  The initial state of a system-bath composite is needed as the input for
prediction from any quantum evolution equation to describe subsequent
system-only reduced dynamics or the noise on the system from joint evolution of
the system and the bath. The conventional wisdom is to write down an
uncorrelated state as if the system and the bath were prepared in the absence
of each other; yet, such a factorized state cannot be the exact description in
the presence of system-bath interactions. Here, we show how to go beyond the
simplistic factorized-state prescription using ideas from quantum tomography:
We employ the maximum-entropy principle to deduce an initial system- bath state
consistent with the available information. For the generic case of weak
interactions, we obtain an explicit formula for the correction to the
factorized state. Such a state turns out to have little correlation between the
system and the bath, which we can quantify using our formula. This has
implications, in particular, on the subject of subsequent
non-completely-positive dynamics of the system. Deviation from predictions
based on such an almost uncorrelated state is indicative of accidental control
of hidden degrees of freedom in the bath.
