  A correlated many-body calculation is presented to characterize the Shannon
information entropy of trapped interacting bosons. We reformulate the one-body
Shannon information entropy in terms of the one-body probability density. The
minimum limit of the entropy uncertainty relation (EUR) is approached by making
$N$ very small in our numerical work. We examine the effect of correlations in
the calculation of information entropy. Comparison with the mean-field result
shows that the correlated basis function is indeed required to characterize the
important features of the information entropies. We also accurately calculate
the point of critical instability of an attractive BEC, which is in close
agreement with the experimental value. Next we calculate two-body entropies in
position and momentum spaces and study quantum correlations in the attractive
BEC.
