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Optimal processes for probabilistic work extraction beyond the second law

According to the second law of thermodynamics, for every transformation performed on a system which is in contact with an environment of fixed temperature, the extracted work is bounded by the decrease of the free energy of the system. However, in a single realization of a generic process, the extracted work is subject to statistical fluctuations which may allow for probabilistic violations of the previous bound. We are interested in enhancing this effect, i.e. we look for thermodynamic processes that maximize the probability of extracting work above a given arbitrary threshold. For any process obeying the Jarzynski identity, we determine an upper bound for the work extraction probability that depends also on the minimum amount of work that we are willing to extract in case of failure, or on the average work we wish to extract from the system. Then we show that this bound can be saturated within the thermodynamic formalism of quantum discrete processes composed by sequences of unitary quenches and complete thermalizations. We explicitly determine the optimal protocol which is given by two quasi-static isothermal transformations separated by a finite unitary quench.