Maximum a posteriori path for open quantum systems

In this work, the Stratonovich stochastic differential equation is considered in the derivation of the maximum a posteriori path (MAP) for an open quantum system. We have derived the modified Onsager-Machlup (OM) function for a stochastic process, which plays an analogous role of the Lagrangian in the conventional classical mechanics. Variational method is applied to obtain the Euler-Lagrange equations. We show that the MAP trajectory is close to the quantum most-likely path obtained by Chantasri et al. [1], but in general not exactly the same. For an open quantum system subject to a no-knowledge measurement, both methods produce the same most-likely path. Numerical simulations for both the modified OM and the quantum most-likely path methods agree very well with theoretical predictions, regardless of the evolution time and different post-selected states.