Quantum Central Limit Theorems, Emergence of Classicality and Time-Dependent Differential Entropy

We derive some quantum central limit theorems for the expectation values of macroscopically coarse-grained observables, which are functions of coarse-grained Hermitian operators consisting of non-commuting variables. Thanks to the Hermiticity constraints, we obtain positive-definite distributions for the expectation values of observables. These probability distributions open some pathway for the emergence of classical behaviours in the limit of an infinitely large number of identical and non-interacting quantum constituents. This is in contradistinction to other mechanisms of classicality emergence due to environmental decoherence and consistent histories. The probability distributions thus derived also enable us to evaluate the non-trivial time-dependence of certain differential entropies.