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arXiv:2602.07887

Quantum Evolution of Hopf Algebra Hamiltonians

Michele Arzano, Antonio Del Prete, Domenico Frattulillo

quant-phgr-qchep-thSubmitted Feb 8, 2026• Announced Feb 10, 2026• 0 citations
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Abstract

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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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.
Abstract Understanding whether quantum gravitational effects can lead to a fundamental decoherence, affecting all systems regardless of their environment, is a long standing open challenge. Here we provide a rigorous derivation of decoherence within a full-fledged model of quantum spacetime, encoded by noncommutativity at the Planck scale. Specifically, we obtain a generalized time evolution of quantum systems in which pure states can evolve into mixed states. This takes the form of a Lindblad-like time evolution for the density operator when the action of time translations generator is deformed by the effects of spacetime noncommutativity. The decoherence time for the evolution of a free particle is used to show that the Planck mass is the maximum allowed mass for elementary quantum systems.
We argue that Hopf-algebra deformations of symmetries -- as encountered in non-commutative models of quantum spacetime -- carry an intrinsic content of operatoroperatoroperator entanglemententanglemententanglement that is enforced by the coproduct-defined notion of composite generators. As a minimal and exactly solvable example, we analyze the Uq(su(2))U_q(\mathfrak{su}(2))Uq​(su(2)) quantum group and a two-qubit realization obtained from the coproduct of a qqq-deformed single-spin Hamiltonian. Although the deformation is invisible on a single qubit, it resurfaces in the two-qubit sector through the non-cocommutative coproduct, yielding a family of intrinsically nonlocal unitaries. We compute their operator entanglement in closed form and show that, for Haar-uniform product inputs, their entangling power is fully determined by the latter. This provides a concrete mechanism by which non-commutative symmetries enforce a baseline of entanglement at the algebraic level, with implications for information dynamics in quantum-spacetime settings and quantum information processing.
A bstract The construction of exactly-solvable models has recently been advanced by considering integrable TT‾ T\overline{T} TT T T ¯ deformations and related Hamiltonian deformations in quantum mechanics. We introduce a broader class of non-Hermitian Hamiltonian deformations in a nonrelativistic setting, to account for the description of a large class of open quantum systems, which includes, e.g., arbitrary Markovian evolutions conditioned to the absence of quantum jumps. We relate the time evolution operator and the time-evolving density matrix in the undeformed and deformed theories in terms of integral transforms with a specific kernel. Non-Hermitian Hamiltonian deformations naturally arise in the description of energy diffusion that emerges in quantum systems from time-keeping errors in a real clock used to track time evolution. We show that the latter can be related to an inverse TT‾ T\overline{T} TT T T ¯ deformation with a purely imaginary deformation parameter. In this case, the integral transforms take a particularly simple form when the initial state is a coherent Gibbs state or a thermofield double state, as we illustrate by characterizing the purity, Rényi entropies, logarithmic negativity, and the spectral form factor. As the dissipative evolution of a quantum system can be conveniently described in Liouville space, we further study the spectral properties of the Liouvillians, i.e., the dynamical generators associated with the deformed theories. As an application, we discuss the interplay between decoherence and quantum chaos in non-Hermitian deformations of random matrix Hamiltonians and the Sachdev-Ye-Kitaev model.
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Theories with Planck-scale deformed symmetries exhibit quantum time evolution in which purity of the density matrix is not preserved. In particular we show that the non-trivial structure of momentum space of these models is reflected in a deformed action of translation generators on operators. Such action in the case of time translation generators leads to a Lindblad-like evolution equation for density matrices when expanded at leading order in the Planckian deformation parameter. This evolution equation is covariant under the deformed realization of Lorentz symmetries characterizing these models.
This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachistochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calculated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of infinite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The finding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commutation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of differential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our efforts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear differential equations for which the presence of hidden symmetries allows one to assemble exact solutions. We use [4] the Painleve test to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions. Finally we study [6] the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT -symmetric way.