  Smoothing is an estimation method whereby a classical state (probability
distribution for classical variables) at a given time is conditioned on
all-time (both past and future) observations. Here we define a smoothed quantum
state for a partially monitored open quantum system, conditioned on an all-time
monitoring-derived record. We calculate the smoothed distribution for a
hypothetical unobserved record which, when added to the real record, would
complete the monitoring, yielding a pure-state "quantum trajectory". Averaging
the pure state over this smoothed distribution yields the (mixed) smoothed
quantum state. We study how the choice of actual unravelling affects the purity
increase over that of the conventional (filtered) state conditioned only on the
past record.
