Quantum conditional mutual information and approximate Markov chains

The Shannon and von Neumann entropies quantify the uncertainty in a system. They are operationally motivated by natural information processing tasks such as compression, channel coding or randomness extraction. In addition to characterizing the fundamental rates at which such tasks can be performed, their additivity properties make them a very valuable tool in applications ranging from complexity theory to many-body systems. A difficulty that arises when dealing with quantum systems is the operational meaning of quantum observers, or the way to interpret conditional entropies. A particularly interesting quantity is the mutual information between two systems conditioned on a third quantum system. What notion of conditional independence does this quantity measure? After an overview of some of the applications of the conditional mutual information, I will show how the quantum conditional mutual information can be related to the task of local recovery. Based on joint work with Renato Renner arXiv:1410.0664.