The mathematical structure of quantum groups, as q-deformations of Lie algebras, has been extensively studied, yet their deep connection with the dynamicsof open quantum systems remains to be elucidated. Based on the framework ofQuantum Iterated Function Systems (QIFS), this paper establishes an emergencemechanism from quantum groups to classical Lie groups. We first construct a concrete QIFS whose generators are a family of completely positive maps acting onthe space of quantum states, and prove that the Lie algebra structure of its lowenergy effective theory corresponds to the Drinfeld–Jimbo quantum group Uq(g).Next, we derive the relation between the q-parameter and the decoherence rate Γ,characteristic coherence time τc as q = e−Γτc, revealing the physical origin of theq-deformation. Based on this, we prove that in the limit of vanishing decoherencerate Γ → 0, the quantum group degenerates to the classical Lie group g. Finally, weestablish a correspondence between the representation theory of quantum groupsand the spectrum of QIFS, providing a unified mathematical framework for understanding the emergence mechanism of quantum symmetries. This result links theconcept of decoherence in quantum information theory with the algebraic structureof quantum groups, revealing a deeper picture where classical Lie group symmetriesemerge as the semiclassical limit of open quantum system dynamics.
The mathematical structure of quantum groups, as q-deformations of Lie algebras, has been extensively studied, yet their deep connection with the dynamicsof open quantum systems remains to be elucidated. Based on the framework ofQuantum Iterated Function Systems (QIFS), this paper establishes an emergencemechanism from quantum groups to classical Lie groups. We first construct a concrete QIFS whose generators are a family of completely positive maps acting onthe space of quantum states, and prove that the Lie algebra structure of its lowenergy effective theory corresponds to the Drinfeld–Jimbo quantum group Uq(g).Next, we derive the relation between the q-parameter and the decoherence rate Γ,characteristic coherence time τc as q = e−Γτc, revealing the physical origin of theq-deformation. Based on this, we prove that in the limit of vanishing decoherencerate Γ → 0, the quantum group degenerates to the classical Lie group g. Finally, weestablish a correspondence between the representation theory of quantum groupsand the spectrum of QIFS, providing a unified mathematical framework for understanding the emergence mechanism of quantum symmetries. This result links theconcept of decoherence in quantum information theory with the algebraic structureof quantum groups, revealing a deeper picture where classical Lie group symmetriesemerge as the semiclassical limit of open quantum system dynamics.
The computational advantage of quantum algorithms over classical algorithmshas been partially proven, but its deep connection with the dynamics of open quantum systems has yet to be elucidated. This paper establishes a formal correspondence between quantum algorithm complexity and dissipative dynamics based onthe framework of Quantum Iterated Function Systems (QIFS). We first prove thatthe convergence rate of a QIFS—governed by the spectral gap ∆—directly determines the query complexity and time complexity of the corresponding quantumalgorithm. Second, we establish an exponential relationship between the quantumgroup parameter q and the success probability of the algorithm, revealing the dualrole played by decoherence in quantum search algorithms: moderate decoherenceaccelerates convergence, while excessive decoherence causes the algorithm to fail.Building on this, we propose the “dissipative speedup” theorem: there exists an optimal decoherence rate Γopt such that the quantum algorithm described by the QIFSachieves minimal time complexity, and this optimal value satisfies Γopt = −lnq/tc.Finally, we explore the connections between this framework and adiabatic quantum computation as well as quantum walk algorithms, and propose experimentalschemes to test the theoretical predictions on superconducting quantum processors.This work unifies the theory of open quantum systems, quantum group algebra,and quantum computational complexity into a common generative framework, providing a new perspective for understanding the persistence of quantum advantagein noisy environments.
Iterated Function Systems (IFS) generate invariant measures with fractal structures through the random iteration of a family of contraction mappings, servingas a standard framework for describing scale generation phenomena in classicalnonlinear dynamics. This paper generalizes classical IFS to the quantum domain,establishing a rigorous mathematical formulation of Quantum Iterated FunctionSystems (QIFS): classical contraction mappings are replaced by families of completely positive maps acting on the space of density operators, and the quantumattractor state is defined as the fixed point satisfying a self-similarity equation. Onthis basis, we prove that the rescaled position distribution of discrete-time quantum random walks with decoherence converges to a convex combination of normaldistributions, with the convergence rate controlled by the spectral gap of the superoperator. Furthermore, we establish a quantum-classical correspondence principle:in the limit of vanishing decoherence rate, the support of the Wigner function ofthe quantum attractor state converges in distribution to the classical IFS attractor,and the convergence exponents satisfy a scaling relation with the classical contraction rates. This framework unifies fractal geometry, quantum open systems, anddecoherence theory into a coherent generative language, providing a rigorous mathematical foundation for understanding scaling behavior in quantum informationprocessing.
