Classical inequalities used in information theory such as those of de Bruijn, Fisher, Cramér, Rao, and Kullback carry over in a natural way from Euclidean space to unimodular Lie groups. These are groups that possess an integration measure that is simultaneously invariant under left and right shifts. All commutative groups are unimodular. And even in noncommutative cases unimodular Lie groups share many of the useful features of Euclidean space. The rotation and Euclidean motion groups, which are perhaps the most relevant Lie groups to problems in geometric mechanics, are unimodular, as are the unitary groups that play important roles in quantum computing. The extension of core information theoretic inequalities defined in the setting of Euclidean space to this broad class of Lie groups is potentially relevant to a number of problems relating to information gathering in mobile robotics, satellite attitude control, tomographic image reconstruction, biomolecular structure determination, and quantum information theory. In this paper, several definitions are extended from the Euclidean setting to that of Lie groups (including entropy and the Fisher information matrix), and inequalities analogous to those in classical information theory are derived and stated in the form of fifteen small theorems. In all such inequalities, addition of random variables is replaced with the group product, and the appropriate generalization of convolution of probability densities is employed. An example from the field of robotics demonstrates how several of these results can be applied to quantify the amount of information gained by pooling different sensory inputs.

中文翻译：

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单模李群的信息理论不等式。

信息论中使用的经典不等式，例如 de Bruijn、Fisher、Cramér、Rao 和 Kullback 的不等式，以自然的方式从欧几里得空间延续到单模李群。这些是具有在左移和右移下同时不变的积分测度的组。所有交换群都是单模的。甚至在非交换情况下，单模李群也共享欧几里得空间的许多有用特征。旋转群和欧几里得运动群可能是与几何力学问题最相关的李群，它们是单模的，在量子计算中扮演重要角色的酉群也是如此。将欧几里德空间中定义的核心信息理论不等式扩展到这一大类李群可能与移动机器人、卫星姿态控制、断层图像重建、生物分子结构确定和量子信息论。在本文中，几个定义从欧几里得设置扩展到李群的定义（包括熵和费雪信息矩阵），并以十五个小定理的形式推导出类似于经典信息理论中的不等式。在所有这些不等式中，随机变量的添加被组积代替，并采用了概率密度卷积的适当推广。