Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Quantum Chemistry

Information theoretical measures from cumulative and survival densities in quantum systems

Humberto G. Laguna, Robin P. Sagar

Published Apr 19, 2017• 7 citations
Publisher

Abstract

Abstract Entropic uncertainty and statistical correlation measures, based on survival and cumulative densities, are explored in some representative quantum systems. We illustrate how the cumulative residual entropy in the quantum well system recovers the correct classical behavior for larger quantum numbers while the Shannon entropy does not. Two interacting and noninteracting oscillators are used to examine two‐particle entropies and their related correlation measures. The joint cumulative residual entropy does distinguish between symmetric and antisymmetric wave functions in interacting systems as the interaction is turned on. The joint Shannon entropy does not distinguish between the symmetries even in the presence of interaction. Conversely, the joint Shannon entropy and joint cumulative residual entropy are both unable to distinguish between the symmetries for certain states of the noninteracting oscillators. As measures of statistical correlation, the mutual information and the cross cumulative residual entropy yield similar behaviors as a function of the strength of the interparticle interaction.

DOI
10.1002/qua.25387

Related papers

Higher-Order Information Measures from Cumulative Densities in Continuous Variable Quantum Systems

Saúl J. C. Salazar, Humberto G. Laguna, Robin P. Sagar

A definition of three-variable cumulative residual entropy is introduced, and then used to obtain expressions for higher order or triple-wise correlation measures, that are based on cumulative residual densities. These information measures are calculated in continuous variable quantum systems comprised of three oscillators, and their behaviour compared to the analogous measures from Shannon information theory. There is an overall consistency in the behaviour of the newly introduced measures as compared to the Shannon ones. There are, however, differences in interpretation, in the case of three uncoupled oscillators, where the correlation is due to wave function symmetry. In interacting systems, the cumulative based measures are shown in order to detect salient features, which are also present in the Shannon based ones.

Quantum Reports•Nov 6, 2020•4 citations

Mutual information sums and relations with interaction energies in N -particle quantum systems

Saúl J. C. Salazar, Humberto G. Laguna, Robin P. Sagar

The sums of position- and momentum-space mutual information measures are used to examine the pairwise and higher-order statistical correlation in the ground states of N-particle coupled oscillators. Analytical expressions for these measures are shown to be related to the logarithmic interaction energies of these states, plus those of mirror states where the intensities of the one- and two-body potentials are interchanged, and the nature of the attractive or repulsive interaction is opposite to that in the original state. The measures separate the contributions from the interactions and those from the effective interactions due to marginalization into different terms. The pairwise mutual information sum is linearly related to the Shannon entropy sum in two particle systems, while the total correlation sum exhibits a similar relationship in three particle ones. In the latter instance, the interaction information sum can be related to entropy differences. This illustrates how entropy sums are connected to correlation measure sums in these systems. All measures approach zero with large N, when the magnitudes of the one- and two-body potentials are fixed. The pair mutual information and total correlation sums decay monotonically with N in the presence of an attractive potential and monotonically increase with a repulsive potential. On the other hand, the interaction information sum exhibits a minimum at small N, with an attractive potential. This is a consequence of the higher-order correlations governing behavior at smaller N while the pairwise ones dominate at larger N. Results are presented when the magnitude of the one-body potential is set to N.

Physical review. E•Sep 26, 2024•4 citations

Phase-space quantum distributions and information theory

Saúl J. C. Salazar, Humberto G. Laguna, Robin P. Sagar

The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.

Physical Review A•Apr 14, 2023•17 citations

Statistical Correlation Measures From Higher‐Order Moments in Quantum Oscillator Systems

Saúl J. C. Salazar, Humberto G. Laguna, Robin P. Sagar

Abstract Triple‐wise and higher‐order correlation measures are proposed using analogies from information theory. Two‐ and three‐variable statistical correlation measures, obtained from higher‐order moments, are examined in three‐particle continuous variable quantum oscillator systems. It is illustrated how the mutual information and cokurtosis measures exhibit similar behaviors, as the strength of the attractive or repulsive interparticle potential is varied.

Advanced Theory and Simulations•Mar 24, 2021•2 citations

Entropic Kullback‐Leibler type distance measures for quantum distributions

Humberto G. Laguna, Saúl J. C. Salazar, Robin P. Sagar

Abstract Entropic distance measures for quantum mechanical probability distributions, which are characterized by nodal structure and symmetry holes, are considered. We illustrate how the Kullback‐Leibler (KL) distance is not well defined in some instances and propose instead the use of the cumulative residual Kullback‐Leibler (CRKL) distance. The KL and CRKL measures are compared and contrasted for some representative quantum mechanical systems: The particle in an infinite well, the harmonic oscillator, and hydrogenic systems. We present cases where CRKL can be used to obtain distances whereas KL cannot be used, and also highlight examples where the KL and CRKL measures yield different behaviors and interpretations. An extension of the CRKL definition is obtained for application to harmonic oscillator systems defined over [−∞, ∞]. Distance measures for two‐variable (particle) distributions are also considered to address generalizations of the mutual information correlation measure. The use of CRKL in measuring distances between orbitals is also discussed.

