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We study how Thomson's formulation of the second law of thermodynamics (no work is extracted from an equilibrium ensemble by a cyclic process) emerges in the quantum situation through the averaging over fluctuations of work. The latter concept is carefully defined for an ensemble of quantum systems, the members of which interact with macroscopic sources of work. The approach is based on splitting a mixed quantum ensemble into pure subensembles, which according to quantum mechanics are maximally complete and irreducible. The splitting is done by filtering the outcomes of a measurement process. The approach is corroborated by comparing to relevant experiments in quantum optics. A critical review is given of two other approaches to fluctuations of work proposed in the literature. It is shown that in contrast to those, the present definition (i) is consistent with the physical meaning of the concept of work as mechanical energy lost by the macroscopic sources, or, equivalently, as the average energy acquired by the ensemble; (ii) applies to an arbitrary nonequilibrium state. There is no direct generalization of the classical work-fluctuation theorem to the proper quantum domain. This implies nonclassical scenarios for the emergence of the second law.
Allahverdyan et al. (Thu,) studied this question.