The purpose of this paper is to exploit the geometric structure of quantum mechanics and of statistical manifolds to study the qualitative effect that the quantum properties have in the statistical description of a system. We show that the end points of geodesics in the classical setting coincide with the probability distributions that minimise Shannon’s entropy, i.e. with distributions of zero dispersion. In the quantum setting this happens only for particular initial conditions, which in turn correspond to classical submanifolds. This result can be interpreted as a geometric manifestation of the uncertainty principle.

中文翻译：

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Fisher-Rao 度量的测地运动方面：经典和量子

本文的目的是利用量子力学和统计流形的几何结构来研究量子特性在系统统计描述中的定性影响。我们表明，经典环境中测地线的端点与最小化香农熵的概率分布一致，即与零色散分布一致。在量子设置中，这仅发生在特定的初始条件下，而这些初始条件又对应于经典子流形。这个结果可以解释为不确定性原理的几何表现。