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.
Abstract We establish a rigorous equivalence between quantum coherence measures defined by the sandwiched Rényi relative entropy and generalized conditional entropies associated with auxiliary quantum systems, providing a direct operational interpretation of sandwiched Rényi relative entropy of coherence as the conditional uncertainty of projective measurement outcomes with quantum memory. Motivated by this connection, we develop a coherence-based approach to quantum uncertainty relations. Within this approach, we introduce a coherence measure induced by the two-parameter α - z -Rényi relative entropy and derive corresponding uncertainty relations for both pure and mixed quantum states. The resulting state-independent bounds recover known results in appropriate limits and yield tighter bounds over a broad parameter regime. Our results provide a systematic connection between quantum coherence, generalized conditional entropies, and uncertainty relations, thereby offering a unified perspective on quantum uncertainty in terms of Rényi-type information measures.
Following Csiszár's approach in classical information theory, it is shown that the quantum α-relative entropies with parameter α ∈ (0,1) can be represented as generalized cutoff rates, and hence a direct operational interpretation of the quantum α-relative entropies are provided. It is also shown that various generalizations of the Holevo capacity, defined in terms of the α-relative entropies, coincide for the parameter range α ∈ (0,2], and an upper bound on the one-shot ε-capacity of a classical-quantum channel in terms of these capacities is given.