Under noisy observations, estimation theory allows one to infer the state of the measured system, if its a priori statistics are given. In the continuous time situation, three different types of estimation can be distinguished: filtering, which is estimating of the state at time t from earlier records; retro-filtering, which is estimating the state at time t from later records; and smoothing, which is estimating the state at time t from both earlier and later records. Of the three, smoothing allows the greatest precision. This theory has been well developed in classical systems, but its application to quantum systems has only recently begun to be explored. Previous works have used the term “quantum smoothing” to mean estimating classical parameters, [Tsang, Phys. Rev. Lett 102, 250403 (2009)], which is essentially classical smoothing in which the noisy observation of the classical parameters is mediated by a quantum system. Here we introduce quantum state smoothin g, where the state of a partially observed open quantum system itself is smoothed. We achieve this by applying classical smoothing to a hypothetical unobserved noisy measurement record correlated with the stochastic dynamics ("quantum trajectories") of the system, induced by that hypothetical measurement. Using the formalism of linear quantum trajectories, we simulate quantum state smoothing for a simple system, and study how the choice of unravelling for the true observation of the system affects how well the unobserved results can be estimated, and hence how effective is the quantum state smoothing. Our investigations shed new light on the nature of open quantum systems and the applicability of classical concepts.
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