An Uncertainty Principle for Discrete Groups

Jang Young Bang

Healy David

Jason Crann, Mehrdad Kalantar

We present a generalization of Hirschman's entropic uncertainty principle for locally compact Abelian groups to unimodular locally compact quantum groups. As a corollary, we strengthen a well-known uncertainty principle for compact groups, and generalize the relation to compact quantum groups of Kac type. We also establish the complementarity of finite-dimensional quantum group algebras. In the non-unimodular setting, we obtain an uncertainty relation for arbitrary locally compact groups using the relative entropy with respect to the Haar weight as the measure of uncertainty. We also show that when restricted to q-traces of discrete quantum groups, the relative entropy with respect to the Haar weight reduces to the canonical entropy of the random walk generated by the state.

Sang-Gyun Youn

The uncertainty principle has been established within the framework of locally compact quantum groups in recent years. This paper demonstrates that entropic uncertainty relations can be strengthened under localizations on discrete quantum groups, which is the case if the dual compact quantum group G is the free orthogonal quantum group ON+ with N ≥ 3 or if G admits an infinite Λ(p) set with p > 2. On the other hand, this paper explains the reason why such phenomena do not appear when G is one of the connected semisimple compact Lie groups, O2+, and the quantum SU(2) groups. Also, we discuss the divergence of entropic uncertainty relations together with some explicit explanations.

E. U. Condon