  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.
