Generalized Exact Fractional Quantum Information Model with Memory Effects

Abstract: 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.

• The paper presents a generalized quantum information framework using the Riemann-Liouville fractional calculus to capture nonlocal memory effects.
• It derives fractional extensions of Shannon entropy and Fisher information, showing enhanced uncertainty and heavy-tailed distributions in quantum states.
• Analytic solutions for the quantum harmonic oscillator demonstrate practical insights into memory-induced modifications in quantum measurements.

Generalized Exact Fractional Quantum Information Model with Memory Effects: An Expert Analysis

Introduction and Motivation

This work introduces a comprehensive framework for quantum information measures in the context of fractional quantum mechanics via the Riemann-Liouville (RL) calculus formalism, providing an explicit model for quantum systems with nonlocal memory effects (2606.13525). The central motivation is driven by the limitations of traditional, integer-order quantum mechanics when applied to anomalously diffusive systems, media with fractal or disordered structures, and dynamics exhibiting long-range temporal or spatial memory. While conventional measures such as Shannon entropy and Fisher information have broad utility for quantifying uncertainty and localization in quantum systems, their standard definitions are fundamentally local and, thus, do not adequately represent quantum systems whose evolution is influenced by nonlocal and hereditary effects. This deficiency necessitates a systematic generalization of both the underlying quantum mechanics (via fractional calculus) and the associated information-theoretic descriptors.

RL Fractional Calculus Framework in Quantum Information

The RL fractional calculus extends differentiation to arbitrary order, encoding memory via integral operators with power-law kernels. The key RL operators, including the left-sided RL derivative, replace pointwise differentials with convolutions weighted by history-dependent kernels. For a function $f(x)$, the RL derivative introduces a scale-dependent, nonlocal sensitivity to past states, with the fractional parameter $\alpha$ ($0<\alpha\leq1$) systematically tuning the depth of the memory—recovering the local (Markovian) limit as $\alpha\rightarrow1$. In quantum theory, wavefunctions are promoted to fractional Sobolev spaces, supporting the meaningful definition of these operators in both standard and weak sense.

Generalization of Quantum Shannon Entropy

The RL formalism leads to a nonlocal extension of the probability density, substituting the pointwise density $\rho(x) = |\Psi(x)|^2$ with a fractional probability flux represented via a Volterra-type singular convolution. This fractional probability density is formally normalized, ensuring the probabilistic interpretation remains well-defined throughout the deformation. The RL-fractional Shannon entropy $S_\alpha$ is given by:

$$S_\alpha = -\int \mathcal{P}_\alpha(x)\ln\mathcal{P}_\alpha(x)\, dx,$$

where $\mathcal{P}_\alpha(x)$ is a nonlocal functional of $\rho(x)$ defined through RL integration/differentiation. Critically, for $\alpha<1$, $\alpha$0 explores the entropy content beyond local structures, with increased entropy manifesting from the enhanced effective support supplied by algebraic memory and nontrivial spatial correlations. In the $\alpha$1 limit, this framework consistently collapses to the standard Shannon entropy.

Fractional Fisher Information: Nonlocal Information Geometry

Analogously, the Fisher information is generalized by replacing local gradients with their fractional RL analogues:

$\alpha$2

The fractional gradient $\alpha$3 does not preserve the classical Leibniz rule but remains closely tied to representations of the fractional Laplacian and Dirichlet-type forms in corresponding functional spaces. Variational analysis reveals that Fisher information minimization yields eigenstates of the fractional Laplacian, with ground states interpolating between Gaussian forms $\alpha$4 and $\alpha$5-stable, heavy-tailed distributions for $\alpha$6. This fundamentally alters the measurement sensitivity and the underlying information geometry, deforming the Fisher-Rao metric from a local to a nonlocal, scale-dependent structure.

Analytical Application: Quantum Harmonic Oscillator

Focusing on the quantum harmonic oscillator, the authors obtain exact analytical expressions for both the RL-fractional Shannon entropy and Fisher information of the ground state. The RL memory kernel reweights the Gaussian probability density, inducing algebraic tails and modifying the entropy and sensitivity landscapes. The fractional entropy admits special function (e.g., Meijer-G, Fox-H) and convergent series representations, maintaining analytic tractability.

Key findings include:

• Monotonic Entropy Increase: For $\alpha$7, $\alpha$8, reflecting enhanced uncertainty and delocalization due to memory-induced spreading of probability density, with $\alpha$9 (2606.13525).
• Memory-Induced Nonlocality: The RL operator modifies the scaling of the probability density, generating algebraic decay and broadening information support relative to the classic exponential Gaussian profile as $0<\alpha\leq1$0 decreases.

The fractional Fisher information inherits the structure of the kinetic term for the fractional Laplacian, yielding non-quadratic dispersion and a breakdown of the unique role of the Gaussian as the minimizer. This translates to heavier tails, a denser spectral distribution, and reduced quantum rigidity as $0<\alpha\leq1$1 falls below unity.

Theoretical and Practical Implications

Theoretical Significance

• Unified Information-Theoretic Foundations: The RL formalism establishes a mathematically rigorous bridge between local quantum information theory and the class of nonlocal quantum systems, providing a single parameter $0<\alpha\leq1$2 controlling the entire interpolation.
• Consistent Variational Structure: Both Shannon entropy and Fisher information are recast as variational functionals, admitting extremals that are the corresponding eigenstates of nonlocal operators, thus extending the connection between information theory and quantum spectral theory.
• Statistical Geometry Deformation: The information geometry of state space, including the Fisher-Rao metric and related geometric manifolds, is deformed in a controlled fashion by the fractional parameter, encoding a transition from local (diffusive) to nonlocal (Lévy) regimes.

Practical Significance

• Modeling Memory-Driven Quantum Dynamics: Systems exhibiting non-Markovian dynamics, anomalous transport, or evolution in disordered/fractal environments can be rigorously addressed using these generalized information measures.
• Quantum Information Processing: Fractional framework offers a tool for analyzing quantum states affected by decoherence, long-range quantum correlations, and anomalous diffusive processes, with direct implications for error modeling and quantum statistical inference.
• Numerics and Generalizability: The analytic tractability and solid normalization of the RL generalizations facilitate adoption in computational quantum mechanics, with extension potential to higher dimensions, excited states, and complex potentials.

Future Directions

Potential developments include generalization to non-Gaussian, excited, or multi-dimensional states, and analysis of the thermodynamic meaning of fractional entropy in nonlocal quantum statistical mechanics. Connections to generalized entropy production, equilibrium in open quantum systems, and the interplay between fractional Fisher geometry and quantum phase transitions present promising avenues. Extension to other nonlocal operator frameworks (e.g., Caputo derivatives) or inclusion of variable-order differentiation may further enhance model flexibility.

Conclusion

The manuscript (2606.13525) constructs a systematic and analytically solid extension of quantum information measures within the RL fractional calculus formalism, capturing memory effects and nonlocality in a controlled fashion. The RL-parameterized model continuously connects the landscape of classical (local) and anomalous (memory-driven) quantum systems, affirming both mathematical coherence and practical utility. Applications to the quantum harmonic oscillator demonstrate entropy increase and diminished localization with growing nonlocality, supporting the physical interpretation of entropy as a measure of probabilistic delocalization in systems governed by fractional quantum dynamics. The framework is positioned for direct application to complex quantum systems, anomalous transport, and nonlocal statistical mechanics, with further generalizations likely to substantially impact modern quantum information science.