We consider a theory of quantum thermodynamics with multiple conserved quantities (or charges). To this end, we generalize the seminal results of Sparaciari et al. [PRA 96:052112, 2017] to the case of multiple, in general non-commuting charges, for which we formulate a resource theory of thermodynamics of asymptotically many non-interacting systems. To every state we associate the vector of its expected charge values and its entropy, forming the phase diagram of the system. Our fundamental result is the Asymptotic Equivalence Theorem (AET), which allows us to identify the equivalence classes of states under asymptotic approximately charge-conserving unitaries with the points of the phase diagram. Using the phase diagram of a system and its bath, we analyze the first and the second laws of thermodynamics. In particular, we show that to attain the second law, an asymptotically large bath is necessary. In the case that the bath is composed of several identical copies of the same elementary bath, we quantify exactly how large the bath has to be to permit a specified work transformation of a given system, in terms of the number of copies of the elementary bath systems per work system (bath rate). If the bath is relatively small, we show that the analysis requires an extended phase diagram exhibiting negative entropies. This corresponds to the purely quantum effect that at the end of the process, system and bath are entangled, thus permitting classically impossible transformations. For a large bath, system and bath may be left uncorrelated and we show that the optimal bath rate, as a function of how tightly the second law is attained, can be expressed in terms of the heat capacity of the bath. Our approach, solves a problem from earlier investigations about how to store the different charges under optimal work extraction protocols in physically separate batteries.

翻译：我们把量子热力学理论与多节制数量(或电量)联系起来。为此,我们将Sparaciari等人[PRA 96:052112,2017年]的创性结果推广到多个(一般非通融性)电荷,为此,我们用一个系统及其浴室的阶段图来分析热力学的资源理论。我们对每一个州都将其预期电荷值的矢量及其增温性联系起来,形成系统的阶段图。我们的基本结果是AET(AET),这使我们得以将Sparaciari et al.[PRA 96:052112,2017年]的创性结果推广到多个(PRA96:05:05211212,2017年 )的创性电荷。为此,我们用一个系统不设充充电量的直线性电动理论,我们分析了热力动力系统的第一和第二定律。特别是,为了达到第二定律, 需要从负式大的洗, 最优的洗方法是由几个不相像的基浴,我们量化的平底的系统, 我们用最短的电算的电算法分析, 使浴法的电压系统能够让一个更深的电压的电流的电压的电算法的电压的电压的电压的电压的电算。