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arXiv:2606.19462

Quantum deformations of $\mathcal{U}(\mathfrak{sl}(2, \mathbb{R}))$. Part I: Fidelity and experimental benchmarking

V. Mariscal, J. J. Relancio, L. Santamaría-Sanz

quant-phmath-phSubmitted Jun 17, 2026• Announced Jun 19, 2026• 0 citations
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Abstract

This work explores the effects of both the standard quantum $q$-deformation and the non-standard $h$-deformation of the Hopf algebra $\mathcal{U}(\mathfrak{sl}(2, \mathbb{R}))$ on multi-qubit systems. By constructing the states of a Hilbert space of $N$ qubits through the Clebsch-Gordan coefficients associated with the deformed algebras, we show that these states naturally coincide with the eigenstates of the Hamiltonian of the $q$- and $h$-deformed Kittel-Shore models. We compare the resulting deformed states with those typically targeted in quantum information experiments, providing a bridge between algebraic constructions and experimentally relevant quantum resources. Fidelities with respect to the undeformed states are computed to establish how the quantum correlations are affected, both for few-qubit systems (including Dicke and non-Dicke states), and in the macroscopic limit ($N \to \infty$) through closed-form formulas derived for arbitrary Dicke states. The results reveal different behaviors between the two deformations. The $q$-deformation smoothly modifies the states and maintains a residual overlap with the original configurations, while the $h$-deformation rapidly makes the states orthogonal to their undeformed counterparts. Both models demand a standard $N^{-1}$ rescaling to preserve fidelity stability in the macroscopic limit.

Comments
36 pages, 13 figures

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Quantum deformations of U(sl(2,R))\mathcal{U}(\mathfrak{sl}(2, \mathbb{R}))U(sl(2,R)). Part I: Fidelity and experimental benchmarking
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In recent years, growing attention has been devoted to the possibility that theories with deformed symmetries, associated with certain models of non-commutative spacetime, may encode a fundamental form of decoherence. This effect should be described by a Lindblad-like evolution governed by the non-trivial Hopf algebra structure of the time-evolution generators. In this work we provide a detailed analysis of such possibility for similar Hopf algebra deformations of the Hamiltonian of a qubit. Starting from a critical examination of the very definition of time evolution through the generalized adjoint action, we explore whether a coherent and physically viable framework can be established. In particular, our analysis shows that a more general combination of adjoint actions always guarantees a von Neumann dynamics and, also in the case of deformed spacetime symmetries considered in the literature, a physically viable Lindblad evolution cannot be established.
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