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, quantum mutual 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 quantum mutual 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, quantum mutual 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 obtained. Applying Kelly's criterion, we will give a practical game theoretic interpretation, in the model to identify quantum state, to relative entropy, mutual information, and asymptotical information, during which we will find that the decrement in the doubling rate achieved with true knowledge of the distribution F over that achieved with incorrect knowledge G is bounded by relative entropy R(F;G) of F relative to G; the increment [Delta] in the doubling rate resulting from side information Y is less than or equal to the mutual information I(X,Y); a good sequence to identify the true quantum state leads to asymptotically optimal growth rate of utility; and applying the asymptotic behavior of classical relative entropy, the utility of the Bayes' strategy will be bounded below in terms of the optimal utility. The first two main parts are to extend classical entropy econometrics, in the view of strategic game theory, to non-commutative data, for example, quantum data in physical implementation, while the third main part is to intrinsically and practically give a game theoretic interpretation of classical relative entropy, mutual information, and asymptotical information, in the model to identify quantum state, upon which a pregnant financial stock may be designed, which may be called "quantum" stock, for its physical implementation.