‘Nonlinear’ positive mappings for density matrices

In quantum physics one uses density matrices on separable Hilbert spaces for the mathematical description of states. Often it is important to know when the image of a density matrix under a mapping is again a density matrix. We propose some new classes of mappings which map the space of density matrices on one Hilbert space H into the space of density matrices on some other Hilbert space K where K=H naturally is allowed. Such mappings may be ‘linear’ or ‘nonlinear’. In the case of finite dimensional Hilbert spaces we find all mappings of this type.