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

Suplemento de la Revista Mexicana de Física

Construction and analysis of statistical correlation measures through Diophantine equations

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

Published May 21, 2025• 0 citations
Publisher

Abstract

In this work, we explore the connection between Diophantine equations and the construction of informational measures, particularly mutual information, total correlation, and higher-order interaction information. These information measures are calculated in continuous variable quantum systems comprised of three to fifty harmonic oscillators, and their behaviour was compared among them. By analyzing the ground state of quantum harmonic oscillators, we establish a mathematical framework where Diophantine constraints emerge naturally in the computation of these quantities. There is an overall consistency in the behaviour of the introduced measures as function of the parameters of pairwise potential and the number of oscillators. Our results provide new insights into the interplay between number theory and quantum information, suggesting novel approaches to quantifying higher-order correlations in many-body quantum systems.

DOI
10.31349/suplrevmexfis.6.011307

Related papers

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

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

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

Higher-order statistical correlations in three-particle quantum systems with harmonic interactions

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

Interaction information, a measure of higher-order statistical correlations, is examined in quantum systems consisting of three coupled and uncoupled oscillators, in position and in momentum space. The magnitudes of the interaction information are greater than zero, and equal in both spaces, in the uncoupled case. This is interpreted as a dominance of synergic effects due to the indistinguishability of particles, and is larger for symmetric wave functions compared to antisymmetric ones. In symmetric wave functions, the magnitude of interaction information increases with the coupling strength, and the inclusion of coupling reveals differences in behavior between the spaces. With attractive potentials, interaction information is positive in momentum space (synergic) and negative in position space (redundant). This is reversed with repulsive potentials. It is now position space which yields positive values, while momentum space gives negative ones. The inclusion of spin provides antisymmetric wave functions which yield that the overall tendencies are maintained. However, there are regions for both attractive and repulsive potentials where the sign of interaction information changes, with the interpretation that the balance changes from a dominance of synergic to that of redundant interactions, and vice versa. This suggests that the nature of the interactions in these systems can be tuned with the coupling strength.

Physical Review A•Apr 6, 2020•15 citations

Tuning Pair and Higher‐Order Mutual Information Measures in Oscillator Systems with N$N$‐Dependent Frequencies

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

Abstract Analytical expressions for pair and higher‐order mutual information measures in position and momentum space are examined for many‐particle () coupled oscillator systems in their ground states. These relations reveal that the magnitudes of the statistical correlations monotonically increase with the strength of the particle interaction for all particle numbers (). On the other hand, the magnitudes of the measures monotonically decrease as increases, when the intensities of the one‐ and two‐body potentials are fixed. However, extremal points appear in the measures as is varied, when the intensity of the one‐body potential is ‐dependent, and set at . Fractional values of lead to the appearance of extremal points. These results illustrate how the intensity of the one‐body potential can be used to tune the behavior of the statistical correlation measures in these systems.

Advanced Theory and Simulations•Oct 17, 2024•0 citations

Quantum State Multi-Modal Correlation Analysis

Jincheng Zhang

This paper investigates the intricate correlations within multi-modal quantum states, presenting a novel theoretical framework and analytical methodology. The core of the research lies in applying quantum information theory and statistical physics to model and understand the complex behaviors inherent in multi-modal quantum systems. We aim to move beyond traditional approaches by establishing a robust mathematical framework for quantifying and characterizing multi-modal entanglement and correlations, specifically focusing on scenarios where multiple quantum systems are linked. The developed methods offer a new perspective on manipulating and controlling these systems, with potential implications for quantum computation, quantum communication, and fundamental quantum information theory. The key contributions include a generalized correlation function, a multi-modal entanglement measure, and an approach to analyzing the dynamics of multi-modal quantum systems. The work explores the connections between classical statistical mechanics and quantum entanglement in these systems, providing insights into the emergence of correlations at the quantum level. ---

•Sep 5, 2026•0 citations
In this work, we explore the connection between Diophantine equations and the construction of informational measures, particularly mutual information, total correlation, and higher-order interaction information. These information measures are calculated in continuous variable quantum systems comprised of three to fifty harmonic oscillators, and their behaviour was compared among them. By analyzing the ground state of quantum harmonic oscillators, we establish a mathematical framework where Diophantine constraints emerge naturally in the computation of these quantities. There is an overall consistency in the behaviour of the introduced measures as function of the parameters of pairwise potential and the number of oscillators. Our results provide new insights into the interplay between number theory and quantum information, suggesting novel approaches to quantifying higher-order correlations in many-body quantum systems.
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.
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.
Interaction information, a measure of higher-order statistical correlations, is examined in quantum systems consisting of three coupled and uncoupled oscillators, in position and in momentum space. The magnitudes of the interaction information are greater than zero, and equal in both spaces, in the uncoupled case. This is interpreted as a dominance of synergic effects due to the indistinguishability of particles, and is larger for symmetric wave functions compared to antisymmetric ones. In symmetric wave functions, the magnitude of interaction information increases with the coupling strength, and the inclusion of coupling reveals differences in behavior between the spaces. With attractive potentials, interaction information is positive in momentum space (synergic) and negative in position space (redundant). This is reversed with repulsive potentials. It is now position space which yields positive values, while momentum space gives negative ones. The inclusion of spin provides antisymmetric wave functions which yield that the overall tendencies are maintained. However, there are regions for both attractive and repulsive potentials where the sign of interaction information changes, with the interpretation that the balance changes from a dominance of synergic to that of redundant interactions, and vice versa. This suggests that the nature of the interactions in these systems can be tuned with the coupling strength.
Tuning Pair and Higher‐Order Mutual Information Measures in Oscillator Systems with NNNN‐Dependent Frequencies
Abstract Analytical expressions for pair and higher‐order mutual information measures in position and momentum space are examined for many‐particle () coupled oscillator systems in their ground states. These relations reveal that the magnitudes of the statistical correlations monotonically increase with the strength of the particle interaction for all particle numbers (). On the other hand, the magnitudes of the measures monotonically decrease as increases, when the intensities of the one‐ and two‐body potentials are fixed. However, extremal points appear in the measures as is varied, when the intensity of the one‐body potential is ‐dependent, and set at . Fractional values of lead to the appearance of extremal points. These results illustrate how the intensity of the one‐body potential can be used to tune the behavior of the statistical correlation measures in these systems.
This paper investigates the intricate correlations within multi-modal quantum states, presenting a novel theoretical framework and analytical methodology. The core of the research lies in applying quantum information theory and statistical physics to model and understand the complex behaviors inherent in multi-modal quantum systems. We aim to move beyond traditional approaches by establishing a robust mathematical framework for quantifying and characterizing multi-modal entanglement and correlations, specifically focusing on scenarios where multiple quantum systems are linked. The developed methods offer a new perspective on manipulating and controlling these systems, with potential implications for quantum computation, quantum communication, and fundamental quantum information theory. The key contributions include a generalized correlation function, a multi-modal entanglement measure, and an approach to analyzing the dynamics of multi-modal quantum systems. The work explores the connections between classical statistical mechanics and quantum entanglement in these systems, providing insights into the emergence of correlations at the quantum level. ---