Physical information entropy and probability Shannon entropy

In a previous work by one of us (R. Urigu) concerning open quantum systems it was remarked that in processes of the type\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$w \to (\bar p_i \bar w_i )$$ \end{document}, when evaluating the information entropy of the environment as the Shannon entropy of the outcome probabilities\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bar p_i $$ \end{document} in the channels\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bar w_i $$ \end{document}, the total information entropy may decrease. We remark here that this decrease is easily excluded by requiring a condition of quantum modelizability of the environment even with respect to Shannon entropy (“cybernetic interpretability” of the environment). Further conditions on the quantum model of the environment are defined (“maximal observability” and “Boolean interpretability”), which are proved to be equivalent, and it turns out that, once satisfied in one model, they also are in any model with pure initial state; furthermore, these conditions turn out to be equivalent to the condition that the process consists of pure operations of the first kind. The relevance to the concept of macroscopicity and to the “von Neumann chain” is discussed.