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Quantum Information Processing

Quantum information measures of conformal fractional non-relativistic systems in a Coulomb potential

Quezia R. D. S. Moreira, A. R. P. Moreira, Abdelmalek Bouzenada, João Batista Rosa Silva, R. Horchani, Faizuddin Ahmed

Published Jun 9, 2026• 0 citations
Publisher

Abstract

Abstract In this work, we examine the quantum information characteristics of a hydrogen-like system under a Coulomb potential by using previously derived exact solutions of the generalized Schrödinger equation formulated within the conformable fractional framework. The approach introduces a fractional parameter $$(0 < \alpha \le 1)$$ ( 0 < α ≤ 1 ) , which permits a continuous transition between standard quantum mechanics and a fractional description. Utilizing these available analytical solutions, we determine the associated energy spectrum and wavefunctions and employ them to quantify the informational structure of the system. In particular, we calculate the Shannon entropy in both position and momentum spaces, along with the Fisher information, as fundamental indicators of uncertainty, spatial spreading, and sensitivity to local variations of the probability density. The results show that the fractional parameter $$(\alpha )$$ ( α ) produces nontrivial corrections to these quantum information measures, indicating modifications in localization properties and momentum profiles. Also, the formalism recovers the conventional hydrogenic limit when $$(\alpha \rightarrow 1)$$ ( α → 1 ) , which validates the consistency of the formulation. In this context, these results show that the conformable fractional framework constitutes a consistent extension for analyzing quantum systems via information-theoretic quantities without modifying the exact solutions, providing a systematic approach to investigate deviations from integer-order dynamics.

