McMillan type convergence for quantum Gibbs states (Q1345851)

It is well-known that for translation-invariant states of infinite tensor products of matrix algebras the entropy density exists. This fact is the consequence of the subadditivity of the von Neumann entropy. In the classical case of shift invariant measures, the Shannon-McMillan theorem tells about the almost sure convergence of the entropy functions under very general conditions. The paper considers the simplest non-commutative extension of McMillan's theorem to quantum Gibbs states. The main result is the almost uniform and strong operator convergence of the entropy operators in the GNS space. Hence there is an operator convergence behind the existence of the entropy density.