Approximate Quantum Markov Chains

Introduction.- Classical Markov chains.- Quantum Markov chains.- Outline.- Preliminaries.- Notation.- Schatten norms.- Functions on Hermitian operators.- Quantum channels.- Entropy measures.- Background and further reading.- Tools for non-commuting operators.- Pinching.- Complex interpolation theory.- Background and further reading.- Multivariate trace inequalities.- Motivation.- Multivariate Araki-Lieb-Thirring inequality.- Multivariate Golden-Thompson inequality.- Multivariate logarithmic trace inequality.- Background and further reading.- Approximate quantum Markov chains.- Quantum Markov chains.- Sufficient criterion for approximate recoverability.- Necessary criterion for approximate recoverability.- Strengthened entropy inequalities.- Background and further reading.- A A large conditional mutual information does not imply bad recovery.- B Example showing the optimality of the Lmax-term.- C Solutions to exercises.- References.- Index.