We introduce an axiomatic approach to entropies and relative that relies only on minimal information-theoretic axioms, namely monotonicity under mixing data-processing as well additivity for product distributions. find these axioms induce sufficient structure establish continuity in the interior of probability simplex meaningful upper lower bounds, e.g., we every entropy satisfying must lie bet...

In this exposition we prove an existence theorem for a quantum mechanical dynamical entropy based on von-Neumann’s measurement theory. To that end we introduced a Shannon type of information associated with a quantum channel or measurement based on Araki’s relative entropy. This is an invariance for the dynamics which generalizes Kolmogorov-Sinai ’s notion of dynamical entropy of a measure pres...