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Markovian quantum master equations (MQMEs) were established nearly half acentury ago. They have often been used in the study of irreversiblethermodynamics. However, the previous results were mainly concerned aboutensemble averages; the stochastic thermodynamics of these systems wentunnoticed for a very long time. This situation remained unchanged until avariety of fluctuation theorems in classical and quantum regimes were found inthe past two decades. In this paper, we systematically summarize the currentunderstanding on the stochastic heat and work in MQMEs using two distinctstrategies. One strategy is to treat the system and its surrounding heat bathas a closed quantum system, to suppose that the evolution of the compositesystem is unitary under a time-dependent total Hamiltonian and to define theheat and work as the changes in energy by applying two energy measurementsscheme to the composite system. The other strategy is to unravel these MQMEsinto random quantum jump trajectories (QJTs) and to define the stochastic heatand work along the individual trajectories. Many important physical concepts,mathematical techniques, and fluctuation theorems at different descriptivelevels are given in as detailed a manner as possible. We also use concretemodels to illustrate these results.

Heat and work in Markovian quantum master equations: concepts, fluctuation theorems, and computations