Li Daili, Zheng Chao
Department of Physics, College of Science, North China University of Technology, Beijing 100144, China.
Entropy (Basel). 2022 Oct 30;24(11):1563. doi: 10.3390/e24111563.
From their conception to present times, different concepts and definitions of entropy take key roles in a variety of areas from thermodynamics to information science, and they can be applied to both classical and quantum systems. Among them is the Rényi entropy. It is able to characterize various properties of classical information with a unified concise form. We focus on the quantum counterpart, which unifies the von Neumann entropy, max- and min-entropy, collision entropy, etc. It can only be directly applied to Hermitian systems because it usually requires that the density matrices is normalized. For a non-Hermitian system, the evolved density matrix may not be normalized; i.e., the trace can be larger or less than one as the time evolution. However, it is not well-defined for the Rényi entropy with a non-normalized probability distribution relevant to the density matrix of a non-Hermitian system, especially when the trace of the non-normalized density matrix is larger than one. In this work, we investigate how to describe the Rényi entropy for non-Hermitian systems more appropriately. We obtain a concisely and generalized form of α-Rényi entropy, which we extend the unified order-α from finite positive real numbers to zero and infinity. Our generalized α-Rényi entropy can be directly calculated using both of the normalized and non-normalized density matrices so that it is able to describe non-Hermitian entropy dynamics. We illustrate the necessity of our generalization by showing the differences between ours and the conventional Rényi entropy for non-Hermitian detuning two-level systems.
从熵的概念诞生至今，不同的熵的概念和定义在从热力学到信息科学等众多领域都发挥着关键作用，并且它们可应用于经典系统和量子系统。其中包括雷尼熵。它能够用一种统一简洁的形式刻画经典信息的各种性质。我们关注的是其量子对应物，它统一了冯·诺依曼熵、最大熵和最小熵、碰撞熵等。它只能直接应用于厄米系统，因为通常要求密度矩阵是归一化的。对于非厄米系统，演化后的密度矩阵可能不归一化；也就是说，随着时间演化，迹可能大于或小于1。然而，对于与非厄米系统密度矩阵相关的非归一化概率分布的雷尼熵，其定义并不明确，特别是当非归一化密度矩阵的迹大于1时。在这项工作中，我们研究如何更恰当地描述非厄米系统的雷尼熵。我们得到了一种简洁且广义的α - 雷尼熵形式，将统一的α阶从有限正实数扩展到零和无穷大。我们的广义α - 雷尼熵可以直接使用归一化和非归一化密度矩阵来计算，从而能够描述非厄米熵动力学。我们通过展示我们的结果与非厄米失谐二能级系统的传统雷尼熵之间的差异，来说明我们进行推广的必要性。
