  Quantum state smoothing is a technique for estimating the quantum state of a
partially observed quantum system at time $\tau$, conditioned on an entire
observed measurement record (both before and after $\tau$). However, this
smoothing technique requires an observer (Alice, say) to know the nature of the
measurement records that are unknown to her in order to characterize the
possible true states for Bob's (say) systems. If Alice makes an incorrect
assumption about the set of true states for Bob's system, she will obtain a
smoothed state that is suboptimal, and, worse, may be unrealizable (not
corresponding to a valid evolution for the true states) or even unphysical (not
represented by a state matrix $\rho\geq0$). In this paper, we review the
historical background to quantum state smoothing, and list general criteria a
smoothed quantum state should satisfy. Then we derive, for the case of linear
Gaussian quantum systems, a necessary and sufficient constraint for
realizability on the covariance matrix of the true state. Naturally, a
realizable covariance of the true state guarantees a smoothed state which is
physical. It might be thought that any putative true covariance which gives a
physical smoothed state would be a realizable true covariance, but we show
explicitly that this is not so. This underlines the importance of the
realizabilty constraint.
