Conditional R-norm entropy and R-norm divergence in quantum logics

Abstract

‎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.‎‎‎