The study of quantum information entropy in confined quantum systems plays a pivotal role in quantum information science. In this work, we investigate the quantum information entropy of a particle in a quantum well under an external electric field by calculating the Rényi entropy (), Tsallis entropy () and the Onicescu information energy (). Our findings reveal that these entropic measures exhibit a strong dependence on the entropy order ( α ). Although both and demonstrate a decreasing trend with increasing α for any given quantum state, they differ significantly in their sensitivity to α variations, displaying distinct behaviors that provide unique insights into quantum uncertainty. For the Onicescu information energy (), a sharp transition around α = 1 appears. This characteristic could be significant for designing and analyzing quantum systems, particularly in enhancing their information processing capabilities. By further examining how these entropic measures vary with quantum well width and electric field strength, we gain a deeper understanding of the system's controllability. This empowers researchers to select optimal metrics for specific applications and improve the accuracy of quantum analyses. This study underscores the importance of entropy‐based analysis in quantum mechanics for advancing quantum control, information theory, and technology development.
The study of quantum information entropy in confined quantum systems plays a pivotal role in quantum information science. In this work, we investigate the quantum information entropy of a particle in a quantum well under an external electric field by calculating the Rényi entropy (), Tsallis entropy () and the Onicescu information energy (). Our findings reveal that these entropic measures exhibit a strong dependence on the entropy order ( α ). Although both and demonstrate a decreasing trend with increasing α for any given quantum state, they differ significantly in their sensitivity to α variations, displaying distinct behaviors that provide unique insights into quantum uncertainty. For the Onicescu information energy (), a sharp transition around α = 1 appears. This characteristic could be significant for designing and analyzing quantum systems, particularly in enhancing their information processing capabilities. By further examining how these entropic measures vary with quantum well width and electric field strength, we gain a deeper understanding of the system's controllability. This empowers researchers to select optimal metrics for specific applications and improve the accuracy of quantum analyses. This study underscores the importance of entropy‐based analysis in quantum mechanics for advancing quantum control, information theory, and technology development.
With the development of quantum technology and information science, significant attention is focused on studying the quantum information properties in confined systems. In this work, we present a systematic investigation of quantum information measures – including position-momentum uncertainty (Δx,Δp), Shannon entropy (Sx, Sp), and Onicescu energy (Ox, Op) in a confined Robin harmonic oscillator (CRHO) with symmetric boundary conditions λL = λR = λ. It is interesting to find that the Heisenberg uncertainty relation ΔxΔp≥ℏ/2, which holds under the Dirichlet boundary condition (λ = 0), is violated for the ground state of the CRHO. In contrast, the entropic uncertainty relation (Sx + Sp ≥ 1 + lnπ) remains satisfied at any Robin length λ and quantum state, demonstrating its fundamental advantage and broader applicability in quantum systems. For a fixed Robin length, we observe that as the quantum well width increases within a finite range, the position-space Shannon entropy Sₓ increases, while the Onicescu energy Oₓ decreases. Despite their opposing tendencies, both Shannon entropy and Onicescu energy reflect the same physical insight regarding the localisation-delocalisation property for the CRHO system. In addition, the dependence of these information measures on the Robin length is discussed, and their physical implications are highlighted. This study demonstrates that boundary conditions play a critical role in quantum information metrics, thereby informing the design and optimisation of quantum devices such as quantum dots, nanoscale resonators and atom-based sensors.
Studying the localisation-delocalisation property of confined quantum systems is crucial for advancements in quantum technology and information science. Here, we investigate the localisation-delocalisation property for a particle in a quantum well under the influence of an electric field by calculating position-momentum space standard deviations (Δx, Δp) and the Shannon entropies (Sx, Sp) as a function of the quantum well width (L) or the electric field strength (F). Our results suggest that when the width L or the strength F changes, both the standard deviations and the Shannon entropies can lead to consistent interpretations in measuring the localisation-delocalisation property of this system. Specifically, for a fixed magnitude of F, Δx and Sx increases with the width L, suggesting an extensive delocalisation, while they diminishes with increasing F for a given width of L, revealing the localisation property of this system. By comparing the behaviours of position-momentum uncertainty relation (ΔxΔp) and the Shannon entropy uncertainty relation (St = Sx + Sp), we find that the behaviour of ΔxΔp with L or F varies for different quantum state. In contrast, the variation of St with L or F is consistent across different quantum states. The underlying physical reasons for the effects of quantum well width and electric field strength on the uncertainty relations of this system are analysed from the differentiation perspective. This study provides deep insights into localisation-delocalisation property of a particle in a quantum well under external perturbations, and has significant applications in the field of quantum physics, condensed physics, semiconductor physics and material science.
