  We consider the transformation of Hamilton operators under various sets of
quantum operations acting simultaneously on all adjacent pairs of particles. We
find mappings between Hamilton operators analogous to duality transformations
as well as exact characterizations of ground states employing non-Hermitean
eigenvalue equations and use this to motivate a generalization of the
stabilizer formalism to non-Hermitean operators. The resulting class of states
is larger than that of standard stabilizer states and allows for example for
continuous variation of local entropies rather than the discrete values taken
on stabilizer states and the exact description of certain ground states of
Hamilton operators.
