  We explore the relation between entanglement entropy of quantum many body
systems and the distribution of corresponding, properly selected, observables.
Such a relation is necessary to actually measure the entanglement entropy. We
show that in general, the Shannon entropy of the probability distribution of
certain symmetry observables gives a lower bound to the entropy. In some cases
this bound is saturated and directly gives the entropy. We also show other
cases in which the probability distribution contains enough information to
extract the entropy: we show how this is done in several examples including BEC
wave functions, the Dicke model, XY spin chain and chains with strong
randomness.
