In this study, we extend the concept of the quantum Gamma function (q-Gamma) by introducing a new (q,\tau)-deformed Gamma function. This generalization allows us to construct an enriched family of Hadamard-type fractional operators, which we then apply to the analysis of memory and decoherence in open quantum systems. The inclusion of the deformation parameter q together with the delay-like scaling parameter \tau makes the proposed (q,\tau)-Hadamard framework particularly suited to capture nonlocal and non-Markovian effects, thereby offering a flexible tool for describing structured reservoirs and anomalous dissipation. To investigate the analytical consequences, we employ a (q,\tau)-Mittag-Leffler function, through which explicit solutions are obtained for a class of fractional differential equations, with particular emphasis on population dynamics models. These solutions reveal a variety of memory-driven features, including sub-exponential decay of coherence and the occurrence of revival phenomena. Both the fractional order alpha and the deformation parameters play a decisive role in shaping the temporal behavior. Beyond population models, the framework also provides insights into fractional quantum master equations, quantum walks with fractional memory, and noise effects in quantum information processing.
In this study, we extend the concept of the quantum Gamma function (q-Gamma) by introducing a new (q,\tau)-deformed Gamma function. This generalization allows us to construct an enriched family of Hadamard-type fractional operators, which we then apply to the analysis of memory and decoherence in open quantum systems. The inclusion of the deformation parameter q together with the delay-like scaling parameter \tau makes the proposed (q,\tau)-Hadamard framework particularly suited to capture nonlocal and non-Markovian effects, thereby offering a flexible tool for describing structured reservoirs and anomalous dissipation. To investigate the analytical consequences, we employ a (q,\tau)-Mittag-Leffler function, through which explicit solutions are obtained for a class of fractional differential equations, with particular emphasis on population dynamics models. These solutions reveal a variety of memory-driven features, including sub-exponential decay of coherence and the occurrence of revival phenomena. Both the fractional order alpha and the deformation parameters play a decisive role in shaping the temporal behavior. Beyond population models, the framework also provides insights into fractional quantum master equations, quantum walks with fractional memory, and noise effects in quantum information processing.
In this paper, we propose a fractional time extension of the Quan tum Master Equation. We introduce a Caputo-type fractional derivative in time as an extension of the exponential decay of the Lindblad framework through the incorporation of fractional derivatives into the Lindblad framework. We show that the analytical and numerical results of our analytical and numerical models, demonstrate that fractional dynamics produces long-memory coherence decay naturally and provides an interpretable and flexible model of non-Markovianity.
We propose and investigate a generalized Schrödinger equation by introducing a fractional Riesz derivative to account for anomalous transport and a memory kernel to describe temporal nonlocal effects. Additionally, we include a long-range interactions term, modeled by an integral operator, which captures spatially extended interactions. Using the Green function approach, we derive analytical solutions and explore their implications in the time-space domain. Our findings reveal novel quantum phenomena arising from the interplay of fractional dynamics, nonlocal potentials, and memory effects, including the emergence of new local maxima in the evolution of Green's functions and distinct localization behaviors.
Abstract Our work explores the time evolution of entanglement, local quantum uncertainty, and correlated coherence within a system modeled by two coupled double quantum dots. The dynamics is described using a time-fractional Schrödinger equation, which is treated here as a phenomenological effective framework introducing temporal nonlocality and memory-like effects, rather than as a first-principles open-system reduction. We vary the fractional parameter τ , the tunneling amplitudes δ A and δ B , and the inter-dot Coulomb interaction strength V , in order to investigate how these key parameters govern the generation, stabilization, and decay of quantum resources within the system. The obtained results reveal that, for both initial states (separable and partially entangled), fractional dynamics with a small τ rapidly generates entanglement reaching maximal values LN ≈ 1 together with strong non-classical correlations quantified by the local quantum uncertainty. Conversely, when τ is closer to unity, entanglement builds up more slowly but the memory-like effects allow quantum resources to remain significant for a longer time, with the negativity remaining above ≈ 0.6 . We also find that higher Coulomb-interaction frequencies V accelerate the build-up of correlations and stabilize coherence, while a strong tunneling asymmetry degrades entanglement and coherence despite the initial benefits of increasing quantum resources. We find that the standard unitary Schrödinger evolution is recovered analytically at τ = 1 , where the Mittag–Leffler functions reduce to ordinary exponentials, while decreasing τ progressively introduces fractional-time nonlocal (memory-like) effects.
Open quantum systems exhibit dynamics ranging from unitary evolution to irreversible dissipation. While the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation uniquely characterizes Markovian CPTP evolution, many physical platforms display non-Markovian features such as algebraic relaxation and coherence backflow. Fractional calculus provides a natural way to model such long-memory behavior through power-law temporal kernels introduced by fractional time derivatives. Here we develop a unified framework that embeds fractional master equations within the broader hierarchy of open-system formalisms. The fractional equation forms a structured subclass of memory-kernel models, reduces to the Lindblad form at unit order, and, through Bochner--Phillips subordination, admits a CPTP representation as an average over Lindblad semigroups. Its resolvent structure further connects fractional dynamics to established non-Markovian approaches, including Nakajima--Zwanzig kernels and hierarchical equations of motion, providing a compact surrogate for long-memory effects. This formulation positions fractional calculus as a rigorous and practical language for quantum dynamics with intrinsic memory, supporting both analytical insight and efficient quantum simulation.
In this paper, we analyze quantum information measures in fractional quantum mechanics using the Riemann-Liouville derivative formalism adopted here. In this case, we initially reconsider the conventional definitions of Shannon entropy and Fisher information, subsequently extending them to fractional quantum systems described by nonlocal differential operator frameworks adopted. Within this generalized formulation, fractional expressions of Shannon entropy and Fisher information are constructed and their mathematical structures examined thoroughly. Also, the formalism is then applied to the quantum harmonic oscillator, yielding explicit analytical expressions derived as functions of the fractional parameter therein. The obtained results demonstrate that fractional derivatives alter the localization properties of probability densities and generate nontrivial variations in information content and sensitivity across system behavior. In this context, the fractional parameter plays a central role in controlling deviations from the standard quantum information measures framework. Also, the study establishes a consistent framework for describing information-theoretic properties of quantum systems governed by nonlocal dynamics.
Fractional calculus provides a natural framework for incorporating memory, dissipation, and temporal nonlocality into quantum dynamics. In this chapter, fractional time evolution is employed to extend conventional quantum gates and entangling operations while preserving Hilbert-space structure, locality, and energetic consistency. Fractional quantum gates are shown to arise from standard local Hamiltonians acting through Mittag Leffler propagators, leading to non-unitary yet physically meaningful operations characterized by Lévy-flight–like temporal statistics. This approach reshapes single- and two-qubit dynamics, producing gradual population transfer, algebraic dephasing, and history-dependent entanglement generation. The framework is further applied to optical and photonic systems, where fractional dynamics naturally emerge through dispersion, mode coupling, and structured propagation, offering insight into phenomena such as PSF broadening and coherence loss. Finally, implications for quantum communication and quantum key distribution are discussed, highlighting how channel memory modifies standard security assumptions and motivates memory-aware protocol design. Overall, fractional quantum dynamics are presented as a unifying theoretical lens for understanding realistic quantum information operations processing in and complex environments.