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数理解析研究所講究録

Operator-valued divergence and entropy(Information and mathematics of non-additivity and non-extensivity : from the viewpoint of functional analysis)

淳一藤井

Published Jun 1, 2007• 0 citations

Abstract

)とすると $\sum\Lambda E_{n}=I$ と、 Jensen type inequality より、 $H_{E}(A; \Lambda)=\eta(\sum_{n}t_{n}\Lambda E_{n})-\sum_{n}t_{n}\eta(\Lambda E_{n})\geq 0$ という非負の量 $H_{E}(A, \Lambda)$ を Holevo 型の相互作用素エントロピーと見ることができる。このとき、これらのトレースを取ると、 Ohya mutual entropy と一致するので、分解についての最大として対応する通信容量も一致することが分かる。実は、このことは、本質的に[17] で示されているが、ここで確認しておこう。定理.密度行列 $A$ と、その Shatten 分解 $E:A= \sum_{n}t_{n}E_{n}$ について、 Tr $H_{E}(A;\Lambda)=RM_{E}(A;\Lambda)$ , $\max_{E}RH_{E}(A;\Lambda)=I(A;\Lambda)$ .

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)とすると

\sum\Lambda E_{n}=I

と、 Jensen type inequality より、

H_{E}(A; \Lambda)=\eta(\sum_{n}t_{n}\Lambda E_{n})-\sum_{n}t_{n}\eta(\Lambda E_{n})\geq 0

という非負の量

H_{E}(A, \Lambda)

を Holevo 型の相互作用素エントロピーと見ることができる。このとき、これらのトレースを取ると、 Ohya mutual entropy と一致するので、分解についての最大として対応する通信容量も一致することが分かる。実は、このことは、本質的に[17] で示されているが、ここで確認しておこう。定理.密度行列

A

と、その Shatten 分解

E:A= \sum_{n}t_{n}E_{n}

について、 Tr

H_{E}(A;\Lambda)=RM_{E}(A;\Lambda)

,

\max_{E}RH_{E}(A;\Lambda)=I(A;\Lambda)

.

The mutual entroy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel.The (semi-classical) mutual entropies for classical input and quantum output were defined by several researchers.The funy quantum mutual entropy, which is called Ohya mutual entroy, for quantum input and output by using the relative entropy was defined by Ohya in 1983.In this paper, we

\infty mpue

With mutual

entropy-WPe

measures and show some resuls for quantum capacity.

The mutual information between the sender of a classical message encoded in quantum carriers and a receiver is fundamentally limited by the Holevo quantity. Using strong subadditivity of entropy, we prove that the Holevo quantity is not larger than an exchange entropy. This implies an upper bound for coherent information. Moreover, restricting our attention to classical information, we bound the transmission distance between probability distributions by their entropic distance, which is a concave function of their Hellinger distance.

The mutual entropy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel.The (semi-classical) mutual entropies for classical input and quantum output were defined by several researchers.The fully quantum mutual entropy, which is called Ohya mutual entropy, for quantum input and output by using the relative entropy was defined by Ohya in 1983.In this paper, we

\mathrm{c}o

mpare with mutual

\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}_{\Psi}

-type measures and show some resuls for quantum capacity.

A formulation of Rényi entropy on

C^*

-algebras