For partitions on quantum logic, the Rényi and Tsallis conditional entropies are introduced. Several relations between the conditional entropies of such partitions are derived.
We further examine some properties of the conditional Rényi and Tsallis–Havrda–Charvát (THC) entropies. Such properties are interesting from the viewpoint of applications in studying protocols of quantum information science and foundations of quantum mechanics. In particular, we consider properties of the conditional Rényi and THC entropies with respect to conditioning on more. We also exemplify that the desired property can be violated with the conditional min-entropy. Applications of such results to the TCH entropy rate are considered. Connections between generalized conditional entropies and error probability are examined. Several relations between various conditional entropies are obtained. It is shown that such relations can be used for bounding the conditional Rényi and TCH entropies.
Recently a new quantum generalization of the Rényi divergence and the corresponding conditional Rényi entropies was proposed. Here, we report on a surprising relation between conditional Rényi entropies based on this new generalization and conditional Rényi entropies based on the quantum relative Rényi entropy that was used in previous literature. Our result generalizes the well-known duality relation H(A|B) + H(A|C) = 0 of the conditional von Neumann entropy for tripartite pure states to Rényi entropies of two different kinds. As a direct application, we prove a collection of inequalities that relate different conditional Rényi entropies and derive a new entropic uncertainty relation.
This contribution deals with the mathematical modeling of R-norm entropy and R-norm divergence in quantum logics. We extend some results concerning the R-norm entropy and conditional R-norm entropy given in (Inf. Control 45, 1980), to the quantum logics. Firstly, the concepts of R-norm entropy and conditional R-norm entropy in quantum logics are introduced. We prove the concavity property for the notion of R-norm entropy in quantum logics and we show that this entropy measure does not have the property of sub-additivity in a true sense. It is proven that the monotonicity property for the suggested type of conditional version of R-norm entropy, holds. Furthermore, we introduce the concept of R-norm divergence of states in quantum logics and we derive basic properties of this quantity. In particular, a relationship between the R-norm divergence and the R-norm entropy of partitions is provided.