Quantum source coding and data compression

This lecture is intended to be an easily accessible first introduction to quantum information theory. The field is large and it is not completely covered even by the recent monograph [15]. Therefore the simple topic of data compression is selected to present some ideas of the theory. Classical information theory is not a prerequisite, we start with the basics of Shannon theory to give a feeling for Shannon entropy and for the informational divergence or relative entropy. The aim is to present Schumacher’s compression theorem and to demonstrate that the von Neumann entropy, introduced in the 1920’s by thermodynamical considerations, is a measure of quantum information exactly in the way as the Shannon entropy is that for classical information. Our discussion makes clear that the compression theorem depends heavily on the existence of the high-probability subspace. At the end of the lecture quantum sources with memory and some related questions are briefly discussed. This part could be skipped by new-comers in the field.