Axiomatic approach to measures of total correlations

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 aim to discuss the challenge of defining a reliable measure of total correlations. We first outline essential properties that an effective correlation measure should satisfy and review existing measures, including quantum mutual information, the p-norm of the correlation matrix, and the recently defined quantum Pearson correlation coefficient. Additionally, we introduce new measures based on R\'enyi and Tsallis relative entropies, as well as the Kullback-Leibler divergence. Our analysis reveals 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 argue that quantum mutual information remains a valid measure of total correlations.