This work is an application of game theory to quantum information. In a state estimate, we are given observations distributed according to an unknown distribution $P_{\theta}$ (associated with award $Q$), which Nature chooses at random from the set $\{P_{\theta}: \theta \in \Theta \}$ according to a known prior distribution $\mu$ on $\Theta$, we produce an estimate $M$ for the unknown distribution $P_{\theta}$, and in the end, we will suffer a relative entropy cost $\mathcal{R}(P;M)$, measuring the quality of this estimate, therefore the whole utility is taken as $P \cdot Q -\mathcal{R}(P; M)$. In an introduction to strategic game, a sufficient condition for minimax theorem is obtained; An estimate is explored in the frame of game theory, and in the view of convex conjugate, we reach one new approach to quantum relative entropy, correspondingly quantum mutual entropy, and quantum channel capacity, which are more general, in the sense, without Radon-Nikodym (RN) derivatives. Also the monotonicity of quantum relative entropy and the additivity of quantum channel capacity are investigated.

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