Axiomatic Approach to Measures of Total Correlations

• Published in: Entropy (Basel, Switzerland) Vol. 26; no. 12; p. 1098
• Main Authors: Moraes, Gabriel L., Angelo, Renato M., Costa, Ana C. S.
• Format: Journal Article
• Language: English
• Published: Switzerland MDPI AG 15.12.2024; MDPI
• Subjects: Axiomatic set theory; Correlation (Statistics); Correlation analysis; Correlation coefficients; Hilbert space; Mathematical research; Methods; quantum correlations; Quantum mechanics; quantum mutual information; Qubits (quantum computing); System reliability; total correlations; Brazil; quantum correlations; quantum mutual information; total correlations
• ISSN: 1099-4300; 1099-4300
• DOI: 10.3390/e26121098

Correlations play a pivotal role in various fields of science, particularly in quantum mechanics, yet their proper quantification remains a subject of debate. In this work, we aimed to discuss the challenge of defining a reliable measure of total correlations. We first outlined the essential properties that an effective correlation measure should satisfy and reviewed existing measures, including quantum mutual information, the p-norm of the correlation matrix, and the recently defined quantum Pearson correlation coefficient. Additionally, we introduced new measures based on Rényi and Tsallis relative entropies, as well as the Kullback–Leibler divergence. Our analysis revealed that while quantum mutual information, the p-norm, and the Pearson measure exhibit equivalence for two-qubit systems, they all suffer from an ordering problem. Despite criticisms regarding its reliability, we argued that QMI remains a valid measure of total correlations.