量子相对熵的第二定律型不等式：导论
Second Law-Like Inequalities with Quantum Relative Entropy: An Introduction
AI总结：
本文以量子相对熵的正定性和单调性为核心，系统介绍其与热力学第二定律相关的不等式，并推广到量子涨落定理、Hatano-Sasa不等式、数据处理不等式及带反馈控制的广义第二定律。
摘要：
我们回顾了有限维希尔伯特空间中量子相对熵的基本性质。特别地，我们重点关注若干与热力学第二定律相关的不等式，其中量子相对熵的正定性和单调性起着关键作用；这些性质可直接应用于第二定律的推导（例如克劳修斯不等式）。此外，正定性与量子涨落定理密切相关，而单调性则导出非平衡稳态的Hatano-Sasa不等式的量子版本。基于单调性，我们还讨论了量子互信息的数据处理不等式，其数学结构与第二定律的数学结构相似。此外，我们推导了带量子反馈控制的广义第二定律。另外，我们回顾了与Petz一致的单调性证明。
英文摘要：
We review the fundamental properties of the quantum relative entropy for finite-dimensional Hilbert spaces. In particular, we focus on several inequalities that are related to the second law of thermodynamics, where the positivity and the monotonicity of the quantum relative entropy play key roles; these properties are directly applicable to derivations of the second law (e.g., the Clausius inequality). Moreover, the positivity is closely related to the quantum fluctuation theorem, while the monotonicity leads to a quantum version of the Hatano-Sasa inequality for nonequilibrium steady states. Based on the monotonicity, we also discuss the data processing inequality for the quantum mutual information, which has a similar mathematical structure to that of the second law. Moreover, we derive a generalized second law with quantum feedback control. In addition, we review a proof of the monotonicity in line with Petz.
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