  Flip and exchange symmetric (FES) many-qubit states, which can be obtained
from a state with the same symmetries by means of invertible local operations
(ILO), constitute a one-parameter family of curves in the Hilbert space.
Eigenstates of FES ILOs correspond to vectors that cannot be transformed to
other FES states. This means equivalence classes of states under ILO can be
determined in a systematic way for an arbitrary number of qubits. More
important, for entangled states, at the boundaries of neighboring equivalence
classes, one can show that when the fidelity between the final state after an
ILO and a state of the neighboring class approaches unity, probability of
success decreases to zero. Therefore, the classes are stable under ILOs.
