A Formulation of R\'enyi Entropy on $C^*$-Algebras

The entropy of probability distribution defined by Shannon has several extensions. R\'enyi entropy is one of the general extensions of Shannon entropy and is widely used in engineering, physics, and so on. On the other hand, the quantum analogue of Shannon entropy is von Neumann entropy. Furthermore, the formulation of this entropy was extended to on $C^*$-algebras by Ohya ($\mathcal{S}$-mixing entropy). In this paper, we formulate Renyi entropy on $C^*$-algebras based on $\mathcal{S}$-mixing entropy and prove several inequalities for the uncertainties of states in various reference systems.