Entanglement classes of permutation-symmetric qudit states: Symmetric operations suffice

We analyze entanglement classes for permutation-symmetric states for n qudits (i.e., d-level systems), with respect to local unitary operations (LU equivalence) and stochastic local operations and classical communication (SLOCC equivalence). In both cases, we show that the search can be restricted to operations where the same local operation acts on all qudits, and we provide an explicit construction for it. Stabilizers of states in the form of one-particle operations preserving permutation symmetry are shown to provide a coarse-grained classification of entanglement classes. We prove that the Jordan form of such one-particle operators is a SLOCC invariant. We find, as representatives of those classes, a discrete set of entangled states that generalize the Greenberger-Horne-Zeilinger and W states for the many-particle qudit case. In the latter case, we introduce the excitation state as a natural generalization of the W state for d>2.