Stochastic thermodynamics can be applied to driven (i.e. open) quantum systems whenever the effects of quantum coherence can be ignored. The dynamics of an open quantum system is then equivalent to a classical stochastic one.  However, this is sometimes at the cost of requiring unrealistic measurements at the beginning and end of a process.
Understanding non-equilibrium quantum thermodynamics more broadly is an important and active area of research. The efficiency of some computing and information theory tasks can be greatly enhanced when using quantum correlated states; quantum correlations can be used not only as a valuable resource in quantum computation, but also in the realm of quantum thermodynamics. New types of quantum devices in non-equilibrium states function very differently to their classical counterparts.  For example, it has been theoretically shown that non-equilibrium quantum ratchet systems function far more efficiently then that predicted by classical thermodynamics.  It has also been shown that quantum coherence can be used to enhance the efficiency of systems beyond the classical Carnot limit. This is because it could be possible to extract work, in the form of photons, from a single heat bath. Quantum coherence can be used in effect to play the role of Maxwell's demon though the broader information theory based interpretation of the second law of thermodynamics is not violated.
Quantum versions of stochastic thermodynamics have been studied for some time and the past few years have seen a surge of interest in this topic. Quantum mechanics involves profound issues around the interpretation of reality (e.g. the Copenhagen interpretation, many-worlds, de Broglie-Bohm theory etc are all competing interpretations that try to explain the unintuitive results of quantum theory) . It is hoped that by trying to specify the quantum-mechanical definition of work, dealing with open quantum systems, analyzing exactly solvable models, or proposing and performing experiments to test non-equilibrium predictions, important insights into the interpretation of quantum mechanics and the true nature of reality will be gained.
Applications of non-equilibrium work relations, like the Jarzynski equality, have recently been proposed for the purposes of detecting quantum entanglement  and to improving optimization problems (minimize or maximize a function of multivariables called the cost function) via quantum annealing .