  The properties of deterministic LOCC transformations of three qubit pure
states are studied. We show that the set of states in the GHZ class breaks into
an infinite number of disjoint classes under this type of transformation. These
classes are characterized by the value of a quantity that is invariant under
these transformations, and is defined in terms of the coefficients of a
particular canonical form in which only states in the GHZ class can be
expressed. This invariant also imposes a strong constraint on any POVM that is
part of a deterministic protocol. We also consider a transformation generated
by a local 2-outcome POVM and study under what conditions it is deterministic,
i.e., both outcomes belong to the same orbit. We prove that for real states it
is always possible to find such a POVM and we discuss analytical and numerical
evidence that suggests that this result also holds for complex states. We study
the transformation generated in the space of orbits when one or more parties
apply several deterministic POVMs in succession and use these results to give a
complete characterization of the real states that can be obtained from the GHZ
state with probability 1.
