  Quantum state smoothing is a technique to estimate an unknown true state of
an open quantum system based on partial measurement information both prior and
posterior to the time of interest. In this paper, we show that the smoothed
quantum state is an optimal state estimator; that is, it minimizes a risk
(expected cost) function. Specifically, we show that the smoothed quantum state
is optimal with respect to two cost functions: the trace-square deviation from
and the relative entropy to the unknown true state. However, when we consider a
related risk function, the linear infidelity, we find, contrary to what one
might expect, that the smoothed state is not optimal. For this case, we derive
the optimal state estimator, which we call the lustrated smoothed state. It is
a pure state, the eigenstate of the smoothed quantum state with the largest
eigenvalue.
