Quantum Martingale Theory and Entropy Production

After introducing thermodynamics along quantum jump trajectories, I will present recent results conerning the development of a martingale theory to describe fluctuations of entropy production for open quantum systems in nonequilbrium steady states. In particular, we identify a decomposition of entropy production into an exponential martingale and a purely quantum term, both obeying integral fluctuation theorems. An important consequence of this approach is the derivation of a set of genuine universal results for stopping-time and infimum statistics of stochastic entropy production. We complement the general formalism with numerical simulations of a qubit system.