The Quantum Iterated Function Systems (QIFS) theory establishes an exponential relation q = e−Γtc linking the quantum group deformation parameter q withthe open system decoherence rate Γ and characteristic coherence time tc, offering anew perspective for understanding the emergence mechanism of quantum symmetries. This paper proposes concrete experimental schemes to implement the QIFSof su(2) generators in superconducting qubit and ion trap systems, aiming to testthis theoretical relation. First, we construct QIFS generators based on amplitudedamping and phase damping channels, and present their physical realizations insuperconducting circuits and ion trap systems. Second, we define two methodsto extract the quantum group parameter qexp from experimental data—quantumstate tomography based on entanglement spectra and time-series analysis based ondynamical correlation functions—and elaborate on their operational steps and error control. Finally, we establish a falsification criteria system including absolutedeviation criterion, goodness-of-fit criterion, self-consistency criterion, and limitingbehavior test. It is explicitly stated that if the deviation between qexp and thetheoretical prediction qth = e−Γtc exceeds 5%, the core proposition of the QIFSframework regarding quantum group emergence is invalidated. This experimentaldesign directly connects abstract quantum group algebraic structures with observable quantum open system dynamics, providing operable experimental criteria fortesting generative quantum theories.
This thesis presents a comprehensive study of phase-space measurements, decoherence, and the emergence of classicality in quantum systems. Motivated by limitations of position-based localisation in standard decoherence models, we develop a phase-space decoherence formalism using coherent-state positive operator-valued measures (POVMs). These generalised measurements, constructed from overcomplete sets of coherent states, provide a natural and physically motivated framework for modelling coarse-grained quantum observations. We develop an iterative measurement scheme for phase-space decoherence in systems with Heisenberg-Weyl, SU(2), SU(1,1), and SU(N) symmetries. In each case, repeated weak measurements gradually suppress quantum interference, driving the system toward a maximally mixed state. A universal structure is shown to emerge: a single iteration maps the Wigner function to the Husimi Q-function, removing negativity, while full decoherence maps the singular Glauber-Sudarshan P-function to the Q-function, eliminating all singularities. This establishes a group-independent mechanism by which phase-space measurements induce classicality. We compare this model to Lindblad-type dynamics, which describe open quantum systems via completely positive, trace-preserving semigroups. For Heisenberg-Weyl and tomographic SU(N) systems, the measurement channel coincides exactly with isotropic Lindblad evolution. In SU(2) and SU(1,1) cases, the correspondence is only partial, with exact agreement in asymptotic regimes. A universal decoherence timescale, t∗, emerges beyond which quasi-probability distributions become strictly positive. We then construct a stochastic unravelling based on Brownian motion on phase space, offering a geometric alternative to standard stochastic Schr\"odinger equations. This reveals that decoherence occurs without GKSL-type dynamics in noncompact systems. Such unravellings preserve the intrinsic phase-space symmetry and provide a natural setting for studying semiclassical limits and quantum-to-classical transitions. Finally, we extend the framework to relativistic quantum systems by defining a covariant coherent-state POVM on the future tube and constructing a Bargmann transform to the forward light cone. Preliminary results on relativistic phase-space decoherence are presented, along with possible directions for future research.
This work presents a structural reformulation of quantum dynamics based on a scalar coherence functional defined over a general state space. The framework distinguishes two mathematically natural classes of dynamics generated by such a functional:(i) gradient flows, which monotonically increase coherence and correspond to stabilization or selection processes, and(ii) antisymmetric flows, which conserve coherence and generate reversible dynamics. We show that standard unitary quantum mechanics arises as a special case when the state space is a complex Hilbert space, the coherence functional is chosen as the negative energy expectation, and the antisymmetric structure corresponds to the canonical complex structure. In this setting, the Schrödinger equation follows directly from the formalism, without postulating unitarity as an independent axiom. The framework is extended to open systems using the density-operator formalism and completely positive trace-preserving maps. Under standard assumptions (Markovianity, continuity, and semigroup structure), the reduced dynamics takes the Gorini–Kossakowski–Sudarshan–Lindblad form, with dissipative terms interpreted as coherence-stabilizing dynamics. A supporting lemma demonstrates exponential suppression of off-diagonal density matrix elements under dephasing Lindblad operators. Using the Wigner phase-space representation, we further show that conservative coherence dynamics reduces to classical Liouville transport in the semiclassical regime, with quantum corrections governed by the Moyal bracket expansion. This work does not propose a modification of quantum theory. Rather, it offers a structural reorganization of existing formalisms. All results remain fully compatible with standard quantum mechanics while providing a unified perspective on unitary evolution, decoherence, and the emergence of classical behavior. Related work. A companion technical note, Structural Decomposition of SRG Recursion into Flow, Phase, and Closure (Halldórsson & ChatGPT, 2025, Zenodo), develops a discrete dynamical-systems formulation of SRG-style recursion and connects it explicitly to Lemmas 2.5, 2.7, and 2.9 of the present work.
We present a group-theoretic reconstruction of quantum mechanics in which the standard formalism emerges from the representation theory of symmetry groups rather than being introduced axiomatically. Starting from physically motivated symmetry principles, we show how the conventional postulates of quantum mechanics including the Hilbert space structure of states, observables as generators of symmetry transformations, the Born probability rule, unitary time evolution, the composition of multipartite systems, and the measurement framework can be recovered within a unified representation-theoretic framework. We then examine the measurement problem by interpreting wave-function collapse as the scattering of quantum states, thereby providing an account of the apparent loss of information. Finally, we investigate the origin of Planck's constant, proposing that it arises as the conserved charge of the Lorentz symmetry group. In this formulation, ℏ is an invariant that relates Lie algebra generators to observable quantities and fixes the scale of quantum action. This framework offers a conceptually unified perspective on the foundations of quantum mechanics, connecting its postulates, measurement theory, and the emergence of the quantum of action to a common symmetry-based origin while identifying conditions under which the conventional quantum formalism is recovered.