  The complexity of representation of operators in quantum mechanics can be
characterized by the operator space entanglement entropy (OSEE). We show that
in the homogeneous Heisenberg XY spin 1/2 chains the OSEE for initial local
operators grows at most logarithmically with time. The prefactor in front of
the logarithm generally depends only on the number of stationary points of the
quasi-particle dispersion relation and for the XY model changes from 1/3 to 2/3
exactly at the point of quantum phase transition to long-range magnetic
correlations in the non-equilibrium steady state. In addition, we show that the
presence of a small disorder triggers a saturation of the OSEE.
