Conditional R-norm Entropy and R-norm Divergence in Quantum Logics

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.