Do uncertainty minimizers attain minimal uncertainty?

The uncertainty principle is a fundamental concept in quantum mechanics, harmonic analysis and signal and information theory. It is rooted in the framework of quantum mechanics, where it is known as the Heisenberg uncertainty principle. In general, the uncertainty principle gives a lower bound on the product of variances for any state f with respect to two self-adjoint operators: The functions that attain the lower bound of the inequality have been investigated extensively, and are known as uncertainty minimizers. However, in information theory, uncertainty is measured in terms of the localization properties, which in turn are defined via the product of the variances. Hence, uncertainty is minimized by the states f, that we call variance minimizers, that attain the minimum of v_f(A)v_f(B). In this paper, we investigate the differences and relations between the uncertainty minimizers and the variance minimizers. We provide a mechanism for obtaining the variance minimizers, and show that the classical uncertainty principle can be violated. Examples that account for the Weyl-Heisenberg, the affine and the affine Weyl-Heisenberg groups are given.