🤖 AI Summary
This work proposes a generalized Rényi relative entropy that satisfies fundamental information-theoretic axioms, particularly the data processing inequality, and provides a tighter upper bound for the Tsallis relative entropy. To this end, the authors introduce for the first time the r-logarithm function and construct a three-parameter family of r-deformed α-z-Rényi relative entropies. Through rigorous analysis of quantum divergences, characterization of the parameter space, and comparison via operator inequalities, they systematically establish the conditions under which this quantity constitutes a valid divergence. The study precisely identifies the parameter region where the data processing inequality holds and demonstrates that, in the setting of density operators, the derived upper bound strictly improves upon existing results.
📝 Abstract
In this article, we consider the $r$-logarithm for defining three-parameter family of Rényi relative entropies that are generalization of the $α$-$z$-Rényi relative entropies. All the members of $r$-deformed $α$-$z$-Rényi relative entropies satisfy the necessary axioms to be a divergence. We expose the range of parameters $α$, $z$ and $r$ for which the data processing inequality holds. We also establish that $r$-deformed $α$-$z$-Rényi relative entropy is an upper bound of the Tsallis relative entropy. Now, we have two upper bounds of the Tsallis relative entropy, which are $r$-deformed $α$-$z$-Rényi relative entropy and the other one, which is discussed in literature. We investigate the order relationship between these two upper bounds of the Tsallis relative entropy. We observe that our new upper bound is more tighter when applicable to the density operators.