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On the Conditional Rényi Entropy

机译：关于条件Rényi熵

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摘要

The Rényi entropy of general order unifies the well-known Shannon entropy with several other entropy notions, like the min-entropy or collision entropy. In contrast to the Shannon entropy, there seems to be no commonly accepted definition for the conditional Rényi entropy: several versions have been proposed and used in the literature. In this paper, we reconsider the definition for the conditional Rényi entropy of general order as proposed by Arimoto in the seventies. We show that this particular notion satisfies several natural properties. In particular, we show that it satisfies monotonicity under conditioning, meaning that conditioning can only reduce the entropy, and (a weak form of) chain rule, which implies that the decrease in entropy due to conditioning is bounded by the number of bits one conditions on. None of the other suggestions for the conditional Rényi entropy satisfies both these properties. Finally, we show a natural interpretation of the conditional Rényi entropy in terms of (unconditional) Rényi divergence, and we show consistency with a recently proposed notion of conditional Rényi entropy in the quantum setting.
机译：广义Rényi熵将众所周知的Shannon熵与其他几个熵概念统一起来，例如min熵或碰撞熵。与Shannon熵相反，对于条件Rényi熵似乎没有公认的定义：在文献中已经提出并使用了几种形式。在本文中，我们重新考虑了有本七十年代提出的一般顺序的条件Rényi熵的定义。我们证明了这个特定的概念满足了几种自然属性。特别是，我们表明它在条件下满足单调性，这意味着条件只能减少熵和链规则的（弱形式），这意味着由于条件而导致的熵的减少受一个条件的位数限制上。关于条件Rényi熵的其他建议均不能同时满足这两个属性。最后，我们用（无条件的）Rényi散度显示了对条件Rényi熵的自然解释，并且我们展示了与量子环境中最近提出的条件Rényi熵概念的一致性。

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