1$-neighbourhood of the identity matrix $O_1(\mathbf{I}) \subset \mathcal{U}(n)$. First, we prove a representation theorem that yields an exact additive decomposition inside the matrix exponent of anti-Hermitian-valued maps. Second, due to the non-commutative nature of quantum operators, we derive a factorised version expressing the target unitary map as a finite sequential product of univariate matrix exponentials. Finally, we provide a concrete topological counterexample based on the lifting property of $\mathcal{SU}(2)$ to demonstrate that these local representation theorems cannot be globally extended to the entire unitary group $\mathcal{U}(n)$ without encountering fundamental structural obstructions.

quant-ph↗

arXiv2026

Kolmogorov--Arnold against bounded translations

Historically originating from Hilbert's 13th problem, the Kolmogorov-Arnold representation theorem (KART) has recently experienced a major revitalisation through its applications to neural networks, specifically Kolmogorov-Arnold Networks (KANs). While the exact representation is well established, its stability under continuous adversarial perturbations of the hidden layer remains a critical open question. In this paper, we investigate the robustness of KART against bounded adversarial translations. We provide an explicit, self-contained, and constructive proof of an approximate representation using fixed, piecewise linear inner functions. Crucially, our construction employs a single outer function that remains invariant for all summands and is independent of the specific adversarial translation, provided its maximum bound is known a priori.

cs.LG↗

arXiv2026

Singularising preserving countable additivity quantum channels on quantum measurable cardinals

We investigate the structural and dynamical properties of a class of quantum channels on von Neumann algebras induced by averaging over operator groups via the Pettis integral. Utilizing the classical Yosida--Hewitt decomposition, we focus on the interplay between the set-theoretic properties of the underlying space and the topological nature of the resulting quantum states. We establish a suffucient condition under which a channel preserves $σ$-additivity and exhibits a singularising property, completely suppressing the normal component of any incoming state. In conjunction with the theory of Ulam real-valued measurable cardinals, this framework reveals a novel phenomenon: the existence of quantum channels that transform normal states into singular yet strictly $σ$-additive states. Furthermore, we analyze the structural constraints on the preservation of state purity imposed by the cardinality of the continuum, and extend our constructions to invariant measures on groups and their unitary representations, establishing the convergence of their Cesàro averages in the strong operator topology.

quant-ph↗

arXiv2026

Kolmogorov--Arnold stability

Regarding the representation theorem of Kolmogorov and Arnold (KA) as an algorithm for representing or < > functions, we test its robustness by analyzing its stability to withstand re-parameterizations of the hidden space. One may think of such re-parameterizations as the work of an adversary attempting to foil the construction of the KA outer function. We find KA to be stable under countable collections of continuous re-parameterizations, but unearth a question about the equi-continuity of the outer functions that, so far, obstructs taking limits and defeating continuous groups of re-parameterizations. This question on the regularity of the outer functions is relevant to the debate over the applicability of KA to the general theory of NNs.

cs.LG↗