Large Deviations at Level 2.5 for Markovian Open Quantum Systems: Quantum Jumps and Quantum State Diffusion

We consider quantum stochastic processes and discuss a level 2.5 large deviation formalism providing an explicit complete characterisation of fluctuations time-averaged quantities, in the large-time limit. analyse two classes dynamics, within this framework. The first class consists jump trajectories related to photon detection; second is state diffusion homodyne detection. For both processes, we present functional starting from corresponding Schr\"odinger equation connections these functionals optimal control theory.