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

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 ${G}^{(3)}$.