基于玻恩定则的香农熵的几何解释
Geometrical Interpretation of Shannon's Entropy Based on the Born Rule
摘要：
在本文中，我们将分析离散概率分布，其中某些实验（微观状态）的特定结果的概率可以由自然数的比率表示（换句话说，概率由有限表示长度的数字表示）。我们将介绍几个基于最近提出的操纵杆概率的结果，它代表了基于玻恩规则的概率的几何解释。引入了离散分布的类空间、类维数以及有效维数等概念。它将显示如何这个简单的几何表示可以导致一个最佳的代码长度编码的信号序列。然后，我们将给出一个新的，几何，解释的香农熵的离散分布。我们将建议香农熵表示分布的有效维数的对数。提出的香农熵的几何解释可以用来初等地证明一些信息不等式。
英文摘要：
In this paper we will analyze discrete probability distributions in which probabilities of particular outcomes of some experiment (microstates) can be represented by the ratio of natural numbers (in other words, probabilities are represented by digital numbers of finite representation length). We will introduce several results that are based on recently proposed JoyStick Probability Selector, which represents a geometrical interpretation of the probability based on the Born rule. The terms of generic space and generic dimension of the discrete distribution, as well as, effective dimension are going to be introduced. It will be shown how this simple geometric representation can lead to an optimal code length coding of the sequence of signals. Then, we will give a new, geometrical, interpretation of the Shannon entropy of the discrete distribution. We will suggest that the Shannon entropy represents the logarithm of the effective dimension of the distribution. Proposed geometrical interpretation of the Shannon entropy can be used to prove some information inequalities in an elementary way.
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