Local Deterministic Transformations of Three-Qubit Pure States

Abstract

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.