A Game Theoretic Approach to Quantum Information

TL;DR

This paper applies game theory to quantum information, introducing a new approach to quantum entropy and channel capacity that does not rely on Radon-Nikodym derivatives, and explores properties like monotonicity and additivity.

Contribution

It develops a game-theoretic framework for quantum information, providing novel formulations of quantum relative entropy and channel capacity without Radon-Nikodym derivatives.

Findings

Established a sufficient condition for the minimax theorem in game theory.

Proposed a new approach to quantum relative entropy and mutual entropy.

Investigated monotonicity of quantum relative entropy and additivity of quantum channel capacity.

Abstract

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.