DOI
10.1007/s11128-026-05229-7

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Abstract In this work, we examine the quantum information characteristics of a hydrogen-like system under a Coulomb potential by using previously derived exact solutions of the generalized Schrödinger equation formulated within the conformable fractional framework. The approach introduces a fractional parameter (0<α≤1)(0 < \alpha \le 1)(0<α≤1) ( 0 < α ≤ 1 ) , which permits a continuous transition between standard quantum mechanics and a fractional description. Utilizing these available analytical solutions, we determine the associated energy spectrum and wavefunctions and employ them to quantify the informational structure of the system. In particular, we calculate the Shannon entropy in both position and momentum spaces, along with the Fisher information, as fundamental indicators of uncertainty, spatial spreading, and sensitivity to local variations of the probability density. The results show that the fractional parameter (α)(\alpha )(α) ( α ) produces nontrivial corrections to these quantum information measures, indicating modifications in localization properties and momentum profiles. Also, the formalism recovers the conventional hydrogenic limit when (α→1)(\alpha \rightarrow 1)(α→1) ( α → 1 ) , which validates the consistency of the formulation. In this context, these results show that the conformable fractional framework constitutes a consistent extension for analyzing quantum systems via information-theoretic quantities without modifying the exact solutions, providing a systematic approach to investigate deviations from integer-order dynamics.
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.
ABSTRACT We investigate the quantum information entropy for a class of single hyperbolic potentials within the context of the fractional Schrödinger equation (FSE). We find that as the derivative variable decreases, the position entropy density function becomes more localized, and its peak heightens. However, there are differences in the degree of localization between the position entropy density functions for each hyperbolic potential, which can be attributed to the varying sizes of the potentials. Conversely, in momentum space, the momentum entropy density function becomes more delocalized, and its peak lowers as the derivative variable decreases for both hyperbolic potentials studied. Our analysis also examines the BBM inequality, demonstrating that it is satisfied for different values of the potential depths. Finally, we explore the Fisher entropy and observe that it increases in position space while decreasing in momentum space as the depth of the wells increases. Our findings provide new insights into the behavior of quantum systems governed by hyperbolic potentials within the fractional Schrödinger framework. The observed localization effects in position space, delocalization in momentum space, and the validation of the BBM inequality highlight the role of fractional derivatives in modifying quantum entropy measures. These results deepen our understanding of quantum information entropy in non‐local quantum systems. They may have implications for fields such as quantum transport in disordered media, semiconductor physics, and the study of anomalous diffusion processes in quantum mechanics.
ABSTRACT In this work, we investigate quantum information entropy for double hyperbolic well potentials within the framework of the fractional Schrödinger equation (FSE). Specifically, we analyze the position and momentum Shannon entropies for two hyperbolic potentials, and , as a function of the fractional derivative order . Our findings reveal that decreasing enhances wave function localization in position space, thereby reducing spatial uncertainty while simultaneously increasing momentum uncertainty. We confirm the validity of the Beckner–Bialynicki‐Birula–Mycielski inequality for both potentials, demonstrating its robustness across different degrees of nonlocality. Furthermore, we explore the behavior of Fisher information, observing that it increases in position space while decreases in momentum space as the well depths grow.
This study presents a comprehensive investigation of quantum information measures for the one-dimensional Ultra Generalized Exponential–Hyperbolic Potential (UGEHP) within the framework of the Schrödinger equation. Analytical solutions for the energy eigenvalues and corresponding wave functions are obtained using the Nikiforov–Uvarov method. These solutions serve as the basis for evaluating Shannon entropy and Fisher information in both position and momentum spaces. The numerical results show that the total Shannon entropy satisfies the Beckner–Białynicki-Birula–Mycielski (BBM) inequality, while the Fisher information adheres to the Stam–Cramér–Rao bounds, thereby confirming consistency with fundamental quantum uncertainty principles. In addition, the energy spectrum exhibits a pronounced dependence on the screening parameter and quantum numbers, indicating the tunable confinement properties of the potential. The combined analysis establishes a clear connection between energy quantization, wavefunction localization, and information-theoretic measures, offering deeper insight into the effectiveness of the UGEHP model in describing complex quantum systems.
Abstract Information entropy has played a key role in a wide range of disciplines, for instance, classical and quantum information processing, quantum computing, quantum dynamics and quantum metrology. Here, we develop an information theoretic formalism using Shannon entropy, to investigate the quantum dynamics of Hamiltonian systems with position-dependent mass. Such systems are of fundamental interest in many areas, for instance, condensed matter, mathematical physics and foundations of quantum mechanics. We explore the phenomenon of fractional revivals for the temporal evolution of wave-packet solutions of Schrödinger equation with position-dependent mass by studying, analytically and numerically, the time-development of Shannon information entropy in position and momentum spaces. It is shown by our numerical results that the effect of spatially varying mass on the fractional revivals can not be fully harnessed using conventional measures, for instance, autocorrelation function. However, based on our numerical analysis it is concluded that information entropy is not only more sensitive to identify the fractional revivals but it also better elucidates the effect of position-dependent mass on the structure of fractional revivals in the form of symmetry breaking.
Fisher information (FI) and Shannon entropy (SP), as fundamental theoretical constructs, play crucial roles in quantum information, offering significant insights into communication systems. This work addresses the radial Schrödinger equation for a cotangent hyperbolic potential, solved using the parametric Nikiforov–Uvarov method. The derived wave functions are utilized to explore FI and SP within a one-dimensional system. The analysis focuses on the influence of the screening parameter and potential depth on these quantities, examining their behavior for the ground state and the first excited state. The study complies with key theoretical principles, including the Cramer–Rao inequality, the Beckner-Bialynicki-Birula-Mycielski (BBM) inequality, and the Heisenberg uncertainty principle. The results shed light on the dynamics of quantum states, emphasizing the relationship between quantum uncertainties and information measures. Significant variations in FI and SP emerge with changes in the screening parameter and potential depth, enhancing the understanding of quantum mechanics in confined systems. The findings also reveal the phenomenon of entropy squeezing, underscoring the complex nature of quantum uncertainties. The model demonstrates realistic behavior, with distinct transition points in SP as a function of potential depth, highlighting unique and intriguing characteristics. These contributions enrich the theoretical framework for investigating quantum systems, providing valuable perspectives on their intricate behaviors.