This question developed out of conversation between myself and Joe Fitzsimons. Is there a succinct stabilizer representation for symmetric states, on systems of n spin-1/2 or (more generally) n higher spin particles?
By a "stabilizer representation", I mean that:
every symmetric state (or some notable, non-trivial family of them which contains more than just product states) is represented as the unique +1-eigenstate of some operator or the unique joint +1-eigenstate of a list of operators, where
each element of this set of stabilizing operators can be succinctly described, as an operator on the larger Hilbert space (i.e. not only as a transformation restricted to the symmetric subspace itself), and
where the stabilizing operators transform in a nice way in the Heisenberg picture under symmetric local unitaries (i.e. unitary transformations of the form U⊗n).
Ideally, one would be able to efficiently describe all sorts of transformations between various symmetric states; but one cannot have everything.
The constraint of being a unique +1-eigenstate of the list of stabilizing operators could also be made subject to the constraint of being a symmetric state. (For instance, many states on n spin-1/2 particles are stabilized by a σz operator on a single spin, but exactly one symmetric state is stabilized by that operator. Not that I would expect such an operator necessarily to arise in the formalism...)
Does a representation with the above properties (or one close to it) exist?