  Models that provide experimentally testable violations of ordinary Quantum
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 stronger condition on those maps,
known as complete positivity. It turns out that experimental data on the
neutral kaon system giving upper bounds to the parameters characterizing
positive maps, also give bounds to those determining completely positive ones.
