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We study stochastic thermodynamics for a quantum system of interest whosedynamics are described by a completely positive trace-preserving (CPTP) map asa result of its interaction with a thermal bath. We define CPTP maps withequilibrium as CPTP maps with an invariant state such that the entropyproduction due to the action of the map on the invariant state vanishes.Thermal maps are a subgroup of CPTP maps with equilibrium. In general, for CPTPmaps, the thermodynamic quantities, such as the entropy production or workperformed on the system, depend on the combined state of the system plus itsenvironment. We show that these quantities can be written in terms of systemproperties for maps with equilibrium. The relations that we obtain are validfor arbitrary coupling strengths between the system and the thermal bath. Thefluctuations of thermodynamic quantities are considered in the framework of atwo-point measurement scheme. We derive the entropy production fluctuationtheorem for general maps and a fluctuation relation for the stochastic work ona system that starts in the Gibbs state. Some simplifications for theprobability distributions in the case of maps with equilibrium are presented.We illustrate our results by considering spin 1/2 systems under thermal maps,non-thermal maps with equilibrium, maps with non-equilibrium steady states andconcatenations of them. Finally, we consider a particular limit in which theconcatenation of maps generates a continuous time evolution in Lindblad formfor the system of interest, and we show that the concept of maps with andwithout equilibrium translates into Lindblad equations with and without quantumdetailed balance, respectively. The consequences for the thermodynamicquantities in this limit are discussed.

Stochastic thermodynamics of quantum maps with and without equilibrium