The Shannon-McMillan theorem and related results for ergodic quantum spin lattice systems and applications in quantum information theory

The aim of this thesis is to formulate and prove quantum extensions of the famous Shannon-McMillan theorem and its stronger version due to Breiman. In ergodic theory the Shannon-McMillan-Breiman theorem is one of the fundamental limit theorems for classical discrete dynamical systems. It can be interpreted as a special case of the individual ergodic theorem. In this work, we consider spin lattice systems, which can be interpreted as dynamical systems under the action of the translation group. The Shannon-McMillan-Breiman theorem states that the Shannon entropy rate of an ergodic lattice system is the asymptotical rate of exponential decrease of probability of almost each individual spin configuration. In information theory, information sources are usually modeled by time-discrete stochastic processes or equivalently by 1-dimensional spin lattice systems. There, this theorem plays an important role, giving an interpretation to the Shannon entropy rate as the asymptotically mean information per signal. At the same time the entropy rate is an achievable lower bound for the compression rate of asymptotically error-free data compression algorithms. It turns out, that there are analogues of the classical Shannon-McMillan theorem and Breiman’s extension for quantum spin lattice systems, modeled as C∗-dynamical systems with respect to the action of the translation group on a quasi-local C∗-algebra. There, the concept of Shannon entropy for discrete probability distributions is generalized by the von Neumann entropy for density operators. A number of results, related to the quantum Shannon-McMillan(Breiman) theorem are presented in this work. Similarly to classical information theory, the existence of asymptotically error-free data compression schemes for ergodic quantum sources is proven based on the quantum Shannon-McMillan theorem. There, the achievable lower bound on the compression rate is given by the von Neumann entropy rate of the quantum source. Furthermore, a structure theorem is proven, which describes the convex decomposition of ergodic states on quantum spin lattice systems into components which are ergodic with respect to some subgroup of the whole translation group. Subsuming, we may say, that the presented results about quantum spin lattice systems establish the von Neumann entropy rate as the generalization of the Shannon entropy rate to the quantum case.