  This book is an introduction to quantum Markov chains and explains how this
concept is connected to the question of how well a lost quantum mechanical
system can be recovered from a correlated subsystem. To achieve this goal, we
strengthen the data-processing inequality such that it reveals a statement
about the reconstruction of lost information. The main difficulty in order to
understand the behavior of quantum Markov chains arises from the fact that
quantum mechanical operators do not commute in general. As a result we start by
explaining two techniques of how to deal with non-commuting matrices: the
spectral pinching method and complex interpolation theory. Once the reader is
familiar with these techniques a novel inequality is presented that extends the
celebrated Golden-Thompson inequality to arbitrarily many matrices. This
inequality is the key ingredient in understanding approximate quantum Markov
chains and it answers a question from matrix analysis that was open since 1973,
i.e., if Lieb's triple matrix inequality can be extended to more than three
matrices. Finally, we carefully discuss the properties of approximate quantum
Markov chains and their implications.
