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Springer eBooks

Relation between Channel Capacity and Quantum Minimax Decision in Quantum Information Theory

Kentaro Kato, Masao Osaki, Tomohiro Suzuki, Masashi Ban, Osamu Hirota

Published Jan 1, 1997• 1 citation
Publisher
DOI
10.1007/978-1-4615-5923-8_7

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Theory of Probability and Its Applications•Jan 1, 2008•15 citations
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 PθP_θPθ​ (associated with award QQQ), which Nature chooses at random from the set {Pθ:θ∈Θ}\{P_θ: θ\in Θ\}{Pθ​:θ∈Θ} according to a known prior distribution μμμ on ΘΘΘ, we produce an estimate MMM for the unknown distribution PθP_θPθ​, and in the end, we will suffer a relative entropy cost R(P;M)\mathcal{R}(P;M)R(P;M), measuring the quality of this estimate, therefore the whole utility is taken as P⋅Q−R(P;M)P \cdot Q -\mathcal{R}(P; M)P⋅Q−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.
This paper is devoted to further study of the Holevo capacity of infinite-dimensional quantum channels. The existence of a unique optimal average state for a quantum channel constrained by an arbitrary convex set of states is shown. The minimax expression for the Holevo capacity of a constrained channel is obtained. The χ\chiχ-function and the convex closure of the output entropy of an infinite-dimensional quantum channel are considered. It is shown that the χ\chiχ-function of an arbitrary channel is lower semicontinuous on the set of all states and has continuous restrictions to subsets of states with continuous output entropy. The explicit expression for the convex closure of the output entropy of an infinite-dimensional quantum channel is obtained and its properties are explored. It is shown that the convex closure of the output entropy coincides with the convex hull of the output entropy on the set of states with finite output entropy and, similarly to the χ\chiχ-function, it has continuous restrictions to subsets of states with continuous output entropy. The applications of the obtained results to the theory of entanglement are considered. The properties of the convex closure of the output entropy make it possible to generalize some results related to the additivity problem to the infinite-dimensional case.