Upper continuity bounds on the relative $q$-entropy for $q>1$

Abstract

Generalized entropies and relative entropies are the subject of active research. Similar to the standard relative entropy, the relative $q$-entropy is generally unbounded for $q>1$. Upper bounds on the quantum relative $q$-entropy in terms of norm distances between its arguments are obtained in finite-dimensional context. These bounds characterize a continuity property in the sense of Fannes.

7 pages, no figures. The statements and the proofs from Section III are corrected. The results of Section IV are improved. Appendix A has been excluded