Rigorous properties and uncertainty-like relationships on product-complexity measures: Application to atomic systems

Most of research on complexities and the corresponding conclusions have been obtained by numerically quantifying their values, and little attention has been paid to their theoretical properties and the exact meaning within an statistical framework, valid for any arbitrary n-dimensional distribution. In the present work, different product-complexities, constructed as a product of two relevant information quantities associated to the concepts of uncertainty and localization are investigated. Different results on product-complexities, such as rigorous bounds, uncertainty-like inequalities, relationships among themselves and also statistical interpretations are given. For the sake of completeness, some of these analytical results on product-complexities are numerically checked for the one-particle densities of atomic systems in both conjugated spaces.