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arXiv:0903.2304 · Physical Review A

Distinction of tripartite Greenberger-Horne-Zeilinger and W states entangled in time (or energy) and space

Jianming Wen, Morton H. Rubin

Published Feb 24, 2009• 23 citations
arXivHTMLPDF

Abstract

In tripartite discrete systems, two classes of genuine tripartite entanglement have been discovered, namely, the Greenberger-Horne-Zeilinger (GHZ) class and the $W$ class. To date, much research effort has been concentrated on the polarization entangled three-photon GHZ and $W$ states. Most studies of continuous-variable multiparticle entanglement have been focused on Gaussian states. In this Brief Report, we examine two classes of three-photon entangled states in space and time. One class is a three-mode three-photon entangled state and the other is a two-mode triphoton state. These states show behavior similar to the GHZ and $W$ states when one of the photons is not detected. The three-mode entangled state resembles a $W$ state, while a two-mode three-photon state resembles a GHZ state when one of the photons is traced away. We characterize the distinction between these two states by comparing the second-order correlation functions ${G}^{(2)}$ with the third-order correlation function ${G}^{(3)}$.

DOI
10.1103/physreva.79.025802

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In tripartite discrete systems, two classes of genuine tripartite entanglement have been discovered, namely, the Greenberger-Horne-Zeilinger (GHZ) class and the WWW class. To date, much research effort has been concentrated on the polarization entangled three-photon GHZ and WWW states. Most studies of continuous-variable multiparticle entanglement have been focused on Gaussian states. In this Brief Report, we examine two classes of three-photon entangled states in space and time. One class is a three-mode three-photon entangled state and the other is a two-mode triphoton state. These states show behavior similar to the GHZ and WWW states when one of the photons is not detected. The three-mode entangled state resembles a WWW state, while a two-mode three-photon state resembles a GHZ state when one of the photons is traced away. We characterize the distinction between these two states by comparing the second-order correlation functions G(2){G}^{(2)}G(2) with the third-order correlation function G(3){G}^{(3)}G(3).
The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides three-qubit states into separable, biseparable, WWW, and Greenberger-Horne-Zeilinger (GHZ) classes. The WWW and GHZ are two subclasses of genuine tripartite entanglement. We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have a complete characterization of SLOCC classes. In the biseparable region (with bipartite entanglement), there exist GHZ-symmetric states that are found to be fully local. In addition, there are bilocal states in both the WWW and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.
The existence and nature of tripartite entanglement of a noninteracting Fermi gas (NIFG) is investigated. Three classes of parametrized entanglement witnesses (EWs) are introduced with the aim of detecting genuine tripartite entanglement in the three-body reduced density matrix and discriminating between the presence of the two types of genuine tripartite entanglement, $W\B$ and $\mathrm{GHZ}\W$ (the convex set of BBB states is comprised of mixed states of product and biseparable states; that of WWW states is comprised of mixed states of BBB states and WWW-type pure entangled states; and the GHZ (Greenberger-Horne-Zeilinger) set contains generic mixtures of any kind for a tripartite system). By choosing appropriate EW operators, the problem of finding GHZ and WWW EWs is reduced to linear programming. Specifically, we devise WWW EWs based on a spin-chain model with periodic boundary conditions, and we construct a class of parametrized GHZ EWs by linearly combining projection operators corresponding to all the different state-vector types arising for a three-fermion system. A third class of EWs is provided by a GHZ stabilizer operator capable of distinguishing $W\B$ from $\mathrm{GHZ}\B$ entanglement, which is not possible with WWW EWs. Implementing these classes of EWs, it is found that all states containing genuine tripartite entanglement are of WWW type, and hence states containing $\mathrm{GHZ}\W$ genuine tripartite entanglement do not arise. Some genuine tripartite entangled states that have a positive partial transpose (PPT) with respect to some bipartition are detected. Finally, it is demonstrated that a NIFG does not exhibit ``pure'' $W\B$ genuine tripartite entanglement: three-party entanglement without any separable or biseparable admixture does not occur.
Abstract This chapter takes a closer look at entanglement in multipartite systems. An examination of pure tripartite systems serves a starting point from which we discover the equivalence classes of Greenberger-Horne-Zeilinger (GHZ) states and W states. We then continue with the mixed-state case and discuss the notions of full separability, partition-separability and biseparability versus genuine tripartite entanglement, before stating the GHZ theorem in the formulation following Mermin. For systems of three or more parties we then formulate the definitions for k-separability and genuine multipartite entanglement (GME), as well as for k-producibility and entanglement depth, followed by prominent examples for GME states such as the generalized GHZ states and Dicke states. We give an overview of the problem of detecting GME using various linear and non-linear GME witnesses, lifted witnesses, PPT mixers, before discussing challenges for the characterization and quantification of multipartite entanglement. We close by analysing the phenomenon of entangled entanglement.
As stated by S. Ghose et al. [Phys. Rev. Lett. 102, 250404 (2009)], there are certain relationships between tripartite entanglement and tripartite nonlocality for three-qubit Greenberger-Horne-Zeilinger (GHZ) class states. In the present work, we have experimentally demonstrated the theoretical results of Ghose et al. by using both three-photon generalized GHZ (GGHZ) states and maximal slice (MS) states with a count of $\ensuremath{\sim}$10/s. From the data, we have verified the agreement of the experimental violation of the Svetlichny inequality with the one predicted by quantum mechanics given the reconstructed density matrix. For the MS states, it is demonstrated that the amount of violation increases linearly following the increase of the degree of tripartite entanglement. In contrast, for GGHZ states, there is a minimal value of the violation when the degree of tripartite entanglement is 1/3. Both of the results are consist with the theoretical prediction.
Evaluation of the Greenberger-Horne-Zeilinger (GHZ) theory requires an entangled nnn-particle state with n> 3n>~3n> 3. To date, experimental achievement of a suitable GHZ state has proved elusive. This paper presents an approach to generate a three-photon entangled state and presents the theory of its space-time structure. Since an accurate picture of the space-time structure is important in understanding the two-photon state, an accurate picture of the three-photon state should be equally important.