Quantum Hypothesis Testing Mutual Information

von Neumann mutual information is a generalization of Shannon mutual information in quantum information theory, and it has been found to have useful applications in the channel capacity. The many classical quantifiers can be extended for pairs of quantum states in various inequivalent ways, due to the non-commutativity of quantum states. Quantum hypothesis testing relative entropy comes from the hypothesis testing question, and it is one of the most fundamental primitives in quantum information processing. We discuss the quantum mutual information with respect to quantum hypothesis testing relative entropy. We give some properties of quantum hypothesis testing relative entropy, and give the relationships between it and other quantum generalized entropies. Using relative entropy, we define quantum hypothesis testing mutual information, and exhibit some properties, such as, data processing inequality. Using the sum between mutual information and condition entropy, we discuss the chain rules, which are generally an important technical tool in information theory.