Abstract Shannon entropy (S), Rényi entropy (R), Tsallis entropy (T), Fisher information (I), and Onicescu energy (E) have been explored extensively in both free H atom (FHA) and confined H atom (CHA). For a given quantum state, accurate results are presented by employing respective exact analytical wave functions in r space. The p‐space wave functions are generated from respective Fourier transforms—for FHA these can be expressed analytically in terms of Gegenbauer polynomials, whereas in CHA these are computed numerically. Exact mathematical expressions of , are derived for circular states of a FHA. Pilot calculations are done taking order of entropic moments (α, β) as in r and p spaces. A detailed, systematic analysis is performed for both FHA and CHA with respect to state indices n, l, and with confinement radius (rc) for the latter. In a CHA, at small rc, kinetic energy increases, whereas decrease with growth of n, signifying greater localization in high‐lying states. At moderate rc, there exists an interplay between two mutually opposing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of n (delocalization). Most of these results are reported here for the first time, revealing many new interesting features. Comparison with literature results, wherever possible, offers excellent agreement.
The entropy of a quantum state measures the information or uncertainty contained in the quantum system and plays a crucial role in quantum information theory. Many entropic properties, such as the subadditivity, have been studied for the discrete system with a finite-dimensional state space, while much less is explored in the infinite-dimensional system. In this work, we study the entropic properties of the continuous system that models a collection of Gaussian modes or fields. In particular, we identify the parameter range for the Rényi, Tsallis, and Unified entropies of Gaussian states that they are additive, subadditive, and strong subadditive. We show that the Rényi entropies are subadditive, further giving rise to the subadditivity of the Tsallis and Unified entropies in certain parameter range. We also find that the Tsallis and Unified entropies are not strong subadditive, while the Rényi entropies with the order ranging from 1 to 2 are conjectured to admit the strong subadditivity. Our results are useful in studying quantum correlations and channel capacities in Gaussian systems.
Abstract In this research work, the quantum information‐theoretic analysis of the static screened Coulomb potential has been carried out by studying both analytically and numerically the entropic measures, Fisher information as well as the Onicescu information energy of its wave function. Explicit expressions of these information‐theoretic measures were obtained. Using the Srivastava–Daoust linearization formula in terms of the multivariate Lauricella hypergeometric function, the Rényi entropy, Tsallis entropy, Onicescu information energy were analytically obtained. From the results obtained, it is observed that some of the Shannon entropies are negative, indicating that, negative entropies exists for the probability densities that are highly localized. The trends in the variation of the information‐theoretic measures with the potential screening parameter a for this atomic model are discussed. The Bialynicki‐Birula, Mycielski inequality (BBM), and the Fisher information based uncertainty relation are also verified.
Abstract Entropic uncertainty relations are studied from the conceptual viewpoint and in the context of various applications. Providing as tight inequalities as possible is thus an important issue. Measurements of special types are indispensable in quantum information science. Their structure allows us to estimate the index of coincidence. Uncertainty relations follow an estimation of entropies at the given index of coincidence. Relations of such a kind are a known issue in information theory. It concerns links between different information measures. Information diagrams are a tool to study relations between two utilized characteristics. A search for improved inequalities uses the index of coincidence as the abscissa and the entropy of interest as the ordinate. This approach before dealt with standard information functions assigned to the Shannon entropy. But generalized entropies often reveal new links between information measures. This paper applies the method of information diagrams to the Rényi and Tsallis entropies. Such entropies and related information functions have found use in quantum information theory. Improved estimates of the Tsallis and Rényi entropies at the given index of coincidence are obtained. Hence, new entropic uncertainty relations for specific types of measurements follow. They include mutually unbiased bases, symmetric informationally complete measurements, and equiangular tight frames.