Action principle for continuous quantum measurement and quantum trajectories with pre and post-selection

We apply an action principle to continuous quantum measurement by introducing a joint probability density function of measurement outcomes and quantum state trajectories in a path integral form. Using a modified principle of least action, we find the paths of maximum likelihood connecting boundary states between any two points in time, at which we call the most-likely paths. We present, as an example, the most-likely paths for a continuous qubit measurement with pre and post-selected states, along with a preliminary comparison to data from a superconducting qubit coupled to a microwave cavity. We, furthermore, introduce other interesting statistical characterizations of the quantum trajectories such as mean paths, variances and most-likely times, that can be derived from our path integral formalism.