Entropy Densities for Algebraic States

Algebraic (or finitely correlated) states are translation-invariant states on an infinite tensor product C*-algebra, whose construction is rather general, including quantum Markov chains. For a strongly mixing algebraic state, we obtain the relation between the mean entropy and another entropy density, which means the macroscopic uniformity under the relevant state in the statistical mechanical sense. For this purpose an outstanding property of approximately product type is shown for strongly mixing algebraic states. We also obtain similar relations of the mean relative entropy to other entropy densities for two translation-invariant states when the reference one is a strongly mixing algebraic state.