Distinction of Tripartite Greenberger-Horne-Zeilinger and W States Entangled in Time-Energy and Space

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 multi-particle entanglement have been focused on Gaussian states. In this paper, 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 with the third-order correlation function G.