  Based on local unitary operators acting on a n-dimensional Hilbert-space, we
investigate selective and collective operator basis sets for N-particle quantum
networks. Selective cluster operators are used to derive the properties of
general cat-states for any n and N. Collective operators are conveniently used
to account for permutation symmetry: The respective Hilbert-space dimension is
then only polynomial in N and governed by strong selection rules. These
selection rules can be exploited for the design of decoherence-free subspaces
as well as for the implementation of efficient routes to entanglement if
suspended switching between states of different symmetry classes could be
realized.
