  In contrast to interacting systems, the ground state of free systems has a
highly ordered pattern of quantum correlations, as witnessed by Wick's
decomposition. Here, we quantify the effect of interactions by measuring the
violation they cause on Wick's decomposition. In particular, we express this
violation in terms of the low entanglement spectrum of fermionic systems.
Moreover, we establish a relation between the Wick's theorem violation and the
interaction distance, the smallest distance between the reduced density matrix
of the system and that of the optimal free model closest to the interacting
one. Our work provides the means to quantify the effect of interactions in
physical systems though measurable quantum correlations.
