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arXiv:1903.02604 · European Journal of Physics

Information and quantum theories: an analysis in one-dimensional systems

Wallas S. Nascimento, Marcos M de Almeida, Frederico V. Prudente

Published Dec 6, 2019• 9 citations
arXivHTMLPDF

Abstract

Abstract We pedagogically present the information theory as originally established, explaining its essential ideas and paying attention to the expression employed to measure the amount of information. We also discuss relationships between information and quantum theories. In this context we determine the information entropies on the position, S x , and momentum, S p , spaces, in addition to the entropy sum, S t . Here, we use the modified entropic expressions that are dimensionally consistent. We provide original explanations for the behaviors of the S x , S p and S t values analyzing the probability densities and by means of the normalization constants and properties of Fourier transform. We validate the entropic uncertainty relation and inspect the standard deviation and information entropies as measures of quantum uncertainty. The systems of interest are unidimensional one-particle quantum systems in the ground and excited states.

DOI
10.1088/1361-6404/ab5f7d

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Abstract We pedagogically present the information theory as originally established, explaining its essential ideas and paying attention to the expression employed to measure the amount of information. We also discuss relationships between information and quantum theories. In this context we determine the information entropies on the position, S x , and momentum, S p , spaces, in addition to the entropy sum, S t . Here, we use the modified entropic expressions that are dimensionally consistent. We provide original explanations for the behaviors of the S x , S p and S t values analyzing the probability densities and by means of the normalization constants and properties of Fourier transform. We validate the entropic uncertainty relation and inspect the standard deviation and information entropies as measures of quantum uncertainty. The systems of interest are unidimensional one-particle quantum systems in the ground and excited states.
The article intertwines the study of wave packet dynamics with information-theoretic measurements in one dimensional (D =1) system. A localized wave packet at time t=0 has been considered here and its change at later instant of time t is calculated for the position space wave function. The momentum space wave function is obtained by taking the Fourier transform of the position space wave function at time t= 0 . These wave functions are then employed to construct the wave packet's respective probability densities in position and momentum space, and later have been used to compute the corresponding position and momentum space Shannon (S) and Fisher information (I) entropies. It has been observed that although the Shannon and Fisher information entropies explicitly depend on the standard deviation, neither the Shannon entropy sum nor the product of Fisher information entropies is. Moreover, for D dimensional systems, the computed values of the Shannon and Fisher entropies are found to satisfy the lower bounds of the Bialynicki-Birula and Myceilski (BBM) inequality relation and the Stam-Cramer-Rao inequalities better known as the Fisher based uncertainty relation. Thus, our theoretical study explores the validation of information-theoretic measurements for wave packet dynamics using the basic formulations of information theory.
This is intended to be an introductory book on information entropy. The concept of information theory is too broad to be captured by single definition. The term entropy is used in information theory to describe the amount of uncertainty. The information entropy is a fundamental quantity, related to thermodynamical entropy, which measures the extent of single particle density. The information and entropy both depend upon the choice of probability distribution. Quantum information science will lead to the creation of a quantum computer to solve problem that could not be solved by classical computer. The ideas from information theory have been applied to many problems of science and technology and in other different fields. Quantum Information entropy plays a crucial role in stronger formulation of uncertainty relation. The quantum information entropy of some important potentials is discussed which are widely used in different fields. The information entropy of these potentials is studied in position as well as momentum space and uncertainty relations are verified.
Calculations of the quantum information entropy have been extended to a non-analytically solvable situation. Specifically, we have investigated the information entropy for a one-dimensional system with a schematic “Landau” potential in a numerical way. Particularly, it is found that the phase transitional behavior of the system can be well expressed by the evolution of quantum information entropy. The calculated results also indicate that the position entropy S x and the momentum entropy S p at the critical point of phase transition may vary with the mass parameter M but their sum remains as a constant independent of M for a given excited state. In addition, the entropy uncertainty relation is proven to be robust during the whole process of the phase transition.
