  We numerically analyse the behavior of the full distribution of collective
observables in quantum spin chains. While most of previous studies of quantum
critical phenomena are limited to the first moments, here we demonstrate how
quantum fluctuations at criticality lead to highly non-Gaussian distributions
thus violating the central limit theorem. Interestingly, we show that the
distributions for different system sizes collapse after scaling on the same
curve for a wide range of transitions: first and second order quantum
transitions and transitions of the Berezinskii-Kosterlitz-Thouless type. We
propose and carefully analyse the feasibility of an experimental reconstruction
of the distribution using light-matter interfaces for atoms in optical lattices
or in optical resonators.
