  Given an arbitrary measurement over a system of interest, the outcome of a
posterior measurement can be used for improving the statistical estimation of
the system state after the former measurement. Here, we realize an
informational-entropic study of this kind of (Bayesian) retrodicted quantum
measurement formulated in the context of quantum state smoothing. We show that
the (average) entropy of the system state after the retrodicted measurement
(smoothed state) is bounded from below and above by the entropies of the first
measurement when performed in a selective and non-selective standard predictive
ways respectively. For bipartite systems the same property is also valid for
each subsystem. Their mutual information, in the case of a former single
projective measurement, is also bounded in a similar way. The corresponding
inequalities provide a kind of retrodicted extension of Holevo bound for
quantum communication channels. These results quantify how much information
gain is obtained through retrodicted quantum measurements in quantum state
smoothing. While an entropic reduction is always granted, in bipartite systems
mutual information may be degraded. Relevant physical examples confirm these
features.