International Journal of Quantum Chemistry•May 23, 2019•15 citations

Correlation and Information‐Theoretic Approaches in Quantum Coupled Double‐Well Systems

Kirtee Kumar, Vinod Prasad

Abstract This study investigates the quantum correlation and information‐theoretic measures of coupled double‐well systems, modeled using Razavy's hyperbolic potential. By employing numerical techniques, including the finite difference method and Kronecker tensor product, the time‐independent equation is solved to determine energy eigenvalues and eigenfunctions for quantum states, such as , , , and of the bipartite quantum system. The analysis focuses on quantum information measures, such as Shannon entropy, mutual information, and statistical correlation coefficients, to understand how potential parameters and and the coupling strength influence correlations. The results reveal intricate dependencies between quantum states, highlighting symmetry and asymmetry in correlation measures for mixed states under opposite coupling potentials. Further, the different behaviors of the statistical correlation coefficients are found in position and momentum space and their information‐theoretic measures as functions of potential parameters, which is very helpful for understanding the dynamics of quantum particles in the coupled system. Additionally, the study investigates the entanglement properties of bipartite quantum states, employing von Neumann, and linear entropy to measure quantum correlations. The results provide critical insights into the dynamics of quantum particles in confined systems, connecting quantum mechanics and information theory. This work opens new avenues to exploring quantum information dynamics in Razavy double‐well systems, which gives novel insights into the role of spatial and momentum correlations in particle interactions.

Advanced Theory and Simulations•Oct 28, 2025•1 citation
Abstract Entropic uncertainty and statistical correlation measures, based on survival and cumulative densities, are explored in some representative quantum systems. We illustrate how the cumulative residual entropy in the quantum well system recovers the correct classical behavior for larger quantum numbers while the Shannon entropy does not. Two interacting and noninteracting oscillators are used to examine two‐particle entropies and their related correlation measures. The joint cumulative residual entropy does distinguish between symmetric and antisymmetric wave functions in interacting systems as the interaction is turned on. The joint Shannon entropy does not distinguish between the symmetries even in the presence of interaction. Conversely, the joint Shannon entropy and joint cumulative residual entropy are both unable to distinguish between the symmetries for certain states of the noninteracting oscillators. As measures of statistical correlation, the mutual information and the cross cumulative residual entropy yield similar behaviors as a function of the strength of the interparticle interaction.
A definition of three-variable cumulative residual entropy is introduced, and then used to obtain expressions for higher order or triple-wise correlation measures, that are based on cumulative residual densities. These information measures are calculated in continuous variable quantum systems comprised of three oscillators, and their behaviour compared to the analogous measures from Shannon information theory. There is an overall consistency in the behaviour of the newly introduced measures as compared to the Shannon ones. There are, however, differences in interpretation, in the case of three uncoupled oscillators, where the correlation is due to wave function symmetry. In interacting systems, the cumulative based measures are shown in order to detect salient features, which are also present in the Shannon based ones.
The sums of position- and momentum-space mutual information measures are used to examine the pairwise and higher-order statistical correlation in the ground states of N-particle coupled oscillators. Analytical expressions for these measures are shown to be related to the logarithmic interaction energies of these states, plus those of mirror states where the intensities of the one- and two-body potentials are interchanged, and the nature of the attractive or repulsive interaction is opposite to that in the original state. The measures separate the contributions from the interactions and those from the effective interactions due to marginalization into different terms. The pairwise mutual information sum is linearly related to the Shannon entropy sum in two particle systems, while the total correlation sum exhibits a similar relationship in three particle ones. In the latter instance, the interaction information sum can be related to entropy differences. This illustrates how entropy sums are connected to correlation measure sums in these systems. All measures approach zero with large N, when the magnitudes of the one- and two-body potentials are fixed. The pair mutual information and total correlation sums decay monotonically with N in the presence of an attractive potential and monotonically increase with a repulsive potential. On the other hand, the interaction information sum exhibits a minimum at small N, with an attractive potential. This is a consequence of the higher-order correlations governing behavior at smaller N while the pairwise ones dominate at larger N. Results are presented when the magnitude of the one-body potential is set to N.
The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.
Abstract Entropic distance measures for quantum mechanical probability distributions, which are characterized by nodal structure and symmetry holes, are considered. We illustrate how the Kullback‐Leibler (KL) distance is not well defined in some instances and propose instead the use of the cumulative residual Kullback‐Leibler (CRKL) distance. The KL and CRKL measures are compared and contrasted for some representative quantum mechanical systems: The particle in an infinite well, the harmonic oscillator, and hydrogenic systems. We present cases where CRKL can be used to obtain distances whereas KL cannot be used, and also highlight examples where the KL and CRKL measures yield different behaviors and interpretations. An extension of the CRKL definition is obtained for application to harmonic oscillator systems defined over [−∞, ∞]. Distance measures for two‐variable (particle) distributions are also considered to address generalizations of the mutual information correlation measure. The use of CRKL in measuring distances between orbitals is also discussed.
Abstract This study investigates the quantum correlation and information‐theoretic measures of coupled double‐well systems, modeled using Razavy's hyperbolic potential. By employing numerical techniques, including the finite difference method and Kronecker tensor product, the time‐independent equation is solved to determine energy eigenvalues and eigenfunctions for quantum states, such as , , , and of the bipartite quantum system. The analysis focuses on quantum information measures, such as Shannon entropy, mutual information, and statistical correlation coefficients, to understand how potential parameters and and the coupling strength influence correlations. The results reveal intricate dependencies between quantum states, highlighting symmetry and asymmetry in correlation measures for mixed states under opposite coupling potentials. Further, the different behaviors of the statistical correlation coefficients are found in position and momentum space and their information‐theoretic measures as functions of potential parameters, which is very helpful for understanding the dynamics of quantum particles in the coupled system. Additionally, the study investigates the entanglement properties of bipartite quantum states, employing von Neumann, and linear entropy to measure quantum correlations. The results provide critical insights into the dynamics of quantum particles in confined systems, connecting quantum mechanics and information theory. This work opens new avenues to exploring quantum information dynamics in Razavy double‐well systems, which gives novel insights into the role of spatial and momentum correlations in particle interactions.