Logarithmic Sobolev Inequalities for Finite Dimensional Quantum Markov Chains

This is an expository chapter on the use of entropic and functional inequalities for bounding rates of convergence of quantum Markov chains on matrix algebras to their stationary distributions. In the classical setting, these inequalities complement eigenvalue techniques and often provide much sharper bounds on the convergence rate. The situation is more intricate in the quantum setting where concepts like reversibility and tensorization become more subtle. Nevertheless, the last decade has seen a fast development of a consistent theory on the subject, currently culminating with the existence of a tensor stable modified log-Sobolev constant for GNS symmetric chains. For this reason, a survey of the results and methods currently known with an emphasis on the problem of tensorization appears timely. Alongside the theory, these notes also contain important examples of applications in quantum information theory, including sharp results for natural chains like the n qubit depolarizing channel, Gibbs samplers over quantum spin chains and approximate unitary k designs.