论文标题
量子力学的逻辑熵和负概率
Logical Entropy and Negative Probabilities in Quantum Mechanics
论文作者
论文摘要
逻辑熵的概念，$ s_l = 1- \ sum_ {i = 1}^n p_i^2 $，其中$ p_i $是归一化的概率，由David Ellerman在一系列最近的论文中介绍。尽管数学公式本身并不是什么新鲜事，但Ellerman提供了对$ S_L $的声音概率解释，以衡量给定集合上分区的区别。相同的公式是量子力学中熵的有用定义，它与量子状态的纯度概念相关。逻辑熵的二次形式将自己的概率概括为包括负值，这一想法可以追溯到Feynman和Wigner。在这里，我们根据逻辑熵的概念来分析和重新解释负面概率。逻辑熵的几种有趣的量子样特性是在有限维空间中得出和讨论的。对于无限维空间（连续性），我们表明，在唯一的假设中，逻辑熵和总概率是在时间上保留的，一个人获得了概率密度的进化方程，而概率密度基本上与wigner函数在相位空间中的量子演化基本相同，至少是当一个人认为动量变量时，它才是一个相同的进化方程。该结果表明，逻辑熵在制定量子物理学的特殊规则中起着重要作用。
The concept of Logical Entropy, $S_L = 1- \sum_{i=1}^n p_i^2$, where the $p_i$ are normalized probabilities, was introduced by David Ellerman in a series of recent papers. Although the mathematical formula itself is not new, Ellerman provided a sound probabilistic interpretation of $S_L$ as a measure of the distinctions of a partition on a given set. The same formula comes across as a useful definition of entropy in quantum mechanics, where it is linked to the notion of purity of a quantum state. The quadratic form of the logical entropy lends itself to a generalization of the probabilities that include negative values, an idea that goes back to Feynman and Wigner. Here, we analyze and reinterpret negative probabilities in the light of the concept of logical entropy. Several intriguing quantum-like properties of the logical entropy are derived and discussed in finite dimensional spaces. For infinite-dimensional spaces (continuum), we show that, under the sole hypothesis that the logical entropy and the total probability are preserved in time, one obtains an evolution equation for the probability density that is basically identical to the quantum evolution of the Wigner function in phase space, at least when one considers only the momentum variable. This result suggest that the logical entropy plays a profound role in establishing the peculiar rules of quantum physics.
