Prediction and retrodiction with continuously monitored Gaussian states

Gaussian states of quantum oscillators are fully characterized by the mean values and the covariance matrix of their quadrature observables. We consider the dynamics of a system of oscillators subject to interactions, damping, and continuous probing which maintain their Gaussian state property. Such dynamics is found in many physical systems that can therefore be efficiently described by the ensuing effective representation of the density matrix $\rho(t)$. Our probabilistic knowledge about the outcome of measurements on a quantum system at time $t$ is not only governed by $\rho(t)$ conditioned on the evolution and measurement outcomes obtained until time $t$, but is also modified by any information acquired after $t$. It was shown in [Phys. Rev. Lett. 111, 160401 (2013)] that this information is represented by a supplementary matrix, $E(t)$. We show here that the restriction of the dynamics of $\rho(t)$ to Gaussian states implies that the matrix $E(t)$ is also fully characterized by a vector of mean values and a covariance matrix. We derive the dynamical equations for these quantities and we illustrate their use in the retrodiction of measurements on Gaussian systems.