不确定性、熵和信息的通用几何方法
Universal geometric approach to uncertainty, entropy and information
摘要：
结果表明，对于任何合奏，无论是经典的或量子的，连续的或离散的，只有一个措施的“体积”的合奏，是兼容的几个基本几何假设。因此，该体积测量是用于表征集合的固有扩散、分散、定位等的优选和通用选择。值得注意的是，这个独特的“系综体积”是一个简单的函数的系综熵，因此提供了一个新的几何表征后者的数量。应用包括统一的，基于体积的推导的Holevo和香农界限在量子和经典信息理论;一个精确的几何解释热力学熵的平衡系综;几何推导的半经典不确定性关系;一个新的手段定义经典和量子局域化的任意演化过程;几何解释的相对熵;以及一种新提出的光束光斑尺寸的定义。的合奏体积比其他本地化措施（均方根偏差，Renyi熵，和逆参与比）的优势进行了讨论。
英文摘要：
It is shown that for any ensemble, whether classical or quantum, continuous or discrete, there is only one measure of the "volume" of the ensemble that is compatible with several basic geometric postulates. This volume measure is thus a preferred and universal choice for characterising the inherent spread, dispersion, localisation, etc, of the ensemble. Remarkably, this unique "ensemble volume" is a simple function of the ensemble entropy, and hence provides a new geometric characterisation of the latter quantity. Applications include unified, volume-based derivations of the Holevo and Shannon bounds in quantum and classical information theory; a precise geometric interpretation of thermodynamic entropy for equilibrium ensembles; a geometric derivation of semi-classical uncertainty relations; a new means for defining classical and quantum localization for arbitrary evolution processes; a geometric interpretation of relative entropy; and a new proposed definition for the spot-size of an optical beam. Advantages of the ensemble volume over other measures of localization (root-mean-square deviation, Renyi entropies, and inverse participation ratio) are discussed.
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