Models that provide experimentally testable violations of ordinary qua
ntum mechanics have been recently proposed. These models are based on
non-unitary time evolutions of density matrices that are generated by
linear positive maps, We discuss the consequences of imposing a strong
er condition on those maps, known as complete positivity. It turns out
that experimental data on tile neutral kaon system giving upper bound
s to the parameters characterizing positive maps, also give bounds to
those determining completely positive ones.