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arXiv:quant-ph/0510142 · Physics Letters A

Entanglement-assisted local transformations between inequivalent classes of three-particle entangled states

Iulia Ghiu, Mohamed Bourennane, Anders Karlsson

Published Aug 1, 2001• 16 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(01)00446-7

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Local operations assisted by classical communication (LOCC) constitute the free operations in entanglement theory. Hence, the determination of LOCC transformations is crucial for the understanding of entanglement. We characterize here almost all LOCC transformations among pure multipartite multilevel states. Combined with the analogous results for qubit states shown by Gour \emph{et al.} [J. Math. Phys. 58, 092204 (2017)], this gives a characterization of almost all local transformations among multipartite pure states. We show that nontrivial LOCC transformations among generic, fully entangled, pure states are almost never possible. Thus, almost all multipartite states are isolated. They can neither be deterministically obtained from local-unitary-inequivalent (LU-inequivalent) states via local operations, nor can they be deterministically transformed to pure, fully entangled LU-inequivalent states. In order to derive this result, we prove a more general statement, namely, that, generically, a state possesses no nontrivial local symmetry. We discuss further consequences of this result for the characterization of optimal, probabilistic single copy and probabilistic multi-copy LOCC transformations and the characterization of LU-equivalence classes of multipartite pure states.
Entanglement is a resource to overcome the natural restriction of operations used for state manipulation to Local Operations assisted by Classical Communication (LOCC). Hence, a bipartite maximally entangled state is a state which can be transformed deterministically into any other state via LOCC. In the multipartite setting no such state exists. There, rather a whole set, the Maximally Entangled Set of states (MES), which we recently introduced, is required. This set has on the one hand the property that any state outside of this set can be obtained via LOCC from one of the states within the set and on the other hand, no state in the set can be obtained from any other state via LOCC. Recently, we studied LOCC transformations among pure multipartite states and derived the MES for three and generic four qubit states. Here, we consider the non-generic four qubit states and analyze their properties regarding local transformations. As already the most coarse grained classification, due to Stochastic LOCC (SLOCC), of four qubit states is much richer than in case of three qubits, the investigation of possible LOCC transformations is correspondingly more difficult. We prove that most SLOCC classes show a similar behavior as the generic states, however we also identify here three classes with very distinct properties. The first consists of the GHZ and W class, where any state can be transformed into some other state non-trivially. In particular, there exists no isolation. On the other hand, there also exist classes where all states are isolated. Last but not least we identify an additional class of states, whose transformation properties differ drastically from all the other classes. Although the possibility of transforming states into local-unitary inequivalent states by LOCC turns out to be very rare, we identify those states (with exception of the latter class) which are in the MES and those, which can be obtained (transformed) non-trivially from (into) other states respectively. These investigations do not only identify the most relevant classes of states for LOCC entanglement manipulation, but also reveal new insight into the similarities and differences between separable and LOCC transformations and enable the investigation of LOCC transformations among arbitrary four qubit states.
We will review ongoing work at KTH on quantum communication, discussing in particular two projects; long wavelength quantum cryptography, and entangled state manipulations. In order to extend quantum communications to longer distances it is essential to be able to use the low loss telecommunication window at 1550 nm. We, and recently others have for some time worked on quantum cryptography systems in this wavelength region. Much of the effort has been devoted to the investigation of the InGaAs avalanche photo diodes used as single-photon counters, and examples of the performances and systems implications will be given. As a second topic, for quantum communication beyond point-to-point transmission, it is of some interest to investigate the possibilities offered by multi-particle entangled states. Recently, very interesting results are emerging on purification using linear optics manipulations. We will discuss recent theory work on three-particle entangled states of the inequivalent GHZ-classes and the W-classes and the possibility to do entangled-state-assisted transformations between these classes, as well as to consider possible applications.