在热操作（thermal operations，TO）下，能量守恒由生成这些操作的酉算子与总哈密顿量之间的对易关系来保证。然而，在现实情形中，系统不可避免地会受到扰动或干扰，这反过来可能改变对易关系，并继而影响由 TO 支配的物理过程。我们将改变后的操作集合称为近似热操作（approximate thermal operations，TO_ε），其中 ε 表示干扰程度。我们给出了此类操作下的状态变换条件，提供了可称为近似第二定律的结果。我们表明，当系统哈密顿量中存在微弱扰动时，状态的变换会使系统状态的对角元不仅彼此发生作用，还开始与非对角元发生作用。同时，非对角元的变换也会使其开始与对角元以及其他非对角元相连接。这种交叉耦合在未受扰动的第二定律中是不允许的。作为应用，我们证明近似热操作可能从量子电池中提取有限的 ergropy（可提取功），而这是精确热操作无法实现的。
英文摘要：
Conservation of energy under thermal operations, \textbf{TO}, is ensured by commutation of the unitary generating such operations with the total Hamiltonian. However in realistic scenarios, perturbations or disturbances in the system are unavoidable, which in turn may alter the commutation relation and hence in succession may affect the physical processes governed by \textbf{TO}. We call the altered set of operations as approximate thermal operations, \textbf{TO}$_ε$, where $ε$ denotes a degree of disturbance. We provide state transformation conditions under such operations, providing what can be referred to as approximate second laws. We show that in presence of feeble perturbations in the system's Hamiltonian, the states transform in such a way that diagonal elements of the system states start talking not only with each other but also with the off-diagonal elements. In parallel, the off-diagonal elements transform in a way such that they start connecting with diagonal elements and other off-diagonal elements. Such cross-talk is disallowed in the unperturbed second laws. As an application, we show that approximate thermal operations may lead to finite ergotropy extraction from quantum batteries, something that the exact ones are unable to.
Boosting Work Extraction in Quantum Batteries via Continuous Environment Monitoring
Università degli Studi dell’Insubria（因苏布里亚大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Milano（意大利国家核物理研究所米兰分部）；Università di Genova（热那亚大学）；CNR-SPIN（意大利国家研究委员会自旋物理中心）；Università degli Studi di Milano（米兰大学）
Geometric power capacity of coherent ergotropy in quantum batteries
Changchun University of Technology（长春工业大学）；Yanbian University（延边大学）；Capital Normal University（首都师范大学）；Max-Planck-Institute for Mathematics in the Sciences（马克斯·普朗克科学促进部数理研究所）
System versus charger in performance optimization of quantum batteries
Bar-Ilan University（巴伊兰大学）；Indian Institute of Technology (Banaras Hindu University)（印度理工学院（贝拿勒斯印度教大学））；Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute（哈里什-昌德拉研究所，霍米·巴巴国立学院成员机构）
Charging Quantum Batteries via Dissipative Quenches
Università di Genova（热那亚大学）；CNR-SPIN（意大利国家研究委员会自旋物理学研究所）；Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Napoli（意大利国家核物理研究所，那不勒斯分部）
Activating thermally charged quantum batteries in finite time: Thermodynamic trade-offs between correlations, work, and information
Chalmers University of Technology（查尔姆斯理工大学）；Indian Institute of Science Education and Research Kolkata（印度科学教育研究所加尔各答分校）；Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
Positive and non-positive measurements in energy distillation from quantum batteries
Harish-Chandra Research Institute（哈里什-昌德拉研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Departamento de Física Teórica, Universidad Complutense de Madrid（马德里康普顿斯大学理论物理系）
Collective Quantum Batteries and Charger-Battery Setup in Open Quantum Systems: Impact of Inter-Qubit Interactions, Dissipation, and Quantum Criticality
Indian Institute of Technology Jodhpur（印度技术学院焦特布尔分校）
Bridging continuous control and Floquet driving for charging many-body spin chains
Quantum Advanced Research Center (QuARC), CSIC（CSIC量子先进研究中心）；Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC（CSIC马德里材料科学研究所）；Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid（马德里自治大学凝聚态物理理论系）
