Estimation of the probabilistic description of quantum physical systems

Finding the analytic solution of the Liouville-von Neumann equation is mathematically very challenging. Thus, in this paper, we propose an alternative method that estimates and predicts the dynamics of quantum systems through the estimation of the probability distribution describing the vectorisation of the associated density operator involved in the Liouville-von Neumann equation. In particular, we show that the dynamics of the density operator and measurement states of a quantum system can be characterised by complex Gaussian probability density functions. Unlike conventional methods that provide a deterministic estimate of the density operator, our probabilistic framework provide a general estimation method that also characterises the uncertainty and imperfections in quantum systems due to noises and other sources. Comparison of exact and estimated solutions for a real molecule interacting with an external electric field demonstrates the effectiveness of the proposed estimation method.