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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Min entropy ： ウィキペディア英語版 Min entropy The min entropy, in information theory, is the smallest of the Rényi family of entropies, corresponding to the most conservative way of measuring the unpredictability of a set of outcomes, as the negative logarithm of the probability of the ''most likely'' outcome. The various Rényi entropies are all equal for a uniform distribution, but measure the unpredictability of a nonuniform distribution in different ways. The min entropy is never greater than the ordinary or Shannon entropy (which measures the average unpredictability of the outcomes) and that in turn is never greater than the Hartley or max entropy, defined as the logarithm of the ''number'' of outcomes. As with the classical Shannon entropy and its quantum generalization, the von Neumann entropy, one can define a conditional versions of min entropy. The conditional quantum min entropy is a one-shot, or conservative, analog of conditional quantum entropy. To interpret a conditional information measure, suppose Alice and Bob were to share a bipartite quantum state $\backslash rho\_$. Alice has access to system A and Bob to system B. The conditional entropy measures the average uncertainty Bob has about Alice's state upon sampling from his own system. The min entropy can be interpreted as the distance of a state from a maximally entangled state. This concept is useful in quantum cryptography, in the context of privacy amplification (See for example 〔Vazirani, Umesh, and Thomas Vidick. "Fully device independent quantum key distribution." (2012)〕). == Definitions == Definition: Let $\backslash rho\_$ be a bipartite density operator on the space $\backslash mathcal\_A\; \backslash otimes\; \backslash mathcal\_B$. The min-entropy of A conditioned on B is defined to be :::$H\_(A|B)\_\; \backslash equiv\; -\backslash inf\_D\_(\backslash rho\_||I\_A\; \backslash otimes\; \backslash sigma\_B)$ where the infimum ranges over all density operators $\backslash sigma\_B$ on the space $\backslash mathcal\_B$. The measure $D\_$ is the maximum relative entropy defined as :::$D\_(\backslash rho||\backslash sigma)\; =\; \backslash inf\_\backslash$ The smooth min entropy is defined in terms of the min entropy. :::$H\_^(A|B)\_\; =\; \backslash sup\_\; H\_(A|B)\_$ where the sup and inf range over density operators $\backslash rho\text{'}\_$ which are $\backslash epsilon$-close to $\backslash rho\_$ . This measure of $\backslash epsilon$-close is defined in terms of the purified distance :::$P(\backslash rho,\backslash sigma)\; =\; \backslash sqrt$ where $F(\backslash rho,\backslash sigma)$ is the fidelity measure. These quantities can be seen as generalizations of the von Neumann entropy. Indeed, the von Neumann entropy can be expressed as :::$S(A|B)\_\; =\; \backslash lim\_\backslash lim\_\backslash fracH\_^(A^n|B^n)\_^(A|B)\_\; \backslash geq\; H\_^(A|BC)\_~.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Min entropy」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.