Construction and analysis of statistical correlation measures through Diophantine equations

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.