  Various parameterizations for the orbits under local unitary transformations
of three-qubit pure states are analyzed. The interconvertibility, symmetry
properties, parameter ranges, calculability and behavior under measurement are
looked at. It is shown that the entanglement monotones of any multipartite pure
state uniquely determine the orbit of that state under local unitary
transformations. It follows that there must be an entanglement monotone for
three-qubit pure states which depends on the Kempe invariant defined in [Phys.
Rev. A 60, 910 (1999)]. A form for such an entanglement monotone is proposed. A
theorem is proved that significantly reduces the number of entanglement
monotones that must be looked at to find the maximal probability of
transforming one multipartite state to another.
