Construction and analysis of statistical correlation measures through Diophantine equations

Authors

DOI:

https://doi.org/10.31349/SuplRevMexFis.6.011307

Keywords:

Diophantine equations, Shannon entropies, higher-order correlation measures, quantum harmonic oscillators

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.

References

C. E. Shannon, A Mathematical Theory of Communication, Bell Syst. Tech. J. 27 (1948) 379, https://doi.org/10.1002/j.1538-7305.1948.tb01338.x

T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed. (John Wiley and Sons, New York, 2006), pp. 13–55.

A. U. J. Lode, B. Chakrabarti, and V. K. B. Kota, Many-body entropies, correlations, and emergence of statistical relaxation in interaction quench dynamics of ultracold bosons, Phys. Rev. A 92 (2015) 033622, https://link.aps.org/doi/10.1103/PhysRevA.92.033622

K. Tanabe, Mutual information in coupled double quantum dots, J. Stat. Mech. 2020 (2020) 093209, https://dx.doi.org/10.1088/1742-5468/abb233

J.-J. Dong, D. Huang, and Y. f. Yang, Mutual information, quantum phase transition, and phase coherence in Kondo systems, Phys. Rev. B 104 (2021) L081115, https://doi.org/10.1103/PhysRevB.104.L081115

G. Nicoletti and D. M. Busiello, Mutual information disentangles interactions from changing environments, Phys. Rev. Lett. 127 (2021) 228301, https://link.aps.org/doi/10.1103/PhysRevLett.127.228301

W. S. Nascimento, M. M. de Almeida, and F. V. Prudente, Coulomb correlation and information entropies in confined helium-like atoms, Eur. Phys. J. D 75 (2021) 171, https://doi.org/10.1140/epjd/s10053-021-00177-6

S. Mondal, K. Sen, and J. K. Saha, He atom in a quantum dot: Structural, entanglement, and information-theoretical measures, Phys. Rev. A 105 (2022) 032821, https://link.aps.org/doi/10.1103/PhysRevA.105.032821

D. Park, Thermal entanglement phase transition in coupled harmonic oscillators with arbitrary time-dependent frequencies, Quantum Inf. Process. 19 (2020), https://doi.org/10.48550/arXiv.1903.03297

J. Faba, V. Martín, and L. Robledo, Correlation energy and quantum correlations in a solvable model, Phys. Rev. A 104 (2021) 032428, https://link.aps.org/doi/10.1103/PhysRevA.104.032428

J. C. Angulo and S. López-Rosa, Mutual information in conjugate spaces for neutral atoms and ions, Entropy 24 (2022) 233, https://doi.org/10.3390/e24020233

P. Schürger and V. Engel, Differential Shannon entropies and correlation measures for Born-Oppenheimer electron-nuclear dynamics: numerical results and their analytical interpretation, Phys. Chem. Chem. Phys. 25 (2023) 28373, https://doi.org/10.3390/e24020233

K. Kumar and V. Prasad, Control of correlation using confinement in case of quantum system, Ann. Phys. 535 (2023) 2300166, https://doi.org/10.1002/andp.202300166

R. Roy, et al., Information theoretic measures for interacting bosons in optical lattice, Phys. Rev. E 107 (2023) 024119, https://link.aps.org/doi/10.1103/PhysRevE.107.024119

C. R. Estañón, et al., The confined helium atom: An information-theoretic approach, Int. J. Quantum Chem. 124 (2024) e27358, https://doi.org/10.1002/qua.27358

J. O. Ernst and F. Tennie, Mode entanglement in fermionic and bosonic harmonium, New J. Phys. 26 (2024) 033042, https://dx.doi.org/10.1088/1367-2630/ad240f

S. J. C. Salazar, H. G. Laguna, and R. P. Sagar, Higher-order statistical correlations in three-particle quantum systems with harmonic interactions, Phys. Rev. A 101 (2020) 042105, https://link.aps.org/doi/10.1103/PhysRevA.101.042105

S. Panzeri, et al., Correlations and the encoding of information in the nervous system, Proc. R. Soc. London B 266 (1999) 1001, https://doi.org/10.1098/rspb.1999.0736

E. Schneidman, W. Bialek, and M. J. Berry, Synergy, redundancy, and independence in population codes, J. Neurosci. 23 (2003) 11539, https://doi.org/10.1523/JNEUROSCI.23-37-11539.2003

P. E. Latham and S. Nirenberg, Synergy, redundancy, and independence in population codes, revisited, J. Neurosci. 25 (2005) 5195, https://doi.org/10.1523/JNEUROSCI.5319-04.2005

B. J. Killian, J. Y. Kravitz, and M. K. Gibson, Extraction of configurational entropy from molecular simulations via an expansion approximation, J. Chem. Phys. 127 (2007) 024107, https://doi.org/10.1063/1.2746329

P. A. Bouvrie, et al., Quantum entanglement in exactly soluble atomic models: the Moshinsky model with three electrons, and with two electrons in a uniform magnetic field, Eur. Phys. J. D 66 (2012), https://doi.org/10.1140/epjd/e2011-20417-4

H. Peng and Y. Ho, Statistical Correlations of the N-particle Moshinsky Model, Entropy 17 (2015) 1882, https://doi.org/10.3390/e17041882

S. Watanabe, Information theoretical analysis of multivariate correlation, IBM J. Res. Dev. 4 (1960) 66, https://doi.org/10.1147/rd.41.0066

T. S. Han, Multiple mutual information and multiple interactions in frequency data, Inf. Control 46 (1980) 26, https://doi.org/10.1016/S0019-9958(80)90478-7

W. J. McGill, Multivariate information transmission, Psychometrika 19 (1954) 97, https://doi.org/10.1007/BF02289159

N. J. Cerf and C. Adami, Entropic Bell inequalities, Phys. Rev. A 55 (1997) 3371, https://link.aps.org/doi/10.1103/PhysRevA.55.3371

H. Matsuda, Physical nature of higher-order mutual information: Intrinsic correlations and frustration, Phys. Rev. E 62 (2000) 3096, https://link.aps.org/doi/10.1103/PhysRevE.62.3096

H. Matsuda, Information theoretic characterization of frustrated systems, Physica A 294 (2001) 180, https://doi.org/10.1016/S0378-4371(01)00039-5

S. J. C. Salazar, H. G. Laguna, and R. P. Sagar, Tuning Pair and Higher-Order Mutual Information Measures in Oscillator Systems with N-Dependent Frequencies, Adv. Theory Simul. 8 (2024) 2400777, https://doi.org/10.1002/adts.202400777

Downloads

Published

2025-05-21

How to Cite

1.
Salazar SJC, Laguna Galindo H, Sagar RP. Construction and analysis of statistical correlation measures through Diophantine equations. Supl. Rev. Mex. Fis. [Internet]. 2025 May 21 [cited 2025 Jul. 17];6(1):011307 1-6. Available from: https://rmf.smf.mx/ojs/index.php/rmf-s/article/view/7997

Issue

Vol. 6 No. 1 (2025): Suplemento de la Revista Mexicana de Física. Commemorating the 50th Anniversary of UAM: Applications of Information Theory in Natural Sciences

Section

13 50th Anniv. of UAM: Applications of Information Theory in Natural Sciences

License

Copyright (c) 2025 S. J. C. Salazar, H. G. Laguna, R. P. Sagar

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Authors retain copyright and grant the Suplemento de la Revista Mexicana de Física right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.