Uncertainty principle, which was first introduced by Werner Heisenberg in 1927, forms a fundamental component of quantum mechanics. A graceful aspect of quantum mechanics is that the uncertainty relations between incompatible observables allow for succinct quan- titative formulations of this revolutionary idea: it is impossible to simultaneously measure two complementary variables of a particle in precision. In particular, information theory offers two basic ways to express the Heisenberg’s principle: variance-based uncertainty relations and entropic uncertainty relations. We first investigate the uncertainty relations based on the sum of variances and derive a family of weighted uncertainty relations to provide an optimal lower bound for all situations. Our work indicates that it seems unreasonable to assume a priori that incompatible observables have equal contribution to the variance-based sum form uncertainty relations. We also study the role of mutually exclusive physical states in the recent work and generalize the variance-based uncertainty relations to mutually exclusive uncertainty relations. Next, we develop a new kind of entanglement detection criteria within the framework of marjorization theory and its matrix representation. By virtue of majorization uncertainty bounds, we are able to construct the entanglement criteria which have advantage over the scalar detect- ing algorithms as they are often stronger and tighter. Furthermore, we explore various expression of entropic uncertainty relations, including sum of Shannon entropies, majorization uncer- tainty relations and uncertainty relations in presence of quantum memory. For entropic uncertainty relations without quantum side information, we provide several tighter bounds for multi-measurements, with some of them also valid for Rényi and Tsallis entropies besides the Shannon entropy. We employ majorization theory and actions of the symmetric group to obtain an admixture bound for entropic uncertainty relations with multi-measurements. Comparisons among existing bounds for multi-measurements are also given. However,classical entropic uncertainty relations assume there has only classical side information. For modern uncertainty relations, those who allowed for non-trivial amount of quantum side information, their bounds have been strengthened by our recent result for both two and multi- measurements. Finally, we propose an approach which can extend all uncertainty relations on Shannon entropies to allow for quantum side information and discuss the applications of our entropic framework. Combined with our uniform entanglement frames, it is possible to detect entanglement via entropic uncertainty relations even if there is no quantum side in- formation. With the rising of quantum information theory, uncertainty relations have been established as important tools for a wide range of applications, such as quantum cryptography, quantum key distribution, entanglement detection, quantum metrology, quantum speed limit and so on. It is thus necessary to focus on the study of uncertainty relations.
In this Thesis we use the notions of information entropy, statistical complexity and\nquantum entanglement, together with the methods of Classical and Quantum Information\nand the algebraic and asymptotic techniques of the theory of orthogonal polynomials and special functions of Applied Mathematics and Mathematical Physics, to study and quantify the multiple facets of the spatial delocalization of\nthe charge and matter distributions of the multidimensional quantum systems of bosonic\nand fermionic character. These facets, which determine both the uncertainty measures of\nthe systems and their physical and chemical properties, are manifested in the enormous\ndiversity of multidimensional geometries of the single-particle densities which characterize\nthe non-relativistic mechano-quantically allowed states of such systems, according to\nthe Density Functional Theories based on the Hohenberg-Kohn theorem and generalizations.\nThe aim of this Thesis is to contribute to the emergent informational representation\nof the quantum systems, which complements the standard representation\nbased on energetic concepts. The analytical determination of the entropy-,\ncomplexity- and entanglement-like measures and their associated uncertainty relations is a fundamental task (not yet solved except for the ground state and\na few low-lying excited states) even for the very small number of elementary quantummechanical\npotentials which are utilized to approach and model the self-consistent mean-\n eld of the physical systems (atoms, molecules,...), as this work shows. Basically, this\nis because the quantum manifestations of the multidimensional physical systems are\ngenerally inaccessible as the associated Schr odinger equation cannot be exactly solved\nexcept for a few cases where the quantum-mechanical potential has a known symmetry.
ABSTRACT The dimensional analysis of the position and momentum variances‐based quantum mechanical Heisenberg uncertainty measures was carried out for generalized Yukawa‐type potential. The entropic information measures given by Shannon entropy sums and Fisher information products were obtained out for generalized Yukawa‐type potential. It is interesting to observe that the Shannon entropy in position space, , increases and then decreases with the quantum number , suggesting a growth in delocalization due to a rapid increase in fluctuations. Likewise, the Shannon entropy in momentum space, , decreases and then increases with the quantum number , and vice versa. This unusual phenomenon may be explained by the parity restrictions inherent in the system. The Bialynicki‐Birula and Mycielski (BBM) entropic uncertainty relation was verified to be saturated; however, the sum of the entropies increases with the quantum number up to a certain point and then decreases as continues to increase. We also found that the Fisher information accurately increases with the quantum number . The numerical results that support the validity of the scaling properties for Shannon entropy sum and Fisher information product were presented. The existence of a squeezed phenomenon in the ground and first excited state for some values of the potential screening parameter in momentum space was found. The Heisenberg uncertainty relation is met when the squeeze in one coordinate is compensated by an increase in the other coordinate. The results obtained exceed the theoretical lower bounds for all the information‐theoretic measures considered, confirming their physical validity and accuracy.