A Game Theoretic Approach to Quantum Information

In this project, bridging entropy econometrics, game theory and information theory, a game theoretic approach will be investigated to quantum information, during which new mathematical definitions for quantum relative entropy, quantummutual information, and quantum channel capacity will be given and monotonicity of entangled quantum relative entropy and additivity of quantum channel capacity will be obtained rigorously in mathematics; also quantum state will be explored in Kelly criterion, during which game theoretic interpretations will be given to classical relative entropy, mutual information, and asymptotical information. In specific, after briefly introducing probability inequalities, C∗-algebra, including von Neumann algebra, and quantum probability, we will overview quantum entanglement, quantum relative entropy, quantum mutual information, and entangled quantum channel capacity in the direction of R. L. Stratonovich and V. P. Belavkin, and upon the monotonicity property of quantummutual information of Araki-Umegaki type and Belavkin-Staszewski type, we will prove the additivity property of entangled quantum channel capacities, extending the results of V. P. Belavkin to products of arbitrary quantum channel to quantum relative entropy of both Araki-Umegaki type and Belavkin-Staszewski type. We will obtain a sufficient condition for minimax theorem in an introduction to strategic game, after which, in the exploration of classical/quantum estimate (here we still use the terminology of quantum estimate in the sense of game theory in accordance to classical estimate, but NOT in the sense of quantum physics or quantum probability), we will find the existence of the minimax value of this game and its minimax strategy, and applying biconjugation in convex analysis, we will arrive at one new approach to quantum relative entropy, quantummutual entropy, and quantum channel capacity, in the sense, independent on Radon-Nikodym derivative, also the monotonicity of quantum relative entropy and the additivity of quantum communication channel capacity will be