  Proposals for nonlinear extenstions of quantum mechanics are discussed. Two
different concepts of "mixed state" for any nonlinear version of quantum theory
are introduced: (i) >genuine mixture< corresponds to operational "mixing" of
different ensembles, and (ii) a mixture described by single density matrix
without having a canonical operational possibility to pick out its specific
convex decomposition is called here an >elementary mixture<. Time evolution of
a class of nonlinear extensions of quantum mechanics is introduced. Evolution
of an elementary mixture cannot be generally given by evolutions of components
of its arbitrary convex decompositions. The theory is formulated in a
"geometric form": It can be considered as a version of Hamiltonian mechanics on
infinite dimensional space of density matrices. A quantum interpretation of the
theory is sketched.
