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.
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