arXiv:quant-ph/0110179v1 31 Oct 2001 Local Deterministic Transformations of Three-Qubit Pure States Federico M. Spedalieri Institute for Quantum Information California Institute of Technology, MC 452-48, Pasadena, CA 91125, USA (November 23, 2018) 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. I. INTRODUCTION A very important part of the study of the entangled states of spatially separated systems, is the study of the transformations that are possible when using only local operations and classical communication (LOCC), since it allows us to classify entangled states and it can be used as one way of quantifying this resource. Two states that are related by local unitary transformations are consid- ered equivalent as far as entanglement is concerned, since both states can be obtained from each other and local operations cannot increase entanglement. The action of the group of local unitaries breaks the space of states into orbits [1]. Then, to transform a pure state into another state in a different orbit by local operations, we need to allow each party to apply a local generalized measure- ment, i.e., a POVM , on her part of the state. For bipartite pure states, the problem of deterministi- cally transforming a state into another has been solved by Nielsen [2], who gave necessary and sufficient condi- tions for a given transformation to be achievable with probability 1. Later Vidal [3] extended this result by cal- culating the maximal probability of success of any LOCC transformation of bipartite pure states. For more than two parties, this problem is still unsolved. The bipartite case seems to be very special due to the existence of the Schmidt decomposition. Any pure bipartite state can be transformed by applying local unitaries into a state of the form |ψ〉 = n  i λ i |ii〉, (1) where the λ i are positive real numbers, |ii〉 = |i〉 A ⊗|i〉 B and {|i〉} are orthonormal vectors on each subsystem. This greatly simplifies the analysis of LOCC transforma- tions: it gives a canonical expression for states in a given orbit, and allows the reduction of an arbitrary LOCC protocol to a protocol in which one party applies local unitaries and local POVMs, and the other party only has to apply a local unitary, conditional on the results obtained by the first party [4]. For multipartite states with three parties or more there is no known reduction of LOCC protocols. For a system of three qubits, several Schmidt-like de- compositions have been proposed [5,6], all based on the idea of using local unitaries to get rid of as many coeffi- cients as possible. One interesting property that emerges from these decompositions is that in general it is not pos- sible to make all the coefficients real. In particular there are states that have at least one coefficient that is com- plex for any local basis, and this has as a consequence that these states are not locally unitarily equivalent to their conjugates (the states obtained by taking the com- plex conjugate of the coefficients). This contrasts with the bipartite case in which, since the Schmidt decompo- sition has only real coefficients, every state is in the same orbit as its conjugate. A POVM applied to a state has, in general, outcomes that belong to different orbits. However, a protocol that transforms a state into another with probability 1, has to include at least one POVM for which all outcomes are in the same orbit. For instance, this has to be the case for the last POVM of the protocol: if its outcomes are not in the same orbit, then the protocol has not achieved the transformation with probability 1. We will call a POVM with this property a deterministic POVM, because we can use such a POVM and suitable local unitaries, to obtain any state in the orbit of the outcomes with prob- ability 1, attaining a deterministic transformation. Since any local POVM can be replaced by a sequence of 2- outcome POVMs, it is then interesting to study the case of a deterministic 2-outcome POVM. In this paper we will study some properties of deter- ministic LOCC protocols and deterministic POVMs ap- plied to 3-qubit pure states. We will only be interested in transformations between states that have genuine tri- 1