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.